A New Class of Accretive Mappings in Semi-Inner Product Space with an Application to Solve Variational Inclusion

Abstract

This work aims to develop a new class of accretive mappings and investigate its associated class of proximal mappings. This new class of accretive mappings is known as generalized $(H^k,\phi )$-$\eta$-accretive mappings. Further, the research work includes a discussion of its application, leading to set-valued variational-like inclusions (SVVLI, in short) in the semi-inner product spaces. The corresponding fixed point problem for SVVLI is given. This fixed point problem is used to provide an iterative scheme for solving SVVLI. Furthermore, we study the convergence analysis of the suggested iterative method. An illustration is built in support of our result.

1. Introduction and Preliminaries

In recent years, the interpolation theory for nonlinear operators between Banach spaces has seen significant progress. Although it has been emphasised, a similar development of interpolation theorems for specific kinds of nonlinear operators is far less widely known. In this continuation, the idea of accretive mapping is extremely significant because it enables the treatment of a large number of functional differential equations and partial differential equations, e.g., heat and wave equations from mathematical physics, (see [1,2,3,4,5]). The development in the theory of accretive mappings has been shown to be quite useful in the theory of variational inequalities and variational inclusion.

Variational inclusion theory includes a multitude of new ideas and approaches that may be considered as a novel extension and generalization of a variational inequality. It is outstanding that a wide range of unrelated problems can be studied in the cohesive structure of variational inclusions. An efficient and workable iterative technique is developed through the investigation of variational inclusions. To determine the solutions for variational inclusions, a variety of iterative techniques have been proposed and debated. Researchers frequently employ the proximal approach, one of the most intriguing and significant tools, to resolve variational inclusions. Detailed research results can be found in the relevant references [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33].

In 2014, Sahu et al. [6] introduced a class of nonlinear implicit variational inclusions with A-monotone mapping and discussed its existence of solutions in semi-inner product spaces. In 2017, Bhat and Zahoor [7] introduced a new system of variational inclusions with $(H,\phi )$-$\eta$-monotone mapping and discussed its approximate solution in the semi-inner product spaces. Their outcomes are an extension and improvement of the results of Sahu et al. [34]. Moreover, they developed the iterative methods by using the resolvent operator approaches to discover an approximate solution to variational inclusion problems.

Recently, Luo and Huang [35] introduced $(H,\phi )$-$\eta$-monotone mappings in Banach spaces, which provided a cohesive framework for the various classes of monotone (accretive) operators. As applications, they solved the various classes of variational inclusions and showed the convergence of iterative schemes involving proximal mappings (resolvent operators). See the references for further applications [6,7,8,13,14,17,20,21,22,23,29,31,32,33,35].

This work provides further insight into existing research. This research aims to develop a new class of accretive mappings known as generalized $(H^k,\phi )$-$\eta$-accretive mappings and investigates its associated class of proximal mappings. This class of accretive mappings is a representation of an addition of two $\eta$-symmetric mappings. It can be viewed as an extension of the class of accretive mappings, which are the generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,\dots )$-accretive mapping [36,37,38] and generalized $H(.,.)$-accretive mapping [8]. Some characteristics of the proximal mapping related to generalized $(H^k,\phi )$-$\eta$-accretive mappings are studied.

Application of the proximal mapping linked with the generalized $(H^k,\phi )$-$\eta$-accretive mapping leads to set-valued variational-like inclusions (SVVLI, in short) in the semi-inner product spaces. Further, the equivalent fixed point problem for SVVLI is given. With the help of this alternative problem, we provide an iterative scheme for solving SVVLI. Furthermore, we study the convergence analysis of the suggested iterative method. An illustration is built in support of our result. For detailed studies, see the related references [8,10,11,15,16,20,22,36].

Before giving necessary definitions for the presentation of this research work, we provide the following well-known definitions and results.

Definition 1

([6,39]). Let $\mathcal{B}$ be a vector space over the field $\mathcal{K}$ of real or complex numbers. A functional $\left[.,.\right]:\mathcal{B}\times \mathcal{B}\to K$ is called a semi-inner product if:

(i) 

$\left[z^{\prime }+z^{″},z\right]=\left[z^{\prime },z\right]+\left[z^{″},z\right],\forall z,z^{\prime },z^{″}\in \mathcal{B},$

(ii) 

$\left[\gamma z^{\prime },z\right]=\gamma [z^{\prime },z],\forall \gamma \in K,z,z^{\prime }\in \mathcal{B},$

(iii) 

$\left[z^{\prime },z^{\prime }\right]>0,for{z}^{\prime }\ne 0,$

(iv) 

$|\left[z^{\prime },z^{″}\right]{|}^2\le \left[z^{\prime },z^{\prime }\right]\left[z^{″},z^{″}\right],\forall z^{\prime },z^{″}\in \mathcal{B}$.

Then, the pair $\left(\mathcal{B},\left[.,.\right]\right)$ is known as a semi-inner product space.

Since $\parallel z\parallel ={\left[z,z\right]}^{1/2}$ is a norm on vector space $\mathcal{B}$, each semi-inner product space becomes a normed linear space. A semi-inner product in a normed linear space can be generated in infinitely many ways. Giles [40] proved that the semi-inner product can be obtained uniquely if the considered space $\mathcal{B}$ becomes a uniform convex smooth Banach space. A comprehensive analysis and key findings on semi-inner product spaces may be found in [39,40,41].

Example 1

([6,40]). Let $L^{\tilde{p}}(\mathcal{B},\alpha )$, where $p\in (1,\infty )$ is a real Banach space. Then, $L^{\tilde{p}}(\mathcal{B},\alpha )$ is a semi-inner product space whose semi-inner product is given by

$$\left[g_1,g_2\right]=\frac{1}{\parallel g_2{\parallel }_{\tilde{p}}^{\tilde{p}-2}}{\int }_{\mathcal{B}}{g}_1\left(t\right){\left|g_2\left(t\right)\right|}^{\tilde{p}-1}sgn\left(g_2\left(t\right)\right)d\alpha ,g_1,g_2\in L^{\tilde{p}}.$$

Definition 2

([6,42]). Let $\mathcal{B}$ be a real Banach space.

(i) 

Then, the modulus of smoothness of Banach space $\mathcal{B}$ is a function ${\rho }_{\mathcal{B}}:[0,\infty )\to [0,\infty )$, given as

$$\begin{array}{c}\hfill {\rho }_{\mathcal{B}}\left(u\right)=\sup \left\{\frac{\parallel z+z^{\prime }\parallel +\parallel z-z^{\prime }\parallel }{2}-1:\parallel z\parallel =1,\parallel z^{\prime }\parallel =u,u>0\right\};\end{array}$$

(ii) 

If ${lim}_{u\to 0}\frac{{\rho }_{\mathcal{B}}\left(u\right)}{u}=0$, then $\mathcal{B}$ is called uniformly smooth;

(iii) 

If there exists non-negative constant c as ${\rho }_{\mathcal{B}}\left(u\right)\le c{u}^q,q>1,$ then $\mathcal{B}$ is called q-uniformly smooth;

(iv) 

If there exists non-negative constant c as ${\rho }_{\mathcal{B}}\left(u\right)\le c{u}^2$, then $\mathcal{B}$ is called 2-uniformly smooth.

Lemma 1

([6,42]). Let $\overline{p}>1$ be a real number and $\mathcal{B}$ be a smooth Banach space. Then, the following statements are equivalent:

(i) 

$\mathcal{B}$ is 2-uniformly smooth;

(ii) 

There is a non-negative constant c such that for each $z,z^{\prime }\in \mathcal{B}$, the following inequality holds:

$$\begin{array}{ccc}\hfill & & \parallel z+z^{\prime }{\parallel }^2\le {\parallel z\parallel }^2+2\langle z^{\prime },f_z\rangle +c{\parallel z^{\prime }\parallel }^2;\hfill \end{array}$$

(1)

where $f_z\in J\left(z\right)$ and $J\left(z\right)$$=\{z^{*}\in {\mathcal{B}}^{*}:\langle z,z^{*}\rangle =\parallel z{\parallel }^2\mathrm{and}\parallel z^{*}\parallel =\parallel z\parallel \}$ is the normalized duality mapping.

Each normed linear space $\mathcal{B}$ is a semi-inner product space (see [39]). In fact, by the Hahn–Banach theorem, for each $z\in \mathcal{B}$, there exists at least one functional $f_z\in {\mathcal{B}}^{*}$ such that $\langle z,f_z\rangle ={\parallel z\parallel }^2$.

Given any such mapping $f:\mathcal{B}\to {\mathcal{B}}^{*}$, we can verify that $\left[z^{\prime },z\right]=\langle z^{\prime },f_z\rangle$ defines a semi-inner product. Consequently, we may express the inequality (1) as

$$\begin{array}{ccc}\hfill & & \parallel z+z^{\prime }{\parallel }^2\le {\parallel z\parallel }^2+2\left[z^{\prime },f_z\right]+c{\parallel z^{\prime }\parallel }^2,\forall z,z^{\prime }\in \mathcal{B},\hfill \end{array}$$

(2)

where the constant c is taken as the best minimum value and known as the smoothness constant.

Example 2

([6]). The functional spaces $L^{\tilde{p}}$ is 2-uniformly smooth for $\tilde{p}\ge 2$ and is $\tilde{p}$-uniformly smooth for $\tilde{p}\in (1,2)$. If $\tilde{p}\in [2,\infty )$, then we have for all $z,z^{\prime }\in L^{\tilde{p}}$:

$$\begin{array}{ccc}& & \parallel z+z^{\prime }{\parallel }^2\le {\parallel z\parallel }^2+2\left[z^{\prime },f_z\right]+(\tilde{p}-1){\parallel z^{\prime }\parallel }^2,\forall z,z^{\prime }\in L^{\tilde{p}},\hfill \end{array}$$

where $\tilde{p}-1$ is the smoothness constant.

It is important to remember the fact that the function space $L^{\tilde{p}}$, where $\tilde{p}>1$, is a uniformly convex smooth Banach space, which allows us to provide a unique semi-inner product on the $L^{\tilde{p}}$ space.

Elsewhere in the paper, unless otherwise indicated, we consider that $\mathcal{B}$ is a 2-uniformly smooth Banach space and ${\mathcal{B}}^k=\underset{ktimes}{\underbrace{\mathcal{B}\times \mathcal{B}\times \dots \times \mathcal{B}}}$.

We remind some prerequisites which will be used throughout this paper.

Definition 3.

Let $\mathcal{A}:\mathcal{B}\to \mathcal{B}$ and $\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$ be single-valued mappings. Then, $\mathcal{A}$ is called

(i) 

Accretive if

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),w-v\right]\ge 0,\forall v,w\in \mathcal{B};\end{array}$$

(ii) 

μ-strongly accretive if there exists $\mu >0$ as

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),w-v\right]\ge \mu {\parallel w-v\parallel }^2,\forall v,w\in \mathcal{B};\end{array}$$

(iii) 

γ-relaxed accretive if there exists $\gamma >0$ as

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),w-v\right]\ge -\gamma {\parallel w-v\parallel }^2,\forall v,w\in \mathcal{B};\end{array}$$

(iv) 

η-accretive if

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),\eta (w,v)\right]\ge 0\forall v,w\in \mathcal{B};\end{array}$$

(v) 

$(\tilde{\mu },\eta )$-strongly accretive if there exists $\tilde{\mu }>0$ as

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),\eta (w,v)\right]\ge \tilde{\mu }{\parallel w-v\parallel }^2\forall v,w\in \mathcal{B};\end{array}$$

(vi) 

$(\tilde{\gamma },\eta )$-relaxed accretive if there exists  $\tilde{\gamma }>0$ as

$$\begin{array}{c}\hfill \left[\mathcal{A}\left(w\right)-\mathcal{A}\left(v\right),\eta (w,v)\right]\ge -\tilde{\gamma }{\parallel w-v\parallel }^2\forall v,w\in \mathcal{B};\end{array}$$

(vii) 

 λ-Lipschitz continuous if there exists $\lambda >0$ as

$$\begin{array}{c}\hfill \parallel \mathcal{A}\left(w\right)-\mathcal{A}\left(v\right)\parallel \le \lambda \parallel w-v\parallel ,\forall v,w\in \mathcal{B}.\end{array}$$

Lemma 2

([9]). Let $\left\{p_n\right\}$ and $\left\{q_n\right\}$ be non-negative real sequences, satisfying $p_{n+1}\le r{p}_n+q_n$ with $q_n\to 0$ and $0<r<1$. Then, $p_n\to 0$ as $n\to \infty$.

Definition 4.

For every $r\in \{1,2,\dots ,k\},k\ge 3$, let $H^k:{\mathcal{B}}^k\to \mathcal{B}$, $K_r:\mathcal{B}\to \mathcal{B}$ and $\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$ be the single-valued mappings. Then, $H^k$ is called

(i) 

$({\alpha }_r,\eta )$-strongly accretive with $K_r$ if there exists ${\alpha }_r>0$ as

$$\begin{array}{c}\hfill \left[H^k(\dots ,K_{r}w,\dots )-H^k(\dots ,K_{r}v,\dots ),\eta (w,v)\right]\ge {\alpha }_r{\parallel w-v\parallel }^2,\forall v,w\in \mathcal{B};\end{array}$$

(ii) 

$({\beta }_r,\eta )$-relaxed accretive with $K_r$ if there exists ${\beta }_r>0$ as

$$\begin{array}{c}\hfill \left[H^k(\dots ,K_{r}w,\dots )-H^k(\dots ,K_{r}v,\dots ),\eta (w,v)\right]\ge -{\beta }_r{\parallel w-v\parallel }^2,\forall v,w\in \mathcal{B};\end{array}$$

(iii) 

${\nu }_r$-Lipschitz continuous with $K_r$ if there exists $q_r>0$ as

$$\begin{array}{c}\hfill \parallel H^k(\dots ,K_r\tilde{w},\dots )-H^k(\dots ,K_r\tilde{v},\dots )\parallel \le q_r\parallel w-v\parallel ,\forall v,w\in \mathcal{B};\end{array}$$

(iv) 

$({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4\dots {\alpha }_{k-1}{\beta }_k,\eta )$-symmetric accretive with $K_1,K_2,\dots ,K_k$ iff for $r\in \{1,3,\dots ,k-1\}$, $H^k(\dots ,K_r,\dots )$ is $({\alpha }_r,\eta )$-strongly accretive with $K_r$ and for $r\in \{2,4,\dots ,k\}$, $H^k(\dots ,K_r,\dots )$ is $({\beta }_r,\eta )$-relaxed accretive with $K_r$, when k is even, satisfying

$$\begin{array}{ccc}& & {\alpha }_1+{\alpha }_3+\dots +{\alpha }_{k-1}\ge {\beta }_2+{\beta }_4+\dots +{\beta }_k\hfill \end{array}$$

and ${\alpha }_1+{\alpha }_3+\dots +{\alpha }_{k-1}={\beta }_2+{\beta }_4+\dots +{\beta }_k$iff $w=v.$

(v) 

$({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4,\dots {\beta }_{k-1},{\alpha }_{k,\eta )}$-symmetric accretive with $K_1,K_2,\dots ,K_k$ iff for $r\in \{1,3,\dots ,p\}$, $H^k(\dots ,K_r,\dots )$ is $({\alpha }_r,\eta )$-strongly accretive with $K_r$ and for $r\in \{2,4,\dots ,k-1\}$, $H^k(\dots ,K_r,\dots )$ is $({\beta }_r,\eta )$-relaxed accretive with $K_r$ when k is odd, satisfying

$$\begin{array}{ccc}& & {\alpha }_1+{\alpha }_3+\dots +{\alpha }_k\ge {\beta }_2+{\beta }_4+\dots +{\beta }_{k-1}\hfill \end{array}$$

and ${\alpha }_1+{\alpha }_3+\dots +{\alpha }_k\ge {\beta }_2+{\beta }_4+\dots +{\beta }_{k-1}$ iff $w=v.$

Definition 5

([7]). Let $\mathcal{A}:\mathcal{B}\to \mathcal{B}$ be a mapping and $\mathcal{R},\mathcal{S}:\mathcal{B}\to 2^{\mathcal{B}}$ be the set-valued mappings, then:

(i) 

Graph of $\mathcal{R}$ is given as:

$$\mathrm{Gr}\left(\mathcal{R}\right)=\left\{\right(w,z):z\in \mathcal{R}(w\left)\right\};$$

(ii) 

Domain of $\mathcal{R}$ is given as:

$$\mathrm{D}\left(\mathcal{R}\right)=\{w\in \mathcal{B}:\exists z\in \mathcal{B}:(w,z)\in \mathcal{R}\};$$

(iii) 

Range of $\mathcal{R}$ is given as:

$$\mathrm{RG}\left(\mathcal{R}\right)=\{z\in \mathcal{B}:\exists w\in \mathcal{B}:(w,z)\in \mathcal{R}\};$$

(iv) 

Inverse of $\mathcal{R}$ is given as:

$${\mathcal{R}}^{-1}=\{(z,w):(w,z)\in \mathcal{R}\};$$

(v) 

Set $\mathcal{S}+\mathcal{R}$, $\mathcal{A}+\mathcal{R}$ and $\beta \mathcal{S}$ for any real number β are given as:

$$\mathcal{S}+\mathcal{R}=\left\{\right(u,w+z):(u,w)\in \mathcal{S},(u,z)\in \mathcal{R}\},$$

$$A+\mathcal{R}=\left\{\right(u,w+z):Au=w,(u,z)\in \mathcal{R}\},$$

$$\beta \mathcal{S}=\left\{\right(w,\beta z):(w,z)\in \mathcal{R}\}.$$

Definition 6.

For every $r\in \{1,2,\dots ,k\},k\ge 3$, let $g_r:\mathcal{B}\to \mathcal{B},\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$ be single-valued mappings and $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ be a set-valued mapping. Then, $\mathcal{M}$ is called

(i) 

$({\overline{\mu }}_r,\eta )$-strongly accretive with $g_r$ if there exists ${\overline{\mu }}_r>0$ as

$$\begin{array}{ccc}& & \left[{\tilde{w}}_r-{\tilde{v}}_r,\eta (w,v)\right]\ge {\overline{\mu }}_r{\parallel w-v\parallel }^2,\forall v,w\in \mathcal{B},{\tilde{v}}_r\in \mathcal{M}(.,\dots ,g_r\left(v\right),\dots ,.),\hfill \\ & & {\tilde{w}}_r\in \mathcal{M}(.,\dots ,g_r\left(w\right),\dots ,.);\hfill \end{array}$$

(ii) 

$({\overline{\gamma }}_r,\eta )$-relaxed accretive with $g_r$ if there exists ${\overline{\gamma }}_r>0$ as

$$\begin{array}{ccc}& & \left[{\tilde{w}}_r-{\tilde{v}}_r,\eta (w,v)\right]\ge -{\overline{\gamma }}_r{\parallel w-v\parallel }^2,\forall v,w\in \in B,{\tilde{v}}_r\in \mathcal{M}(.,\dots ,g_r\left(v\right),\dots ,.),\hfill \\ & & {\tilde{w}}_r\in \mathcal{M}(.,\dots ,g_r\left(w\right),\dots ,.);\hfill \end{array}$$

(iii) 

$({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\mu }}_{k-1}{\overline{\gamma }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$ iff for $r\in \{1,3,\dots ,k-1\}$, $\mathcal{M}(\dots ,g_r,\dots )$ is $({\overline{\mu }}_r,\eta )$-strongly accretive with $g_r$ and for $r\in \{2,4,\dots ,k\}$, $\mathcal{M}(\dots ,g_r,\dots )$ is $({\overline{\gamma }}_r,\eta )$-relaxed accretive with $g_r$, when k is even, satisfying

$$\begin{array}{ccc}& & {\overline{\mu }}_1+{\overline{\mu }}_3+\dots +{\overline{\mu }}_{k-1}\ge {\overline{\gamma }}_2+{\overline{\gamma }}_4+\dots +{\overline{\gamma }}_k\hfill \end{array}$$

and ${\overline{\mu }}_1+{\overline{\mu }}_3+\dots +{\overline{\mu }}_{k-1}={\overline{\gamma }}_2+{\overline{\gamma }}_4+\dots +{\overline{\gamma }}_k\mathrm{iff}w=v.$

(iv) 

$({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4,\dots {\overline{\mu }}_k,{\overline{\gamma }}_{k-1},\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$ iff for $r\in \{1,3,\dots ,k\}$, $\mathcal{M}(\dots ,g_r,\dots )$ is $({\overline{\mu }}_r,\eta )$-strongly accretive with $g_r$ and for $r\in \{2,4,\dots ,k-1\}$, $\mathcal{M}(\dots ,g_r,\dots )$ is $({\overline{\gamma }}_r,\eta )$-relaxed accretive with $g_r$, when k is odd, satisfying

$$\begin{array}{ccc}& & {\overline{\mu }}_1+{\overline{\mu }}_3+\dots +{\overline{\mu }}_k\ge {\overline{\gamma }}_2+{\overline{\gamma }}_4+\dots +{\overline{\gamma }}_{k-1}\hfill \end{array}$$

and ${\overline{\mu }}_1+{\overline{\mu }}_3+\dots +{\overline{\mu }}_k={\overline{\gamma }}_2+{\overline{\gamma }}_4+\dots +{\overline{\gamma }}_{k-1}$ iff $w=v.$

Definition 7.

A single-valued mapping $\mathcal{K}:{\mathcal{B}}^k\to \mathcal{B}$ is called ${\lambda }_r$-Lipschitz continuous in the $r^{th}$-component if there exists ${\lambda }_r>0$ as

$$\begin{array}{ccc}& & \parallel \mathcal{K}(\dots ,w,\dots )-\mathcal{K}(\dots ,v,\dots )\parallel \le {\lambda }_r\parallel w-v\parallel ,\forall v,w\in \mathcal{B}.\hfill \end{array}$$

2. Generalized $({\mathit{H}}^{\mathit{k}},\mathit{\phi })$-$\mathit{\eta }$-Accretive Mappings

For every $r\in \{1,2,\dots ,k\},k\ge 3$, let $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ be a set-valued mapping and $\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B},H^k:{\mathcal{B}}^k\to \mathcal{B}$, $K_r,g_r:\mathcal{B}\to \mathcal{B}$ and $\phi :\mathcal{B}\to \mathcal{B}$ be single-valued mappings. Now, we present and study the following new class of accretive mappings.

Definition 8.

Let $r\in \{1,2,\dots ,k\},k\ge 3$, then $\mathcal{M}$ is called generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$

(i) 

Iff $\phi \circ \mathcal{M}$ is $({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\mu }}_{k-1}{\overline{\gamma }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$ and $[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]$$\left(\mathcal{B}\right)=\mathcal{B}$ if k is even;

 

(ii) 

Iff $\phi \circ \mathcal{M}$ is $({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\gamma }}_{k-1}{\overline{\mu }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$ and $[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]$$\left(\mathcal{B}\right)=\mathcal{B}$ if k is odd.

Remark 1.

(i)   Let $\eta (w,v)=w-v$ and $\phi \left(u\right)=\rho u$, where $\rho >0$.

Then, Definition 8 coincides with the definition of the generalized $H^k(.,.,\dots ,.)$-accretive mapping considered by Gupta and Khan [36,37,38].

(ii) 

In addition to the above case (i), let $H^k(K_1,K_2,\dots ,K_k)=H(K_1,K_2)$ and $\mathcal{M}(g_1,g_2,\dots ,g_k)=\mathcal{M}(g_1,g_2)$.

Then, Definition 8 coincides with the definition of the generalized $H(.,.)$-accretive mapping considered by Kazmi et al. [8].

(iii) 

In addition to the above case (ii), let $\mathcal{M}(.,.,\dots )=\mathcal{M}(.).$

Then, Definition 8 coincides with the definition of $H(.,.)$-accretive considered by Zou and Huang [10].

(iv) 

In addition to the above case (iii) except $\eta (w,v)=w-v$.

Then, Definition 8 coincides with the definition of the $\left(H\right(.,.)$-$\eta )$-accretive mapping considered by Zou and Huang [43].

(v) 

In addition to the above case (iv) except $\phi \left(u\right)=\rho u$.

Then, Definition 8 coincides with the definition of the generalized $H(.,.)$-accretive mapping considered by Ahmad and Dilshad and Mishra and Diwan [11,12].

(vi) 

In addition to the case (i) excluding $\eta (w,v)=w-v$ and $\phi \left(u\right)=\rho u$, let $\mathcal{M}(.,.,\dots )=\mathcal{M}$, $H^k(.,.,..,.)=H$.

Then, Definition 8 coincides with the definition of the $(H,\phi )$-η-accretive mapping considered by Bhat and Zahoor [7] and Lou and Huang [35].

(vii) 

Let $H^k(K_{1}u,K_{2}u,.,..,K_{k}u)=Au$, $u\in \mathcal{B}$ and $\mathcal{M}(.,.,\dots )=\mathcal{M}$.

Then, Definition 8 coincides with the definition of the $(A,\eta )$-accretive mapping considered in [13,14]. Thus, the class of $(H^k,\phi )$-η-accretive mapping in Banach spaces is given a unifying framework for the various classes of accretive mappings (monotone operators). For details, see [6,18,19,20,21,22,23,29,31,32,33].

Now, we are illustrating an example to show that $\mathcal{M}$ is an $(H^k,\phi )$-$\eta$-accretive mapping with $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$.

Example 3.

Let $\mathcal{B}={\mathbb{R}}^2$ and $r\in \{1,2,\dots ,k\}$ with the usual semi-inner product

$$\left[(y_1,y_2),(x_1,x_2)\right]=y_1{x}_1+y_2{x}_2.$$

Let mapping $K_r$ for each $r\in \{1,2,\dots ,k\}$ be defined as

$$\begin{array}{ccc}& & K_1\left(y\right)=\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right),K_3\left(y\right)=\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right),\dots .,K_{k-1}\left(y\right)=\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right),\hfill \\ & & K_2\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right),K_4\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right),\dots .,K_k\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right).\hfill \end{array}$$

Assume that mapping $H^k$ is defined by

$$\begin{array}{ccc}& & H(K_1,K_2,\dots ,K_{k-1},K_k)\left(y\right)=K_1\left(y\right)-K_2\left(y\right)+\dots +K_{k-1}\left(y\right)-K_k\left(y\right).\hfill \end{array}$$

Let for any $u_2,u_3,\dots .u_k\in {\mathbb{R}}^2$,

$$\begin{array}{ccc}& & \left[H^k(K_1\left(y\right),u_2,\dots ,u_{k-1})-H^k(K_1\left(x\right),u_2,\dots ,u_{k-1}),\eta (y,x)\right]\hfill \\ & & =\left[(K_1\left(y\right)-u_2+\dots -u_k)-(K_1\left(x\right)-u_2+\dots -u_k),y-x\right]\hfill \\ & & =\left[K_1\left(y\right)-K_1\left(x\right),y-x\right]\hfill \\ & & =\left[\left(\genfrac{}{}{0pt}{}{y_1-x_1}{\frac{3(y_2-x_2)}{2}}\right),\left(\genfrac{}{}{0pt}{}{y_1-x_1}{y_2-x_2}\right)\right]\hfill \\ & & ={(y_1-x_1)}^2+\frac{3}{2}{(y_2-x_2)}^2\hfill \\ & & \ge {1\parallel y-x\parallel }^2.\hfill \end{array}$$

Thus, $H^k$ is $(1,\eta )$-strongly accretive with $K_1$. Similarly, we can check for each $r\in \{1,3,\dots ,k-1\}$.

Let for any $u_1,u_3,\dots .u_k\in {\mathbb{R}}^2$,

$$\begin{array}{ccc}& & \left[H^k(u_1,K_2\left(y\right),\dots ,u_{k-1})-H^k(u_1,K_2\left(x\right),\dots ,u_{k-1}),\eta (y,x)\right]\hfill \\ & & =\left[(u_1-K_2\left(y\right)+\dots -u_k)-(u_1-K_2\left(x\right)+\dots -u_k),y-x\right]\hfill \\ & & =-\left[K_2\left(y\right)-K_2\left(x\right),y-x\right]\hfill \\ & & =-\left[\left(\genfrac{}{}{0pt}{}{\frac{(y_1-x_1)}{2}}{\frac{(y_2-x_2)}{3}}\right),\left(\genfrac{}{}{0pt}{}{y_1-x_1}{y_2-x_2}\right)\right]\hfill \\ & & =-\left[\frac{1}{2}{(y_1-x_1)}^2+\frac{1}{3}{(y_2-x_2)}^2\right]\hfill \\ & & \ge -\frac{1}{2}{\parallel y-x\parallel }^2.\hfill \end{array}$$

Thus, $H^k$ is $(\frac{1}{2},\eta )$-relaxed accretive with $K_2$. Similarly, we can check for each $r\in \{2,4,\dots ,k\}$.

Assume that the mapping $g_r$ for each $r\in \{1,2,\dots ,k\}$ is given as

$$\begin{array}{ccc}& & g_1\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right),g_3\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right),\dots .,g_{k-1}\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right),\hfill \\ & & g_2\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{4}}\right),g_4\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{4}}\right),\dots .,g_k\left(y\right)=\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{2}}\right).\hfill \end{array}$$

Let set-valued mapping $\mathcal{M}$ be defined as

$$\begin{array}{ccc}& & \mathcal{M}(g_1,g_2,\dots ,g_{k-1},g_k)\left(y\right)=g_1\left(y\right)-g_2\left(y\right)+\dots +g_{k-1}\left(y\right)-g_k\left(y\right).\hfill \end{array}$$

Let for any $v_2,v_3,\dots .v_k\in {\mathbb{R}}^2$:

$$\begin{array}{ccc}& & \left[\phi \circ \mathcal{M}(g_1\left(y\right),v_2,\dots ,v_{k-1})-\phi \circ \mathcal{M}(g_1\left(x\right),v_2,\dots ,v_{k-1}),\eta (y,x)\right]\hfill \\ & & =\left[(g_1\left(y\right)-v_2+\dots -v_k)-(g_1\left(x\right)-y_2+\dots -y_k),y-x\right]\hfill \\ & & =\left[g_1\left(y\right)-g_1\left(x\right),y-x\right]\hfill \\ & & =\left[\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{x_1}{2}+\frac{x_2}{2}}{-\frac{x_1}{2}+\frac{x_2}{3}}\right),\left(\genfrac{}{}{0pt}{}{y_1}{y_2}\right)-\left(\genfrac{}{}{0pt}{}{x_1}{x_2}\right)\right]\hfill \\ & & =\frac{1}{2}{(y_1-x_1)}^2+\frac{1}{3}{(y_2-x_2)}^2\hfill \\ & & \ge \frac{1}{3}{\parallel y-x\parallel }^2.\hfill \end{array}$$

Thus, $\phi \circ \mathcal{M}$ is $(\frac{1}{3},\eta )$-strongly accretive with $g_1$. Similarly, we can check for each $r\in \{1,3,\dots ,k-1\}$.

Let for any $v_1,v_3,\dots .v_k\in {\mathbb{R}}^2$:

$$\begin{array}{ccc}& & \left[\phi \circ \mathcal{M}(v_1,g_2\left(y\right),\dots ,v_{k-1})-\phi \circ \mathcal{M}((v_1,g_2\left(x\right),\dots ,v_{k-1}),\eta (y,x)\right]\hfill \\ & & =\left[(v_1-g_2\left(y\right)+\dots -v_k)-(v_1-g_2\left(x\right)+\dots -y_k),y-x\right]\hfill \\ & & =-\left[g_2\left(y\right)-g_2\left(x\right),y-x\right]\hfill \\ & & =-\left[\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{4}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{x_1}{3}-x_2}{x_1+\frac{x_2}{4}}\right),\left(\genfrac{}{}{0pt}{}{y_1}{y_2}\right)-\left(\genfrac{}{}{0pt}{}{x_1}{x_2}\right)\right]\hfill \\ & & =-\left[\left(\genfrac{}{}{0pt}{}{\frac{(y_1-x_1)}{3}-(y_2-x_2)}{(y_1-x_1)+\frac{(y_2-x_2)}{4}}\right),\left(\genfrac{}{}{0pt}{}{y_1-x_1}{y_2-x_2}\right)\right]\hfill \\ & & =-\left[\frac{1}{3}{(y_1-x_1)}^2+\frac{1}{4}{(y_2-x_2)}^2\right]\hfill \\ & & \ge -\frac{1}{3}{\parallel y-x\parallel }^2.\hfill \end{array}$$

Thus, $\phi \circ \mathcal{M}$ is $(\frac{1}{3},\eta )$-accretive with $g_2$. Similarly, we can check for each $r\in \{2,4,\dots ,k\}$:

$$\begin{array}{ccc}\hfill & & [H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]\left(y\right)\hfill \\ & & =[H^k(K_1\left(y\right),K_2\left(y\right),\dots ,K_k\left(y\right))+\phi \circ \mathcal{M}(g_1\left(y\right),g_2\left(y\right),\dots ,g_k\left(y\right))]\hfill \\ & & =[K_1\left(y\right)-K_2\left(y\right)+\dots -K_k\left(y\right))+\phi (g_1\left(y\right)-g_2\left(y\right)+\dots -g_k\left(y\right))]\hfill \\ & & =[K_1\left(y\right)-K_2\left(y\right)+\dots -K_k\left(y\right))+\lambda (g_1\left(y\right)-g_2\left(y\right)+\dots -g_k\left(y\right))]\hfill \\ & & =\left[\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right)+\dots +\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right)\right]\hfill \\ & & +\lambda \left[\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{4}}\right)+\dots +\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{y_1}{2}+\frac{y_2}{3}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{y_2}{4}}\right)\right]\hfill \\ & & =\frac{k}{2}\left[\left(\genfrac{}{}{0pt}{}{y_1}{\frac{3y_2}{2}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}}{\frac{y_2}{3}}\right)\right]+\frac{k\lambda }{2}\left[\left(\genfrac{}{}{0pt}{}{\frac{y_1}{2}+\frac{y_2}{2}}{-\frac{z_1}{2}+\frac{y_2}{3}}\right)-\left(\genfrac{}{}{0pt}{}{\frac{y_1}{3}-y_2}{y_1+\frac{z_2}{4}}\right)\right]\hfill \\ & & =\frac{k}{2}\left[\left(\genfrac{}{}{0pt}{}{(3+\frac{\lambda }{6})y_1+3\frac{\lambda }{2}{y}_2}{-\frac{3\lambda }{2}{y}_1+\frac{14+\lambda }{12}{y}_2}\right)\right]\hfill \end{array}$$

(3)

It is easy to show that the whole ${\mathbb{R}}^2$ is generated by the right-hand side of (3).

$$\begin{array}{ccc}& & [H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]\left({\mathbb{R}}^2\right)={\mathbb{R}}^2.\hfill \end{array}$$

To discuss various aspects of generalized $(H^k,\phi )$-$\eta$-accretive mappings and move on to the subsequent sections, let us take into account the following assumptions $I_1$–$I_5$. 

• If k is even:
• ${\mathbf{I}}_{\mathbf{1}}$: Mapping $H^k$ is $({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4\dots {\alpha }_{k-1}{\beta }_k,\eta )$-symmetric accretive with $K_1,K_2,\dots ,K_k$.
• ${\mathbf{I}}_{\mathbf{2}}$: Mapping $\phi \circ \mathcal{M}$ is $({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\mu }}_{k-1}{\overline{\gamma }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$.
• If k is odd:
• ${\mathbf{I}}_{\mathbf{3}}$: Mapping $H^k$ is $({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4\dots {\beta }_{k-1}{\alpha }_k,\eta )$-symmetric accretive with $K_1,K_2,\dots ,K_k$.
• ${\mathbf{I}}_{\mathbf{4}}$: Mapping $\phi \circ \mathcal{M}$ is (${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\gamma }}_{k-1}{\overline{\mu }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$.
• ${\mathbf{I}}_{\mathbf{5}}$: Let $\eta$ be ℏ-Lipschitz continuous.

Theorem 1.

If assumptions $I_1$–$I_4$ are true for $r\in \{1,2,\dots k\}$, $k\ge 3$, and $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ is a generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$, then  $[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}$$(g_1,g_2,\dots ,g_k){]}^{-1}$ is single-valued.

Proof. 

For any given $u\in \mathcal{B}$, let $x,y\in {[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]}^{-1}\left(u\right)$. It follows that

$$\begin{array}{ccc}& & -H^k(K_{1}x,K_{2}x,\dots ,K_{k}x)+u\in \phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)x,\hfill \\ & & -H^k(K_{1}y,K_{2}y,\dots ,K_{k}y)+u\in \phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)y.\hfill \end{array}$$

If k is even: Since $\phi \circ \mathcal{M}$ is $({\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4\dots {\overline{\mu }}_{k-1}{\overline{\gamma }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$ and $H^k$ is $({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4\dots {\alpha }_{k-1}{\beta }_k,\eta )$-symmetric accretive with $K_1,K_2,\dots ,K_k$, we obtain:

$$\begin{array}{ccc}& & ({\overline{\mu }}_1-{\overline{\gamma }}_1+\dots +{\overline{\mu }}_{k-1}-{\overline{\gamma }}_k){\parallel x-y\parallel }^2\hfill \\ & & \le [-H^k(K_{1}x,K_{2}x,\dots ,K_{k}x)+u\hfill \\ & & -(-H^k(K_{1}y,K_{2}y,\dots ,K_{k}y)+u),\eta (x,y)]\hfill \\ & & \le -\left[H^k(K_{1}x,K_{2}x,\dots ,K_{k}x)-H^k(K_{1}y,K_{2}y,\dots ,K_{k}y),\eta (x,y)\right]\hfill \\ & & =-\left[H^k(K_{1}x,K_{2}x,\dots ,K_{k}x)-H^k(K_{1}y,K_{2}x,\dots ,K_{k}x),\eta (x,y)\right]\hfill \\ & & -\left[H^k(K_{1}y,K_{2}x,\dots ,K_{k}x)-H^k(K_{1}y,K_{2}y,\dots ,K_{k}x),\eta (x,y)\right]\hfill \\ & & :\hfill \\ & & :\hfill \\ & & -\left[H^k(K_{1}y,K_{2}y,\dots ,K_{k}x)-H^k(K_{1}y,K_{2}y,\dots ,K_{k}y),\eta (x,y)\right]\hfill \\ & & \le -\left[{\alpha }_1-{\beta }_2+{\alpha }_3-{\beta }_4+\dots +{\alpha }_{k-1}-{\beta }_k\right]{\parallel x-y\parallel }^2.\hfill \end{array}$$

It implies that

$$\begin{array}{ccc}\hfill & & \left[\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r\right]{\parallel x-y\parallel }^2\le 0.\hfill \end{array}$$

(4)

Since $\sum {\overline{\mu }}_r>\sum {\overline{\gamma }}_r$, $\sum {\alpha }_r>\sum {\beta }_r$, we have $\parallel x-y\parallel \le 0$. It implies that $x=y$. Thus, ${(H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k))}^{-1}$ is single-valued. Similarly, if k is odd, we can demonstrate the outcome.    □

Next, we use the Theorem 1 to define the following proximal mapping $R_{\phi ,\mathcal{M}(.,.,..,.)}^{\eta ,H^k(.,.,..,.)}$.

Definition 9.

If assumptions $I_1$–$I_4$ are true for $r\in \{1,2,\dots k\}$, $k\ge 3$, and $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ is a generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$, then a proximal mapping $R_{\phi ,\mathcal{M}(.,.,..,.)}^{\eta ,H^k(.,.,..,.)}:\mathcal{B}\to \mathcal{B}$ is defined as

$$\begin{array}{ccc}\hfill & & R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(x\right)={[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]}^{-1}\left(x\right),\forall x\in \mathcal{B}.\hfill \end{array}$$

(5)

Now, we discuss the Lipschitz continuity.

Theorem 2.

If assumptions $I_1$–$I_4$ are true for $r\in \{1,2,\dots k\}$, $k\ge 3$, and $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ is a generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$, then, the proximal mapping $R_{\phi ,\mathcal{M}(.,.,..,.)}^{\eta ,H^k(.,.,..,.)}:\mathcal{B}\to \mathcal{B}$ is $\frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}$-Lipschitz continuous,  where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\sum {\alpha }_r-\sum {\beta }_r={\alpha }_1-{\beta }_2+{\alpha }_3-{\beta }_4+\dots -{\alpha }_{k-1}+{\beta }_k,\hfill \\ \sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r={\overline{\mu }}_1-{\overline{\gamma }}_2+{\overline{\mu }}_3-{\overline{\gamma }}_4+\dots -{\overline{\mu }}_{k-1}+{\overline{\gamma }}_k,ifkiseven,\hfill \end{array}\right.\end{array}$$

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\sum {\alpha }_r-\sum {\beta }_r={\alpha }_1-{\beta }_2+{\alpha }_3-{\beta }_4+\dots -{\beta }_{k-1}+{\alpha }_k,\hfill \\ \sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r={\overline{\mu }}_1-{\overline{\gamma }}_2+{\overline{\mu }}_3-{\overline{\gamma }}_4+\dots -{\overline{\gamma }}_{k-1}+{\overline{\mu }}_k,ifkisodd.\hfill \end{array}\right.\end{array}$$

Proof. 

Let $x,y\in \mathcal{B}$ and from (5), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{R}_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(y\right)={(H^k(K_1,K_2,...,K_k)+\phi \circ \mathcal{M}(g_1,g_2,...,g_k))}^{-1}\left(z\right),\hfill \\ R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(z\right)={(H^k(K_1,K_2,...,K_k)+\phi \circ \mathcal{M}(g_1,g_2,...,g_k))}^{-1}\left(z\right).\hfill \end{array}\right.\end{array}$$

It follows that

$$\begin{array}{ccc}& & \left(y-H^k(K_1\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(y\right)\right),K_2\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(y\right)\right),\dots ,K_k\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(y\right)\right))\right)\hfill \\ & & \in \phi \circ \mathcal{M}\left(R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(y\right)\right),\hfill \\ & & \left(z-H^k(K_1\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(z\right)\right),K_2\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(z\right)\right),\dots ,K_k\left(R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}\left(z\right)\right))\right)\hfill \\ & & \in \phi \circ \mathcal{M}\left(R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(z\right)\right).\hfill \end{array}$$

Let $\widehat{y}=R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(y\right)$ and $\widehat{z}=R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(z\right).$

If k is even: Since $\phi \circ \mathcal{M}$ is $({\overline{\mu }}_1{\overline{\gamma }}_2\dots {\overline{\mu }}_{k-1}{\overline{\gamma }}_k,\eta )$-symmetric accretive with $g_1,g_2,\dots ,g_k$, we have

$$\begin{array}{ccc}& & ({\overline{\mu }}_1-{\overline{\gamma }}_1+\dots +{\overline{\mu }}_{k-1}-{\overline{\gamma }}_k){\parallel \widehat{y}-\widehat{z}\parallel }^2\hfill \\ & & \le \left[\widehat{y}-\widehat{z}-(H^k(K_1,K_2,\dots ,K_k)\left(\widehat{y}\right)-H^k(K_1,K_2,\dots ,K_k)\left(\widehat{z}\right)),\eta (\widehat{y},\widehat{z})\right].\hfill \end{array}$$

Using the fact that $H^k$ is $({\alpha }_1{\beta }_2{\alpha }_3{\beta }_4\dots {\alpha }_{k-1}{\beta }_k,\eta )$-symmetric accretive with $K_1,K_2,\dots ,K_k$ and $\eta (.,.)$ is ℏ Lipschitz continuous in the above inequality, we have

$$\begin{array}{ccc}& & \parallel y-z\parallel \hslash \parallel \widehat{y}-\widehat{z}\parallel \ge \parallel \widehat{y}-\widehat{z}\parallel \parallel \eta \left(\widehat{y},\widehat{z}\right)\parallel \ge \left[\widehat{y}-\widehat{z},\eta \left(\widehat{y},\widehat{z}\right)\right]\hfill \\ & & \ge \left[H^k(K_1,K_2,\dots ,K_k)\left(\widehat{y}\right)-H^k(K_1,K_2,\dots ,K_k)\left(\widehat{z}\right),\eta \left(\widehat{y},\widehat{z}\right)\right]\hfill \\ & & +\left(\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r\right)\parallel \widehat{y}-\widehat{z}{\parallel }^2\hfill \\ & & \ge {\alpha }_1\parallel \widehat{y}-\widehat{z}{\parallel }^2-{\beta }_2\parallel \widehat{y}-\widehat{z}{\parallel }^2+{\alpha }_3\parallel \widehat{y}-\widehat{z}{\parallel }^2-\dots {\beta }_k\parallel \widehat{y}-\widehat{z}{\parallel }^2\hfill \\ & & +\left(\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r\right)\parallel \widehat{y}-\widehat{z}{\parallel }^2\hfill \\ & & =\left[\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r\right]\parallel \widehat{y}-\widehat{z}{\parallel }^2\hfill \end{array}$$

or

$$\begin{array}{ccc}& & \parallel y-z\parallel \hslash \parallel \widehat{y}-\widehat{z}\parallel \ge \left[\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r\right]\parallel \widehat{y}-\widehat{z}{\parallel }^2.\hfill \end{array}$$

That is,

$$\begin{array}{ccc}& & \parallel R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(y\right)-R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(z\right)\parallel \le \frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}\parallel y-z\parallel ,\forall y,z\in \mathcal{B}.\hfill \end{array}$$

If k is odd: Similarly, we can demonstrate that $R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}$ is $\frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}$-Lipschitz continuous, where

$$\begin{array}{ccc}& & \sum {\alpha }_r-\sum {\beta }_r={\alpha }_1-{\beta }_2+{\alpha }_3-{\beta }_4+\dots -{\beta }_{k-1}+{\alpha }_k,\hfill \\ & & \sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r={\overline{\mu }}_1-{\overline{\gamma }}_2+{\overline{\mu }}_3-{\overline{\gamma }}_4+\dots -{\overline{\gamma }}_{k-1}+{\overline{\mu }}_k.\hfill \end{array}$$

   □

3. An Application

For every $r\in \{1,2,\dots ,k\},k\ge 3$, let $H^k,\mathcal{K}:{\mathcal{B}}^k\to \mathcal{B}$, $\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$, and $\mathcal{A},K_r,g_r,p,\phi :\mathcal{B}\to \mathcal{B}$ be single-valued mappings, ${\mathcal{N}}_r:\mathcal{B}\to 2^{\mathcal{B}}$ be set-valued mappings. Let set-valued mapping $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ be a generalized $(H^k,\phi )$-$\eta$-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$.

We study the set-valued variational-like inclusions problem (SVVLIP, in short): for any $a\in \mathcal{B},$ find $t\in \mathcal{B},{\tilde{v}}_1\in {\mathcal{N}}_1\left(t\right),{\tilde{v}}_2\in {\mathcal{N}}_2\left(t\right),\dots ,{\tilde{v}}_k\in {\mathcal{N}}_k\left(t\right)$ such that

$$a\in \mathcal{A}(t-p\left(t\right))+\mathcal{M}(g_1\left(t\right),g_2\left(t\right),\dots ,g_k\left(t\right))-\mathcal{K}({\tilde{v}}_1,{\tilde{v}}_2,\dots ,{\tilde{v}}_k).$$

(6)

SVVLIP (6) coincides with the following general variational inclusion (GVIP): find $t\in \mathcal{B},v\in \mathcal{N}\left(t\right)$ such that

$$0\in \mathcal{A}\left(t\right)+\mathcal{M}\left(v\right),$$

(7)

when $a=0$, $\mathcal{K},p=0$, $\mathcal{M}(g_1\left(t\right),g_2\left(t\right),\dots ,g_k\left(t\right))=M$, $N_1=\dots =N_k=N$, $\phi \left(u\right)=\rho u,\rho >0$, $\eta (u,v)=u-v$ for each $u,v\in \mathcal{B}$.

GVIP (7) was studied by Zou and Huang [10] if M is a $H(.,.)$-accretive mapping in Banach spaces.

In addition to the above case, let $\mathcal{B}$ be a Hilbert space X and $\mathcal{M}$ is a H-monotone mapping, then GVIP (7) was studied by Fang and Huang in [15]. GVIP (7) contains complementarity problems and some variational inclusions (inequalities) as special cases; see the references [44,45].

Note: For a given appropriate selection of $K_r,g_r,{\mathcal{N}}_r,\mathcal{M}\phi ,\eta$, $r\in \{1,2,\dots ,k\}$, and the spaces $\mathcal{B}$, the SGVLIP (6) coincides with different classes of variational inclusions and variational inequalities. See the relevant references [13,16,36].

Definition 10.

A set-valued mapping $\mathcal{N}:\mathcal{B}\to CB\left(\mathcal{B}\right)$ is called $\tilde{\mathcal{H}}$-Lipschitz continuous with $\zeta >0$, if

$$\begin{array}{c}\hfill \tilde{\mathcal{H}}(\mathcal{N}y,\mathcal{N}z)\le \zeta \parallel y-z\parallel ,\forall y,z\in \mathcal{B}.\end{array}$$

Lemma 3.

Let $\phi :\mathcal{B}\to \mathcal{B}$ be a mapping with $\phi ({\tilde{u}}_1+{\tilde{u}}_2)=\phi \left({\tilde{u}}_1\right)+\phi \left({\tilde{u}}_2\right)$ and $Kernel\left(\phi \right)=\left\{0\right\}$, where $Kernel\left(\phi \right)=\{\tilde{w}\in \mathcal{B}:\phi \left(\tilde{w}\right)=0\}$. For any $(t,{\tilde{v}}_1,{\tilde{v}}_2,\dots ,{\tilde{v}}_k)$, where $t\in \mathcal{B}$, ${\tilde{v}}_1\in {\mathcal{N}}_1\left(t\right),{\tilde{v}}_2\in {\mathcal{N}}_2\left(t\right),\dots ,{\tilde{v}}_k\in {\mathcal{N}}_k\left(t\right)$ is a solution of SVVLIP (6) iff $(t,{\tilde{v}}_1,{\tilde{v}}_2,\dots ,{\tilde{v}}_k)$ satisfies:

$$\begin{array}{ccc}\hfill & & t=R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}[H^k(K_1,K_2,\dots ,K_k)\left(t\right)-\mathcal{A}(t-p\left(t\right))+a\hfill \\ & & +\phi \circ \mathcal{K}({\tilde{v}}_1,{\tilde{v}}_2,\dots ,{\tilde{v}}_k)],\hfill \end{array}$$

(8)

where $R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}\left(t\right)={[H^k(K_1,K_2,\dots ,K_k)+\phi \circ \mathcal{M}(g_1,g_2,\dots ,g_k)]}^{-1}\left(t\right),$$\forall t\in \mathcal{B}$.

Proof. 

The proof of the above lemma is a direct consequence of the generalized $(H^k,\phi )$-$\eta$-accretive mapping; therefore, the proof is omitted.    □

In order to obtain an approximate solution to the SVVLIP (6), we now propose the iterative scheme based on Lemma 3.

Here, we provide sufficient convergence criteria for iterative sequences developed by Algorithm 1.

Algorithm 1

For any given $t_0\in \mathcal{B}$, select ${\tilde{v}}_0^1\in {\mathcal{N}}_1\left(t_0\right),{\tilde{v}}_0^2\in {\mathcal{N}}_2\left(t_0\right),\dots ,{\tilde{v}}_0^k\in {\mathcal{N}}_k\left(t_0\right)$ and obtain $\left\{t_n\right\}$, $\left\{{\tilde{v}}_n^1\right\}$, $\left\{{\tilde{v}}_n^2\right\}$,..., $\left\{{\tilde{v}}_n^k\right\}$, by the following iterative scheme: $$\begin{array}{ccc}\hfill & & t_{n+1}=R_{\phi ,\mathcal{M}(.,.,...)}^{\eta ,H^k(.,.,...)}[H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-\mathcal{A}(t_n-p\left(t_n\right))+a\hfill \\ & & +\phi \circ K({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^p)],\hfill \\ & & {\tilde{v}}_n^1\in {\mathcal{N}}_1\left(t_n\right):\parallel {\tilde{v}}_{n+1}^1-{\tilde{v}}_n^1\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{H}}({\mathcal{N}}_1\left(t_{n+1}\right),{\mathcal{N}}_1\left(t_n\right)),\hfill \\ & & {\tilde{v}}_n^2\in {\mathcal{N}}_2\left(t_n\right):\parallel {\tilde{v}}_{n+1}^2-{\tilde{v}}_n^2\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{H}}({\mathcal{N}}_2\left(t_{n+1}\right),{\mathcal{N}}_2\left(t_n\right)),\hfill \\ & & :\hfill \\ & & :\hfill \\ & & {\tilde{v}}_n^k\in {\mathcal{N}}_k\left(t_n\right):\parallel {\tilde{v}}_{n+1}^k-{\tilde{v}}_n^k\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{H}}({\mathcal{N}}_k\left(t_{n+1}\right),{\mathcal{N}}_k\left(t_n\right)),\hfill \end{array}$$ $n=0,1,2,\dots .$ and $\tilde{\mathcal{H}}(.,.)$ is the Hausdorff metric on $CB\left(\mathcal{B}\right)$.

Theorem 3.

For every $r\in \{1,2,\dots ,k\}$, $k\ge 3$, let $g_r,K_r:\mathcal{B}\to \mathcal{B}$, $\eta :\mathcal{B}\times \mathcal{B}\to \mathcal{B}$, $\mathcal{K}:{\mathcal{B}}^k\to \mathcal{B}$ be single-valued mappings, $H^k:{\mathcal{B}}^k\to \mathcal{B}$ be a $q_r$-Lipschitz continuous with $K_r$ and $p:\mathcal{B}\to \mathcal{B}$ be a δ-relaxed accretive and ${\lambda }_p$-Lipschitz continuous mapping. Let $\mathcal{A}:\mathcal{B}\to \mathcal{B}$ be a ${\lambda }_{\mathcal{A}}$-Lipschitz continuous, and for every $r\in \{1,2,\dots ,k\}$, ${\mathcal{N}}_r:\mathcal{B}\to 2^{\mathcal{B}}$ be ${\zeta }_i$-$\mathcal{H}$-Lipschitz continuous. Let $\phi :\mathcal{B}\to \mathcal{B}$ be a mapping with $\phi ({\tilde{u}}_1+{\tilde{u}}_2)=\phi \left({\tilde{u}}_1\right)+\phi \left({\tilde{u}}_2\right)$ and $Kernel\left(\phi \right)=\left\{0\right\}$, where $Kernel\left(\phi \right)=\{\tilde{w}\in \mathcal{B}:\phi \left(\tilde{w}\right)=0\}$ such that $\phi \circ \mathcal{K}$ is ${\lambda }_r$-Lipschitz continuous in the $r^{th}$-component. The set-valued mapping $\mathcal{M}:{\mathcal{B}}^k\to 2^{\mathcal{B}}$ is a generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$. In addition, the following conditions are satisfied:

$$\begin{array}{ccc}\hfill & & 0<\hslash \frac{\left[(q_1+q_2+\dots +q_k)+\left({\lambda }_1{\zeta }_1+{\lambda }_2{\zeta }_2+\dots +{\lambda }_k{\zeta }_k\right)+{\lambda }_{\mathcal{A}}\sqrt{1+2\delta +c{\lambda }_p}\right]}{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}<1.\hfill \end{array}$$

(9)

Then, iterative sequences $(\left\{t_n\right\},\left\{{\tilde{v}}_n^1\right\},\left\{{\tilde{v}}_n^2\right\},\dots .,\left\{{\tilde{v}}_n^k\right\})$ constructed from Algorithm 1 converge strongly to $(t,{\tilde{v}}^1,{\tilde{v}}^2,..,{\tilde{v}}^k)$, a solution of SVVLIP (6).

Proof. 

We now argue that $t_n$ strongly converges to t. From Theorem 2 and Algorithm 1, we have

$$\begin{array}{ccc}& & \parallel t_{n+1}-t_n\parallel =\parallel R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}[H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-\mathcal{A}(t_n-p\left(t_n\right))\hfill \\ & & +a+\phi \circ \mathcal{K}({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^k)]\hfill \\ & & -R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}[H^k(K_1,K_2,\dots ,K_k)\left(t_{n-1}\right)-\mathcal{A}(t_{n-1}-p\left(t_{n-1}\right))\hfill \\ & & +a+\phi \circ \mathcal{K}({\tilde{v}}_{n-1}^1,{\tilde{v}}_{n-1}^2,\dots ,{\tilde{v}}_{n-1}^k)]\parallel \hfill \\ & & \le \frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}\parallel [H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-\mathcal{A}(t_n-p\left(t_n\right))\hfill \\ & & +a+\phi \circ \mathcal{K}({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^k)]\hfill \\ & & -[H^k(K_1,K_2,\dots ,K_k)\left(t_{n-1}\right)-\mathcal{A}(t_{n-1}-p\left(t_{n-1}\right))\hfill \\ & & +a+\phi \circ \mathcal{K}({\tilde{v}}_{n-1}^1,{\tilde{v}}_{n-1}^2,\dots ,{\tilde{v}}_{n-1}^k)]\parallel \hfill \\ & & =\frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}\parallel [H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-H^k(K_1,K_2,\dots ,K_k)\left(t_{n-1}\right)]\hfill \\ & & -[\mathcal{A}(t_n-p\left(t_n\right))-\mathcal{A}(t_{n-1}-p\left(t_{n-1}\right))]\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +[\phi \circ \mathcal{K}({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^k)-\phi \circ \mathcal{K}({\tilde{v}}_{n-1}^1,{\tilde{v}}_{n-1}^2,\dots ,{\tilde{v}}_{n-1}^k)\parallel \hfill \\ & & \le \frac{\hslash }{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}\parallel H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-H^k(K_1,K_2,\dots ,K_k)\left(t_{n-1}\right)\parallel \hfill \\ & & +\parallel \mathcal{A}(t_n-p\left(t_n\right))-\mathcal{A}(t_{n-1}-p\left(t_{n-1}\right))\parallel \hfill \\ & & +\parallel \phi \circ \mathcal{K}({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^k)-\phi \circ \mathcal{K}({\tilde{v}}_{n-1}^1,{\tilde{v}}_{n-1}^2,\dots ,{\tilde{v}}_{n-1}^k)\parallel \hfill \end{array}$$

(10)

As $H^k$ is $q_r$-Lipschitz continuous with $K_r$, we have

$$\begin{array}{ccc}\hfill & & \parallel H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-H^k(K_1,K_2,\dots ,K_k)\left(t_{n-1}\right)\parallel \hfill \\ & & \le \parallel H^k(K_1{t}_n,K_2{t}_n,\dots ,K_k{t}_n)-H^k(K_1{t}_{n-1},K_2{t}_n,\dots ,K_k{t}_n)\parallel \hfill \\ & & +\parallel H^k(K_1{t}_{n-1},K_2{t}_n,\dots ,K_k{t}_n)-H^k(K_1{t}_{n-1},K_2{t}_{n-1},\dots ,K_k{t}_n)\parallel \hfill \\ & & :\hfill \\ & & :\hfill \\ & & +\parallel H^k(K_1{t}_{n-1},K_2{t}_{n-1},\dots ,K_k{t}_n)-H^k(K_1{t}_{n-1},K_2{t}_{n-1},\dots ,K_k{t}_{n-1})\parallel \hfill \\ & & \le q_1\parallel t_n-t_{n-1}\parallel +q_2\parallel t_n-t_{n-1}\parallel +\dots +q_k\parallel t_n-t_{n-1}\parallel |\hfill \\ & & =\left(q_1+q_2+\dots +q_k\right)\parallel t_n-t_{n-1}\parallel .\hfill \end{array}$$

(11)

Using the ${\lambda }_r$-Lipschitz continuity of $\phi \circ \mathcal{K}$ and ${\zeta }_i$-$\tilde{\mathcal{H}}$-Lipschitz continuity of ${\mathcal{N}}_r$, we have

$$\begin{array}{ccc}\hfill & & \parallel \phi \circ \mathcal{K}({\overline{v}}_n^1,{\overline{v}}_n^2,\dots ,{\overline{v}}_n^k)-\phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_{n-1}^2,\dots ,{\overline{v}}_{n-1}^k)\parallel \hfill \\ & & \le \parallel \phi \circ \mathcal{K}({\overline{v}}_n^1,{\overline{v}}_n^2,\dots ,{\overline{v}}_n^k)-\phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_n^2,\dots ,{\overline{v}}_n^k)\parallel \hfill \\ & & +\parallel \phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_n^2,\dots ,{\overline{v}}_n^k)-\phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_{n-1}^2,\dots ,{\overline{v}}_n^k)\parallel \hfill \\ & & :\hfill \\ & & :\hfill \\ & & +\parallel \phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_{n-1}^2,\dots ,{\overline{v}}_{n-1}^k)-\phi \circ \mathcal{K}({\overline{v}}_{n-1}^1,{\overline{v}}_{n-1}^2,\dots ,{\overline{v}}_n^k)\parallel \hfill \\ & & \le {\lambda }_1\parallel {\overline{v}}_n^1-{\overline{v}}_{n-1}^1\parallel +{\lambda }_2\parallel {\overline{v}}_n^2-{\overline{v}}_{n-1}^2\parallel +\dots .+{\lambda }_k\parallel {\overline{v}}_n^k-{\overline{v}}_{n-1}^k\parallel \hfill \\ & & \le {\lambda }_1\left(1+n^{-1}\right)\tilde{\mathcal{H}}({\mathcal{N}}_1\left(t_n\right),{\mathcal{N}}_1\left(t_{n-1}\right))+{\lambda }_2\left(1+n^{-1}\right)\tilde{\mathcal{H}}({\mathcal{N}}_2\left(t_n\right),{\mathcal{N}}_2\left(t_{n-1}\right))\hfill \\ & & +\dots +{\lambda }_k\left(1+n^{-1}\right)\tilde{\mathcal{H}}({\mathcal{N}}_k\left(t_n\right),{\mathcal{N}}_k\left(t_{n-1}\right))\hfill \\ & & \le \left(1+\frac{1}{n}\right)\left[{\lambda }_1{\zeta }_1+{\lambda }_2{\zeta }_2+\dots +{\lambda }_k{\zeta }_k\right]\parallel t_n-t_{n-1}\parallel \hfill \end{array}$$

(12)

We compute the following by using the ${\lambda }_{\mathcal{A}}$-Lipschitz continuity of $\mathcal{A}$:

$$\begin{array}{ccc}\hfill & & \parallel [\mathcal{A}(t_n-p\left(t_n\right))-\mathcal{A}(t_{n-1}-p\left(t_{n-1}\right))\parallel \le {\lambda }_{\mathcal{A}}\parallel t_n-p\left(t_n\right)-(t_{n-1}-p\left(t_{n-1}\right))\parallel \hfill \\ & & ={\lambda }_{\mathcal{A}}\parallel t_n-t_{n-1}-(p\left(t_n\right)-p\left(t_{n-1}\right))\parallel .\hfill \end{array}$$

(13)

Using the $\delta$-relaxed accretivity, the ${\lambda }_p$ Lipschitz continuity of p, and Lemma 1, we have

$$\begin{array}{ccc}\hfill & & \parallel t_n-t_{n-1}-(p\left(t_n\right)-p\left(t_{n-1}\right)){\parallel }^2={\parallel t_n-t_{n-1}\parallel }^2-2[p\left(t_n\right)-p\left(t_{n-1}\right),t_n-t_{n-1}]\hfill \\ & & +c\parallel p\left(t_n\right)-p\left(t_{n-1}\right){\parallel }^2\hfill \\ & & \le \parallel t_n-t_{n-1}{\parallel }^2-2[-\delta \parallel t_n-t_{n-1}{\parallel }^2]+c{\lambda }_p{\parallel t_n-t_{n-1}\parallel }^2\hfill \\ & & \le \left(1+2\delta +c{\lambda }_p\right)\parallel t_n-t_{n-1}{\parallel }^2.\hfill \end{array}$$

(14)

So, we have

$$\begin{array}{ccc}\hfill & & \parallel t_n-t_{n-1}-(p\left(t_n\right)-p\left(t_{n-1}\right))\parallel \le \sqrt{1+2\delta +c{\lambda }_p}\parallel t_n-t_{n-1}\parallel .\hfill \end{array}$$

(15)

Using (11)–(15) in (10), we obtain

$$\begin{array}{ccc}\hfill & & \parallel t_{n+1}-t_n\parallel ={\theta }_n\parallel t_n-t_{n-1}\parallel ,\hfill \end{array}$$

(16)

where

$$\begin{array}{ccc}\hfill & & {\theta }_n=\hslash \frac{\left[(q_1+q_2+\dots +q_k)+\left(1+\frac{1}{n}\right)\left({\lambda }_1{\zeta }_1+{\lambda }_2{\zeta }_2+\dots +{\lambda }_k{\zeta }_k\right)+{\lambda }_{\mathcal{A}}\sqrt{1+2\delta +c{\lambda }_p}\right]}{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}.\hfill \end{array}$$

We can obtain ${\theta }_n⟶\theta$ as $n⟶\infty$, where

$$\begin{array}{ccc}\hfill & & \theta =\hslash \frac{\left[(q_1+q_2+\dots +q_k)+\left({\lambda }_1{\zeta }_1+{\lambda }_2{\zeta }_2+\dots +{\lambda }_k{\zeta }_k\right)+{\lambda }_{\mathcal{A}}\sqrt{1+2\delta +c{\lambda }_p}\right]}{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}.\hfill \end{array}$$

It follows from (9) that $\theta \in (0,1)$. Thus, $\left\{t_n\right\}$ is a Cauchy sequence. Then, $t\in \mathcal{B}$ exists in such a way that $\left\{t_n\right\}⟶x$ as $n\to \infty$. Next, we show that ${\tilde{v}}_n^1\to {\tilde{v}}^1\in {\mathcal{N}}_1\left(t\right)$. From Algorithm 1 and the Lipschitz continuity of ${\mathcal{N}}_1$, we have

$$\begin{array}{ccc}& & \parallel {\tilde{v}}_n^1-{\tilde{v}}_{n-1}^1\parallel \le \left(1+\frac{1}{n}\right)\tilde{\mathcal{H}}({\mathcal{N}}_1\left(t_n\right),{\mathcal{N}}_1\left(t_{n-1}\right))\le \left(1+\frac{1}{n}\right){\zeta }_1\parallel t_n-t_{n-1}\parallel .\hfill \end{array}$$

It shows that $\left\{{\tilde{v}}_n^1\right\}$ is a Cauchy sequence. Similarly, we can prove that $\left\{{\tilde{v}}_n^2\right\},\dots ,\left\{{\tilde{v}}_n^k\right\}$ are Cauchy sequences. Then, there exist ${\tilde{v}}^1,{\tilde{v}}^2,\dots {\tilde{v}}^k$ such that ${\tilde{v}}_n^1\to {\tilde{v}}^1,{\tilde{v}}_n^2\to {\tilde{v}}^2,\dots ,{\tilde{v}}_n^k\to {\tilde{v}}^k,$ as $n\to \infty$. Now, we argue that ${\tilde{v}}^1\in {\mathcal{N}}_1\left(t\right)$. Since ${\tilde{v}}_n^1\in {\mathcal{N}}_1\left(t\right)$, we have

$$\begin{array}{ccc}& & d({\tilde{v}}^1,{\mathcal{N}}_1\left(t\right))\le \parallel {\tilde{v}}^1-{\tilde{v}}_n^1\parallel +\tilde{d}({\tilde{v}}_n^1,{\mathcal{N}}_1\left(t\right))\hfill \\ & & \le \parallel {\tilde{v}}^1-{\tilde{v}}_n^1\parallel +\tilde{\mathcal{H}}({\mathcal{N}}_1\left(t_n\right),{\mathcal{N}}_1\left(t\right))\hfill \\ & & \le \parallel {\tilde{v}}^1-{\tilde{v}}_n^1\parallel +{\zeta }_1\parallel t_n-t\parallel \mathrm{as}n\to \infty .\hfill \end{array}$$

Since ${\mathcal{N}}_1\left(t\right)$ is closed, thus ${\tilde{v}}^1\in {\mathcal{N}}_1\left(t\right)$. Through the same procedure, we can show that ${\tilde{v}}^2\in {\mathcal{N}}_2\left(t\right),{\tilde{v}}^3\in {\mathcal{N}}_3\left(t\right),\dots ,{\tilde{u}}^k\in {\mathcal{N}}_k\left(t\right)$. By the continuity of $R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}$, $\phi \circ \mathcal{K},K_r,$$g_r,{\mathcal{N}}_r,\mathcal{M}$ and Algorithm 1, it follows that $(t,{\tilde{v}}^1,{\tilde{v}}^2,\dots ,{\tilde{v}}^k)$ satisfies:

$$\begin{array}{ccc}& & t_{n+1}=R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}[H^k(K_1,K_2,\dots ,K_k)\left(t_n\right)-\phi \circ \mathcal{K}({\tilde{v}}_n^1,{\tilde{v}}_n^2,\dots ,{\tilde{v}}_n^k)]\hfill \\ & & \to t=R_{\phi ,\mathcal{M}(.,.,\dots )}^{\eta ,H^k(.,.,\dots )}[H^k(K_1,K_2,\dots ,K_k)\left(t\right)-\phi \circ \mathcal{K}({\tilde{v}}^1,{\tilde{v}}^2,\dots ,{\tilde{v}}^k)].\hfill \end{array}$$

By Lemma 3, SVVLIP (6) has a solution $(t,{\tilde{v}}^1,{\tilde{v}}^2,\dots ,{\tilde{v}}^k)$. □

Remark 2.

(i) Theorem 3 coincides with Theorem 4.1 in [10] if

$a=0$, $\mathcal{K},p=0$, $\mathcal{M}(g_1\left(t\right),g_2\left(t\right),\dots ,g_k\left(t\right))=M$, $\phi \left(u\right)=\rho u,\rho >0$, for each $u\in \mathcal{B}$, $\eta (u,v)=u-v$ for each $u,v\in \mathcal{B}$ each $v\in \mathcal{B}$.

(ii) 

In addition to the above case, let $\mathcal{M}$ is a H-monotone mapping, and $\mathcal{B}$ be a Hilbert space X. Then, Theorem 3 coincides with Theorem 3.1 in [15].

The functional space $L^{\tilde{p}}$ is a 2-uniformly smooth space when $\tilde{p}\in [2,\infty )$ is as given in Example 2. Let $\mathcal{B}=L^{\tilde{p}}$, where $\tilde{p}\in [2,\infty )$. Then, Theorem 3 yields the following outcome:

Corollary 1.

For every $r\in \{1,2,\dots ,k\}$, $k\ge 3$, let $g_r,K_r:L^{\tilde{p}}\to L^{\tilde{p}}$, $\eta :L^{\tilde{p}}\times L^{\tilde{p}}\to L^{\tilde{p}}$, $\mathcal{K}:{L^{\tilde{p}}}^k\to L^{\tilde{p}}$ be single-valued mappings, $H^k:{L^{\tilde{p}}}^k\to L^{\tilde{p}}$ be a $q_r$-Lipschitz continuous with $K_r$ and $p:L^{\tilde{p}}\to L^{\tilde{p}}$ be a δ-relaxed accretive and ${\lambda }_p$-Lipschitz continuous mapping. Let $\mathcal{A}:L^{\tilde{p}}\to L^{\tilde{p}}$ be a ${\lambda }_{\mathcal{A}}$-Lipschitz continuous, and for every $r\in \{1,2,\dots ,k\}$, ${\mathcal{N}}_r:L^{\tilde{p}}\to 2^{L^{\tilde{p}}}$ be ${\zeta }_i$-$\mathcal{H}$-Lipschitz continuous. Let $\phi :L^{\tilde{p}}\to L^{\tilde{p}}$ be a mapping with $\phi ({\tilde{u}}_1+{\tilde{u}}_2)=\phi \left({\tilde{u}}_1\right)+\phi \left({\tilde{u}}_2\right)$ and $Ker\left(\phi \right)=\left\{0\right\}$, where $Ker\left(\phi \right)=\{\tilde{w}\in L^{\tilde{p}}:\phi \left(\tilde{w}\right)=0\}$ such that $\phi \circ \mathcal{K}$ is ${\lambda }_r$-Lipschitz continuous in the $r^{th}$-component. The set-valued mapping $\mathcal{M}:{L^{\tilde{p}}}^k\to 2^{L^{\tilde{p}}}$ is a generalized $(H^k,\phi )$-η-accretive mapping with mappings $(K_1,K_2,\dots ,K_k)$ and $(g_1,g_2,\dots ,g_k)$. In addition, the following conditions are satisfied:

$$\begin{array}{c}\hfill 0<\hslash \frac{\left[(q_1+q_2+\dots +q_k)+\left({\lambda }_1{\zeta }_1+{\lambda }_2{\zeta }_2+\dots +{\lambda }_k{\zeta }_k\right)+{\lambda }_{\mathcal{A}}\sqrt{1+2\delta +(\tilde{p}-1){\lambda }_p}\right]}{\sum {\alpha }_r-\sum {\beta }_r+\sum {\overline{\mu }}_r-\sum {\overline{\gamma }}_r}<1,\end{array}$$

where $\tilde{p}-1$ is the smoothness constant. Then, iterative sequences $(\left\{t_n\right\},\left\{{\tilde{v}}_n^1\right\},\left\{{\tilde{v}}_n^2\right\},$$\dots .,\left\{{\tilde{v}}_n^k\right\})$ constructed from Algorithm 1 converge strongly to $(t,{\tilde{v}}^1,{\tilde{v}}^2,..,{\tilde{v}}^k)$, a solution of SVVLIP (6).

4. Conclusions

In this paper, we have investigated a new class of accretive mappings, known as generalized $(H_k,\phi )$-$\eta$-accretive mappings and their proximal mapping. As its significance role, we have studied a set-valued variational-like inclusion problem in the frame of semi-inner product spaces. An iterative scheme involving a generalized $(H_k,\phi )$-$\eta$ accretive mapping was given. Further, we have concentrated on the convergence of the proposed iterative scheme. An example is constructed in support of our result. The obtained result as “Theorem 3” in this paper unifies, improves, and extends the main results obtained in [10,15].