A New Class of ψ-Caputo Fractional Viscoelastic Contact Problems with Adhesion in Symmetric Dual Spaces

Abstract

This paper is dedicated to the discussion of a new long memory $\psi$-Caputo fractional viscoelastic friction contact problem with adhesion in symmetric dual spaces. In this contact model, the contact condition is described by Clarke-generalized gradient of nonconvex and non-smooth function involving adhesion. Firstly, we investigate a new system involving a $\psi$-Caputo fractional variational–hemivariational inequalities with history-dependent operators coupled with a nonlinear differential equation. By employing the Rothe method in conjunction with surjectivity results for multivalued pseudomonotone operators, the solvability of weak solutions to $\psi$-Caputo fractional differential variational–hemivariational inequality system is obtained. Furthermore, we applied the abstract results obtained to the history-dependent viscoelastic contact problem considering adhesion phenomena and provided the existence of solution for this contact problem.

1. Introduction

Fractional calculus is an extension of integer calculus and plays a crucial role in people’s lives and industrial production. In materials science, particularly in the study of viscoelastic materials, fractional calculus has demonstrated substantial potential and advantages. By introducing fractional-order models, researchers can more accurately describe the dynamic behavior and memory characteristics of materials. Recently, fractional differential equations have been widely applied in mechanics, chemistry, engineering, and fluid flow models, such as [1,2,3,4,5,6]. In recent years, research on $\psi$-fractional differential systems can be found in, for instance [7,8,9].

As an extension of variational inequality and hemivariational inequality, variational–hemivariational inequality is a special type of variational inequality that includes both convex and nonconvex energy functionals. They play a crucial role in describing many mechanical contact problems in solid and fluid mechanics. Many efforts have been discovered and many excellent works have emerged in this area (see [10,11,12,13,14]). In [15], the authors studied the existence of weak solutions for a class of history-dependent variational–hemivariational inequalities and applied the results to contact problems. Recently, the application of differential variational inequalities in contact mechanics can be found in [16,17,18]. There have been research achievements on differential variational–hemivariational inequalities, see [19,20]. For example, in [20], the authors used the generalized penalty method to study the existence and uniqueness of solutions for a class of differential variational–hemivariational inequalities. For research on contact problems considering adhesive, see [21,22,23].

In [24], Han et al. established the solvability of a class of fractional hemivariational inequalities using the Rothe method. In [25], Zeng et al. successfully applied a class of fractional elliptic hemivariational inequalities to the contact problem, and obtained the existence and uniqueness of the solutions for this contact problem. Recently, the author studied a class of viscoelastic contact models within the framework of $\psi$-fractional calculus and established the existence of solutions (see [26]).

Inspired by the aforementioned studies, this paper investigates viscoelastic contact problems with history-dependent operators within the framework of $\psi$-fractional calculus. A new $\psi$-fractional variational–hemivariational inequalities is derived, and the solvability of the contact model is established. Compared to [24,25], our fractional model better captures viscoelastic memory effects. Unlike [26], we use non-smooth contact potentials to model adhesion realistically. Our research model exhibits greater generality, serving as an extension of the models presented in [24,25,26].

There are three main innovative points in the current article. Firstly, we consider a $\psi$-Caputo fractional viscoelastic frictional contact model featuring a long-term memory constitutive law. This model is entirely new. Second, we account for the adhesion phenomenon occurring during the contact process, where the adhesion field is characterized by a nonlinear differential equation. The outcome of our study is a coupled system comprising a $\psi$-Caputo fractional variational–hemivariational inequality and a nonlinear differential equation. Our results demonstrate enhanced generality and constitute an extension of the findings in some research, which also represents the novel contribution of this paper.

The rest of this paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, a class of $\psi$-Caputo fractional viscoelastic contact problems with long-term memory is introduced, in which adhesion is considered. And a variational form of the contact problem is provided. In Section 4, the existence of solution for $\psi$-Caputo fractional differential variational–hemivariational inequalities with history-dependent operators is obtained. In Section 5, by using the abstract results, the existence of solutions to the contact problem proposed in Section 3 is established.

2. Preliminary Work

Let $(E,\parallel ·{\parallel }_E)$ be a Banach space and let $E^{\ast }$ represent the dual of E. Let V be a reflexive Banach space and $V^{\ast }$ represent the dual of V. Let X and Y be two separable and reflexive Banach spaces. The dual spaces of X and Y are denoted by $X^{\ast }$ and $Y^{\ast }$, respectively.

Definition 1

([27]). A locally Lipschitz function $\mathcal{P}:E\to \mathbb{R}$. Then, the Clarke directional derivative of $\mathcal{P}$ in the $l\in E$ direction at the point $\pi \in E$ is defined as

$${\mathcal{P}}^0(\pi ;l)=\underset{w\to \pi ,\lambda \downarrow 0}{\mathrm{lim\; sup}}{\displaystyle \frac{\mathcal{P}(w+\lambda l)-\mathcal{P}(w)}{\lambda }}.$$

The generalized gradient of $\mathcal{P}$ at $u\in E$ is defined by the set

$$\partial \mathcal{P}(\pi )=\{\xi \in E^{\ast }\mid {\mathcal{P}}^0(\pi ;l)\ge \langle \xi ,l\rangle \mathrm{for}\mathrm{all}l\in E\}.$$

Definition 2

([27]). Let $\Psi :U\to \mathbb{R}$ be a convex function and U be an open convex subset of E. ${\vartheta }^{\ast }$ is called a sub-gradient of Ψ if the inequality

$$\Psi (u)\ge \Psi (\vartheta )+{\langle {\vartheta }^{\ast },u-\vartheta \rangle }_{E^{\ast }\times E}$$

holds for all $\vartheta \in E$. The convex subdifferential of Ψ at u is given by the set

$${\partial }_c\Psi (u)=\{{\vartheta }^{\ast }\in E^{\ast }\mid \Psi (u)\ge \Psi (\vartheta )+{\langle {\vartheta }^{\ast },u-\vartheta \rangle }_{E^{\ast }\times E}\}.$$

Definition 3

([28]). A nonempty, bounded, closed, and convex multivalued operator $T:V\to 2^{V^{\ast }}$ is said to be pseudomonotone, if $v_n\rightharpoonup v$ weakly in V and $v_n^{\ast }\rightharpoonup v^{\ast }$ weakly in $V^{\ast }$ with $v_n^{\ast }\in T{v}_n$ and ${\mathrm{lim\; sup}}_{n\to \infty }{\langle v_n^{\ast },v_n-v\rangle }_{V^{\ast }\times V}\le 0,$ then $v^{\ast }\in Tv$ and ${\langle v_n^{\ast },v_n\rangle }_{V^{\ast }\times V}\to {\langle v^{\ast },v\rangle }_{V^{\ast }\times V}.$

Theorem 1

([28]). Let the multivalued operator $T:V\to 2^{V^{\ast }}$ be pseudomonotone and coercive. Then, T is surjective, i.e., for every $\pi \in V^{\ast },$ there is $u\in V$ such that $Tu\ni \pi .$

Lemma 1

([29]). Nonnegative sequences $\left\{a_n\right\},\left\{b_n\right\}$ and $\left\{c_n\right\}$ satisfy

$$a_n\le b_n+\sum _{k=1}^{n-1}{c}_k{a}_k\mathrm{for}n\ge 1.$$

Then, we have

$$a_n\le b_n+\sum _{k=1}^{n-1}{c}_k{b}_k\exp \left(\sum _{j=k+1}^{n-1}{c}_j\right)\mathrm{for}n\ge 1.$$

For $m\ge 0$, if $\left\{a_n\right\}$ and $\left\{c_n\right\}$ satisfy

$$\begin{array}{c}\hfill a_n\le m+\sum _{k=1}^{n-1}{c}_k{a}_k\mathrm{for}n\ge 1.\end{array}$$

Then, the following inequality holds

$$a_n\le m\exp \left(\sum _{k=1}^{n-1}{c}_k\right).$$

Definition 4

([7]). Let $\alpha >0$, $I=(0,T)(T>0)$, $x\in L^1(I,X)$, and let $\psi \in C^1(I)$ be an increasing function such that ${\psi }^{\prime }(t)\ne 0$ on I. The $\alpha >0$ order ψ-fractional integral of x is defined by

$$\begin{array}{c}\hfill I_{0,t}^{\alpha ;\psi }x(t)={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_0^t{\psi }^{\prime }(s){(\psi (t)-\psi (s))}^{\alpha -1}x(s)\mathrm{d}s,t\in I,\end{array}$$

where $\Gamma (·)$ is the Gamma function.

Definition 5

([7]). Let $\alpha >0$, $n\in \mathbb{N}$, $x\in W^{1,1}(I,X)$ and let $\psi \in C^n(I)$ be an increasing function such that ${\psi }^{\prime }(t)\ne 0$, for all $t\in I$, the ψ-Caputo fractional derivative of order α for x is defined by

$$\begin{array}{c}\hfill {}^{C}D_{0,t}^{\alpha ;\psi }x(t)=I_{0,t}^{n-\alpha ;\psi }{\left({\displaystyle \frac{1}{{\psi }^{\prime }(t)}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\right)}^{n}x(t),t\in I,\end{array}$$

where $n=\left[\alpha \right]+1$ for $\alpha \ne \mathbb{N}$, $n=\alpha$ for $\alpha \in \mathbb{N}$. In particular, if $0<\alpha \le 1$, we have

$$\begin{array}{c}\hfill {}^{C}D_{0,t}^{\alpha ;\psi }x(t)=\left\{\begin{array}{cc}\hfill {\displaystyle \frac{1}{\Gamma (1-\alpha )}{\int }_0^t{(\psi (t)-\psi (s))}^{-\alpha }{x}^{\prime }(s)\mathrm{d}s,}& \hfill if\alpha \in (0,1),\\ \hfill {\displaystyle \frac{x^{\prime }(t)}{{\psi }^{\prime }(t)}},& \hfill if\alpha =1.\end{array}\right.\end{array}$$

More knowledge about $\psi$-fractional calculus can be found in [7,8,9,26].

3. The Mechanical Model

There is a deformable body in a domain $\Omega \subset {\mathbb{R}}^d(d=2,3)$ with the Lipschitz continuous boundary $\partial \Omega$ and $\partial \Omega ={\Gamma }_D\cup {\Gamma }_N\cup {\Gamma }_N$ with $\mathrm{meas}({\Gamma }_D)>0$. Here, ${\Gamma }_D,{\Gamma }_N$ and ${\Gamma }_N$ are three disjoint measurable parts. ${\mathbb{S}}^d$ is the space second-order symmetric tensors on ${\mathbb{R}}^d$. We provide the following norms in spaces ${\mathbb{R}}^d$ and ${\mathbb{S}}^d$ as follows:

$$\mathit{u}·\mathit{\lambda }=u_i{\lambda }_i,{\parallel \mathit{u}\parallel }_{{\mathbb{R}}^d}=\sqrt{(\mathit{u}·\mathit{u})},\forall \mathit{u}=(u_i),\mathit{\lambda }=({\lambda }_i)\in {\mathbb{R}}^d,$$

$$\mathit{\sigma }:\mathit{\tau }={\sigma }_{ij}{\tau }_{ij},{\parallel \mathit{\sigma }\parallel }_{{\mathbb{S}}^d}=\sqrt{(\mathit{\sigma }:\mathit{\sigma })},\forall \mathit{\sigma }\in ({\sigma }_{ij}),\mathit{\tau }=({\tau }_{ij})\in {\mathbb{S}}^d.$$

We use $\mathit{\nu }$ to represent the outword unit normal of boundary $\partial \Omega$. For any $\mathit{\vartheta }\in {\mathbb{R}}^d$, the normal and tangential traces of $\mathit{\lambda }$ are denoted by ${\lambda }_{\nu }=\mathit{\lambda }·\mathit{\nu }$ and ${\mathit{\lambda }}_{\tau }=\mathit{\lambda }-{\lambda }_{\nu }\mathit{\nu }$ on $\partial \Omega$, respectively. Here, $\mathit{\sigma }=\mathit{\sigma }(\mathit{x},t)$ and $\mathit{u}=\mathit{u}(\mathit{x},t)$ denotes the stress field and the displacement field, respectively. Here, $\mathit{x}\in \Omega$ and $t\in I=[0,T]$. Let $\mathcal{D}=I\times \Omega$, ${\mathcal{T}}_D=I\times {\Gamma }_D$, ${\mathcal{T}}_C=I\times {\Gamma }_C$ and ${\mathcal{T}}_N=I\times {\Gamma }_N$. We use $\mathit{\epsilon }$ to represent the small strain tensor on the displacement field $\mathit{u}$, with $\mathit{\epsilon }(\mathit{u})=({\mathit{\epsilon }}_{ij}(\mathit{u}))$ and

$$({\mathit{\epsilon }}_{ij}(\mathit{u}))={\displaystyle \frac{1}{2}}(u_{i,j}+u_{j,i})(i,j=1,\dots ,d).$$

Next, we will present the contact mechanics model discussed in this article.

Problem 1.

Find a displacement field $\mathit{u}:\mathcal{D}\to {\mathbb{R}}^d$, a stress field $\mathit{\sigma }:\mathcal{D}\to {\mathbb{S}}^d$ and a bonding field $\lambda :{\mathcal{T}}_C\to [0,1]$ such that

$$\begin{array}{}\mathrm{(1)}& \mathit{\sigma }(t)=\tilde{\mathcal{A}}(\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t))+\tilde{\mathcal{B}}(\mathit{\epsilon }(\mathit{u}(t)))+{\int }_0^t\tilde{\mathcal{C}}(t-s)\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t))\mathrm{d}s\mathrm{in}\mathcal{D},\mathrm{(2)}& \mathrm{Div}\mathit{\sigma }(t)+{\mathit{f}}_0(t)=\mathbf{0}\mathrm{in}\mathcal{D},\mathrm{(3)}& \mathit{u}(t)=0\mathrm{on}{\mathcal{T}}_D,\mathrm{(4)}& \mathit{\sigma }(t)\mathit{\nu }={\mathit{f}}_N(t)\mathrm{on}{\mathcal{T}}_N,\mathrm{(5)}& -{\sigma }_{\nu }(t)\in \partial j_{\nu }(\lambda (t),u_{\nu }(t))\mathrm{on}{\mathcal{T}}_C,\mathrm{(6)}& \left\{\begin{array}{c}\parallel {\sigma }_{\tau }{\left(t\right)\parallel }_{{\mathbb{S}}^d}\le F_b\left(t\right)\mathrm{if}{\mathbf{u}}_{\tau }\left(t\right)={\mathbf{0}}_{{\mathbb{R}}^d},\hfill \\ {\sigma }_{\tau }\left(t\right)=-F_b\left(t\right){\displaystyle \frac{{\mathit{u}}_{\tau }\left(t\right)}{\Vert {\mathit{u}}_{\tau }{\left(t\right)\Vert }_{{\mathbb{R}}^d}}}\mathrm{if}{\mathit{u}}_{\tau }\left(t\right)\ne {\mathbf{0}}_{{\mathbb{R}}^d}\hfill \end{array}\right.\mathrm{on}{\mathcal{T}}_C.\mathrm{(7)}& {\lambda }^{\prime }(t)=\mathfrak{F}(t,\lambda (t),\mathit{u}(t))\mathrm{on}{\mathcal{T}}_C,\mathrm{(8)}& \lambda (0)={\lambda }_0\mathrm{on}{\Gamma }_C,\mathrm{(9)}& \mathit{u}(0)={\mathit{u}}_0\mathrm{in}\Omega .\end{array}$$

To aid reader comprehension, we provide a concise mechanical interpretation of the governing equations and constraints in Problem 1. Specifically, Equation (1) incorporates three key operators: $\tilde{\mathcal{A}}$(viscosity), $\tilde{\mathcal{B}}$ (elasticity), and $\tilde{\mathcal{C}}$ (relaxation), each contributing to the material dynamic response. Equation (2) formulates the motion balance, where ${\mathit{f}}_0$ denotes the body force density and $\mathrm{Div}$ represents the divergence operator. On the boundary ${\Gamma }_D$, the body is fixed, imposing the displacement constraint specified in condition (3). Conversely, Equation (4) defines the traction boundary condition on ${\Gamma }_N$, with ${\mathit{f}}_N$ denoting applied surface tractions. The normal contact behavior in Equation (5) is characterized via the Clarke subdifferential of a nonconvex potential $j_{\mathit{\nu }}$, which is dependent on the time-varying adhesion field $\lambda (t)$. Notably, $\lambda$ evolves according to a $\psi$-fractional differential Equation (7), with its dynamics coupled to the displacement field and contact surface ${\Gamma }_C$. Frictional effects are addressed in Equation (6). Finally, initial conditions (8) and (9) specify the initial adhesion $(\lambda (0)={\lambda }_0)$ and displacement ($\mathit{u}(0)={\mathit{u}}_0$) fields, respectively.

Next, we will derive the variational form of Problem 1. In order to do so, we first define some necessary function spaces as follows.

The following assumptions are made regarding $\tilde{\mathcal{A}}$, $\tilde{\mathcal{B}}$, $\tilde{\mathcal{C}}$, $\mathcal{Q}$, $j_{\nu }$, ${\mathit{f}}_D$, and ${\mathit{f}}_N$ in the Problem 1.

H₀:

${\mathit{f}}_0\in L^2(I;L^2(\Omega ;{\mathbb{R}}^d)),{\mathit{f}}_N\in L^2(I;L^2({\Gamma }_N;{\mathbb{R}}^d)),{\mathit{u}}_0\in \mathcal{Y}$ and ${\lambda }_0\in L^2({\Gamma }_C)$.

H₁:

$\tilde{\mathcal{A}}:\Omega \times {\mathbb{S}}^d\to {\mathbb{S}}^d$ is such that

$$\left\{\begin{array}{c}(i)\tilde{\mathcal{A}}=(a_{ijkl})\in Q_{\infty },0\le i,j,k,l\le d;\hfill \\ (\mathrm{ii})\mathrm{there} \mathrm{exists}{L}_{\mathcal{A}}>0\mathrm{such} \mathrm{that}\tilde{\mathcal{A}}\mathit{\tau }:\mathit{\tau }\ge L_{\tilde{\mathcal{A}}}{\parallel \mathit{\tau }\parallel }_{{\mathbb{S}}^d}^2\mathrm{for} \mathrm{all}\mathit{\tau }\in {\mathbb{S}}^d.\hfill \end{array}\right.$$

H₂:

$\tilde{\mathcal{B}}:\Omega \times {\mathbb{S}}^d\to {\mathbb{S}}^d$ is such that

$$\left\{\begin{array}{c}(i)\tilde{\mathcal{B}}=(b_{ijkl})\in Q_{\infty },0\le i,j,k,l\le d;\hfill \\ (\mathrm{ii})\tilde{\mathcal{B}}\mathit{\tau }:\mathit{\tau }\ge \mathbf{0}\mathrm{for} \mathrm{all}\mathit{\tau }\in {\mathbb{S}}^d.\hfill \end{array}\right.$$

H₃:

$\tilde{\mathcal{C}}:\Omega \times (0,T)\times {\mathbb{S}}^d\to {\mathbb{S}}^d$ is such that

$$\left\{\begin{array}{c}(i)\tilde{\mathcal{C}}=(c_{ijkl})\in Q_{\infty },0\le i,j,k,l\le d;\hfill \\ (\mathrm{ii})\tilde{\mathcal{C}}\mathit{\tau }:\mathit{\tau }\ge \mathbf{0}\mathrm{for} \mathrm{all}\mathit{\tau }\in {\mathbb{S}}^d;\hfill \\ (\mathrm{iii})\tilde{\mathcal{C}}\mathrm{is} \mathrm{Lipschitz} \mathrm{continuous} \mathrm{with} \mathrm{Lipschitz} \mathrm{constant}{L}_{\tilde{\mathcal{C}}}>0.\hfill \end{array}\right.$$

H₄:

$j_{\nu }:{\Gamma }_C\times I\times \mathbb{R}\to \mathbb{R}$ with the following conditions

$$\left\{\begin{array}{c}(i)\mathrm{for} \mathrm{all}\mu \in \mathbb{R},j_{\nu }(·,·,\mu )\mathrm{is} \mathrm{measurable} \mathrm{on}{\mathcal{T}}_C;\hfill \\ (\mathrm{ii})j_{\nu }(\mathit{x},t,·)\mathrm{is} \mathrm{locally} \mathrm{Lipschitz} \mathrm{a}.\mathrm{e}.(\mathit{x},t)\in {\mathcal{T}}_C;\hfill \\ (\mathrm{iii})|\partial j_{\nu }(\mathit{x},t,\mu )|\le c_{j\nu }(1+|\mu |)\mathrm{with}{c}_{j\nu }>0;\hfill \\ (\mathrm{iv})j_{\nu }^0(\mathit{x},.,\mu )\mathrm{is} \mathrm{upper} \mathrm{semi-continuous} \mathrm{on}I\mathrm{for} \mathrm{all}\mu \in \mathbb{R}\mathrm{a}.\mathrm{e}.\mathit{x}\in {\Gamma }_C.\hfill \end{array}\right.$$

H₅:

$\mathfrak{F}:{\Gamma }_C\times I\times \mathbb{R}\times {\mathbb{R}}^d\to \mathbb{R}$ is such that

$$\left\{\begin{array}{c}(i)\mathfrak{F}(·,·,\varpi ,\mathit{\chi })\mathrm{is} \mathrm{measurable} \mathrm{on}{\Gamma }_C\times I,(\varpi ,\mathit{\chi })\in \mathbb{R}\times {\mathbb{R}}^d;\hfill \\ (\mathrm{ii})|\mathfrak{F}(\mathit{x},t,{\varpi }_1,{\mathit{\chi }}_1)-\mathfrak{F}(\mathit{x},t,{\varpi }_2,{\mathit{\chi }}_2)|\le L_{\mathfrak{F}}(|{\varpi }_1-{\varpi }_2|+\parallel {\mathit{\chi }}_1-{\mathit{\chi }}_2\parallel )\hfill \\ \mathrm{for} \mathrm{all}{\varpi }_1,{\varpi }_2\in \mathbb{R},{\mathit{\chi }}_1,{\mathit{\chi }}_2\in {\mathbb{R}}^d\mathrm{and}a.e.(\mathit{x},t)\in ({\Gamma }_C\times I)\mathrm{with}{L}_{\mathfrak{F}}>0;\hfill \\ (\mathrm{iii})\mathfrak{F}(\mathit{x},t,0,\mathit{\chi })=0,\mathfrak{F}(\mathit{x},t,\varpi ,\mathit{\chi })\ge 0\mathrm{for}\varpi \le 0,\mathrm{and}\mathfrak{F}(\mathit{x},t,\varpi ,\mathit{\chi })\le 0\mathrm{for}\varpi \ge 1\hfill \\ \mathrm{for} \mathrm{all}\mathit{\chi }\in {\mathbb{R}}^d,a.e.(\mathit{x},t)\in ({\Gamma }_C\times I).\hfill \end{array}\right.$$

H₆:

Assume that $G:I\times X\times Y\to Y$ satisfies that

$$\left\{\begin{array}{c}\left(\mathrm{i}\right)\mathrm{for} \mathrm{all}\varpi \in X,\varrho \in Y,\mathrm{the} \mathrm{mapping}t\mapsto G(t,\varpi ,\varrho )\mathrm{is} \mathrm{measurable} \mathrm{on} I;\hfill \\ \left(\mathrm{ii}\right)\parallel G(t,{\varpi }_1,{\varrho }_1)-G(t,{\varpi }_2,{\varrho }_2){\parallel }_Y\le L_G(\parallel {\varpi }_1-{\varpi }_2{\parallel }_X+(\parallel {\varrho }_1-{\varrho }_2{\parallel }_Y)\mathrm{with}{L}_G>0;\hfill \\ \left(\mathrm{iii}\right)t\mapsto G(t,0,0)\mathrm{belongs} \mathrm{to}{L}^2(I;Y).\hfill \end{array}\right.$$

Now let us derive the variational formula for Problem 1. By multiplying $\mathit{\vartheta }\in \mathcal{Y}$ on both sides of Equation (2), we can obtain

$${\int }_{\Omega }\mathrm{Div}\mathit{\sigma }·\mathit{\vartheta }\mathrm{d}\mathit{x}=-{\langle {\mathit{f}}_0(t),\mathit{\vartheta }\rangle }_{\mathcal{V}\times \mathcal{Y}}.$$

Furthermore, by virtue of following Green’s formula

$${\int }_{\Omega }\mathit{\sigma }·\mathit{\epsilon }(\mathit{\vartheta })\mathrm{d}\mathit{x}+{\int }_{\Omega }\mathrm{Div}\mathit{\sigma }·\mathit{\vartheta }\mathrm{d}\mathit{x}={\int }_{\partial \Omega }\mathit{\sigma }\mathit{\nu }·\mathit{\vartheta }\mathrm{d}\Gamma$$

we obtain

$$\begin{array}{c}\hfill {\langle \mathit{\sigma }(t),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}={\langle {\mathit{f}}_0(t),\mathit{\vartheta }\rangle }_{\mathcal{V}\times \mathcal{Y}}+{\int }_{{\Gamma }_D}\mathit{\sigma }\mathit{\nu }·\mathit{\vartheta }\mathrm{d}\Gamma +{\int }_{{\Gamma }_N}\mathit{\sigma }\mathit{\nu }·\mathit{\vartheta }\mathrm{d}\Gamma +{\int }_{{\Gamma }_C}\mathit{\sigma }\mathit{\nu }·\mathit{\vartheta }\mathrm{d}\Gamma .\end{array}$$

Applying the (3) and (4), we have

$$\begin{array}{c}\hfill {\langle \mathit{\sigma }(t),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}={\langle {\mathit{f}}_0(t),\mathit{\vartheta }\rangle }_{\mathcal{V}\times \mathcal{Y}}+{\langle {\mathit{f}}_N(t),\mathit{\vartheta }\rangle }_{L^2({\Gamma }_N;{\mathbb{R}}^d)\times \mathcal{Y}}+{\int }_{{\Gamma }_C}({\mathit{\sigma }}_{\nu }(t){\mathit{\vartheta }}_{\nu }+{\mathit{\sigma }}_{\tau }(t)·{\mathit{\vartheta }}_{\tau })\mathrm{d}\Gamma .\end{array}$$

(10)

Furthermore, by the definition of the sub-gradient and (5), we have

$$\begin{array}{c}\hfill {\sigma }_{\nu }(t){\vartheta }_{\nu }\le -j_{\nu }^0(\lambda (t),u_{\nu }(t);{\vartheta }_{\nu }).\end{array}$$

(11)

By utilizing the Riesz representation principle, we know that there exists an element $\mathit{f}\in {\mathcal{Y}}^{\ast }$ such that

$$\begin{array}{c}\hfill {\langle \mathit{f}(t),\mathit{\vartheta }\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}={\langle {\mathit{f}}_0(t),\mathit{\vartheta }\rangle }_{\mathcal{V}\times \mathcal{Y}}+{\langle {\mathit{f}}_N(t),\mathit{\vartheta }\rangle }_{L^2({\Gamma }_N;{\mathbb{R}}^d)}\end{array}$$

(12)

for all $\mathit{\vartheta }\in \mathcal{Y}$, a.e. $t\in I$, where ${\mathcal{Y}}^{\ast }$ denotes the dual space of $\mathcal{Y}$. By substituting inequality (11) into (10) and combining (1) and (12), we can obtain

$$\begin{array}{cc}\hfill & \langle \tilde{\mathcal{A}}{(\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t)),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}+{\langle \tilde{\mathcal{B}}(\mathit{\epsilon }(\mathit{u}(t))),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}+\langle {\int }_0^t\tilde{\mathcal{C}}(t-s)\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t))\mathrm{d}s,\mathit{\epsilon }(\mathit{\vartheta })\rangle \hfill \\ \hfill & {\displaystyle +{\int }_{{\Gamma }_C}{j}_{\nu }^0(\lambda (t),u_{\nu }(t);{\vartheta }_{\nu })\mathrm{d}\Gamma +{\int }_{{\Gamma }_C}{F}_b(t)(\parallel {\mathit{\vartheta }}_{\tau }(t){\parallel }_{{\mathbb{R}}^d})\mathrm{d}\Gamma \ge {\langle \mathit{f}(t),\mathit{\vartheta }\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}}\hfill \end{array}$$

for a.e. $t\in I$. Combining the last inequality and (7)–(9), we obtain the variational formulation of Problem 1.

Problem 2.

Find $\lambda \in W^{1,2}(I,L^2({\Gamma }_C)),\mathit{u}\in W^{1,2}(I;\mathcal{Y})$ such that

$$\left\{\begin{array}{c}\langle \tilde{\mathcal{A}}{(\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t)),\mathit{\epsilon }(\mathit{\vartheta }-\mathit{u}(t))\rangle }_{\mathcal{X}}+{\langle \tilde{\mathcal{B}}(\mathit{\epsilon }(\mathit{u}(t))),\mathit{\epsilon }(\mathit{\vartheta }-\mathit{u}(t))\rangle }_{\mathcal{X}}\hfill \\ {\displaystyle +\langle {\int }_0^t\tilde{\mathcal{C}}(t-s)\mathit{\epsilon }{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t))\mathrm{d}s,\mathit{\epsilon }(\mathit{\vartheta }-\mathit{u}(t))\rangle }\hfill \\ {\displaystyle +{\int }_{{\Gamma }_C}{j}_{\nu }^0(\lambda (t),{\mathit{u}}_{\nu }(t);{\vartheta }_{\nu }-{\mathit{u}}_{\nu }(t))\mathrm{d}\Gamma +{\int }_{{\Gamma }_C}{F}_b(t)(\parallel {\mathit{\vartheta }}_{\tau }{(t)\parallel }_{{\mathbb{R}}^d}-\parallel {\mathit{u}}_{\tau }(t){\parallel }_{{\mathbb{R}}^d})\mathrm{d}\Gamma }\hfill \\ {\ge \langle \mathit{f}(t),\mathit{\vartheta }-\mathit{u}(t))\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}\mathrm{for}\mathrm{all}\mathit{\vartheta }\in \mathcal{Y},\mathrm{a}.\mathrm{e}.\mathrm{t}\in \mathrm{I},\hfill \\ {\lambda }^{\prime }(t)=\mathfrak{F}(t,\lambda (t),\mathit{u}(t))\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,\hfill \\ \lambda (0)={\lambda }_0\mathrm{on}{\Gamma }_C,\hfill \\ \mathit{u}(0)={\mathit{u}}_0\mathrm{in}\Omega .\hfill \end{array}\right.$$

(13)

4. Main Result

Some main conclusions about abstract variational–hemivariational inequality systems will be given below.

Lemma 2.

Suppose that ${\lambda }_0\in Y$, $\varphi \in W^{1,2}(I;X)$, and the condition $(H_6)$ is satisfied, then the following Cauchy problem

$$\left\{\begin{array}{c}{\lambda }^{\prime }(t)=G(t,\lambda (t),\varphi (t))\mathrm{for}a.e.t\in I,\hfill \\ \lambda (0)={\lambda }_0.\hfill \end{array}\right.$$

(14)

has a unique solution $\lambda \in W^{1,2}(I;Y)$, and given ${\varphi }_i\in W^{1,2}(I;X)$, we have

$$\begin{array}{c}\hfill {\displaystyle \parallel {\lambda }_1(t)-{\lambda }_2{(t)\parallel }_Y\le L_{\lambda }{\int }_0^t{\parallel {\varphi }_1(t)-{\varphi }_2(t)\parallel }_X\mathrm{d}s}\end{array}$$

(15)

where $L_{\lambda }>0,{\lambda }_i$ denotes the unique solution corresponding to ${\varphi }_i$, for $i=1,2$.

Proof. 

Given $\varphi \in W^{1,2}(I;X)$, let

$$\begin{array}{c}\hfill {\mathcal{G}}_{\varphi }(t,\varrho )=G(t,\varrho ,\varphi (t))\mathrm{for}\mathrm{all}\varrho \in Y,a.e.t\in I.\end{array}$$

(16)

According to $(H_G)(iii)$, for all $\varrho ,\varpi \in (Y,X)$, we have

$$\begin{array}{cc}{\parallel G(t,\varrho ,\varpi )\parallel }_Y\hfill & \le {\parallel G(t,\varrho ,\varpi )-G(t,0,0)\parallel }_Y+{\parallel G(t,0,0)\parallel }_Y\hfill \\ & \le L_G{(\parallel \varrho \parallel }_Y+{\parallel \varpi \parallel }_X{)+\parallel G(t,0,0)\parallel }_Y.\hfill \end{array}$$

Furthermore, by using $(H_G)(i)$ and $(H_G)(ii)$, the mapping $t\to G(t,\varphi (t),\varrho )\in L^2(I;Y)$ for all $\varphi \in W^{1,2}(I,X)$ and $\varrho \in Y.$ Thus, we conclude that for all $\varrho \in Y$$,\mathcal{G}(t,\varrho )\in L^2(I;Y)$. From $(H_G)(ii)$, one has

$$\begin{array}{c}\hfill \parallel {\mathcal{G}}_{\varphi }(t,{\varrho }_1)-{\mathcal{G}}_{\varphi }(t,{\varrho }_2){\parallel }_Y\le \parallel G(t,\varphi (t),{\varrho }_1)-G(t,\varphi (t),{\varrho }_2){\parallel }_Y\le L_G{\parallel {\varrho }_1-{\varrho }_2\parallel }_Y\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I.\end{array}$$

By applying ([27], Theorem 9.9), we conclude that (14) has unique solution $\lambda \in W^{1,2}(I,Y)$. On the other hand, for any given $\varphi \in W^{1,2}(I;X)$, we get

$$\begin{array}{c}\hfill {\displaystyle \lambda (t)={\lambda }_0+{\int }_0^{t}G(t,\lambda (t),\varphi (t))\mathrm{d}s\mathrm{for}\mathrm{all}t\in I.}\end{array}$$

For ${\varphi }_i\in W^{1,2}(I;X)$, ${\lambda }_i\in W^{1,2}(I;Y)$, we have

$$\begin{array}{cc}\parallel {\lambda }_1(t)-{\lambda }_2{(t)\parallel }_Y\hfill & {\displaystyle \le {\int }_0^t{\parallel G(s,{\lambda }_1(s),{\varphi }_1(s))-G(s,{\lambda }_2(s),{\varphi }_2(s))\parallel }_Y\mathrm{d}s}\hfill \\ & \le L_G\left({\displaystyle {\int }_0^t\parallel {\lambda }_1(s)-{\lambda }_2{(s)\parallel }_Y+{\int }_0^t{\parallel {\varphi }_1(s)-{\varphi }_2(s)\parallel }_X}\right)\mathrm{d}s.\hfill \end{array}$$

For the inequality above, using the Gronwall inequality, we can know that

$$\begin{array}{c}\hfill {\displaystyle \parallel {\lambda }_1(t)-{\lambda }_2{(t)\parallel }_Y\le L_G(1+T{L}_G{e}^{T{L}_G}){\int }_0^t{\parallel {\varphi }_1(t)-{\varphi }_2(t)\parallel }_X\mathrm{d}s\mathrm{for}\mathrm{all}t\in I.}\end{array}$$

which implies that (15) holds with $L_{\lambda }=L_G(1+T{L}_G{e}^{T{L}_G})$. So the proof of the Lemma 2 is completed. □

The existence of solutions for a class of abstract variational–hemivariational inequalities is summarized below.

Let W be separable and reflexive Banach spaces. $W^{\ast }$ and ${\mathcal{W}}^{\ast }$ represent the dual spaces of W and $\mathcal{W}$, respectively. Define the following function spaces

$$\begin{array}{c}\hfill \mathcal{W}=L^p(I;W),{\mathcal{W}}^{\ast }=L^q(I;W^{\ast })\mathrm{and}\mathcal{V}=\{w\in \mathcal{W}{\mid }^C{D}_{0,t}^{\beta ;\psi }w\in \mathcal{W}\},\end{array}$$

where $0<\beta \le 1,{\displaystyle \frac{1}{\beta }}<p<+\infty$ and $q={\displaystyle \frac{p}{p-1}}.$

Problem 3.

Find $(\lambda ,h)\in W^{1,2}(I,X)\times \mathcal{V}$ such that

$$\left\{\begin{array}{c}{\displaystyle \lambda (t)={\int }_0^{t}G(s,\lambda (s),\mathcal{M}(h(t))\mathrm{d}s+{\lambda }_0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,}\hfill \\ {\langle \mathcal{A}{(}^C{D}_{0,t}^{\beta ;\psi }h(t))+\mathcal{B}h(t)-f(t),v-h(t)\rangle }_{Y\times Y^{\ast }}+\langle (\mathcal{Q}{(}^C{D}_{0,t}^{\beta ;\psi }h))(t),v-h(t)\rangle \hfill \\ +{\mathcal{J}}^0(\lambda (t),\mathcal{M}h(t);\mathcal{M}v-\mathcal{M}h(t))+\Psi (\mathcal{M}v)-\Psi (\mathcal{M}h(t))\ge 0\mathrm{for}v\in \mathcal{W},\mathrm{a}.\mathrm{e}.t\in I,\hfill \\ \lambda (0)={\lambda }_0,h(0)=h_0.\hfill \end{array}\right.$$

(17)

The following hypothesis are proposed regarding the data in Problem 3.

H₇:

$\mathcal{A}\in \mathcal{L}(W,W^{\ast })$ and there exists a constant $m_{\mathcal{A}}>0$ such that $\langle \mathcal{A}u,u\rangle \ge m_{\mathcal{A}}{\parallel u\parallel }_W^2$ for all $u\in W$.

H₈:

$\mathcal{B}\in \mathcal{L}(W,W^{\ast })$ and $\langle \mathcal{B}u,u\rangle \ge 0$ for all $u\in W$.

H₉:

The compact operator $\mathcal{M}\in \mathcal{L}(W,Y)$.

H₁₀:

The function $\mathcal{J}:X\times Y\to \mathbb{R}$ is such that

$$\left\{\begin{array}{c}(i)\varpi \mapsto \mathcal{J}(\varpi ,\varrho )\mathrm{is} \mathrm{measurable} \mathrm{for} \mathrm{all} \varrho \in Y;\hfill \\ (\mathrm{ii})\varrho \mapsto \mathcal{J}(\varpi ,\varrho )\mathrm{is} \mathrm{locally} \mathrm{Lipschitz} \mathrm{for} \mathrm{all}\varpi \in X;\hfill \\ {(\mathrm{iii})\parallel \partial \mathcal{J}(\varpi ,\varrho )\parallel }_{Y^{\ast }}\le L_{\mathcal{J}}{(1+\parallel \varrho \parallel }_Y)\mathrm{for} \mathrm{all}\varpi \in X\mathrm{and}\varrho \in Y,\mathrm{where} \mathrm{contact}{L}_{\mathcal{J}}>0;\hfill \\ (\mathrm{iv})(\varpi ,\varrho )\mapsto {\mathcal{J}}^0(\varpi ,\varrho ;v)\mathrm{is} \mathrm{upper} \mathrm{semi-continuous} \mathrm{from}X\times Y\mathrm{into}\mathbb{R}\mathrm{for} \mathrm{all}v\in Y.\hfill \end{array}\right.$$

H₁₁:

$\Psi :Y\to \mathbb{R}$ is such that

$$\left\{\begin{array}{c}(i)\Psi (y)\mathrm{is} \mathrm{proper}, \mathrm{convex}\mathrm{and}\mathrm{lower}\mathrm{semi-continuous} \mathrm{l}.\mathrm{s}.\mathrm{c}.\mathrm{on}Y;\hfill \\ (\mathrm{ii})\parallel {\partial }_c{\Psi (y)\parallel }_{Y^{\ast }}\le L_{\Psi }{(1+\parallel y\parallel }_Y)\mathrm{for} \mathrm{all}y\in Y,\mathrm{where} \mathrm{contact}{L}_{\Psi }>0.\hfill \end{array}\right.$$

H₁₂:

The operator $\widehat{k}\in C(I\times I,\mathcal{L}(W,W^{\ast })$ satisfies

$$\parallel \widehat{k}(t_1,s)-\widehat{k}(t_2,s)\parallel \le L_{\widehat{k}}|t_1-t_2|\mathrm{for} \mathrm{all}s,t_1,t_2\in I,$$

with $L_{\widehat{k}}>0$ and $m_{\widehat{k}}=\underset{t,s\in I\times I}{\max }\parallel \widehat{k}(t,s)\parallel .$

H₁₃:

The operator $\mathcal{Q}\in \mathcal{L}(W,W^{\ast })$ is defined by

$$(\mathcal{Q}u)(t)=\mathcal{H}\left({\int }_0^t\widehat{k}(t,s)u(s)\mathrm{d}s+{ϵ}_{\mathcal{Q}}\right),$$

and there exists constant $L_{\mathcal{Q}}>0$ such that

$$\parallel (\mathcal{Q}{u}_1)(t)-(\mathcal{Q}{u}_2)(t){\parallel }_{{\mathcal{W}}^{\ast }}\le L_{\mathcal{Q}}{\int }_0^t{\parallel u_1(s)-u_2(s)\parallel }_{\mathcal{W}}\mathrm{d}s$$

for all $u_1(t),u_2(t)\in C(I;W),$ where $\mathcal{H}\in \mathcal{L}(W,W^{\ast })$, ${ϵ}_{\mathcal{Q}}\in W$, and $L_{\mathcal{Q}}=m_{\widehat{k}}\parallel \mathcal{H}\parallel .$

H₁₄:

$f\in {\mathcal{W}}^{\ast }$ and $h_0\in W$, $M_0=\underset{t\in I}{\max }\left\{{\psi }^{\prime }(t)\right\}$.

Remark 1.

In assumption $H_{13}$, assuming $\mathcal{Q}$ satisfies the Lipschitz condition, ensuring the convergence of the operator is crucial.

Furthermore, let $w(t)={}^{C}D_{0,t}^{\alpha ;\psi }h(t)$ in Problem 3, we get

$$\begin{array}{c}\hfill h(t)=I_{0,t}^{\beta ;\psi }w(t)+h_0\mathrm{for} \mathrm{a}.\mathrm{e}.t\in I.\end{array}$$

Therefore, Problem 3 can be rewritten in the following form.

Problem 4.

Find $(\lambda ,w)\in W^{1,2}(I,X)\times \mathcal{W}$ such that

$$\left\{\begin{array}{c}{\displaystyle \lambda (t)={\int }_0^{t}G(s,\lambda (s),\mathcal{M}(I_{0,s}^{\beta ;\psi }w(s)+h_0))\mathrm{d}s+{\lambda }_0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,}\hfill \\ \langle \mathcal{A}(w(t))+\mathcal{B}(I_{0,t}^{\beta ;\psi }w(t)+h_0),v-(I_{0,t}^{\beta ;\psi }w(t)+h_0)\rangle +\langle (\mathcal{Q}w)(t),v-(I_{0,t}^{\beta ;\psi }w(t)+h_0)\rangle \hfill \\ +{\mathcal{J}}^0(\lambda (t),\mathcal{M}(I_{0,t}^{\beta ;\psi }w(t)+h_0);\mathcal{M}v-\mathcal{M}(I_{0,t}^{\beta ;\psi }w(t)+h_0))\hfill \\ +\Psi (\mathcal{M}v)-\Psi (\mathcal{M}(I_{0,t}^{\beta ;\psi }w(t)+h_0))\ge \langle f(t),v-(I_{0,t}^{\beta ;\psi }w(t)+h_0)\rangle \hfill \\ \mathrm{for}v\in \mathcal{W},\mathrm{a}.\mathrm{e}.t\in I.\hfill \end{array}\right.$$

(18)

Based on (18), we know that the equivalent inclusion of Problem 4 is as follows.

Problem 5.

Find $(\lambda ,w)\in W^{1,2}(I,X)\times \mathcal{W}$ such that

$$\left\{\begin{array}{c}{\displaystyle \lambda (t)={\int }_0^{t}G(s,\lambda (s),\mathcal{M}(I_{0,s}^{\beta ;\psi }w(s)+h_0))\mathrm{d}s+{\lambda }_0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,}\hfill \\ \mathcal{A}(w(t))+\mathcal{B}(I_{0,t}^{\beta ;\psi }w(t)+h_0)+(\mathcal{Q}w)(t)+{\mathcal{M}}^{\ast }\partial \mathcal{J}(\lambda (t),\mathcal{M}(I_{0,t}^{\beta ;\psi }w(t)+h_0))\hfill \\ +{\partial }_c\Psi (\mathcal{M}(I_{0,t}^{\beta ;\psi }w(t)+h_0))\ni f(t)\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I.\hfill \end{array}\right.$$

(19)

Theorem 2.

Assume that the conditions $H_7,H_8,H_9,H_{10},H_{11}$, $H_{12}$, $H_{13}$ and $H_{14}$ hold. Then Problem 5 admits at last one solution $(\lambda ,w)\in W^{1,2}(I;X)\times \mathcal{W}$.

Next, we will use the Rothe method to prove Theorem 2. The proof follows the Rothe method: (1) time discretization, (2) solvability of the discrete problem, (3) taking the limit. Let $N\in {\mathbb{N}}^{+}$, $\tau =T/N$, and the equidistant nodes of the interval $[0,T]$ are denoted by ${\left\{t_k\right\}}_{k=0}^N={\left\{k\tau \right\}}_{k=0}^N$, and ${\displaystyle f_{\tau }(t)=f_{\tau }^k=\frac{1}{\tau }{\int }_{t_{k-1}}^{t_k}f(s)\mathrm{d}s}$ for $t\in (t_{k-1},t_k],k=1,2,\dots ,N$. Next, we consider the discretized format of Problem 5.

Problem 6.

Find ${\left\{w_{\tau }^k\right\}}_{k=0}^N\in W$, ${\lambda }_{\tau }\in W^{1,2}(I;X)$ and ${\left\{{\xi }_{\tau }^k\right\}}_{k=0}^N\in Y^{\ast }$ such that $h_{\tau }^0=h_0$, and

$$\begin{array}{}\mathrm{(20)}& {\lambda }_{\tau }(t)={\int }_0^{t}G(s,{\lambda }_{\tau }(s),\mathcal{M}{\widehat{h}}_{\tau }(s))\mathrm{d}s+{\lambda }_0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in (0,t_k),\mathrm{(21)}& \mathcal{A}{w}_{\tau }^k+\mathcal{B}{h}_{\tau }^k+v_{\tau }^k+{\partial }_c\Psi (\mathcal{M}{h}_{\tau }^k)+{\mathcal{M}}^{\ast }{\xi }_{\tau }^k\ni f_{\tau }^k\end{array}$$

with ${\xi }_{\tau }^k\in \partial \mathcal{J}({\lambda }_{\tau }(t_k),\mathcal{M}{h}_{\tau }^k)$ for $k=1,2,\dots ,N$, where $h_{\tau }^k$, $v_{\tau }^k$ and ${\widehat{h}}_{\tau }(t)$ for $t\in (0,t_k)$ are defined by

$$\begin{array}{cc}{h}_{\tau }^k\hfill & =h_0+{\displaystyle \frac{1}{\Gamma (\beta )}}\sum _{j=1}^k{w}_{\tau }^j{\int }_{t_{j-1}}^{t_j}{\psi }^{\prime }(s){(\psi (t_k)-\psi (s))}^{\beta -1}\mathrm{d}s\hfill \\ & =h_0+{\displaystyle \frac{1}{\Gamma (\beta +1)}}\sum _{j=1}^k{\left({\left((\psi (t_k)-\psi (t_{j-1})\right)}^{\beta }-(\psi (t_k)-\psi (t_j)\right)}^{\beta })w_{\tau }^j\hfill \\ & =h_0+{\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^k\left({({\psi }^{\prime }({\eta }_{j-1}))}^{\beta }{(k-j+1)}^{\beta }-{({\psi }^{\prime }({\eta }_j))}^{\beta }{(k-j)}^{\beta }\right)w_{\tau }^j\hfill \end{array}$$

(22)

where ${\eta }_{j-1}\in (t_k,t_{j-1})$, ${\eta }_j\in (t_k,t_j)$, and

$$\begin{array}{c}\hfill v_{\tau }^k=\mathcal{H}\left(\sum _{j=1}^{k-1}{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_k,s)w_{\tau }^j\mathrm{d}s+{\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)w_{\tau }^k\mathrm{d}s+{ϵ}_{\mathcal{Q}}\right),\end{array}$$

(23)

and

$$\begin{array}{c}\hfill {\widehat{h}}_{\tau }(t)=\left\{\begin{array}{c}\sum _{j=1}^N{\mathcal{X}}_{(t_{j-1},t_j]}(t)h_{\tau }^{j-1},0<t\le T,\hfill \\ h_0,t=0,\hfill \end{array}\right.\end{array}$$

(24)

respectively. Here, ${\mathcal{X}}_{(t_{j-1},t_j]}$ be defined by

$$\begin{array}{c}\hfill {\mathcal{X}}_{(t_{j-1},t_j]}(t)=\left\{\begin{array}{c}1,t\in (t_{j-1},t_j],\hfill \\ 0,otherwise.\hfill \end{array}\right.\end{array}$$

Under the assumptions above, the solvability theorem for Problem 6 is given below.

Theorem 3.

Assume that the conditions $H_7,H_8,H_9,H_{10},H_{11}$, $H_{12}$, $H_{13}$ and $H_{14}$ hold. Then Problem 6 has at least one solution $(u_{\tau }^k,{\lambda }_{\tau })\in W\times W^{1,2}(I;X)$.

Proof. 

For the given $h_{\tau }^0,h_{\tau }^1,\dots ,h_{\tau }^{n-1},$ we will prove the existence of $h_{\tau }^n$, ${\xi }_{\tau }^n$ and ${\lambda }_{\tau }$ that satisfy (20) and (21). Using the definition of $h_{\tau }^k$, we can obtain $h_{\tau }^0,h_{\tau }^1,\dots ,h_{\tau }^{n-1}$. Therefore, ${\widehat{h}}_{\tau }$ is well-defined in $(0,t_n)$. It is easy to know that ${\widehat{h}}_{\tau }\in L^2(0,t_n;W)$. Furthermore, we can verify that ${\widehat{h}}_{\tau }$ satisfies all the conditions of Lemma 2. Now we consider inclusion problem (21) and write it in the following equivalent form

$$\begin{array}{c}\mathcal{A}{w}_{\tau }^k+\mathcal{B}{h}_{\tau }^k+\mathcal{H}({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)w_{\tau }^k\mathrm{d}s)+{\partial }_c\Psi (h_{\tau }^k)\hfill \\ +{\mathcal{M}}^{\ast }\partial \mathcal{J}({\lambda }_{\tau }(t_k),\mathcal{M}{h}_{\tau }^k)\ni f_{\tau }^k+\mathcal{H}(\sum _{j=1}^{k-1}{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_k,s)w_{\tau }^j\mathrm{d}s+{ϵ}_{\mathcal{Q}}).\hfill \end{array}$$

(25)

Let

$$\begin{array}{cc}\hfill & \widehat{h}=h_0+{\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({({\psi }^{\prime }({\eta }_{j-1}))}^{\beta }{(k-j+1)}^{\beta }-{({\psi }^{\prime }({\eta }_j))}^{\beta }{(k-j)}^{\beta }\right)h_{\tau }^j,\hfill \\ \hfill & \widehat{c}={\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}{\left({\psi }^{\prime }({\eta }_{n-1})\right)}^{\beta },{\eta }_{n-1}\in (t_n,t_{n-1}).\hfill \end{array}$$

It is easy to see that inclusion problem (25) is equivalent to the following inclusion problem. Find $\zeta \in W$ such that

$$\begin{array}{c}\mathcal{A}\zeta +\mathcal{B}(\widehat{h}+\widehat{c}\zeta )+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)+{\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}\zeta ))\hfill \\ +{\mathcal{M}}^{\ast }\partial \mathcal{J}\left({\lambda }_{\tau }(t_k),\mathcal{M}(\widehat{h}+\widehat{c}\zeta )\right)\ni {\mathcal{F}}_{\tau }^k\hfill \end{array}$$

(26)

where ${\displaystyle {\mathcal{F}}_{\tau }^k={\displaystyle f_{\tau }^k}+\mathcal{H}({\sum }_{j=1}^{k-1}{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_k,s)w_{\tau }^j\mathrm{d}s+{ϵ}_{\mathcal{Q}}).}$ And we can see that

$$\begin{array}{cc}\langle {\mathcal{F}}_{\tau }^k,u-u_0\rangle \hfill & \le \parallel {\mathcal{F}}_{\tau }^k{\parallel }_{W^{\ast }}{(\parallel u\parallel }_W\parallel u_0{\parallel }_W)\hfill \\ & \le \tilde{c}(\parallel u\parallel +1),u,u_0\in W\hfill \end{array}$$

(27)

where $\tilde{c}=\parallel {\mathcal{F}}_{\tau }^k{\parallel }_{W^{\ast }}\max \{1,\parallel u_0{\parallel }_W\}$.

We will show that the multivalued operator $\mathrm{\Xi }:W\to 2^{W^{\ast }}$ defined by

$$\begin{array}{cc}\mathrm{\Xi }\zeta =\hfill & \mathcal{A}\zeta +\mathcal{B}(\widehat{h}+\widehat{c}\zeta )+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)+{\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}\zeta ))\hfill \\ \hfill & +{\mathcal{M}}^{\ast }\partial \mathcal{J}\left({\lambda }_{\tau }(t_k),\mathcal{M}(\widehat{h}+\widehat{c}\zeta )\right)\hfill \end{array}$$

(28)

is surjective. To this end, we first prove that operator $\mathrm{\Xi }$ is coercive.

Under the assumptions $H_7-H_{13}$ and the definition of $\widehat{c}$, we get

$$\begin{array}{c}|\langle \mathcal{B}(\widehat{h}+\widehat{c}\zeta )+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)+{\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}\zeta ))\hfill \\ +{\mathcal{M}}^{\ast }\partial \mathcal{J}\left({\lambda }_{\tau }(t_k),\mathcal{M}(\widehat{h}+\widehat{c}\zeta )\right),\zeta {\rangle }_{W^{\ast }\times W}|\hfill \\ \le \parallel \mathcal{B}\parallel \parallel \widehat{h}\parallel \parallel \zeta \parallel +\widehat{c}{\parallel \mathcal{B}\parallel \parallel \zeta \parallel }^2+L_{\Psi }\parallel \mathcal{M}\parallel (1+\parallel \mathcal{M}\parallel (\parallel \widehat{h}\parallel +\widehat{c}\parallel \zeta \parallel ))\parallel \zeta \parallel \hfill \\ +L_{\mathcal{J}}\parallel \mathcal{M}\parallel (1+\parallel \widehat{h}\parallel \parallel \mathcal{M}\parallel )\parallel \zeta \parallel ++\parallel \mathcal{H}\parallel {\int }_{t_{k-1}}^{t_k}\parallel \widehat{k}(t_k,s){\parallel \parallel \zeta \parallel }^2\mathrm{d}s+L_{\mathcal{J}}\widehat{c}{\parallel \mathcal{M}\parallel }^2{\parallel \zeta \parallel }^2\hfill \\ \le \left(\parallel \mathcal{B}\parallel \parallel \widehat{h}\parallel +(L_{\Psi }+L_{\mathcal{J}}\parallel \mathcal{M}\parallel )(1+\parallel \widehat{h}\parallel \parallel \mathcal{M}\parallel )\right)\parallel \zeta \parallel +\tau m_{\widehat{k}}{\parallel \mathcal{H}\parallel \parallel \zeta \parallel }^2\hfill \\ +{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\Psi }+L_{\mathcal{J}}){\parallel \mathcal{M}\parallel }^2{)\parallel \zeta \parallel }^2.\hfill \end{array}$$

(29)

By using $H_7$, (27) and (29), one has

$$\begin{array}{cc}{\displaystyle \frac{{\langle \theta -{\mathcal{F}}_{\tau }^k,\zeta \rangle }_{W^{\ast }\times W}}{\parallel \zeta \parallel }}\ge \hfill & (m_{\mathcal{A}}-\tau m_{\widehat{k}}\parallel \mathcal{H}\parallel -{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\Psi }+L_{\mathcal{J}}){\parallel \mathcal{M}\parallel }^2)\parallel \zeta \parallel \hfill \\ & \hfill -\left(\parallel \mathcal{B}\parallel \parallel \widehat{h}\parallel +(L_{\Psi }+L_{\mathcal{J}}\parallel \mathcal{M}\parallel )(1+\parallel \widehat{h}\parallel \parallel \mathcal{M}\parallel )+\parallel {\mathcal{F}}_{\tau }^k{\parallel }_{W^{\ast }}\right)\end{array}$$

for all $\theta \in \mathrm{\Xi }\zeta$ and $0<\tau <{\tau }_0$, where

$$\begin{array}{c}{\tau }_0={\left({\displaystyle \frac{(m_{\mathcal{A}}-\tau m_{\widehat{k}}\parallel \mathcal{H}\parallel )\Gamma (\beta +1)}{M_0^{\beta }(\parallel \mathcal{B}\parallel +(L_{\Psi }+L_{\mathcal{J}}){\parallel \mathcal{M}\parallel }^2)}}\right)}^{{\displaystyle \frac{1}{\beta }}},\hfill \end{array}$$

which implies that the coercivity is fulfilled.

On the other hand, we show that operator $\mathrm{\Xi }$ is pseudomonotone. Under the conditions $H_7$, $H_8$, and $H_{13}$, we obtain that for every $\zeta \in W$, the set $\mathrm{\Xi }\zeta$ is nonempty, bounded, closed, and convex, this means that $\mathrm{\Xi }$ is nonempty, bounded, closed, and convex operator. Let ${\zeta }_n\rightharpoonup \zeta$ in Y as $n\to \infty$, ${\zeta }_n^{\ast }\in \mathrm{\Xi }{\zeta }_n$, ${\zeta }_n^{\ast }\rightharpoonup {\zeta }^{\ast }$ in $W^{\ast }$ as $n\to \infty$, and $\underset{n\to \infty }{\mathrm{lim\; sup}}{\langle {\zeta }_n^{\ast },{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}\le 0$. We know that ${\zeta }^{\ast }\in \mathrm{\Xi }\zeta$ and

$$\begin{array}{c}\underset{n\to \infty }{\lim }{\langle {\zeta }_n^{\ast },{\zeta }_n\rangle }_{W^{\ast }\times W}={\langle {\zeta }^{\ast },\zeta \rangle }_{W^{\ast }\times W}.\hfill \end{array}$$

(30)

Next, the detailed proof process of Equation (30) will be presented.

Firstly, we prove that ${\zeta }^{\ast }\in \mathrm{\Xi }\zeta$. In fact, according to $\mathcal{A},\mathcal{B},\mathcal{H}\in \mathcal{L}(W,W^{\ast })$, and ${\zeta }_n\rightharpoonup \zeta$ in W as $n\to \infty$, we obtain that

$$\begin{array}{}\mathrm{(31)}& A{\zeta }_n\rightharpoonup A\zeta \mathrm{in}{W}^{\ast },\mathrm{(32)}& B(\widehat{h}+\widehat{c}{\zeta }_n)\rightharpoonup B(\widehat{h}+\widehat{c}\zeta )\mathrm{in}{W}^{\ast },\mathrm{(33)}& \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right)\rightharpoonup \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)\mathrm{in}{W}^{\ast }.\end{array}$$

From the compactness of $\mathcal{M}$ and ${\zeta }_n\rightharpoonup \zeta$ in W as $n\to \infty$, one has

$$\begin{array}{c}\mathcal{M}(\widehat{h}+\widehat{c}{\zeta }_n)\to \mathcal{M}(\widehat{h}+\widehat{c}\zeta )\mathrm{in}W.\hfill \end{array}$$

(34)

For a given ${\lambda }_{\tau }(t_k)$, let ${\xi }_n\in \partial \mathcal{J}({\lambda }_{\tau }(t_k),\mathcal{M}(\widehat{h}+\widehat{c}{\zeta }_n))$ and ${\delta }_n\in {\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}{\zeta }_n))$. Based on $H_{10}$ and $H_{11}$, we obtain that $\left\{{\xi }_n\right\}\subset W^{\ast }$ and $\left\{{\delta }_n\right\}\in W^{\ast }$ are bounded. So, there are convergent subsequences, which are still represented by $\left\{{\xi }_n\right\}$ and $\left\{{\delta }_n\right\}$, respectively. So, we have

$$\begin{array}{c}{\xi }_n\rightharpoonup \xi \mathrm{and}{\delta }_n\rightharpoonup \delta \mathrm{in}{W}^{\ast }\mathrm{as}n\to \infty .\hfill \end{array}$$

(35)

From (34) and (35), we get $\xi \in \partial \mathcal{J}(\mathcal{M}(\widehat{h}+\widehat{c}\zeta ))$ and $\delta \in {\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}\zeta ))$. By using (31)–(35), we can obtain that

$$\begin{array}{cc}\underset{n\to \infty }{\lim }{\zeta }_n^{\ast }\hfill & =\underset{n\to \infty }{\lim }\left(\mathcal{A}{\zeta }_n+\mathcal{B}(\widehat{h}+\widehat{c}{\zeta }_n)+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right)+{\partial }_c\Psi (\mathcal{M}(\widehat{h}+\widehat{c}{\zeta }_n))+{\mathcal{M}}^{\ast }{\xi }_n\right)\hfill \\ \hfill & =\mathcal{A}\zeta +\mathcal{B}(\widehat{h}+\widehat{c}\zeta )+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)+M^{\ast }\delta +M^{\ast }\xi .\hfill \end{array}$$

Using ${\zeta }_n^{\ast }\rightharpoonup {\zeta }^{\ast }$ as $n\to \infty$, we get

$$\begin{array}{c}{\zeta }^{\ast }=\mathcal{A}\zeta +\mathcal{B}(\widehat{h}+\widehat{c}\zeta )+\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right)+M^{\ast }\delta +M^{\ast }\xi ,\hfill \end{array}$$

which implies that ${\zeta }^{\ast }\in \mathrm{\Xi }\zeta$.

Subsequently, we prove that $\underset{n\to \infty }{\lim }{\langle {\zeta }_n^{\ast },{\zeta }_n\rangle }_{W^{\ast }\times W}={\langle {\zeta }^{\ast },\zeta \rangle }_{W^{\ast }\times W}.$

• Under assumptions $H_8$ and ${\zeta }_n\rightharpoonup \zeta$ in W, we have

$$\begin{array}{c}\underset{n\to \infty }{\lim \sup }\langle \mathcal{B}(\widehat{h}+\widehat{c}{\zeta }_n),\zeta -{\zeta }_n{\rangle }_{W^{\ast }\times W}\hfill \\ \le \underset{n\to \infty }{\lim \sup }\langle \mathcal{B}(\widehat{h}+\widehat{c}\zeta ),\zeta -{\zeta }_n{\rangle }_{W^{\ast }\times W}=0\hfill \end{array}$$

(36)

and

$$\begin{array}{c}\underset{n\to \infty }{\lim \sup }\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right),\zeta -{\zeta }_n{\rangle }_{W^{\ast }\times W}\hfill \\ \le \underset{n\to \infty }{\lim \sup }\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right),\zeta -{\zeta }_n{\rangle }_{W^{\ast }\times W}=0.\hfill \end{array}$$

(37)

By using ${\zeta }_n^{\ast }\in \mathrm{\Xi }{\zeta }_n$, we have

$$\begin{array}{c}\underset{n\to \infty }{\lim \sup }\langle {\zeta }_n^{\ast },{\zeta }_n\rangle \hfill \\ =\underset{n\to \infty }{\lim \sup }\left({\langle \mathcal{A}{\zeta }_n+\mathcal{B}(\widehat{h}+\widehat{c}{\zeta }_n),{\zeta }_n\rangle }_{X^{\ast }\times X}+{\langle {\mathcal{M}}^{\ast }{\xi }_n+{\mathcal{M}}^{\ast }{\delta }_n,\mathcal{M}{\zeta }_n\rangle }_{W^{\ast }\times W}\right)\hfill \\ +\underset{n\to \infty }{\lim \sup }{〈\mathcal{H}\left({\int }_{t_{k-1}}^{t_k}h(t_k,s){\zeta }_n\mathrm{d}s\right),{\zeta }_n〉}_{W^{\ast }\times W}.\hfill \end{array}$$

(38)

From (34), (36)–(38) and $\underset{n\to \infty }{\lim \sup }{\langle {\zeta }_n^{\ast },{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}\le 0$, we obtain

$$\begin{array}{cc}\hfill & \underset{n\to \infty }{\lim \sup }{\langle \mathcal{A}{\zeta }_n,{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}\hfill \\ \hfill & \le \underset{n\to \infty }{\lim \sup }{\langle {\zeta }_n^{\ast },{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}+\underset{n\to \infty }{\lim \sup }{\langle \mathcal{B}(\widehat{h}+\widehat{c}{\zeta }_n,\zeta -{\zeta }_n)\rangle }_{W^{\ast }\times W}\hfill \\ \hfill & +\underset{n\to \infty }{\lim \sup }\langle {\mathcal{M}}^{\ast }{\delta }_n,\mathcal{M}(\zeta -{\zeta }_n)\rangle +\underset{n\to \infty }{\lim \sup }\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right),\zeta -{\zeta }_n{\rangle }_{W^{\ast }\times W}\hfill \\ \hfill & +\underset{n\to \infty }{\lim \sup }\langle {\mathcal{M}}^{\ast }{\xi }_n,\mathcal{M}(\zeta -{\zeta }_n)\rangle \le 0.\hfill \end{array}$$

On the other hand, by using the monotonicity of $\mathcal{A}$, it follows that

$$\begin{array}{c}\underset{n\to \infty }{\lim \sup }{\langle \mathcal{A}{\zeta }_n,{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}\ge \underset{n\to \infty }{\lim \sup }{\langle \mathcal{A}\zeta ,{\zeta }_n-\zeta \rangle }_{W^{\ast }\times W}=0.\hfill \end{array}$$

Based on the discussion above, we get

$$\begin{array}{c}\underset{n\to \infty }{\lim }\langle \mathcal{A}{\zeta }_n,{\zeta }_n\rangle =\langle \mathcal{A}\zeta ,\zeta \rangle .\hfill \end{array}$$

(39)

By a similar scheme, we can get the following conclusion

$$\begin{array}{c}\underset{n\to \infty }{\lim }{〈\mathcal{B}(\widehat{\vartheta }+\widehat{c}{\zeta }_n),{\zeta }_n〉}_{W^{\ast }\times W}={〈\mathcal{B}(\widehat{h}+\widehat{c}\zeta ),\zeta 〉}_{W^{\ast }\times W}\hfill \end{array}$$

(40)

and

$$\begin{array}{c}\underset{n\to \infty }{\lim }\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right),{\zeta }_n{{\rangle }_{W^{\ast }\times W}=\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s)\zeta \mathrm{d}s\right),\zeta \rangle }_{W^{\ast }\times W}.\hfill \end{array}$$

(41)

Thus, we have

$$\begin{array}{cc} & \underset{n\to \infty }{\lim }{\langle {\zeta }_n^{\ast },{\zeta }_n\rangle }_{W^{\ast }\times W}=\underset{n\to \infty }{\lim }{\langle \mathcal{A}{\zeta }_n,{\zeta }_n\rangle }_{W^{\ast }\times W}+\underset{n\to \infty }{\lim }{\langle \mathcal{B}(\widehat{h}+\widehat{c}{\zeta }_n),{\zeta }_n\rangle }_{W^{\ast }\times W}\hfill \\ \hfill & +\underset{n\to \infty }{\lim }{\langle {\delta }_n,\mathcal{M}{\zeta }_n\rangle }_{W^{\ast }\times W}+\langle \mathcal{H}\left({\int }_{t_{k-1}}^{t_k}\widehat{k}(t_k,s){\zeta }_n\mathrm{d}s\right),{\zeta }_n{\rangle }_{W^{\ast }\times W}\hfill \\ \hfill & +\underset{n\to \infty }{\lim }{\langle {\xi }_n,\mathcal{M}{\zeta }_n\rangle }_{W^{\ast }\times W}={\langle {\zeta }^{\ast },\zeta \rangle }_{W^{\ast }\times W}.\hfill \end{array}$$

So, operator $\mathrm{\Xi }$ is pseudomonotone. Using Theorem 1, we obtain that $\mathrm{\Xi }$ is surjective for all $0<\tau <{\tau }_0$. The proof of the Theorem 3 is completed. □

Next, we will provide results of a priori estimates for the sequence of solution of Problem 6.

Lemma 3.

Assume that the hypotheses $H_8$–$H_{14}$ hold. Then, there exist ${\tilde{\tau }}_0>0$ and ${\tilde{C}}_i>0$ $(i=1,2,3,4,5)$ independent of τ, such that $\tau \in (0,{\tilde{\tau }}_0)$, the solutions of Problem 6 satisfy

$$\begin{array}{c}\underset{k=1,2,\dots ,N}{\max }\parallel w_{\tau }^k\parallel \le {\tilde{C}}_1,\hfill \end{array}$$

(42)

$$\begin{array}{c}\underset{k=1,2,\dots ,N}{\max }\parallel h_{\tau }^k\parallel \le {\tilde{C}}_2,\hfill \end{array}$$

(43)

$$\begin{array}{c}\underset{k=1,2,\dots ,N}{\max }\parallel v_{\tau }^n\parallel \le {\tilde{C}}_3,\hfill \end{array}$$

(44)

$$\begin{array}{c}\underset{k=1,2,\dots ,N}{\max }\parallel {\delta }_{\tau }^k\parallel \le {\tilde{C}}_4,\hfill \end{array}$$

(45)

$$\begin{array}{c}\underset{k=1,2,\dots ,N}{\max }\parallel {\xi }_{\tau }^k\parallel \le {\tilde{C}}_5,\hfill \end{array}$$

(46)

where ${\delta }_{\tau }^k\in {\partial }_c\Psi (\mathcal{M}{h}_{\tau }^k)$, ${\xi }_{\tau }^k\in \partial \mathcal{J}({\lambda }_{\tau }(t_k),\mathcal{M}{h}_{\tau }^k)$ with

$$\begin{array}{c}\mathcal{A}{w}_{\tau }^k+\mathcal{B}{h}_{\tau }^k+v_{\tau }^k+{\mathcal{M}}^{\ast }{\delta }_{\tau }^k+{\mathcal{M}}^{\ast }{\xi }_{\tau }^k\ni f_{\tau }^k(k=1,2,\dots ,N).\hfill \end{array}$$

Proof. 

Based on hypothesis $(H_{\mathcal{B}})$ and (22), we get

$$\begin{array}{cc}\langle \mathcal{B}{h}_{\tau }^n,w_{\tau }^n\rangle \hfill & =\langle \mathcal{B}(h_0+{\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^n({({\psi }^{\prime }({\xi }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }\hfill \\ & -{({\psi }^{\prime }({\xi }_j))}^{\beta }{(n-j)}^{\beta })w_{\tau }^j),w_{\tau }^n\rangle \hfill \\ & \ge -\parallel \mathcal{B}{h}_0{\parallel }_{W^{\ast }}\parallel w_{\tau }^n\parallel -{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}\parallel \mathcal{B}\parallel \sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \parallel w_{\tau }^n\parallel \hfill \\ & -{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}\parallel \mathcal{B}\parallel \parallel w_{\tau }^n{\parallel }^2.\hfill \end{array}$$

(47)

From hypotheses $H_{12}$ and $H_{13}$ and the definition of $v_{\tau }^n$, we obtain

$$\begin{array}{cc}\langle v_{\tau }^n,w_{\tau }^n\rangle \hfill & =〈\mathcal{H}\left({ϵ}_{\mathcal{Q}}+\sum _{j=1}^{n-1}{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_n,s)w_{\tau }^j\mathrm{d}s+{\int }_{t_{n-1}}^{t_n}\widehat{k}(t_n,s)w_{\tau }^n\mathrm{d}s\right),w_{\tau }^n〉\hfill \\ & \ge -\parallel \mathcal{H}\parallel \left(\parallel {ϵ}_{\mathcal{Q}}\parallel +\sum _{j=1}^{n-1}{\int }_{t_{j-1}}^{t_j}\parallel \widehat{k}(t_n,s)\parallel \parallel w_{\tau }^j\parallel \mathrm{d}s\right)\parallel w_{\tau }^n\parallel -\tau \parallel \mathcal{H}\parallel \parallel \widehat{k}(t_n,s)\parallel {\parallel w_{\tau }^n\parallel }^2\hfill \\ & \ge -\parallel \mathcal{H}\parallel \left(\parallel {ϵ}_{\mathcal{Q}}\parallel +L_{\widehat{k}}\sum _{j=1}^{n-1}{\int }_{t_{j-1}}^{t_j}\parallel w_{\tau }^j\parallel \mathrm{d}s\right)\parallel w_{\tau }^n\parallel -T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel {\parallel w_{\tau }^n\parallel }^2.\hfill \end{array}$$

(48)

Using the $H_{10}$ and $H_{11}$, we obtain

$$\begin{array}{cc}\langle {\xi }_{\tau }^n,\mathcal{M}{w}_{\tau }^n\rangle \hfill & \ge -L_{\mathcal{J}}(1+\parallel \mathcal{M}{h}_{\tau }^n\parallel )\parallel \mathcal{M}{w}_{\tau }^n\parallel \hfill \\ & \ge -L_{\mathcal{J}}\parallel \mathcal{M}{w}_{\tau }^n\parallel (1+\parallel \mathcal{M}{h}_0\parallel +{\displaystyle \frac{{\tau }^{\beta }\parallel \mathcal{M}\parallel }{\Gamma (\beta +1)}}\sum _{j=1}^n({({\psi }^{\prime }({\xi }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }\hfill \\ & -{({\psi }^{\prime }({\eta }_j))}^{\beta }{(n-j)}^{\beta })\parallel w_{\tau }^j\parallel )\hfill \\ & \ge -L_{\mathcal{J}}\left(\parallel \mathcal{M}\parallel +{\parallel \mathcal{M}\parallel }^2\parallel h_0\parallel \right)\parallel w_{\tau }^n\parallel -{\displaystyle \frac{L_{\mathcal{J}}{\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}{\parallel w_{\tau }^n\parallel }^2\hfill \\ & -{\displaystyle \frac{L_{\mathcal{J}}{\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \parallel w_{\tau }^n\parallel \hfill \end{array}$$

(49)

and

$$\begin{array}{cc}\langle {\delta }_{\tau }^n,\mathcal{M}{w}_{\tau }^n\rangle \hfill & \ge -L_{\Psi }(1+\parallel \mathcal{M}{h}_{\tau }^n\parallel )\parallel \mathcal{M}{w}_{\tau }^n\parallel \hfill \\ & \ge -L_{\Psi }\parallel \mathcal{M}{w}_{\tau }^n\parallel (1+\parallel \mathcal{M}{h}_0\parallel +{\displaystyle \frac{{\tau }^{\beta }\parallel \mathcal{M}\parallel }{\Gamma (\beta +1)}}\sum _{j=1}^n({({\psi }^{\prime }({\xi }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }\hfill \\ & -{({\psi }^{\prime }({\eta }_j))}^{\beta }{(n-j)}^{\beta })\parallel w_{\tau }^j\parallel )\hfill \\ & \ge -L_{\Psi }\left(\parallel \mathcal{M}\parallel +{\parallel \mathcal{M}\parallel }^2\parallel h_0\parallel \right)\parallel w_{\tau }^n\parallel -{\displaystyle \frac{L_{\Psi }{\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}{\parallel w_{\tau }^n\parallel }^2\hfill \\ & -{\displaystyle \frac{L_{\Psi }{\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \parallel w_{\tau }^n\parallel .\hfill \end{array}$$

(50)

Based on (47)–(50) and the coercivity of $\mathcal{A}$, we have

$$\begin{array}{cc}\langle f_{\tau }^n,w_{\tau }^n\rangle \hfill & =\langle \mathcal{A}{w}_{\tau }^n+\mathcal{B}{h}_{\tau }^n+v_{\tau }^n,w_{\tau }^n\rangle +\langle {\delta }_{\tau }^n,\mathcal{M}{w}_{\tau }^n\rangle +\langle {\xi }_{\tau }^n,\mathcal{M}{w}_{\tau }^n\rangle \hfill \\ \hfill & \ge m_{\mathcal{A}}\parallel w_{\tau }^n{\parallel }^2-\left({\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }\parallel \mathcal{B}\parallel }{\Gamma (\beta +1)}}+{\displaystyle \frac{(L_{\mathcal{J}}+L_{\Psi }){\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}+T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel \right){\parallel w_{\tau }^n\parallel }^2\hfill \\ \hfill & -\left(\parallel \mathcal{B}{h}_0{\parallel }_{W^{\ast }}+(L_{\mathcal{J}}+L_{\Psi })\parallel \mathcal{M}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2\parallel h_0\parallel +\parallel \mathcal{H}\parallel \parallel {ϵ}_{\mathcal{Q}}\parallel \right)\parallel w_{\tau }^n\parallel \hfill \\ \hfill & -L_{\widehat{k}}\parallel \mathcal{H}\parallel \sum _{j=1}^{n-1}\parallel w_{\tau }^j\parallel {\int }_{t_{j-1}}^{t_j}\mathrm{d}s\parallel w_{\tau }^n\parallel \hfill \\ \hfill & -{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }\parallel \mathcal{B}\parallel }{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \parallel w_{\tau }^n\parallel \hfill \\ \hfill & -{\displaystyle \frac{(L_{\mathcal{J}}+L_{\Psi }){\tau }^{\beta }{M}_0^{\beta }{\parallel \mathcal{M}\parallel }^2}{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \parallel w_{\tau }^n\parallel ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \left(m_{\mathcal{A}}-{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)-T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel \right)\parallel w_{\tau }^n\parallel \hfill \\ \hfill & \le {\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel \hfill \\ \hfill & +L_{\widehat{k}}\parallel \mathcal{H}\parallel \sum _{j=1}^{n-1}\parallel w_{\tau }^j\parallel {\int }_{t_{j-1}}^{t_j}\mathrm{d}s+\parallel \mathcal{B}{h}_0{\parallel }_{W^{\ast }}+(L_{\mathcal{J}}+L_{\Psi })\parallel \mathcal{M}\parallel \hfill \\ \hfill & +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2\parallel h_0\parallel +\parallel \mathcal{H}\parallel \parallel {ϵ}_{\mathcal{Q}}\parallel +\parallel f_{\tau }^n{\parallel }_{W^{\ast }}.\hfill \end{array}$$

Let ${\tilde{\tau }}_0={\left({\displaystyle \frac{(m_{\mathcal{A}}-T{L}_h\parallel \mathcal{H}\parallel )\Gamma (\beta +1)}{2M_0^{\beta }(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)}}\right)}^{{\displaystyle \frac{1}{\beta }}}$, we know that

$$m_{\mathcal{A}}-{\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }}{\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)-T{L}_h\parallel \mathcal{H}\parallel \ge {\displaystyle \frac{m_{\mathcal{A}}}{2}}$$

for all $\tau \in (0,{\tilde{\tau }}_0)$. Furthermore, we get

$$\begin{array}{cc}\parallel w_{\tau }^n\parallel \hfill & \le {\displaystyle \frac{2(\parallel \mathcal{B}{h}_0{\parallel }_{W^{\ast }}+(L_{\mathcal{J}}+L_{\Psi })\parallel \mathcal{M}\parallel +(L_{\mathcal{J}}+L_{\Psi })\parallel \mathcal{M}{\parallel }^2\parallel h_0\parallel +\parallel \mathcal{H}\parallel \parallel {ϵ}_{\mathcal{Q}}\parallel )}{m_{\mathcal{A}}}}\hfill \\ \hfill & +{\displaystyle \frac{2L_h\parallel \mathcal{H}\parallel {\sum }_{j=1}^{n-1}\parallel w_{\tau }^j\parallel {\int }_{t_{j-1}}^{t_j}\mathrm{d}s}{m_{\mathcal{A}}}}+{\displaystyle \frac{2\parallel f_{\tau }^n{\parallel }_{W^{\ast }}}{m_{\mathcal{A}}}}\hfill \\ \hfill & +{\displaystyle \frac{2{\tau }^{\beta }{M}_0^{\beta }}{m_{\mathcal{A}}\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)\parallel w_{\tau }^j\parallel .\hfill \end{array}$$

Using $(H_1)$, one has $\parallel f_{\tau }^n\parallel \le L_f$ for all $\tau >0,n\in N$, where $L_f>0$. Let

$$\begin{array}{c}{\tilde{C}}_0={\displaystyle \frac{2(\parallel \mathcal{B}{h}_0{\parallel }_{Y^{\ast }}+(L_{\mathcal{J}}+L_{\Psi })\parallel M\parallel +(L_{\mathcal{J}}+L_{\Psi })\parallel \mathcal{M}{\parallel }^2\parallel h_0\parallel +\parallel \mathcal{H}\parallel \parallel {ϵ}_{\mathcal{Q}}\parallel )}{m_{\mathcal{A}}}}+{\displaystyle \frac{2L_f}{m_{\mathcal{A}}}}.\hfill \end{array}$$

By using Gronwall inequality, we know that

$$\begin{array}{cc}\hfill & \parallel w_{\tau }^n\parallel \hfill \\ \hfill & \le {\tilde{C}}_0\exp \left({\displaystyle \frac{2{\tau }^{\beta }{M}_0^{\beta }(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)}{m_{\mathcal{A}}\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left({(n-j+1)}^{\beta }-{(n-j)}^{\beta }\right)+2T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel \right)\hfill \\ \hfill & \le {\tilde{C}}_0\exp \left({\displaystyle \frac{2M_0^{\beta }}{m_{\mathcal{A}}\Gamma (\beta +1)}}(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)t_n^{\beta }+2T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel \right)\hfill \\ \hfill & \le {\tilde{C}}_0\exp \left({\displaystyle \frac{2M_0^{\beta }(\parallel \mathcal{B}\parallel +(L_{\mathcal{J}}+L_{\Psi }){\parallel \mathcal{M}\parallel }^2)T^{\beta }}{m_{\mathcal{A}}\Gamma (\beta +1)}}+2T{L}_{\widehat{k}}\parallel \mathcal{H}\parallel \right):={\tilde{C}}_1.\hfill \end{array}$$

From (22), we know that

$$\begin{array}{cc}\parallel h_{\tau }^n\parallel \hfill & =∥h_0+{\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^n\left({({\psi }^{\prime }({\xi }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }-{({\psi }^{\prime }({\xi }_j))}^{\beta }{(n-j)}^{\beta }\right)w_{\tau }^j∥\hfill \\ \hfill & \le \parallel h_0\parallel +{\displaystyle \frac{{\tilde{C}}_1{M}_0^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^n(t_{n-j+1}^{\beta }-t_{n-j}^{\beta })\hfill \\ \hfill & \le \parallel h_0\parallel +{\displaystyle \frac{{\tilde{C}}_1{M}_0^{\beta }}{\Gamma (\beta +1)}}{t}_n^{\beta }\hfill \\ \hfill & \le \parallel h_0\parallel +{\displaystyle \frac{{\tilde{C}}_1{M}_0^{\beta }{T}^{\beta }}{\Gamma (\beta +1)}}:={\tilde{C}}_2.\hfill \end{array}$$

Under the hypotheses $(H_{\mathcal{H}})$ and $(H_{\widehat{k}})$, we have

$$\begin{array}{c}\parallel v_{\tau }^n\parallel =\parallel \mathcal{H}\left({ϵ}_{\mathcal{Q}}+\sum _{j=1}^n{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_n,s)w_{\tau }^n\mathrm{d}s\right)\parallel \le \parallel \mathcal{H}\parallel (\parallel {ϵ}_{\mathcal{Q}}\parallel +T{L}_{\widehat{k}}{\tilde{C}}_1):={\tilde{C}}_3.\hfill \end{array}$$

From $(H_{\mathcal{J}})(ii)$ and $(H_{\Psi })(ii)$, we get

$$\begin{array}{c}\parallel {\xi }_{\tau }^n\parallel \le L_{\mathcal{J}}(1+\parallel \mathcal{M}{h}_{\tau }^n\parallel )\le L_{\mathcal{J}}(1+{\tilde{C}}_2\parallel M\parallel ):={\tilde{C}}_4,\hfill \\ \parallel {\delta }_{\tau }^n\parallel \le L_{\Psi }(1+\parallel \mathcal{M}{h}_{\tau }^n\parallel )\le L_{\Psi }(1+{\tilde{C}}_2\parallel M\parallel ):={\tilde{C}}_5.\hfill \end{array}$$

where ${\xi }_{\tau }^n\in \partial \mathcal{J}(x_{\tau }(t_n),\mathcal{M}{h}_{\tau }^n)$ and ${\delta }_{\tau }^n\in {\partial }_c\Psi (\mathcal{M}{h}_{\tau }^n)$.

So far, the proof of the Lemma 3 has been completed. □

Next, in order to provide the existence result of the solution to Problem 3, we define ${\tilde{w}}_{\tau },{\tilde{h}}_{\tau },{\tilde{v}}_{\tau },{\delta }_{\tau },{\xi }_{\tau }$ and $f_{\tau }$ by

$$\begin{array}{c}{\tilde{w}}_{\tau }(t)=w_{\tau }^n,t\in (t_{n-1},t_n],\hfill \\ {\tilde{h}}_{\tau }(t)=h_{\tau }^n,t\in (t_{n-1},t_n],\hfill \\ {\tilde{v}}_{\tau }(t)=v_{\tau }^n,t\in (t_{n-1},t_n],\hfill \\ {\xi }_{\tau }(t)={\xi }_{\tau }^n,t\in (t_{n-1},t_n],\hfill \\ {\delta }_{\tau }(t)={\delta }_{\tau }^n,t\in (t_{n-1},t_n],\hfill \\ f_{\tau }(t)=f_{\tau }^n,t\in (t_{n-1},t_n]\hfill \end{array}$$

for $n=1,2,\dots ,N.$

The theorem on the convergence of sequences will be presented below.

Theorem 4.

Assume that $H_7$–$H_{14}$ hold and the sequence $\left\{{\tau }_n\right\}$ satisfies ${\tau }_n\to 0(n\to +\infty )$. Then, the following conclusion holds:

$$\begin{array}{c}{\tilde{w}}_{\tau }\rightharpoonup w(\tau \to 0),\mathrm{in}{L}^p(0,T;W),\hfill \\ {\xi }_{\tau }\rightharpoonup \xi (\tau \to 0),\mathrm{in}{L}^q(0,T;Y^{\ast }),\hfill \\ {\delta }_{\tau }\rightharpoonup \delta (\tau \to 0),\mathrm{in}{L}^q(0,T;Y^{\ast }),\hfill \\ {\lambda }_{\tau }\to \lambda (\tau \to 0),\mathrm{in}{W}^{1,2}(0,T;X),\hfill \end{array}$$

where $(w,\lambda )\in \mathcal{W}\times W^{1,2}(0,T;X)$ is a solution of Problem 5.

Proof. 

Due to $\parallel w_{\tau }^n\parallel \le {\tilde{C}}_1$, we have

$${\parallel {\tilde{w}}_{\tau }\parallel }_{L^p(0,T;W)}^p={\int }_0^T\parallel {\tilde{w}}_{\tau }(s){\parallel }^p\mathrm{d}s=\sum _{i=1}^N{\int }_{t_{i-1}}^{t_i}\parallel w_{\tau }^i{\parallel }^p\mathrm{d}s=\tau \sum _{i=1}^N{\parallel w_{\tau }^i\parallel }^p\le {\tilde{C}}_6.$$

So, $\left\{{\tilde{w}}_{\tau }\right\}$ is bounded in $L^p(0,T;W)$ which implies that there exists $w\in L^p(0,T;W)$ such that

$$\begin{array}{c}{\tilde{w}}_{\tau }\rightharpoonup w(\tau \to 0),\mathrm{in}{L}^p(0,T;W).\hfill \end{array}$$

(51)

For any $t\in [0,T]$ and $v^{\ast }\in W^{\ast }$, let $\widehat{g}(s)={(\psi (t)-\psi (s))}^{\beta -1}{\psi }^{\prime }(s)v^{\ast }{\chi }_{[0,t]}(s)$ for $s\in (0,t)$. Clearly, $\widehat{g}\in L^q(0,T;W^{\ast })$ because ${\displaystyle \frac{1}{\beta }}<p<+\infty$. So, we have

$$\begin{array}{cc}\hfill & \left|\langle {\mathit{\sigma }}^{\ast },{\displaystyle \frac{1}{\Gamma (\beta )}}{\int }_0^t{\psi }^{\prime }(s){(\psi (t)-\psi (s))}^{\beta -1}{\tilde{w}}_{\tau }(s)\mathrm{d}s-{\int }_0^t{\psi }^{\prime }(s){(\psi (t)-\psi (s))}^{\beta -1}w(s)\mathrm{d}s\rangle \right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (\beta )}}{\int }_0^t\left|\langle {(\psi (t)-\psi (s))}^{\beta -1}{\psi }^{\prime }(s){\mathit{\sigma }}^{\ast },{\tilde{w}}_{\tau }(s)-w(s)\rangle \right|\mathrm{d}s\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (\beta )}}\left|{\langle \widehat{g},{\tilde{w}}_{\tau }-w\rangle }_{L^q(0,T;W^{\ast })\times L^p(0,T;W)}\right|\to 0,\mathrm{as}\tau \to 0.\hfill \end{array}$$

So, we can get

$$\begin{array}{c}{I}_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)\rightharpoonup I_{0,t}^{\beta ;\psi }w(t)\mathrm{in}W,\mathrm{as}\tau \to 0\hfill \end{array}$$

(52)

for all $t\in I.$ Moreover, one has

$$\begin{array}{cc}\hfill & \parallel {\tilde{h}}_{\tau }(t)-h_0-I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)\parallel \hfill \\ \hfill & =\parallel {\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^n\left({({\psi }^{\prime }({\eta }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }-{({\psi }^{\prime }({\eta }_j))}^{\beta }{(n-j)}^{\beta }\right)w_{\tau }^j\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma (\beta )}}{\int }_0^t{\psi }^{\prime }(s){(\psi (t)-\psi (s))}^{\beta -1}{\tilde{w}}_{\tau }(s)\mathrm{d}s\parallel \hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma (\beta )}}\parallel {\int }_0^{t_n}{\left(\psi (t_n)-\psi (s)\right)}^{\beta -1}{\tilde{w}}_{\tau }(s){\psi }^{\prime }(s)\mathrm{d}s\hfill \\ \hfill & -{\int }_0^t{\left(\psi (t_n)-\psi (s)\right)}^{\beta -1}{\tilde{w}}_{\tau }(s){\psi }^{\prime }(s)\mathrm{d}s\parallel \hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (\beta )}}∥{\int }_t^{t_n}{\left(\psi (t_n)-\psi (s)\right)}^{\beta -1}{\tilde{w}}_{\tau }(s){\psi }^{\prime }(s)\mathrm{d}s∥\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\beta )}}∥{\int }_0^t\left[{\left(\psi (t_n)-\psi (s)\right)}^{\beta -1}-{\left(\psi (t)-\psi (s)\right)}^{\beta -1}\right]{\tilde{w}}_{\tau }(s){\psi }^{\prime }(s)\mathrm{d}s∥\hfill \\ \hfill & \le {\displaystyle \frac{{\tilde{C}}_6}{\Gamma (\beta +1)}}\parallel {\left(\psi (t_n)-\psi (t)\right)}^{\beta }\parallel \hfill \\ \hfill & +{\displaystyle \frac{{\tilde{C}}_6}{\Gamma (\beta +1)}}\parallel {\psi }^{\beta }(t_n)-{\left(\psi (t_n)-\psi (t)\right)}^{\beta }-{\psi }^{\beta }(t)\parallel \hfill \end{array}$$

for $t\in (t_{n-1},t_n]$. Therefor, we can deduce that

$$\begin{array}{c}{\tilde{h}}_{\tau }(t)\to h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)\mathrm{in}W,\mathrm{as}\tau \to 0\hfill \end{array}$$

(53)

for all $t\in I$. Using (52), we have

$$\begin{array}{c}{\tilde{h}}_{\tau }(t)\rightharpoonup h_0+I_{0,t}^{\beta ;\psi }w(t)\mathrm{in}W,\mathrm{as}\tau \to 0\hfill \end{array}$$

(54)

for $t\in I$. Based to the compactness of $\mathcal{M}$, we obtain that

$$\begin{array}{c}\mathcal{M}({\tilde{h}}_{\tau }(t))\to \mathcal{M}(h_0+I_{0,t}^{\beta ;\psi }w(t))\mathrm{in}Y,\mathrm{as}\tau \to 0\hfill \end{array}$$

(55)

for $t\in I$. One the other hand, one has

$$\begin{array}{cc}\parallel {\tilde{h}}_{\tau }(t)-{\widehat{h}}_{\tau }(t)\parallel \hfill & ={\displaystyle \frac{{\tau }^{\beta }}{\Gamma (\beta +1)}}\parallel \sum _{j=1}^n\left({({\psi }^{\prime }({\eta }_{j-1}))}^{\beta }{(n-j+1)}^{\beta }-{({\psi }^{\prime }({\eta }_j))}^{\beta }{(n-j)}^{\beta }\right)w_{\tau }^j\hfill \\ & -\sum _{j=1}^{n-1}\left({({\psi }^{\prime }({\eta }_j))}^{\beta }{(n-j)}^{\beta }-{({\psi }^{\prime }({\eta }_{j+1}))}^{\beta }{(n-j-1)}^{\beta }\right)w_{\tau }^j\parallel \hfill \\ & ={\displaystyle \frac{{\tau }^{\beta }{\mathcal{M}}_0^{\beta }}{\Gamma (\beta +1)}}\sum _{j=1}^{n-1}\left|{(n-j+1)}^{\beta }-2{(n-j)}^{\beta }+{(n-j-1)}^{\beta }\right|\parallel u_{\tau }^j\parallel \hfill \\ & +{\displaystyle \frac{{\tau }^{\beta }{\mathcal{M}}_0^{\beta }}{\Gamma (\beta +1)}}\parallel u_{\tau }^n\parallel \hfill \\ & \le {\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }{\tilde{C}}_1}{\Gamma (\beta +1)}}\left(\sum _{j=1}^{n-1}\left|{(n-j+1)}^{\beta }-2{(n-j)}^{\beta }+{(n-j-1)}^{\beta }\right|+1\right)\hfill \\ & \le {\displaystyle \frac{{\tau }^{\beta }{M}_0^{\beta }{\tilde{C}}_1}{\Gamma (\beta +1)}}(1+n^{\beta }-{(n-1)}^{\beta })\to 0,\mathrm{as}\tau \to 0\hfill \end{array}$$

(56)

for $t\in (t_{n-1},t_n]$. Based on (56) and the compactness of operator $\mathcal{M}$, it is true that

$$\begin{array}{c}\mathcal{M}({\widehat{h}}_{\tau }(t))\to \mathcal{M}(h_0+I_{0,t}^{\beta ;\psi }w(t))\mathrm{in}Y,\mathrm{as}\tau \to 0\hfill \end{array}$$

(57)

for $t\in (t_{n-1},t_n]$. Since $w\in L^p(0,T;W)$, it is obvious that $\mathcal{M}(h(t))\in L^2(I;W)$. Lemma 2 implies that there is a unique solution $\lambda \in (I;X)$ that satisfies

$$\begin{array}{c}\lambda (t)={\int }_0^{t}G(s,\lambda (s),\mathcal{M}(h(s)))\mathrm{d}s+{\lambda }_0.\hfill \end{array}$$

(58)

Using condition $H_6$, we have

$$\begin{array}{cc}\parallel {\lambda }_{\tau }(t)-\lambda (t)\parallel \hfill & \le {\int }_0^t\parallel G(s,{\lambda }_{\tau }(s),\mathcal{M}({\widehat{h}}_{\tau }(s)))-G(s,\lambda (s),\mathcal{M}(h(s)))\parallel \mathrm{d}s\hfill \\ \hfill & \le L_G{\int }_0^t\left(\parallel {\lambda }_{\tau }(s)-\lambda (s)\parallel +\parallel \mathcal{M}({\widehat{h}}_{\tau }(s))+\mathcal{M}(h(s))\parallel \right)\mathrm{d}s\hfill \\ \hfill & \le T{L}_G{\parallel \mathcal{M}{\widehat{h}}_{\tau }-\mathcal{M}h\parallel }_{C(0,T;Y)}+L_G{\int }_0^t\parallel {\lambda }_{\tau }(s)-\lambda (s)\parallel \mathrm{d}s\hfill \\ \hfill & =\delta (\tau )+L_G{\int }_0^t\parallel {\lambda }_{\tau }(s)-\lambda (s)\parallel \mathrm{d}s\hfill \end{array}$$

where $\delta (\tau )=T{L}_G{\parallel \mathcal{M}{\widehat{h}}_{\tau }-\mathcal{M}h\parallel }_{C(0,T;Y)}$. Using Gronwall inequality, we have

$$\begin{array}{c}\parallel {\lambda }_{\tau }(t)-\lambda (t)\parallel \le {\tilde{C}}_7\delta (\tau ),\forall t\in I.\hfill \end{array}$$

From (57), we obtain

$$\begin{array}{c}\parallel {\lambda }_{\tau }(t)-\lambda (t)\parallel \le {\tilde{C}}_7\delta (\tau )\to 0,\mathrm{as}\tau \to 0.\hfill \end{array}$$

So, we have ${\lambda }_{\tau }\to \lambda$ in $W^{1,2}(I;X)$ as $\tau \to 0$. On the other hand, based on the boundedness of sequence $\left\{{\xi }_{\tau }\right\}$, it can be concluded that sequences $\left\{{\xi }_{\tau }\right\}$ has a convergent subsequence (still denoted as $\left\{{\xi }_{\tau }\right\}$) and $\left\{{\delta }_{\tau }\right\}$ has a convergent subsequence (still denoted as $\left\{{\delta }_{\tau }\right\}$). So there exist $\xi ,\delta \in Y^{\ast }$, such that

$$\begin{array}{c}{\xi }_{\tau }\rightharpoonup \xi \mathrm{in}{Y}^{\ast }\mathrm{as}\tau \to 0,\hfill \end{array}$$

(59)

$$\begin{array}{c}{\delta }_{\tau }\rightharpoonup \delta \mathrm{in}{Y}^{\ast }\mathrm{as}\tau \to 0.\hfill \end{array}$$

(60)

By applying the conclusion of ([27], Lemma 12), the fact that

$$\begin{array}{c}{\xi }_{\tau }^k\in \partial \mathcal{J}({\lambda }_{\tau }(t_k),\mathcal{M}(h_{\tau }^k)),{\delta }_{\tau }^k\in {\partial }_c\Psi (\mathcal{M}(h_{\tau }^k))\mathrm{for}k=1,2,\dots ,N\hfill \end{array}$$

and combined with ${\lambda }_{\tau }\to \lambda$ in $W^{1,2}(0,T;X)$, (57), (59) and (60), then utilizing ([27], Theorem 3.13), we can derive that

$$\begin{array}{c}\xi (t)\in \partial \mathcal{J}(x(t),\mathcal{M}(h_0+I_{0,t}^{\beta ;\psi }w(t))),\delta (t)\in {\partial }_c\Psi (\mathcal{M}(h_0+I_{0,t}^{\beta ;\psi }w(t)))\hfill \end{array}$$

for a.e. $t\in (0,T)$. From boundedness of sequence $\left\{{\tilde{w}}_{\tau }\right\}$ and the hypothesis $(H_{\widehat{k}})$, it is ensure that

$$\begin{array}{c}∥{\int }_0^t\widehat{k}(t,s){\tilde{w}}_{\tau }(s)\mathrm{d}s-{\int }_0^{t_k}\widehat{k}(t_k,s){\tilde{w}}_{\tau }(s)\mathrm{d}s∥\hfill \\ =∥{\int }_0^t\widehat{k}(t,s){\tilde{w}}_{\tau }(s)\mathrm{d}s-{\int }_0^t\widehat{k}(t_k,s){\tilde{w}}_{\tau }(s)\mathrm{d}s-{\int }_t^{t_k}\widehat{k}(t_k,s){\tilde{w}}_{\tau }(s)\mathrm{d}s∥\hfill \\ \le {\int }_t^{t_k}\parallel \widehat{k}(t_k,s)\parallel \parallel {\tilde{w}}_{\tau }(s)\parallel \mathrm{d}s+{\int }_0^t\parallel \widehat{k}(t,s)-\widehat{k}(t_k,s)\parallel \parallel {\tilde{w}}_{\tau }(s)\parallel \mathrm{d}s\hfill \\ \le \tau {\tilde{C}}_6(m_{\widehat{k}}+L_{\widehat{k}})\to 0(\tau \to 0)\mathrm{for} \mathrm{a}.\mathrm{e}.t\in [t_{k-1},t_k].\hfill \end{array}$$

(61)

Furthermore, we introduce the Nemytskii operator ${\mathcal{E}}_1,{\mathcal{E}}_2:\mathcal{W}\to {\mathcal{W}}^{\ast }$ by

$$({\mathcal{E}}_{1}w)(t)=\mathcal{H}\left({\int }_0^t\widehat{k}(t,s)w(s)\mathrm{d}s\right)\mathrm{and}({\mathcal{E}}_{2}w)(t)=\mathcal{H}(w(t))$$

for all $w\in \mathcal{W}$ and a.e.$t\in I.$ Based on (51), we get

$$\underset{\tau \to 0}{\lim }{\langle {\mathcal{E}}_1{\tilde{w}}_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}={\langle {\mathcal{E}}_{1}w,v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}$$

for all $v\in \mathcal{W}.$ Let

$${\varphi }_{\tau }(t)={ϵ}_{\mathcal{Q}}+\sum _{j=1}^k{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_k,s)w_{\tau }^j\mathrm{d}s,t\in (t_{j-1},t_j].$$

According to $H_{12}$, $H_{13}$, and (61), we have

$$\begin{array}{cc}\hfill & {\mathcal{E}}_2({\varphi }_{\tau }-{ϵ}_{\mathcal{Q}})-{\mathcal{E}}_1({\tilde{w}}_{\tau })\hfill \\ \hfill & =\mathcal{H}\left(\sum _{j=1}^k{\int }_{t_{j-1}}^{t_j}\widehat{k}(t_k,s)w_{\tau }^j\mathrm{d}s\right)-\mathcal{H}\left({\int }_0^t\widehat{k}(t,s){\tilde{w}}_{\tau }(t)\mathrm{d}s\right)\to 0\mathrm{strongly} \mathrm{in}{\mathcal{W}}^{\ast },\mathrm{as}\tau \to 0,\hfill \end{array}$$

which implies

$$\begin{array}{c}\underset{\tau \to 0}{\lim }{\langle {\mathcal{E}}_2{\varphi }_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\hfill \\ =\underset{\tau \to 0}{\lim }\left({\langle {\mathcal{E}}_2({\varphi }_{\tau }-{ϵ}_{\mathcal{Q}})-{\mathcal{E}}_1({\tilde{w}}_{\tau }),v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+{\langle {\mathcal{E}}_1({\tilde{w}}_{\tau }),v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+{\langle {\mathcal{E}}_2({ϵ}_{\mathcal{Q}}),v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\right)\hfill \\ ={\langle {\mathcal{E}}_1(w),v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+{\langle {\mathcal{E}}_2({ϵ}_{\mathcal{Q}}),v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}.\hfill \end{array}$$

(62)

Next, we prove that $(\lambda ,w)\in W^{1,2}(I;X)\times \mathcal{W}$ is the solution to Problem 5. To this end, we define the operators ${\mathcal{A}}_0$, ${\mathcal{B}}_0$ and ${\mathcal{M}}_0$ by

$$\begin{array}{c}({\mathcal{A}}_{0}v)(t)=\mathcal{A}(v(t)),({\mathcal{B}}_{0}v)(t)=\mathcal{B}(y_0+I_{0,t}^{\beta ;\psi }v(t))\mathrm{and}({\mathcal{M}}_{0}v)(t)=\mathcal{M}(v(t))\hfill \end{array}$$

for $v\in \mathcal{W}$, a.e. $t\in I$, respectively. According to (31) and $\mathcal{A}\in \mathcal{L}(W,W^{\ast })$, we have

$$\begin{array}{c}{\mathcal{A}}_0{\tilde{w}}_{\tau }\rightharpoonup {\mathcal{A}}_{0}w\mathrm{in}{\mathcal{W}}^{\ast }\mathrm{as}\tau \to 0.\hfill \end{array}$$

(63)

Under (32) and $H_8$, we obtain

$$\begin{array}{c}\mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t))\rightharpoonup \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }w(t))\mathrm{in}{\mathcal{W}}^{\ast },\mathrm{as}\tau \to 0,\hfill \end{array}$$

for all $t\in I$. Moreover, we have

$$\begin{array}{cc}\langle {\mathcal{B}}_0{\tilde{h}}_{\tau },v\rangle \hfill & =\langle \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)),v(t)\rangle \hfill \\ \hfill & \le \parallel \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t))\parallel \parallel v(t)\parallel \hfill \\ \hfill & \le \left({\displaystyle \frac{2{\psi }^{\beta }(T)\parallel \mathcal{B}\parallel {\tilde{C}}_5}{\Gamma (\beta +1)}}+T\parallel \mathcal{B}\parallel \parallel h_0\parallel \right)\parallel v(t)\parallel .\hfill \end{array}$$

Furthermore, by utilizing above inequality and Lebesgue-dominated convergence theorem, we can obtain

$$\begin{array}{cc}\underset{\tau \to 0}{\lim }{\langle {\mathcal{B}}_0{\tilde{h}}_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\hfill & =\underset{\tau \to 0}{\lim }{\int }_0^T\langle \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)),v(t)\rangle \mathrm{d}t\hfill \\ & ={\int }_0^T\underset{\tau \to 0}{\lim }\langle \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }{\tilde{w}}_{\tau }(t)),v(t)\rangle \mathrm{d}t\hfill \\ & ={\int }_0^T\langle \mathcal{B}(h_0+I_{0,t}^{\beta ;\psi }w(t)),v(t)\rangle \mathrm{d}t\hfill \\ & ={\langle {\mathcal{B}}_{0}h,v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}.\hfill \end{array}$$

(64)

By using the compactness of the Nemytskii operator ${\mathcal{M}}_0$, we find that

$$\begin{array}{c}\underset{\tau \to 0}{\lim }\langle {\xi }_{\tau },{\mathcal{M}}_{0}v\rangle =\langle \xi ,{\mathcal{M}}_{0}v\rangle ,\underset{\tau \to 0}{\lim }\langle {\delta }_{\tau },{\mathcal{M}}_{0}v\rangle =\langle \delta ,{\mathcal{M}}_{0}v\rangle \hfill \end{array}$$

(65)

for all $v\in \mathcal{W}.$ Moreover, according to ([29], Lemma 3.3), we know that

$$\begin{array}{c}{f}_{\tau }\to f\mathrm{strongly} \mathrm{in}{\mathcal{W}}^{\ast },\mathrm{as}\tau \to 0.\hfill \end{array}$$

(66)

From (60)–(66), we obtain the following result

$$\begin{array}{cc}\hfill & \underset{\tau \to 0}{\lim \sup }{\langle {\mathcal{A}}_0{\tilde{w}}_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+\underset{\tau \to 0}{\lim \sup }{\langle {\mathcal{B}}_0{\tilde{h}}_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+\underset{\tau \to 0}{\lim \sup }{\langle {\mathcal{E}}_2{\varphi }_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\hfill \\ \hfill & +\underset{\tau \to 0}{\lim \sup }{\langle {\xi }_{\tau },{\mathcal{M}}_{0}v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}+\underset{\tau \to 0}{\lim \sup }{\langle {\delta }_{\tau },{\mathcal{M}}_{0}v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}-\underset{\tau \to 0}{\lim \inf }{\langle f_{\tau },v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\ge 0,\hfill \end{array}$$

for all $v\in \mathcal{W}$. Thus, we have

$$\begin{array}{c}{\langle {\mathcal{A}}_{0}w+{\mathcal{B}}_{0}h+\mathcal{H}h+{{\mathcal{M}}_0}^{\ast }\xi +{{\mathcal{M}}_0}^{\ast }\delta ,v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}}\ge {\langle f,v\rangle }_{{\mathcal{W}}^{\ast }\times \mathcal{W}},\hfill \end{array}$$

where $\xi (t)\in \partial \mathcal{J}(x(t),{\mathcal{M}}_0(h_0+I_{0,t}^{\beta ;\psi }u(t)))$, $\delta (t)\in {\partial }_c\Psi ({\mathcal{M}}_0(h_0+I_{0,t}^{\beta ;\psi }u(t)))$ for a.e. $t\in I$. This implies $(w,\lambda )\in \mathcal{W}\times W^{1,2}(0,T;X)$ is a solution of Problem 5, which finishes the proof of the theorem. □

5. Solvability of Contact Problems

We now aim to prove the solvability of Problem 2 via Theorem 2.

Theorem 5.

Assume that $H_0$–$H_6$ hold. Then, Problem 2 has at least one solution $(\mathit{u},\lambda )\in W^{1,2}(I;\mathcal{Y})\times W^{1,2}(I,L^2({\Gamma }_C)).$

Proof. 

The proof based on Theorem 2 in the specific case where $p=2$. To this end, we define operators $\mathcal{A}:\mathcal{Y}\to {\mathcal{Y}}^{\ast }$, $\mathcal{B}:\mathcal{Y}\to {\mathcal{Y}}^{\ast }$, $\mathcal{J}:\mathcal{U}\times \mathcal{Z}\to \mathbb{R}$, $\Psi :\mathcal{Y}\to \mathbb{R}$ and $\mathcal{Q}:C(I;\mathcal{Y})\to C(I;{\mathcal{Y}}^{\ast })$ by

$$\begin{array}{}\mathrm{(67)}& {\langle \mathcal{A}\mathit{u},\mathit{\vartheta }\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}={\langle \tilde{\mathcal{A}}(\mathit{\epsilon }(\mathit{u})),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}\mathrm{for}\mathit{u},\mathit{\vartheta }\in \mathcal{Y},\mathrm{(68)}& {\langle \mathcal{B}\mathit{u},\mathit{\vartheta }\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}={\langle \tilde{\mathcal{B}}(\mathit{\epsilon }(\mathit{u})),\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}\mathrm{for}\mathit{u},\mathit{\vartheta }\in \mathcal{Y},\mathrm{(69)}& \mathcal{J}(\lambda ,\mathit{u})={\int }_{{\Gamma }_C}{j}_{\nu }(\mathit{x},\lambda ,u_{\nu })\mathrm{d}\Gamma \mathrm{for}\mathit{u}\in \mathcal{Z},\lambda \in \mathcal{U},\mathrm{(70)}& \Psi (\mathit{u})={\int }_{{\Gamma }_C}{F}_b(t){\parallel {\mathit{u}}_{\tau }\parallel }_{{\mathbb{R}}^d}\mathrm{d}\Gamma \mathrm{for}\mathit{u}\in \mathcal{Z},\mathrm{(71)}& \langle (\mathcal{Q}\mathit{u})(t),\mathit{\vartheta }\rangle ={\langle {\int }_0^t\tilde{\mathcal{C}}(t-s)\mathit{\epsilon }(\mathit{u}(s))\mathrm{d}s,\mathit{\epsilon }(\mathit{\vartheta })\rangle }_{\mathcal{X}}\mathrm{for}\mathit{u},\mathit{\vartheta }\in \mathcal{Y},\mathrm{(72)}& {\langle \mathit{f}(t),\mathit{\vartheta }\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}={\langle {\mathit{f}}_0(t),\mathit{\vartheta }\rangle }_{L^2(\Omega ;{\mathbb{R}}^d)}+{\langle {\mathit{f}}_2(t),\gamma \mathit{\vartheta }\rangle }_{L^2({\Gamma }_N;{\mathbb{R}}^d)}.\end{array}$$

Also, we consider the trace operator $\gamma :\mathcal{Y}\to \mathcal{Z}$, let $\mathcal{M}=\gamma$ and $G:I\times \mathcal{Z}\times \mathcal{U}$ defined by

$$\begin{array}{c}G(t,\mathit{v},\lambda )=\mathfrak{F}(\rho ,t,\lambda (\rho ),\mathit{v}(\rho ))\mathrm{for}\mathit{v}\in \mathcal{Z},\lambda \in \mathcal{U}\mathrm{a}.\mathrm{e}.\rho \in {\Gamma }_C.\hfill \end{array}$$

(73)

According to (67)–(73), Problem 2 can be transformed into the following abstract $\psi$-fractional differential variational–hemivariational inequality: find $\lambda \in W^{1,2}(I,\mathcal{U})$, $\mathit{u}\in W^{1,2}(I;\mathcal{Y})$ such that

$$\left\{\begin{array}{c}{\langle \mathcal{A}{(}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t)),\mathit{\vartheta }-\mathit{u}(t),\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}+{\langle \mathcal{B}(\mathit{u}(t)),\mathit{\vartheta }-\mathit{u}(t)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}\hfill \\ +{\langle ({\mathcal{Q}}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u})(t),\mathit{\vartheta }-\mathit{u}(t)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}+{\mathcal{J}}^0(\lambda (t),\gamma \mathit{u}(t);\gamma \mathit{\vartheta }-\gamma \mathit{u}(t))\hfill \\ +\Psi (\gamma \mathit{\vartheta })-\Psi (\gamma \mathit{u}(t))\ge {\langle \mathit{f}(t),\mathit{\vartheta }-\mathit{u}(t)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}\hfill \\ \mathrm{for}\mathrm{all}\mathit{\vartheta }\in \mathcal{Y},a.e.t\in I,\hfill \\ {\lambda }^{\prime }(t)=G(t,\lambda (t),\mathit{u}(t))\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,\hfill \\ \lambda (0)={\lambda }_0,\mathit{u}(0)={\mathit{u}}_0.\hfill \end{array}\right.$$

(74)

Moreover, we denote $\zeta (t){=}^C{D}_{0,t}^{\alpha ;\psi }\mathit{u}(t)$ for a.e. $t\in I$. Thus, we have

$$\begin{array}{c}\mathit{u}(t)=I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I.\hfill \end{array}$$

(75)

Then, (74) can be rewritten as follows. Find $\zeta \in W^{1.2}(I;\mathcal{Y})$, and $\lambda \in W^{1,2}(I;\mathcal{U})$ such that

$$\left\{\begin{array}{c}{\langle \mathcal{A}(\zeta (t)),\mathit{\vartheta }-(I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}+{\langle \mathcal{B}(I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0),\mathit{\vartheta }-(I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}\hfill \\ +{\langle (\mathcal{Q}\zeta )(t),\mathit{\vartheta }-(I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}+\Psi (\gamma \mathit{\vartheta })-\Psi (\gamma (I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0))\hfill \\ +{\mathcal{J}}^0(x(t),\gamma (I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0(t));\gamma \mathit{\vartheta }-\gamma (I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0))\hfill \\ \ge {\langle \mathit{f}(t),\mathit{\vartheta }-(I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0)\rangle }_{{\mathcal{Y}}^{\ast }\times \mathcal{Y}}\mathrm{for}\mathrm{all}\mathit{\vartheta }\in \mathcal{Y},a.e.t\in I,\hfill \\ {\lambda }^{\prime }(t)=G(t,\lambda (t),\gamma (I_{0,t}^{\alpha ;\psi }\zeta (t)+{\mathit{u}}_0))\mathrm{for}\mathrm{a}.\mathrm{e}.t\in I,\hfill \\ \lambda (0)={\lambda }_0.\hfill \end{array}\right.$$

(76)

Let $W=\mathcal{Y}$, $X=\mathcal{U}$, and $Y=\mathcal{Z}$. Next, we verify that the operators $\mathcal{A}$, $\mathcal{B}$, $\mathcal{J}$, $\Psi$ and $\mathcal{Q}$ defined by (67)–(71) satisfy the assumptions $H_7$, $H_8$, $H_{10}$, $H_{11}$ and $H_{13}$, respectively. Under the assumption $H_1$, operator $\mathcal{A}$ given by (65) satisfies hypothesis $H_7$. Since operator $\tilde{\mathcal{B}}$ satisfies properties $H_2$, this yields that operator $\mathcal{B}$ satisfies $H_8$. Based on assumptions $H_4$ and ([27], Corollary 4.15), we can conclude that the conditions $H_{10}(i)$ and $(ii)$ are satisfied and $L_{\mathcal{J}}=\sqrt{2}{c}_{j\nu }$. The upper semicontinuous of the function $(\lambda ,\mathit{u})\mapsto J^0(\lambda ,\mathit{u};\mathit{\vartheta })$ can be derived from the upper semicontinuous of $j_{\nu }$ and Fatou’s lemma; this is how condition $H_{10}(iii)$ is satisfied. According to $F_b\in L^2({\Gamma }_C;I)$ and the norm function is convex, we obtain that $\Psi$ defined by (70) satisfies $H_{11}(i)$ and $H_{11}(ii)$. From ([27], Theorem 3.47), it follows that the condition $H_{11}(iii)$ holds with $L_{\Psi }=\underset{t\in I}{\max }{\parallel F_b(t)\parallel }_{L^2({\Gamma }_C)}$. According to ([29], Theorem 3.9.34), we conclude that the trace operator $\gamma$ satisfies the $H_9$. Finally, by using hypothesis $H_5$, we know that operator G defined by (73) satisfies condition $H_6$. Moreover, since

$$\begin{array}{cc}\hfill & \langle (\mathcal{Q}{\mathit{u}}_1)(t)-(\mathcal{Q}{\mathit{u}}_2)(t),\mathit{\vartheta }\rangle \hfill \\ \hfill & \le {\int }_{\Omega }{\int }_0^t\parallel \tilde{\mathcal{C}}{(t-s)\parallel }_{Q_{\infty }}\parallel \mathit{\epsilon }({\mathit{u}}_1(s))-\mathit{\epsilon }({\mathit{u}}_2(s)){\parallel }_{{\mathbb{S}}^d}\mathrm{d}s{\parallel \mathit{\epsilon }(\mathit{\vartheta })\parallel }_{{\mathbb{S}}^d}\mathrm{d}\mathit{x}\hfill \\ \hfill & \le \underset{t\in I}{\max }{\parallel \tilde{\mathcal{C}}(t)\parallel }_{Q_{\infty }}{\int }_0^t\parallel {\mathit{u}}_1(s)-{\mathit{u}}_2{(s)\parallel }_{\mathcal{Y}}\mathrm{d}s{\parallel \mathit{\vartheta }\parallel }_{\mathcal{Y}}\hfill \end{array}$$

for all ${\mathit{u}}_1,{\mathit{u}}_2\in C(0,T;\mathcal{Y})$ and $\mathit{\vartheta }\in \mathcal{Y}$. Thus, we have

$$\begin{array}{c}\parallel (\mathcal{Q}{\mathit{u}}_1)(t)-(\mathcal{Q}{\mathit{u}}_2)(t){\parallel }_{{\mathcal{Y}}^{\ast }}\le \underset{t\in I}{\max }\parallel \tilde{\mathcal{C}}(t){\parallel }_{Q_{\infty }}{\int }_0^t{\parallel {\mathit{u}}_1(s)-{\mathit{u}}_2(s)\parallel }_{\mathcal{Y}}\mathrm{d}s.\hfill \end{array}$$

This means that $\mathcal{Q}$ defined by (71) satisfies $(H_{\mathcal{Q}})$ with $L_{\mathcal{Q}}=\underset{t\in I}{\max }{\parallel \tilde{\mathcal{C}}(t)\parallel }_{Q_{\infty }}$. By using Theorem 2, we obtain that Problem 2 has at last one solution $(\mathit{u},\lambda )\in W^{1,2}(I;\mathcal{Y})\times W^{1,2}(I,\mathcal{U})$. □

6. Conclusions

In this paper, we investigate a class of $\psi$-Caputo fractional history-dependent viscoelastic frictional contact problem. The main mathematical contributions include establishing a unified framework combining $\psi$-Caputo fractional derivatives, history-dependent operators, and multivalued pseudomonotone operators for adhesive contact problems. The theoretical results prove the existence of weak solutions for the proposed variational–hemivariational inequality system via the Rothe method. Practical implications include demonstrating the applicability of our results to real-world viscoelastic contact problems with adhesion effects.

In the future, we aim to investigate the numerical approximation and convergence analysis of the proposed variational–hemivariational inequality system. Developing efficient numerical schemes, such as finite element methods combined with iterative algorithms for multivalued pseudomonotone operators, would enable the practical implementation of our theoretical results and facilitate comparisons with experimental data from viscoelastic contact experiments.