Set-Valued Variational Inclusion Governed by Generalized α_iβ_j-H^p(., ., ...)-Accretive Mapping

Abstract

This work is concerned with the new notion called generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping that is the sum of two symmetric accretive mappings. It is an extension of the generalized $\alpha \beta$-$H(., .)$-accretive mapping. The proximal point mapping linked to the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings is defined, and some of its characteristics are discussed. As an application of the new proximal point mapping, we consider a set-valued variational inclusion problem in q-uniformly smooth Banach spaces. Further, we propose an iterative scheme connected with ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-proximal point mapping to find the solution of a variational inclusion problem and discuss its convergence criteria under appropriate assumptions. Some examples are constructed in support of generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings.

1. Introduction and Preliminaries

Variational inequality is a very powerful tool to study a large variety of problems that appear in electricity, mechanics, operation research, optimal control, etc. The projection mapping techniques, shrinking projection mapping techniques, CQ methods, and proximal-point mapping techniques have been widely used to solve variational inequality (inclusions) problems, see [1,2,3,4,5,6,7,8,9]. The accretive property of the underlying proximal-point mapping has a significant role in the field of variational inequalities and their generalizations. Huang and Fang [10] were the first to consider and study m-accretive mappings and related proximal-point mappings in Banach spaces. After that, many researchers introduced and studied different kinds of generalized m-accretive mappings, see [1,4,5,6]. Sun et al. [7] proposed and analyzed M-monotone mappings in Hilbert spaces. A few research works linked to the M-monotone are provided in [3,9].

In particular, the investigation of $H(., .)$-accretive mappings has been very deeply carried out, and there have been many important results on the variational inclusions (or inequalities) for their related mappings, e.g., the characterization of $H(., .)$-monotone operators with applications to variational inclusions [11], the generalized nonlinear set-valued mixed quasi-variational inequalities [12], and the completely generalized set-valued quasi-variational inclusions [13].

In recent years, $H(., .)$-accretive mappings and generalized $\alpha \beta$-$H(., .)$-accretive mappings have been investigated and studied by many researchers, see for example, [3,9,14]. Very recently, Nazemi [15] investigated and studied $C_n$-monotone mappings, and Guan and Hu [16] considered $C_n$-$\eta$-monotone mappings in Banach spaces and studied some classes of variational inclusions involving these mappings.

Impelled by the ongoing research work presented above, in this paper we consider the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping defined on a product set, which is the sum of two symmetric accretive mappings. This notion is a generalized form of generalized $\alpha \beta$-$H^p(., .)$-accretive mapping et al. [3], which is done with the idea of $C_n$ monotone mapping studied and analyzed by Nazemi [15]. We define the proximal-point mapping associated with the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings and prove that it is single-valued and Lipschitz continuous. An iterative algorithm involving a generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping is constructed. We perform the convergence part of an iterative algorithm for the solution of set-valued variational inclusion problems with some suitable assumptions in the setting of q-uniformly smooth Banach spaces. Some examples are constructed in support of generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings. Using the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping technique described in this paper, one can extend and improve the results given in [3,9,14,15,16,17,18,19,20,21,22,23,24].

Let Y be a real Banach space endowed with the norm $\parallel .\parallel$ and an inner product $⟨., .⟩$, which presents the generalized duality pairing between Y and $Y^{*}$. Let $CB\left(Y\right)$ be the family of all nonempty closed and bounded subsets of Y and $2^Y$ be the power set of of Y. We set Y^p = $\stackrel{\underbrace{Y\times Y\times ...\times Y}}{p times}$.

Definition 1

([25]). The generalized duality mapping ${\mathit{j}}_{\mathit{q}}:Y\to 2^{Y^{*}}$ is defined by

$$\begin{array}{c}\hfill {\mathit{j}}_{\mathit{q}}\left(z\right)=\{\tilde{z}\in Y^{*}:⟨z,\tilde{z}⟩={\parallel z\parallel }^q,\parallel \tilde{z}{\parallel =\parallel z\parallel }^{q-1}\},\forall z\in Y,\end{array}$$

where $q>1$ is a constant. In particular, ${\mathit{j}}_2$ is the usual normalized duality mapping. It is known that, in general, ${\mathit{j}}_q\left(z\right)={\parallel z\parallel }^{q-1}\mathit{j}\left(z\right),\forall z\in Y.$ If Y is equivalent to real Hilbert space X, then ${\mathit{j}}_2$ becomes an identity mapping on X.

Definition 2

([25]). A Banach space Y is smooth if, for every $z\in Y$ with $\parallel z\parallel =1,$ there exists a unique $l\in Y^{*}$ such that $\parallel l\parallel =l\left(z\right)=1$.

Definition 3

([25]). Let ${\mathrm{\Omega }}_Y:[0,\infty )\to [0,\infty )$ be a mapping; then, the modulus of smoothness of Y at μ is defined by

$${\mathrm{\Omega }}_Y\left(\mu \right)=\sup \left\{\frac{\parallel z+\tilde{z}\parallel +\parallel z-\tilde{z}\parallel }{2}-1:\parallel z\parallel \le 1,\parallel \tilde{z}\parallel \le \mu \right\}.$$

Definition 4

([25]). A Banach space Y is said to be

(i)

uniformly smooth if ${\lim }_{\mu \to 0}\frac{{\mathrm{\Omega }}_Y\left(\mu \right)}{\mu }=0;$

(ii)

q-uniformly smooth$(q>1)$, ∃$k>0$with ${\mathrm{\Omega }}_Y\left(\mu \right)\le k{\mu }^q,\mu \in [0,\infty ).$

It is observed that ${\mathit{j}}_q$ is single-valued if Y is uniformly smooth.

Lemma 1

([25]). A real uniformly smooth Banach space Y is q-uniformly smooth if and only if there exists a constant $c_q>0$ such that, for every $z,\tilde{z}\in Y,$

$$\parallel z+\tilde{z}{\parallel }^q\le {\parallel z\parallel }^q+q⟨\tilde{z},{\mathit{j}}_q\left(z\right)⟩+c_q{\parallel \tilde{z}\parallel }^q.$$

The following new notions are needed to continue subsequent sections.

Definition 5.

For $i\in \{1,2,...,p\},p\ge 3$, let $H^p:Y^p\to Y$ be a mapping and $A_i:Y\to Y$ be a single-valued mapping. Then, $H^p$ is said to be

(i)

${\alpha }_i$-strongly accretive with $A_i$ if there exists ${\alpha }_i>0$ such that

$$\begin{array}{c}\hfill ⟨H^p(v_1,...,v_{i-1},A_i\tilde{w},v_{i+1},...,v_n)-H^p(v_1,...,v_{i-1},A_i\tilde{v},v_{i+1},...,v_n),J_q(\tilde{w}-\tilde{v})⟩\\ \hfill \ge {\alpha }_i{\parallel \tilde{w}-\tilde{v}\parallel }^q,\forall \tilde{w},\tilde{v},v_1,...,v_{i-1},v_{i+1},...,v_n\in Y;\end{array}$$

(ii)

${\beta }_i$-relaxed accretive with $A_i$ if there exists ${\beta }_i>0$ such that

$$\begin{array}{c}\hfill ⟨H^p(v_1,...,v_{i-1},A_i\tilde{w},v_{i+1},...,v_n)-H^p(v_1,...,v_{i-1},A_i\tilde{v},v_{i+1},...,v_n),J_q(\tilde{w}-\tilde{v})⟩\\ \hfill \ge -{\beta }_i{\parallel \tilde{w}-\tilde{v}\parallel }^q,\forall \tilde{w},\tilde{v},v_1,...,v_{i-1},v_{i+1},...,v_n\in Y;\end{array}$$

(iii)

$s_i$-Lipschitz continuous with $A_i$ if there exists $s_i>0$ such that

$$\begin{array}{c}\hfill \parallel H^p(v_1,...,v_{i-1},A_i\tilde{w},v_{i+1},...,v_n)-H^p(v_1,...,v_{i-1},A_i\tilde{v},v_{i+1},...,v_n)\parallel \le s_i\parallel \tilde{w}-\tilde{v}\parallel ,\\ \hfill \forall \tilde{w},\tilde{v},v_1,...,v_{i-1},v_{i+1},...,v_n\in Y;\end{array}$$

(iv)

${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4...{\alpha }_{p-1}{\beta }_p$-symmetric accretive with $A_1,A_2,...,A_p$ if and only if, for $i\in \{1,3,...,p-1\}$, $H^p(...,$$A_i,...)$ is ${\alpha }_i$-strongly accretive with $A_i$, and for $j\in \{2,4,...,p\}$, $H^p(...,A_j,...)$ is ${\beta }_j$-relaxed accretive with $A_j$, where p is even, satisfying

$${\beta }_2+{\beta }_4+...+{\beta }_p\le {\alpha }_1+{\alpha }_3+...+{\alpha }_{p-1}$$

and ${\beta }_2+{\beta }_4+...+{\beta }_p={\alpha }_1+{\alpha }_3+...+{\alpha }_{p-1}$ if and only if $\tilde{w}=\tilde{v}$;

(v)

${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4,...{\beta }_{p-1},{\alpha }_p$-symmetric accretive with $A_1,A_2,...,A_p$ if and only if, for $i\in \{1,3,...,p\}$,

$H^p(...,A_i,...)$ is ${\alpha }_i$-strongly accretive with $A_i$, and for $j\in \{2,4,...,p-1\}$, $H^p(...,A_j,...)$ is ${\beta }_j$-relaxed accretive $A_j$ where p is odd, satisfying

$${\beta }_2+{\beta }_4+...+{\beta }_{p-1}\le {\alpha }_1+{\alpha }_3+...+{\alpha }_p$$

and ${\beta }_2+{\beta }_4+...+{\beta }_{p-1}={\alpha }_1+{\alpha }_3+...+{\alpha }_p$ iff $\tilde{w}=\tilde{v}$.

Definition 6.

For $i\in \{1,2,...,p\},p\ge 3$, let $\mathcal{N}:Y^p\to 2^Y$ be a set-valued mapping and $f_i:Y\to Y$ be a single-valued mapping. Then, $\mathcal{N}$ is said to be

(i)

${\overline{\mu }}_i$-strongly accretive with $f_i$ if there exists ${\overline{\mu }}_i>0$ such that

$$\begin{array}{ccc}& & ⟨{\tilde{w}}_i-{\tilde{v}}_i,J_q(\tilde{w}-\tilde{v})⟩\ge {\overline{\mu }}_i{\parallel \tilde{w}-\tilde{v}\parallel }^q,\forall \tilde{w},\tilde{v},v_1,...,v_{i-1},v_{i+1},...,v_p\in Y,\hfill \\ & & {\tilde{w}}_i\in \mathcal{N}(v_1,...,v_{i-1},f_i\left(\tilde{w}\right),v_{i+1},...,v_n),{\tilde{v}}_i\in \mathcal{N}(v_1,...,v_{i-1},f_i\left(\tilde{v}\right),v_{i+1},...,v_n);\hfill \end{array}$$

(ii)

${\overline{\gamma }}_i$-relaxed accretive with $f_i$ if there exists ${\overline{\gamma }}_i>0$ such that

$$\begin{array}{ccc}& & ⟨{\tilde{w}}_i-{\tilde{v}}_i,J_q(\tilde{w}-\tilde{v})⟩\ge -{\overline{\gamma }}_i{\parallel \tilde{w}-\tilde{v}\parallel }^q,\forall \tilde{w},\tilde{v},v_1,...,v_{i-1},v_{i+1},...,v_p\in Y,\hfill \\ & & {\tilde{w}}_i\in \mathcal{N}(v_1,...,v_{i-1},f_i\left(\tilde{w}\right),v_{i+1},...,v_n),{\tilde{v}}_i\in \mathcal{N}(v_1,...,v_{i-1},f_i\left(\tilde{v}\right),v_{i+1},...v_n);\hfill \end{array}$$

(iii)

${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$ if and only if, for $i\in \{1,3,...,p-1\}$, $\mathcal{N}(...,f_i,...)$ is ${\overline{\mu }}_i$-strongly accretive with $f_i$, and for $j\in \{2,4,...,p\}$, $\mathcal{N}(...,f_j,...)$ is ${\overline{\gamma }}_j$-relaxed accretive with $f_j$, where p is even, satisfying

$${\overline{\gamma }}_2+{\overline{\gamma }}_4+...+{\overline{\gamma }}_p\le {\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1}$$

and ${\overline{\gamma }}_2+{\overline{\gamma }}_4+...+{\overline{\gamma }}_p={\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1}$ iff $\tilde{w}=\tilde{v}$;

(iv)

${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4,...{\overline{\mu }}_p,{\overline{\gamma }}_{p-1}$-symmetric accretive with $f_1,f_2,...,f_p$ if and only if, for $i\in \{1,3,...,p\}$, $\mathcal{N}(...,f_i,...)$ is ${\overline{\mu }}_i$-strongly accretive with $f_i$, and for $j\in \{2,4,...,p-1\}$, $\mathcal{N}(...,f_j,...)$ is ${\overline{\gamma }}_j$-relaxed accretive with $f_j$, where p is odd, satisfying

$${\overline{\gamma }}_2+{\overline{\gamma }}_4+...+{\overline{\gamma }}_{p-1}\le {\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_p$$

and ${\overline{\gamma }}_2+{\overline{\gamma }}_4+...+{\overline{\gamma }}_{p-1}={\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_p$ if and only if $\tilde{w}=\tilde{v}$.

Definition 7.

For $i\in \{1,2,...,p\},p\ge 3,$ let $\mathcal{N}:Y^p\to 2^Y$, $S_i:Y\to 2^Y$ be set-valued mappings, $H^p,F:Y^p\to Y$ be mappings, and $A_i:Y\to Y$ be single-valued mappings; then, F is said to be

(i)

${\overline{\alpha }}_i$-strongly accretive with $S_i$ and $H^p$ in the $ith$-argument if there exists ${\overline{\alpha }}_i>0$ such that

$$\begin{array}{ccc}& & ⟨F(...,w^i,...)-F(...,v^i,...),J_q(H^p(A_1{w}_1,...,A_p{w}_p)-H^p(A_1{v}_1,...,A_p{v}_p))⟩\hfill \\ & & \ge {\overline{\alpha }}_i\parallel H^p(A_1{w}_1,...,A_p{w}_p)-H^p{(A_1{v}_1,...,A_p{v}_p\parallel }^q,\hfill \\ & & \forall w_1,w_2...,w_p,v_1,v_2,...,v_p\in Y,w^i\in S\left(w_i\right),v^i\in S\left(v_i\right);\hfill \end{array}$$

(ii)

$l_i$-Lipschitz continuous in the $ith$-argument if there exists $l_i>0$ such that

$$\begin{array}{ccc}& & \parallel F(v_1,..,v_{i-1},\tilde{w},v_{i+1}...v_p)-F(v_1,..,v_{i-1},\tilde{v},v_{i+1}...v_p)\parallel \le l_i\parallel \tilde{w}-\tilde{v}\parallel ,\hfill \\ & & \forall \tilde{w},\tilde{v},v_1,..,v_{i-1},v_{i+1}...v_p\in Y.\hfill \end{array}$$

2. Generalized α_iβ_j-H^p$(., ., ..., .)$ Accretive Mappings

At first, we consider some assumptions ($M_1$-$M_4$) to introduce and study the new notion generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ..., .)$-accretive mappings.

For $i\in \{1,2,...,p\},p\ge 3,$ let $\mathcal{N}:Y^p\to 2^Y$ be a set-valued mapping, $H^p:Y^p\to Y$ be a mapping, and $A_i,f_i:Y\to Y$ be single-valued mappings.

• $M_1$: If p is an even number, $H^p$ is ${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4...{\alpha }_{p-1}{\beta }_p$-symmetric accretive with $A_1,A_2,...,A_p$.
• $M_2$: If p is an odd number, $H^p$ is ${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4...{\beta }_{p-1}{\alpha }_p$-symmetric accretive with $A_1,A_2,...,A_p$.
• $M_3$: If p is an even number, N is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$.
• $M_4$: If p is an odd number, N is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\gamma }}_{p-1}{\overline{\mu }}_p$-symmetric accretive with $f_1,f_2,...,f_p$.

Definition 8.

Let $p\ge 3$; then, $\mathcal{N}$ is said to be a generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ..., .)$-accretive mapping with $A_1,A_2,...,A_p$ and $f_1,f_2,...,f_p$

(i)

if and only if $\mathcal{N}$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$ (i.e., assumption $M_3$ holds), and $(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))\left(Y\right)=Y$, for all $\rho >0$, if p is an even number;

(ii)

if and only if $\mathcal{N}$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\gamma }}_{p-1}{\overline{\mu }}_p$-symmetric accretive with $f_1,f_2,...,f_n$ (i.e., assumption $M_4$ holds), and $(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))\left(Y\right)=Y$, for all $\rho >0$, if p is an odd number.

Example 1.

Let p be an even number and $H^p$ be ${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4...{\alpha }_{p-1}{\beta }_p$-symmetric accretive with $A_1,A_2,...,A_p$. Let $f:Y\to \mathbb{R}$ be a functional on Y; then, the vector $\tilde{x}$, such that

$$\begin{array}{c}\hfill f\left(z\right)-f\left(y\right)\ge ⟨\tilde{x},z-y⟩,\forall z\in Y,\end{array}$$

(1)

where $f\left(z\right)$ is finite for each $z\in Y$, is known as the subgradient of f at z. The collections of all such subgradients of f at z holding (1) are known as the subdifferential $\delta f\left(z\right)$ of f at z. Let $f_1,f_2,...,f_p:Y\to Y$ be locally Lipschitz functionals on Y such that subdifferential $\delta (f_1,f_2,...,f_p)$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$. Then, $H^p(.,.,...)+\delta (f_1,f_2,...,f_p)$ is $\mathcal{L}$-strongly accretive, where $\mathcal{L}=\left[\sum {\alpha }_i+\sum {\overline{\mu }}_i-\left(\sum {\beta }_j+\sum {\overline{\gamma }}_j\right)\right]$$>0$; equivalently, we can state that $\delta (f_1,f_2,...,f_p)$ is a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...)$-accretive mapping.

Example 2.

For each $i\in \{1,2,...,p\}$, p is an even number, and $Y=\mathbb{R}$ (see Appendix A). Let $A_i,f_i:\mathbb{R}\to \mathbb{R}$ and $H^p:{\mathbb{R}}^p\to \mathbb{R}$ be the mappings, which are defined as:

$$\begin{array}{ccc}& & A_1\left(x\right)=\frac{x^3}{8}+\frac{2x}{3},A_3\left(x\right)=\frac{x^3}{8}+\frac{2x}{3},....,A_{p-1}\left(x\right)=\frac{x^3}{8}+\frac{2x}{3};\hfill \\ & & A_2\left(x\right)=\frac{x}{2},A_4\left(x\right)=\frac{x}{2},...,=A_p\left(x\right)=\frac{x}{2};\hfill \\ & & f_1\left(x\right)=\frac{x}{4},f_3\left(x\right)=\frac{x}{4},....,f_{p-1}\left(x\right)=\frac{x}{4};\hfill \\ & & f_2\left(x\right)=\frac{5x}{12},f_4\left(x\right)=\frac{5x}{12},...,=f_p\left(x\right)=\frac{5x}{12};\hfill \\ & & H^p(A_1\left(x\right),A_2\left(x\right),...,A_p\left(x\right))=A_1\left(x\right)-A_2\left(x\right)+...+A_{p-1}\left(x\right)-A_p\left(x\right)\hfill \end{array}$$

such that the inequality $xy+x^2+y^2>0$ is satisfied for all $x\in {\mathbb{R}}^2.$

Let $\mathcal{N}:{\mathbb{R}}^p\to 2^{\mathbb{R}}$ be a set-valued mapping defined as:

$$\begin{array}{ccc}& & \mathcal{N}(f_1\left(x\right),f_2\left(x\right),...,f_{p-1}\left(x\right),f_p\left(x\right))=f_1\left(x\right)-f_2\left(x\right)+...+f_{p-1}\left(x\right)-f_p\left(x\right).\hfill \end{array}$$

Then, $H^p(.,.,.,..)$ is $\frac{2}{3}$-strongly accretive with $(A_1,A_3,...,A_{p-1})$ and $\frac{3}{2}$-relaxed accretive with $(A_2,A_4,...,A_p)$, and $\mathcal{N}$ is $\frac{1}{4}$-strongly accretive with $(f_1,f_3,...,f_{p-1})$ and $\frac{17}{12}$-relaxed accretive with $(A_2,A_4,...,A_p)$.

One can easily verify the following for $\rho =1$:

$$\begin{array}{ccc}& & [H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p)]\mathbb{R}=\mathbb{R}.\hfill \end{array}$$

Hence, $\mathcal{N}$ is a generalized ${\alpha }_i{\beta }_j$- $H^p(.,.,...)$-accretive mapping with $(A_1,A_2,...,A_p)$ and $(f_1,f_2,...,f_p).$

Proposition 1.

Let assumptions $M_1$ and $M_2$ hold for every $i\in \{1,2,...p\}$, $p\ge 3$, and let $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(A_1,A_2,...,A_p)$-accretive mapping with mappings $(A_1,A_2,...,A_p)$ and $(f_1,f_2,...,f_p)$ with $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j,\sum {\alpha }_i>\sum {\beta }_j,$ if $⟨\tilde{x}-\tilde{y},J_q(u^{\prime }-v^{\prime })⟩\ge 0$ is satisfied for each $(v^{\prime },\tilde{y})\in \mathrm{Gr}\left(\mathcal{N}(f_1,f_2,...,f_p)\right)$, $\tilde{x}\in \mathcal{N}(f_1,f_2,...,f_p)\left(u^{\prime }\right)$, where $\mathrm{Gr}\left(\mathcal{N}\right(f_1,f_2,$$...,f_p\left)\right)=\{(u^{\prime },\tilde{x}):\tilde{x}\in \mathcal{N}(f_1,f_2,...,f_p)\left(u^{\prime }\right)\}$.

Proof. 

Assume that there exists $(w_0,z_0)\notin \mathrm{Gr}\left(\mathcal{N}(f_1,f_2,...,f_p)\right)$ such that

$$\begin{array}{c}\hfill ⟨z_0-x,J_q(w_0-u)⟩\ge 0,\forall (u,x)\in \mathrm{Gr}\left(\mathcal{N}(f_1,f_2,...,f_p)\right).\end{array}$$

(2)

If p is even: Since $\mathcal{N}$ is a generalized ${\alpha }_i{\beta }_j$-$H^p(A_1,A_2,...,A_p)$-accretive mapping, then $\mathcal{N}$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$, and $(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))\left(Y\right)=Y$ holds for each $\rho >0$; then, there exists $(w_1,z_1)\in \mathrm{Gr}\left(\mathcal{N}(f_1,f_2,...,f_p)\right)$ such that

$$\begin{array}{ccc}\hfill & & H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)+\rho z_0=H^p(A_1{w}_1,A_2{w}_1,...,A_p{w}_1)+\rho z_1\in Y.\hfill \end{array}$$

(3)

From (2) and (3), we have

$$\begin{array}{ccc}& & \rho z_0-\rho z_1=H^p(A_1{w}_1,A_2{w}_1,...,A_p{w}_1)-H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)\in Y,\hfill \\ & & ⟨\rho z_0-\rho z_1,J_q(u_0-w_1)⟩=\hfill \\ & & ⟨H^p(A_1{w}_1,A_2{w}_1,...,A_p{w}_1)-H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0),J_q(u_0-w_1)⟩.\hfill \end{array}$$

Setting $(u,x)=(w_1,z_1)$ in (2) and using $M_3$ in (3), we obtain

$$\begin{array}{ccc}\hfill & & [({\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1})-({\overline{\gamma }}_2+{\overline{\gamma }}_4-...+{\overline{\gamma }}_p)]{\parallel w_0-w_1\parallel }^q\le \rho ⟨z_0-z_1,J_q(w_0-w_1)⟩\hfill \\ & & \le -⟨H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)-H^p(A_1{w}_1,A_2{w}_1,...,A_p{w}_1),J_q(w_0-w_1)⟩\hfill \\ & & =-⟨H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)-H^p(A_1{w}_1,A_2{w}_0,...,A_p{w}_0),J_q(w_0-w_1)⟩\hfill \\ & & -⟨H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)-H^p(A_1{w}_0,A_2{w}_1,...,A_p{w}_0),J_q(w_0-w_1)⟩\hfill \\ & & :\hfill \\ & & :\hfill \\ & & -⟨H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_0)-H^p(A_1{w}_0,A_2{w}_0,...,A_p{w}_1),J_q(w_0-w_1)⟩\hfill \\ & & \le -[{\alpha }_1\parallel w_0-w_1{\parallel }^q-{\beta }_2\parallel w_0-w_1{\parallel }^q]\hfill \\ & & -[{\alpha }_3\parallel w_0-w_1{\parallel }^q-{\beta }_4\parallel w_0-w_1{\parallel }^q]\hfill \\ & & :\hfill \\ & & :\hfill \\ & & -[{\alpha }_{p-1}\parallel w_0-w_1{\parallel }^q-{\beta }_p\parallel w_0-w_1{\parallel }^q]\hfill \\ & & =-[({\alpha }_1+{\alpha }_3+...+{\alpha }_{p-1})-({\beta }_2+{\beta }_4+...+{\beta }_p)]{\parallel w_0-w_1\parallel }^q\hfill \\ & & \left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]{\parallel w_0-w_1\parallel }^q\le 0,\hfill \end{array}$$

(4)

where

$$\begin{array}{ccc}& & \sum {\alpha }_i={\alpha }_1+{\alpha }_3+...+{\alpha }_{p-1},\sum {\beta }_j={\beta }_2+{\beta }_4-...+{\beta }_p,\hfill \\ & & \sum {\overline{\mu }}_i={\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1},\sum {\overline{\gamma }}_j={\overline{\gamma }}_2+{\overline{\gamma }}_4-...+{\overline{\gamma }}_p.\hfill \end{array}$$

Since $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j$, $\sum {\alpha }_i>\sum {\beta }_j$ and $\rho >0$, it implies that $w_0=w_1$. By (2), we have $z_0=z_1$. Thus, $(w_1,z_1)=(w_o,z_o)\in \mathrm{Gr}\left(\mathcal{N}(f_1,f_2,...,f_p)\right)$. Similarly, we can prove the result when p is an odd number. This completes the proof. □

Theorem 1.

Let assumptions $M_1$ and $M_2$ hold for every $i\in \{1,2,...p\}$, $p\ge 3$, and let $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(A_1,A_2,...,A_p)$-accretive mapping with $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j,\sum {\alpha }_i>\sum {\beta }_j$; then, $\left(H^p(A_1,A_2,...,A_p\right)+\rho \mathcal{N}(f_1,f_2,...,f_p){)}^{-1}$ is single-valued.

Proof. 

For any given $u\in Y$, let $x,y\in {(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))}^{-1}\left(u\right)$. It follows that

$$\begin{array}{ccc}& & -H^p(A_{1}x,A_{2}x,...,A_{p}x)+u\in \rho \mathcal{N}(f_1,f_2,...,f_p)x,\hfill \\ & & -H^p(A_{1}y,A_{2}y,...,A_{p}y)+u\in \rho \mathcal{N}(f_1,f_2,...,f_p)y.\hfill \end{array}$$

If p is even: Since $\mathcal{N}$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$, we have

$$\begin{array}{ccc}& & ({\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1}-{\overline{\gamma }}_2-{\overline{\gamma }}_4-...-{\overline{\gamma }}_p){\parallel x-y\parallel }^q\hfill \\ & & \le \frac{1}{\rho }⟨-H^p(A_{1}x,A_{2}x,...,A_{p}x)+u-(-H^p(A_{1}y,A_{2}y,...,A_{p}y)+u),J_q(x-y)⟩\hfill \\ & & \Rightarrow \rho ({\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1}-{\overline{\gamma }}_2-{\overline{\gamma }}_4-...-{\overline{\gamma }}_p){\parallel x-y\parallel }^q\hfill \\ & & \le -⟨H^p(A_{1}x,A_{2}x,...,A_{p}x)-H^p(A_{1}y,A_{2}y,...,A_{p}y),J_q(x-y)⟩\hfill \\ & & =-(⟨H^p(A_{1}x,A_{2}x,...,A_{p}x)-H^p(A_{1}y,A_{2}x,...,A_{p}x),J_q(x-y)⟩\hfill \\ & & -⟨H^p(A_{1}y,A_{2}x,...,A_{p}x)-H^p(A_{1}y,A_{2}y,...,A_{p}x),J_q(x-y)⟩\hfill \\ & & :\hfill \\ & & :\hfill \\ & & -⟨H^p(A_{1}y,A_{2}y,...,A_{p}x)-H^p(A_{1}y,A_{2}y,...,A_{p}y),J_q(x-y)⟩.\hfill \end{array}$$

We proceed through the same process as that to obtain (4), and we have

$$\begin{array}{ccc}\hfill & & \left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]{\parallel x-y\parallel }^q\le 0.\hfill \end{array}$$

(5)

Since $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j$, $\sum {\alpha }_i>\sum {\beta }_j$ and $\rho >0$, we have $\parallel x-y\parallel \le 0$. It implies that $x=y$. Thus, ${(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))}^{-1}$ is single-valued. Similarly, we can prove the result when p is an odd number. This completes the proof. □

Definition 9.

Let assumptions $M_1$ and $M_2$ hold for $p\ge 3$, and let $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...)$-accretive mapping with $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j,\sum {\alpha }_i>\sum {\beta }_j$. A proximal-point mapping $R_{\rho ,\mathcal{N}(.,.,..,.)}^{H^p(.,.,..,.)}:Y\to Y$ is defined by

$$\begin{array}{ccc}& & R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)={[H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p)]}^{-1}\left(x\right),\forall x\in Y,\hfill \end{array}$$

where ρ is a non-negative constant.

Now, we prove the Lipschitz continuity of the proximal-point mapping.

Theorem 2.

Let assumptions $M_1$ and $M_2$ hold for $p\ge 3$, and let $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...)$-accretive mapping with $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j,\sum {\alpha }_i>\sum {\beta }_j$. Then, the proximal-point mapping $R_{\rho ,\mathcal{N}(.,.,..,.)}^{H^p(.,.,..,.)}:Y\to Y$ is Δ-Lipschitz continuous, where

$$\mathrm{\Delta }={\left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]}^{-1}.$$

Proof. 

Let $x,y\in Y$ and by Definition 9, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{R}_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)={(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))}^{-1}\left(x\right),\hfill \\ R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)={(H^p(A_1,A_2,...,A_p)+\rho \mathcal{N}(f_1,f_2,...,f_p))}^{-1}\left(y\right).\hfill \end{array}\right.\end{array}$$

It follows that

$$\begin{array}{ccc}& & \frac{1}{\rho }(x-H^p(A_1\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)\right),A_2\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)\right),...,A_p\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)\right)))\in \mathcal{N}\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)\right),\hfill \\ & & \frac{1}{\rho }(y-H^p(A_1\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)\right),A_2\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)\right),...,A_p\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)\right)))\in \mathcal{N}\left(R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)\right).\hfill \end{array}$$

Let $x^1=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)$ and $y^1=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right).$

If p is even: Since $\mathcal{N}$ is ${\overline{\mu }}_1{\overline{\gamma }}_2...{\overline{\mu }}_{p-1}{\gamma }_p$-symmetric accretive with $(f_1,f_2,...,f_p)$, we have

$$\begin{array}{ccc}& & ⟨(x-H^p(A_1\left(x^1\right),A_2\left(x^1\right),...,A_p\left(x^1\right)))-(y-H^p(A_1\left(y^1\right),A_2\left(y^1\right),...,A_p\left(y^1\right))),J_q(x^1-y^1)⟩\hfill \\ & & \ge \rho ({\overline{\mu }}_1-{\overline{\gamma }}_2+{\overline{\mu }}_3-{\overline{\gamma }}_4+...+{\overline{\mu }}_{p-1}-{\overline{\gamma }}_p){\parallel x^1-y^1\parallel }^q,\hfill \\ & & ⟨x-y-(H^p(A_1\left(x^1\right),A_2\left(x^1\right),...,A_p\left(x^1\right))-H^p(A_1\left(y^1\right),A_2\left(y^1\right),...,A_p\left(y^1\right))),J_q(x^1-y^1)⟩\hfill \\ & & \ge \rho (({\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1}-({\overline{\gamma }}_2+{\overline{\gamma }}_4-...+{\overline{\gamma }}_p))\parallel x^1-y^1{\parallel }^q.\hfill \end{array}$$

Let $\sum {\alpha }_i={\alpha }_1+{\alpha }_3+...+{\alpha }_{p-1}, \sum {\beta }_j={\beta }_2+{\beta }_4-...+{\beta }_p,$$\sum {\overline{\mu }}_i={\overline{\mu }}_1+{\overline{\mu }}_3+...+{\overline{\mu }}_{p-1, }$$\sum {\overline{\gamma }}_j={\overline{\gamma }}_2+{\overline{\gamma }}_4-...+{\overline{\gamma }}_p.$

We have

$$\begin{array}{ccc}& & \parallel x-y\parallel \parallel x^1-y^1{\parallel }^{q-1}\ge ⟨x-y,J_q(x^1-y^1)⟩\hfill \\ & & \ge ⟨H^p(A_1\left(x^1\right),A_2\left(x^1\right),...,A_p\left(x^1\right))-H^p(A_1\left(y^1\right),A_2\left(y^1\right),...,A_p\left(y^1\right)),\hfill \\ & & J_q(x^1-y^1)⟩+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right){\parallel x^1-y^1\parallel }^q\hfill \\ & & \ge {\alpha }_1\parallel x^1-y^1{\parallel }^q-{\beta }_2\parallel x^1-y^1{\parallel }^q+{\alpha }_3\parallel x^1-y^1{\parallel }^q-...{\beta }_p{\parallel x^1-y^1\parallel }^q\hfill \\ & & +\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right){\parallel x^1-y^1\parallel }^q\hfill \\ & & =\left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]{\parallel x^1-y^1\parallel }^q.\hfill \end{array}$$

Hence,

$$\begin{array}{ccc}& & \parallel x-y\parallel \parallel x^1-y^1{\parallel }^{q-1}\ge \left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]{\parallel x^1-y^1\parallel }^q;\hfill \end{array}$$

that is,

$$\begin{array}{c}\Vert R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(x\right)-R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}\left(y\right)\parallel \le \mathrm{\Delta }\parallel x-y\parallel ,\forall x,y\in Y,\hfill \end{array}$$

where $\mathrm{\Delta }={\left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]}^{-1}.$ Similarly, we can prove the result when p is an odd number. This completes the proof. □

3. Set-Valued Variational Inclusions

Let Y be a q-uniformly smooth Banach space. For each $i\in \{1,2,...,p\},p\ge 3$, let $H^p,F:Y^p\to Y$ be the mappings, $A_i,f_i:Y\to Y$ be the single-valued mappings, and $S_i:Y\to 2^Y$ be a set-valued mapping. Let $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...,.)$-accretive mapping.

Now, the problem is to find $x\in Y,u_1\in S_1\left(x\right),u_2\in S_2\left(x\right),...,u_p\in S_p\left(x\right)$ such that

$$\begin{array}{c}\hfill \mathrm{\Theta }\in F(u_1,u_2,...,u_p)+\mathcal{N}(f_1\left(x\right),f_2\left(x\right),...,f_p\left(x\right)).\end{array}$$

(6)

Special cases:

(i)

If $F(u_1,u_2,...,u_p)=F(u_1,u_2)$, and $\mathcal{N}(f_1\left(x\right),f_2\left(x\right),...,f_p\left(x\right))=\mathcal{N}(f_1\left(x\right),f_2\left(x\right))$, then problem (6) reduces to find $x\in Y,u_1\in S_1\left(x\right),u_2\in S_2\left(x\right)$ such that

$$\mathrm{\Theta }\in F(u_1,u_2)+\mathcal{N}(f_1\left(x\right),f_2\left(x\right)).$$

(7)

(ii)

If $f_1=f_2=f,$$S_1=S_2=S$, and $\mathcal{N}(.,.)=\mathcal{N}(.)$, then problem (7) reduces to find $x\in Y,u\in S\left(x\right)$ such that

$$\mathrm{\Theta }\in u+\mathcal{N}\left(f\right(x\left)\right).$$

(8)

Problem (8) has been studied by Huang [26] when $\mathcal{N}$ is a m-accretive mapping.

(iii)

If $F(u_1,u_2)=S\left(x\right)$ and S is single-valued mapping, then problem (7) reduced to find $x\in Y$ such that

$$\mathrm{\Theta }\in S\left(x\right)+\mathcal{N}\left(x\right).$$

(9)

Problem (9) has been studied by Zou and Huang [9] when $\mathcal{N}$ is an $H(., .)$-accretive mapping. For the generalized m-accretive mapping, problem (9) was studied by Bi et al. [27].

Definition 10.

A set-valued mapping $S:Y\to CB\left(Y\right)$ is said to be $\tilde{\mathcal{D}}$- Lipschitz continuous with $\zeta >0$, if

$$\begin{array}{c}\hfill \tilde{\mathcal{D}}(Sy,Sz)\le \zeta \parallel y-z\parallel ,\forall y,z\in Y.\end{array}$$

Theorem 3.

For any given $(\tilde{x},{\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)\in Y\times Y^p$, $(\tilde{x},{\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)$ is a solution of (6) if and only if $(\tilde{x},{\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)$ satisfies

$$\begin{array}{c}\hfill \tilde{x}=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1,A_2,...,A_p)\left(\tilde{x}\right)-\rho F({\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)],\end{array}$$

(10)

where $\rho$ is a non-negative constant.

Proof. 

Let $(\tilde{x},{\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)$ be the solution of problem (6), we have

$$\begin{array}{ccc}& & \mathrm{\Theta }\in F({\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)+\mathcal{N}(f_1\left(\tilde{x}\right),f_2\left(\tilde{x}\right),...,f_p\left(\tilde{x}\right))\hfill \\ & & \iff H^p(A_1\left(\tilde{x}\right),A_2\left(\tilde{x}\right),...,A_p\left(\tilde{x}\right))\in H^p(A_1\left(\tilde{x}\right),A_2\left(\tilde{x}\right),...,A_p\left(\tilde{x}\right))\hfill \\ & & +\rho F({\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)+\mathcal{N}(f_1\left(\tilde{x}\right),f_2\left(\tilde{x}\right),...,f_p\left(\tilde{x}\right))\hfill \\ & & \iff H^p(A_1\left(\tilde{x}\right),A_2\left(\tilde{x}\right),...,A_p\left(\tilde{x}\right))-\rho F({\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)\in H^p(A_1\left(\tilde{x}\right),A_2\left(\tilde{x}\right),...,A_p\left(\tilde{x}\right))\hfill \\ & & +\rho \mathcal{N}(f_1\left(\tilde{x}\right),f_2\left(\tilde{x}\right),...,f_p\left(\tilde{x}\right))\hfill \\ & & \iff \tilde{x}=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1\left(\tilde{x}\right),A_2\left(\tilde{x}\right),...,A_p\left(\tilde{x}\right))-\rho F({\tilde{u}}_1,{\tilde{u}}_2,...,{\tilde{u}}_p)].\hfill \end{array}$$

□

Algorithm 1.

For any given$x_0^1\in Y$, we select$u_0^1\in S_1\left(x_0^1\right),u_0^2\in S_2\left(x_0^1\right),...,u_0^p\in S_p\left(x_0^1\right)$and obtain$\left\{x_n^1\right\}$, $\left\{u_n^1\right\}$, $\left\{u_n^2\right\}$,..., $\left\{u_n^p\right\}$by the following iterative scheme

$$\begin{array}{ccc}& & x_{n+1}^1=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-\rho F_1(u_n^1,u_n^2,...,u_n^p)],\hfill \\ & & u_n^1\in S_1\left(x_n^1\right):\parallel u_{n+1}^1-u_n^1\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_1\left(x_{n+1}^1\right),S_1\left(x_n^1\right)),\hfill \\ & & u_n^2\in S_2\left(x_n^1\right):\parallel u_{n+1}^2-u_n^2\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_2\left(x_{n+1}^1\right),S_2\left(x_n^1\right)),\hfill \\ & & :\hfill \\ & & :\hfill \\ & & u_n^p\in S_p\left(x_n^1\right):\parallel u_{n+1}^p-u_n^p\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_p\left(x_{n+1}^1\right),S_p\left(x_n^1\right)),\hfill \end{array}$$

where n = 0,1,2,... and$\rho >0$.

Theorem 4.

Let problem (6) hold with assumptions $M_1$-$M_4$ and $\mathcal{N}:Y^p\to 2^Y$ be a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...)$-accretive mapping with $\sum {\overline{\mu }}_i>\sum {\overline{\gamma }}_j,\sum {\alpha }_i>\sum {\beta }_j$. For each $i\in \{1,2,...,p\}$, we assume the following:

(i)

$S_i$ is ${\zeta }_i$-$\mathcal{D}$-Lipschitz continuous;

(ii)

$H^p(.,.,...)$ is $s_i$-Lipschitz continuous with $A_i$;

(iii)

F is ${\overline{\alpha }}_i$-strongly accretive with $f_i$ and $H^p(A_1,A_2,..,A_p)$ in the $ith$-argument;

(iv)

F is $l_i$-Lipschitz continuous in $ith$-argument;

(v)

in addition the following condition is satisfied:

$$\begin{array}{ccc}\hfill & & {\mathrm{\Delta }}^q\left[s^q+c_q{\rho }^q{{\rm Y}}^q-\rho q\overline{\alpha }{s}^q\right]<1.\hfill \end{array}$$

(11)

Then, the iterative sequences $(\left\{x_n^1\right\},\left\{u_n^1\right\},\left\{u_n^2\right\},....,\left\{u_n^p\right\})$ developed by Algorithm 1 converge strongly to $(x^1,u^1,u^2,..,u^p)$ a solution to problem (6).

Proof. 

From Algorithm 1 and Theorem 2, we have

$$\begin{array}{ccc}\hfill & & \parallel x_{n+1}^1-x_n^1\parallel =\parallel R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-\rho F(u_n^1,u_n^2,...,u_n^p)]\hfill \\ & & -R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right)-\rho F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)]\parallel \hfill \\ & & \le \mathrm{\Delta }\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-\rho F(u_n^1,u_n^2,...,u_n^p)\hfill \\ & & -(H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right)-\rho F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p))\parallel \hfill \\ & & =\mathrm{\Delta }\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right)\hfill \\ & & -\rho (F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p))\parallel \hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill & & \parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right)\hfill \\ & & -\rho (F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)){\parallel }^q\hfill \\ & & \le \parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right){\parallel }^q\hfill \\ & & -\rho q⟨(F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩\hfill \\ & & +c_q{\rho }^q{\parallel F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)\parallel }^q.\hfill \end{array}$$

(13)

By using the $s_i$-Lipschitz continuity of $H^P(.,...,A_i,..,.)$, we have

$$\begin{array}{ccc}\hfill & & \parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right)\parallel \hfill \\ & & \le \parallel H^p(A_1{x}_n^1,A_2{x}_n^1,...,A_p{x}_n^1))-H^p(A_1{x}_{n-1}^1,A_2{x}_n^1,...,A_p{x}_n^1)\parallel \hfill \\ & & +\parallel H^p(A_1{x}_{n-1}^1,A_2{x}_n^1,...,A_p{x}_n^1)-H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_n^1)\parallel \hfill \\ & & :\hfill \\ & & :\hfill \\ & & +\parallel H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_n^1)-H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_{n-1}^1)\parallel \hfill \\ & & \le s_1\parallel x_{n-1}^1-x_n^1\parallel +s_2\parallel x_{n-1}^1-x_n^1\parallel +...+s_p\parallel x_{n-1}^1-x_n^1\parallel |\hfill \\ & & =s\parallel x_{n-1}^1-x_n^1\parallel ,\mathrm{where}\mathrm{s}={\mathrm{s}}_1+{\mathrm{s}}_2+...+{\mathrm{s}}_{\mathrm{p}}.\hfill \end{array}$$

(14)

By the $l_i$-Lipschitz continuity of F and ${\zeta }_i$-$\mathcal{D}$-Lipschitz continuity of $S_i$, we have

$$\begin{array}{ccc}\hfill & & \parallel F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)\parallel \hfill \\ & & \le \parallel F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_n^2,...,u_n^p)\parallel \hfill \\ & & +\parallel F(u_{n-1}^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_n^p)\parallel \hfill \\ & & :\hfill \\ & & :\hfill \\ & & +\parallel F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_n^p)\parallel \hfill \\ & & \le l_1\parallel u_n^1-u_{n-1}^1\parallel +l_2\parallel u_n^2-u_{n-1}^2\parallel +...+l_p\parallel u_n^p-u_{n-1}^p\parallel \hfill \\ & & \le l_1\left(1+\frac{1}{n}\right)\tilde{\mathcal{D}}(S_1\left(x_n^1\right),S_1\left(x_{n-1}^1\right))+l_2\left(1+\frac{1}{n}\right)\tilde{\mathcal{D}}(S_2\left(x_n^2\right),S_2\left(x_{n-1}^2\right))\hfill \\ & & +...+l_p\left(1+\frac{1}{n}\right)\tilde{\mathcal{D}}(S_p\left(x_n^p\right),S_p\left(x_{n-1}^p\right))\hfill \\ & & \le \left(l_1{\zeta }_1\left(1+\frac{1}{n}\right)+l_2{\zeta }_2\left(1+\frac{1}{n}\right)+...+l_p{\zeta }_p\left(1+\frac{1}{n}\right)\right)\parallel x_n^1-x_{n-1}^1\parallel \hfill \\ & & \le \left(1+\frac{1}{n}\right){\rm Y}\parallel x_n^1-x_{n-1}^1\parallel ,\mathrm{where}{\rm Y}=l_1{\zeta }_1+l_2{\zeta }_2+...+l_p{\zeta }_p.\hfill \end{array}$$

(15)

Now, we compute

$$\begin{array}{ccc}& & -⟨F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩,\hfill \\ & & =-[⟨F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_n^2,...,u_n^p),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩,\hfill \\ & & +⟨F(u_{n-1}^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_n^p),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩,\hfill \\ & & :\hfill \\ & & :\hfill \\ & & +⟨F(u_{n-1}^1,u_{n-1}^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩],\hfill \end{array}$$

we have

$$\begin{array}{ccc}\hfill & & -⟨F(u_n^1,u_n^2,...,u_n^p)-F(u_{n-1}^1,u_{n-1}^2,...,u_{n-1}^p),\hfill \\ & & J_q(H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right))⟩,\hfill \\ & & \le -[{\overline{\alpha }}_1\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right){)\parallel }^q\hfill \\ & & +{\overline{\alpha }}_2\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right){)\parallel }^q\hfill \\ & & :\hfill \\ & & :\hfill \\ & & +{\overline{\alpha }}_p\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right){)\parallel }^q]\hfill \\ & & \le -({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p)\parallel H^p(A_1,A_2,...,A_p)\left(x_n^1\right)-H^p(A_1,A_2,...,A_p)\left(x_{n-1}^1\right){)\parallel }^q\hfill \\ & & \le -\left[\right({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p)\parallel H^p(A_1{x}_n^1,A_2{x}_n^1,...,A_p{x}_n^1))\hfill \\ & & -H^p(A_1{x}_{n-1}^1,A_2{x}_n^1,...,A_p{x}_n^1\parallel \hfill \\ & & +({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p)\parallel H^p(A_1{x}_{n-1}^1,A_2{x}_n^1,...,A_p{x}_n^1)\hfill \\ & & -H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_n^1)\parallel \hfill \\ & & :\hfill \\ & & :\hfill \\ & & +({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p)\parallel H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_n^1)\hfill \\ & & -H^p(A_1{x}_{n-1}^1,A_2{x}_{n-1}^1,...,A_p{x}_{n-1}^1)\parallel ]\hfill \\ & & \le -({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p\left)\right\{s_1\parallel x_n^1-x_{n-1}^1\parallel +s_2\parallel x_n^1-x_{n-1}^1\parallel +...+s_p\parallel x_n^1-x_{n-1}^1{\parallel \}}^q\hfill \\ & & \le -({\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p){(s_1+s_2+...+s_p)}^q{\parallel x_n^1-x_{n-1}^1\parallel }^q\hfill \\ & & =-\overline{\alpha }{s}^q{\parallel x_n^1-x_{n-1}^1\parallel }^q,\mathrm{where}\overline{\alpha }={\overline{\alpha }}_1+{\overline{\alpha }}_2+...+{\overline{\alpha }}_p.\hfill \end{array}$$

(16)

Using Equations (13)–(16) in Equation (12), then Equation (12) becomes

$$\begin{array}{ccc}& & \parallel x_{n+1}^1-x_n^1\parallel \le \mathrm{\Delta }{\left[s^q+c_q{\rho }^q{{\rm Y}}^q{\left(1+\frac{1}{n}\right)}^q-\rho q\overline{\alpha }{s}^q\right]}^{\frac{1}{q}}\parallel x_n^1-x_{n-1}^1\parallel ,\hfill \end{array}$$

which implies that

$$\begin{array}{ccc}\hfill & & \parallel x_{n+1}^1-x_n^1\parallel \le {\mathrm{\Theta }}_n\parallel x_n^1-x_{n-1}^1\parallel ,\mathrm{where}{\mathrm{\Theta }}_n=\mathrm{\Delta }{\left[s^q+c_q{\rho }^q{{\rm Y}}^q{\left(1+\frac{1}{n}\right)}^q-\rho q\overline{\alpha }{s}^q\right]}^{\frac{1}{q}}.\hfill \end{array}$$

(17)

Let $\mathrm{\Theta }=\mathrm{\Delta }{\left[s^q+c_q{\rho }^q{{\rm Y}}^q-\rho q\overline{\alpha }{s}^q\right]}^{\frac{1}{q}}.$ From (11), we have

$$\begin{array}{ccc}& & {\mathrm{\Delta }}^q\left[s^q+c_q{\rho }^q{{\rm Y}}^q-\rho q\overline{\alpha }{s}^q\right]<1,\mathrm{where}\mathrm{\Delta }={\left[\sum {\alpha }_i-\sum {\beta }_j+\rho \left(\sum {\overline{\mu }}_i-\sum {\overline{\gamma }}_j\right)\right]}^{-1}.\hfill \end{array}$$

Thus, we have ${lim}_{n\to \infty }{\mathrm{\Theta }}_n\to \mathrm{\Theta }$. By (11), we have $0\le \mathrm{\Theta }<1$. This implies that $\parallel x_{n+1}^1-x_n^1\parallel \to 0$ as $n\to \infty$; therefore, $\left\{x_n^1\right\}$ is a Cauchy sequence in Y. Then, there exists $x\in Y$ with ${lim}_{n\to \infty }{x}_n^1\to x^1$. By the $\mathcal{D}$-Lipschitz continuity of $S_1,S_2,...,S_p$ and Algorithm 1, we have

$$\begin{array}{ccc}& & \parallel u_{n+1}^1-u_n^1\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_1\left(x_{n+1}^1\right),S_1\left(x_n^1\right))\le \left[1+\frac{1}{n+1}\right]{\zeta }_1\parallel x_{n+1}^1-x_n^1\parallel ,\hfill \\ & & \parallel u_{n+1}^2-u_n^2\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_2\left(x_{n+1}^1\right),S_2\left(x_n^1\right))\le \left[1+\frac{1}{n+1}\right]{\zeta }_2\parallel x_{n+1}^1-x_n^1\parallel ,\hfill \end{array}$$

$$\begin{array}{ccc}& & :\hfill \\ & & \parallel u_{n+1}^p-u_n^p\parallel \le \left[1+\frac{1}{n+1}\right]\tilde{\mathcal{D}}(S_p\left(x_{n+1}^1\right),S_p\left(x_n^1\right))\le \left[1+\frac{1}{n+1}\right]{\zeta }_p\parallel x_{n+1}^1-x_n^1\parallel .\hfill \end{array}$$

It shows that $\left\{u_n^1\right\},\left\{u_n^2\right\},....,\left\{u_n^p\right\}$ are Cauchy sequences; then, there exists $u^1,u^2,...u^p$ such that $u_n^1\to u^1,u_n^2\to u^2,...,u_n^p\to u^p,$ as $n\to \infty$. Now, we show that $u^1\in S_1\left(x^1\right)$. Since $u_n^1\in S_1\left(x^1\right)$, we have

$$\begin{array}{ccc}& & d(u^1,S_1\left(x^1\right))\le \parallel u^1-u_n^1\parallel +d(u_n^1,S_1\left(x^1\right))\hfill \\ & & \le \parallel u^1-u_n^1\parallel +\tilde{\mathcal{D}}(S_1\left(x_n^1\right),S_1\left(x^1\right))\hfill \\ & & \le \parallel u^1-u_n^1\parallel +{\zeta }_1\parallel x_n^1-x^1\parallel .\hfill \end{array}$$

Since $S_1\left(x^1\right)$ is closed, $u^1\in S_1\left(x^1\right)$. Similarly, we can prove $u^2\in S_2\left(x^1\right),u^3\in S_3\left(x^1\right),...,u^p$$\in S_p\left(x^1\right)$. By continuity, we know that $x^1,u^1,u^2,...,u^p$ satisfy

$$\begin{array}{c}\hfill x^1=R_{\rho ,\mathcal{N}(.,.,...)}^{H^p(.,.,...)}[H^p(A_1,A_2,...,A_p)\left(x^1\right)-\rho F(u^1,u^2,...,u^p)].\end{array}$$

Thus, problem (6) has the solution $(x^1,u^1,u^2,...,u^p)$. □

Remark 1.

Let p be an even number. Let $f:Y\to \mathbb{R}$ be a functional on Y; then, the vector $x^{*}$, such that

$$\begin{array}{c}\hfill f\left(z\right)-f\left(y\right)\ge ⟨x^{*},z-y⟩,\forall z\in Y,\end{array}$$

(18)

where $f\left(z\right)$ is finite for each $z\in Y$, is known as the subgradient of f at z. The collections of all such subgradients of f at z holding (18) are known as the subdifferential $\delta f\left(z\right)$ of f at z. Let $\delta (.,.,f_i,..):Y^p\to 2^Y$ be set-valued mapping and consider the inclusion problem to find $x^{*}\in Y$ such that

$$\begin{array}{c}\hfill 0\in F(u_1,u_2,...,u_p)+\delta (f_1,f_2,...,f_p)\left(x^{*}\right).\end{array}$$

(19)

Then, it turns out that $H^p(.,.,...)+\delta (f_1,f_2,...,f_p)$ is $\mathcal{L}$-strongly accretive, where $\mathcal{L}=\left[\sum {\alpha }_i+\sum {\overline{\mu }}_i-\left(\sum {\beta }_j+\sum {\overline{\gamma }}_j\right)\right]>0$, if $H^p$ is ${\alpha }_1{\beta }_2{\alpha }_3{\beta }_4...{\alpha }_{p-1}{\beta }_p$-symmetric accretive with $A_1,A_2,...,A_p$, $f_1,f_2,...,f_p:Y\to Y$ as the locally Lipschitz functionals on Y, and $\delta (f_1,f_2,...,f_p)$ is ${\overline{\mu }}_1{\overline{\gamma }}_2{\overline{\mu }}_3{\overline{\gamma }}_4...{\overline{\mu }}_{p-1}{\overline{\gamma }}_p$-symmetric accretive with $f_1,f_2,...,f_p$; equivalently, we can state that $\delta (...,f_i,..)$ is a generalized ${\alpha }_i{\beta }_j$-$H^p(.,.,...)$-accretive mapping. Now, all the assumptions of Theorem 4 hold, and one can easily find the solution to problem (19) by using Theorem 4.

4. Conclusions

In this article, we considered the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings defined on a product set, which contained the generalized $\alpha \beta$-$H(., .)$-accretive mappings, $H(., .)$-accretive mappings, and $C_n$ monotone mappings as special cases, since variational inclusions, generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings, and proximal-point mappings have applications in physics, economics and management sciences. Therefore, we considered and studied a variational inclusion problem including a generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mapping. We discussed the convergence criteria of an iterative algorithm to solve problem (6) under appropriate assumptions in q-uniformly smooth Banach spaces. In future, the results that are obtained for the proximal-point mapping associated with the generalized ${\alpha }_i{\beta }_j$-$H^p(., ., ...)$-accretive mappings conferred in this article can be continued to the Yosida inclusion problems in the setting of Banach spaces.