Generalized Implicit Set-Valued Variational Inclusion Problem with ⊕ Operation

Abstract

In this paper, we consider a resolvent operator which depends on the composition of two mappings with ⊕ operation. We prove some of the properties of the resolvent operator, that is, that it is single-valued as well as Lipschitz-type-continuous. An existence and convergence result is proven for a generalized implicit set-valued variational inclusion problem with ⊕ operation. Some special cases of a generalized implicit set-valued variational inclusion problem with ⊕ operation are discussed. An example is constructed to illustrate some of the concepts used in this paper.

1. Introduction

Because of applications in optimization problems, mathematical programming, equilibrium problems, engineering, economics and operation research etc., suitable progress has been achieved in both theory and application of various types of variational inequalities (inclusions) and their generalizations. After careful observation, it was noticed that the projection method and its variant forms cannot be applied for solving variational inclusions. This fact motivated researchers to use techniques based on resolvent operators. The resolvent operator and its variant forms represent an important tool for finding the approximate solutions of variational inclusions. The main idea in this technique is to establish the equivalence between the variational inclusions and the fixed point problems using the concept of resolvent. For more details, we refer to [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] and the references therein.

⊕ operation, that is, XOR-operation is a binary operation and behaves like the ADD operation: It takes two arguments and produces one result. This operation is commutative, associative, and self-inverse. In Boolean algebra, it is the same as addition modulo(2). XOR represents the inequality function, i.e., the output is true if the inputs are not alike; otherwise, the output is false. It is interesting to note that if we take the XOR of any number with 1, then we get the complement of the number, and if we take XOR with 0, then we get the same number. XOR terminology is used to generate pseudo-random numbers, to detect error in digital communication, inside CPU it helps in addition operation, etc.

Li and his co-authors [21,22,23] first used the ⊕ operation for solving some classes of variational inclusions and after that, Ahmad and his co-authors [24,25,26] also solved some generalized variational inclusions with ⊕ operation.

In this paper, we consider a resolvent operator with ⊕ operation involving composition of two mappings. We proved some properties of the resolvent operator. An iterative algorithm was constructed to solved a generalized implicit set-valued variational inclusion problem with ⊕ operation in real ordered positive Hilbert spaces. An existence and convergence result was proven for a generalized implicit set-valued inclusion problem with ⊕ operation. Some special cases are discussed and an example is given in support of some of the concepts used in this work.

2. Preliminaries

Let C be a cone with partial ordering “≤”. An ordered Hilbert space with norm $\parallel ·\parallel$ and inner product $⟨·,·⟩$ is called positive if $0\le x$ and $0\le y,$ then $0\le ⟨x,y⟩$ holds. Throughout the paper, ${\mathcal{H}}_p$ is assumed to be a real ordered positive Hilbert space. We denote $2^{{\mathcal{H}}_p}$ (respectively, $C^{*}({\mathcal{H}}_p)$) as the family of nonempty (respectively, compact) subsets of ${\mathcal{H}}_p,$ and d is the metric induced by the norm and $\mathcal{D}(.,.)$ is the Hausdörff metric on $C^{*}({\mathcal{H}}_p).$

Now, we illustrate some known concepts and results which are needed to prove the main result. The following concepts and results can be found in [20,21,22,23,24,25,26,27].

Definition 1.

A nonempty closed convex subset C of ${\mathcal{H}}_p$ is said to be a cone if:

(i) 

for any $x\in C$ and any $\lambda >0,$ $\lambda x\in C$;

(ii) 

if $x\in C$ and $-x\in C,$ then $x=0$.

Definition 2.

Let C be the cone, then:

(i) 

C is called a normal cone if there exists a constant ${\lambda }_N>0$ such that $0\le x\le y$ implies $\parallel x\parallel \le {\lambda }_N\parallel y\parallel$, for all $x,y\in {\mathcal{H}}_p$;

(ii) 

for any $x,y\in {\mathcal{H}}_p,$ $x\le y$ if and only if $y-x\in C;$

(iii) 

x and y are said to be comparative to each other if either $x\le y$ or $y\le x$ holds and is denoted by $x\propto y.$

Definition 3.

For any $x,y\in {\mathcal{H}}_p,$ $lub\{x,y\}$ denotes the least upper bound and $glb\{x,y\}$ denotes the greatest lower bound of the set $\{x,y\}.$ Suppose $lub\{x,y\}$ and $glb\{x,y\}$ exist, then some binary operations are given below:

(i) 

$x\vee y=lub\{x,y\};$

(ii) 

$x\wedge y=glb\{x,y\};$

(iii) 

$x\oplus y=(x-y)\vee (y-x);$

(iv) 

$x\odot y=(x-y)\wedge (y-x).$

The operations $\vee ,\wedge ,$⊕, and ⊙ are called OR, AND, XOR, and XNOR operations, respectively.

Lemma 1.

If $x\propto y,$ then lub{$x,y$} and glb{$x,y$} exist such that $x-y\propto y-x$ and $0\le (x-y)\vee (y-x).$

Lemma 2.

For any natural number $n,$ $x\propto y_n$ and $y_n\to y^{*}$ as $n\to \infty ,$ then $x\propto y^{*}.$

Proposition 1.

Let ⊕ be an XOR operation and ⊙ be an XNOR operation. Then the following relations hold for all $x,y,u,v,w\in {\mathcal{H}}_p$ and $\alpha ,\beta ,\lambda \in \mathbb{R}$:

(i) 

$x\odot x=0,$$x\odot y=y\odot x=-(x\oplus y)=-(y\oplus x);$

(ii) 

if $x\propto 0,$ then $-x\oplus 0\le x\le x\oplus 0;$

(iii) 

$(\lambda x)\oplus (\lambda y)=\left|\lambda \right|(x\oplus y);$

(iv) 

$0\le x\oplus y,$ if $x\propto y;$

(v) 

if $x\propto y,$ then $x\oplus y=0$ if and only if $x=y;$

(vi) 

$(x+y)\odot (u+v)\ge (x\odot u)+(y\odot v);$

(vii) 

$(x+y)\odot (u+v)\ge (x\odot v)+(y\odot u);$

(viii) 

if $x,y$ and w are comparative to each other, then $(x\oplus y)\le (x\oplus w)+(w\oplus y);$

(ix) 

$\alpha x\oplus \beta x=|\alpha -\beta |x=(\alpha \oplus \beta )x,$ if $x\propto 0$.

Proposition 2.

Let C be a normal cone in ${\mathcal{H}}_p$ with constant ${\lambda }_N,$ then for each $x,y\in {\mathcal{H}}_p,$ the following relations hold:

(i) 

$\parallel 0\oplus 0\parallel =\parallel 0\parallel =0;$

(ii) 

$\parallel x\vee y\parallel \le \parallel x\parallel \vee \parallel y\parallel \le \parallel x\parallel +\parallel y\parallel ;$

(iii) 

$\parallel x\oplus y\parallel \le \parallel x-y\parallel \le {\lambda }_N\parallel x\oplus y\parallel ;$

(iv) 

if $x\propto y,$ then $\parallel x\oplus y\parallel =\parallel x-y\parallel .$

Definition 4.

Let $F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be a single-valued mapping, then:

(i) 

F is said to be comparison mapping, if for each $x,y\in {\mathcal{H}}_p,$ $x\propto y$ then $F(x)\propto F(y),$ $x\propto F(x)$ and $y\propto F(y);$

(ii) 

F is said to be strongly comparison mapping, if F is a comparison mapping and $F(x)\propto F(y)$ if and only if $x\propto y,$ for all $x,y\in {\mathcal{H}}_p.$

Definition 5.

A single-valued mapping $F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ is said to be β-ordered compression mapping if F is a comparison mapping and:

$$F(x)\oplus F(y)\le \beta (x\oplus y),for0<\beta <1.$$

Definition 6.

Let $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ be a set-valued mapping. Then:

(i) 

M is said to be a comparison mapping if for any $v_x\in M(x),$ $x\propto v_x,$ and if $x\propto y,$ then for $v_x\in M(x)$ and $v_y\in M(y),$ $v_x\propto v_y,forallx,y\in {\mathcal{H}}_p.$

(ii) 

A comparison mapping M is said to be α-non-ordinary difference mapping if:

$$(v_x\oplus v_y)\oplus \alpha (x\oplus y)=0holds,forallx,y\in {\mathcal{H}}_p,v_x\in M(x)and{v}_y\in M(y);$$

(iii) 

A comparison mapping M is said to be θ-ordered rectangular if there exists a constant $\theta >0$ such that:

$$⟨v_x\odot v_y,-(x\oplus y)⟩\ge \theta {\parallel x\oplus y\parallel }^{2}holds,forallx,y\in {\mathcal{H}}_p,thereexists{v}_x\in M(x)and{v}_y\in M(y).$$

Definition 7.

A set-valued mapping $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is said to be λ-XOR-ordered strongly monotone compression mapping if $x\propto y,$ then there exists a constant $\lambda >0$ such that:

$$\lambda (v_x\oplus v_y)\ge x\oplus y,forallx,y\in {\mathcal{H}}_p,v_x\in M(x),v_y\in M(y).$$

Definition 8.

A set-valued mapping $T:{\mathcal{H}}_p\to C^{*}({\mathcal{H}}_p)$ is said to be $\mathcal{D}$-Lipschitz continuous if for all $x,y\in {\mathcal{H}}_p$, $x\propto y,$ there exists a constant ${\lambda }_T>0$ such that:

$$\mathcal{D}(T(x),T(y))\le {\lambda }_T\parallel x\oplus y\parallel .$$

Definition 9.

A single-valued mapping $F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ is said to be Lipschitz-type-continuous if there exists a constant $\delta >0$ such that:

$$\parallel F(x)\oplus F(y)\parallel \le \delta \parallel x\oplus y\parallel ,forallx,y\in {\mathcal{H}}_p.$$

Let $H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mappings, we consider the composition of H and F as:

$$(H\circ F)(x)=H(F(x)),\mathrm{for}\mathrm{all}x\in {\mathcal{H}}_p.$$

Definition 10.

Let $H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mappings such that $H\circ F$ is strongly comparison and β-ordered compression mapping. Then, a set-valued comparison mapping $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is said to be $(\alpha ,\lambda )$-XOR-NODSM if M is an α-non-ordinary difference mapping and λ-XOR-ordered strongly monotone compression mapping and $[(H\circ F)\oplus \lambda M]({\mathcal{H}}_p)={\mathcal{H}}_p,$ for $\alpha ,\beta ,\lambda >0.$

Definition 11.

Let $H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mapping such that $H\circ F$ is strongly comparison and β-ordered compression mapping. Suppose that the set-valued mapping $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is $(\alpha ,\lambda )$-XOR-NODSM mapping. We define the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}:{\mathcal{H}}_p\to {\mathcal{H}}_p$ by:

$${\mathcal{J}}_{\lambda ,M}^{H,F}(x)={[(H\circ F)\oplus \lambda M]}^{-1}(x),forallx\in {\mathcal{H}}_{p}and\alpha ,\lambda >0.$$

(1)

Now, we present some properties of the resolvent operator defined by (1).

Proposition 3.

Let $H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mappings such that $(H\circ F)$ is β-ordered compression mapping and $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is the set-valued θ-ordered rectangular mapping with $\lambda \theta >\beta$. Then, the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}:{\mathcal{H}}_p\to {\mathcal{H}}_p$ is single-valued.

Proof. 

For any given $u\in {\mathcal{H}}_p$ and $\lambda >0,$ let $x,y\in {[(H\circ F)\oplus \lambda M]}^{-1}(u).$ Then:

$$v_x=\frac{1}{\lambda }(u\oplus (H\circ F)(x))=\frac{1}{\lambda }(u\oplus H(F(x)))\in M(x),$$

and:

$$v_y=\frac{1}{\lambda }(u\oplus (H\circ F)(y))=\frac{1}{\lambda }(u\oplus (H\circ F)(y))\in M(y).$$

Using (i) and (ii) of Proposition 1, we obtain:

$$\begin{array}{ccc}\hfill v_x\odot v_y& =& \frac{1}{\lambda }(u\oplus H(F(x)))\odot \frac{1}{\lambda }(u\oplus H(F(y)))\hfill \\ & =& \frac{1}{\lambda }[(u\oplus H(F(x)))\odot (u\oplus H(F(y)))]\hfill \\ & =& -\frac{1}{\lambda }[(u\oplus H(F(x)))\oplus (u\oplus H(F(y)))]\hfill \\ & =& -\frac{1}{\lambda }[(u\oplus u)\oplus (H(F(x))\oplus H(F(y)))]\hfill \\ & =& -\frac{1}{\lambda }[0\oplus (H(F(x))\oplus H(F(y)))]\hfill \\ & \le & -\frac{1}{\lambda }[H(F(x))\oplus H(F(y))].\hfill \end{array}$$

Thus, we have:

$$v_x\odot v_y\le -\frac{1}{\lambda }[H(F(x))\oplus H(F(y))].$$

(2)

Since M is $\theta$-ordered rectangular mapping, $(H\circ F)$ is $\beta$-ordered compression mapping and using (2), we have:

$$\begin{array}{ccc}\hfill {\theta \parallel x\oplus y\parallel }^2& \le & ⟨v_x\odot v_y,-(x\oplus y)⟩\hfill \\ & \le & ⟨-\frac{1}{\lambda }[H(F(x))\oplus H(F(y))],-(x\oplus y)⟩\hfill \\ & \le & \frac{1}{\lambda }⟨(H\circ F)(x)\oplus (H\circ F)(y),x\oplus y⟩\hfill \\ & \le & \frac{1}{\lambda }⟨\beta (x\oplus y),x\oplus y⟩\hfill \\ & =& \frac{\beta }{\lambda }{\parallel x\oplus y\parallel }^2,\hfill \end{array}$$

i.e.,

$$\begin{array}{ccc}\hfill \left(\theta -\frac{\beta }{\lambda }\right){\parallel x\oplus y\parallel }^2& \le & 0,\mathrm{for}\lambda \theta >\beta ,\hfill \end{array}$$

which shows that:

$$\parallel x\oplus y\parallel =0,\mathrm{which}\mathrm{implies}x\oplus y=0.$$

Therefore $x=y,$ i.e., the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}$ is single-valued, for $\lambda \theta >\beta$.  □

Proposition 4.

Let $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ be $(\alpha ,\lambda )$-XOR-NODSM set-valued mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F}.$ Let $H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mappings such that $(H\circ F)$ is strongly comparison mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F}.$ Then the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}:{\mathcal{H}}_p\to {\mathcal{H}}_p$ is a comparison mapping.

Proof. 

Since M is $(\alpha ,\lambda )$-XOR-NODSM set-valued mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F},$ i.e., M is $\alpha$-non-ordinary difference as well as $\lambda$-XOR-ordered strongly monotone compression mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F}.$ For any $x,y\in {\mathcal{H}}_p,$ let $x\propto y$ and:

$$v_x^{*}=\frac{1}{\lambda }(x\oplus (H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))=\frac{1}{\lambda }(x\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x))))\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(x)),$$

(3)

and:

$$v_y^{*}=\frac{1}{\lambda }(y\oplus (H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(y))=\frac{1}{\lambda }(y\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y))))\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(y)).$$

(4)

Since M is $\lambda$-XOR-ordered strongly monotone compression mapping and using (3) and (4), we have:

$$\begin{array}{ccc}\hfill (x\oplus y)& \le & \lambda (v_x^{*}\oplus v_y^{*})\hfill \\ \hfill (x\oplus y)& \le & \left(x\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\right)\oplus \left(y\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right)\hfill \\ \hfill (x\oplus y)& \le & (x\oplus y)\oplus \left(H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right)\hfill \\ \hfill 0& \le & H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\hfill \\ \hfill 0& \le & \left[H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))-H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right]\hfill \\ & & \vee \left[H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))-H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\right],\hfill \end{array}$$

which implies either:

$$\begin{array}{ccc}\hfill 0& \le & \left[H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))-H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right]or\hfill \\ \hfill 0& \le & \left[H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))-H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\right].\hfill \end{array}$$

Thus, in both cases, we have:

$$(H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(x))\propto (H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(y)).$$

Since $(H\circ F)$ is strongly comparison mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F},$ thus, we have ${\mathcal{J}}_{\lambda ,M}^{H,F}(x)\propto {\mathcal{J}}_{\lambda ,M}^{H,F}(y),$ i.e., the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}$ is a comparison mapping.  □

Proposition 5.

If all the mappings and conditions are the same as those stated in Proposition 3, then the following condition holds:

$$\parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel \le \frac{1}{(\lambda \theta -\beta )}\parallel x\oplus y\parallel ,for\lambda \theta >\beta and\alpha ,\beta ,\lambda >0,$$

i.e., the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}$ is Lipschitz-type-continuous mapping.

Proof. 

Let $x,y\in {\mathcal{H}}_p,$ and:

$$v_x^{*}=\frac{1}{\lambda }(x\oplus (H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))=\frac{1}{\lambda }(x\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x))))\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(x)),$$

(5)

and:

$$v_y^{*}=\frac{1}{\lambda }(y\oplus (H\circ F)({\mathcal{J}}_{\lambda ,M}^{H,F}(y))=\frac{1}{\lambda }(y\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y))))\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(y)).$$

(6)

Since $(H\circ F)$ is $\beta$-ordered compression mapping and using (5) and (6), we have

$$\begin{array}{ccc}\hfill v_x^{*}\oplus v_y^{*}& =& \frac{1}{\lambda }\left[\left(x\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\right)\oplus \left(y\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right)\right]\hfill \\ & =& \frac{1}{\lambda }\left[(x\oplus y)\oplus \left(H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(x)))\oplus H(F({\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\right)\right]\hfill \\ & \le & \frac{1}{\lambda }[(x\oplus y)\oplus \beta ({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y))].\hfill \end{array}$$

(7)

Since M is $\theta$-ordered rectangular mapping and using (7), for any:

$${\mathcal{J}}_{\lambda ,M}^{H,F}(x)\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(x))\mathrm{and}{\mathcal{J}}_{\lambda ,M}^{H,F}(y)\in M({\mathcal{J}}_{\lambda ,M}^{H,F}(y)),\mathrm{we}\mathrm{have}:$$

$$\begin{array}{ccc}\hfill \theta \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}{(y)\parallel }^2& \le & ⟨v_x^{*}\odot v_y^{*},-({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y))⟩\hfill \\ & \le & ⟨v_x^{*}\oplus v_y^{*},({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y))⟩\hfill \\ & \le & \frac{1}{\lambda }[⟨(x\oplus y)\oplus \beta ({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)),{\mathcal{J}}_{\lambda ,M}^{H,F}(x)\hfill \\ & & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)⟩]\hfill \\ & \le & \frac{1}{\lambda }[\parallel (x\oplus y)\oplus \beta ({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\hfill \\ & & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y))\parallel \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel ].\hfill \end{array}$$

Using (iii) of Proposition 2, we have:

$$\begin{array}{ccc}\hfill \theta \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}{(y)\parallel }^2& \le & \frac{1}{\lambda }[\parallel (x\oplus y)\oplus \beta ({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\hfill \\ & & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y))\parallel \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel ]\hfill \end{array}$$

$$\begin{array}{cc}\le & \frac{1}{\lambda }[\parallel (x\oplus y)-(\beta ({\mathcal{J}}_{\lambda ,M}^{H,F}(x)\hfill \\ & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)))\parallel \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel ]\hfill \\ \le & \frac{1}{\lambda }\left[\parallel x\oplus y\parallel \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel \right]\hfill \\ & +\frac{\beta }{\lambda }{\parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel }^2.\hfill \end{array}$$

It follows that:

$$\begin{array}{c}\hfill \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel \le \frac{1}{(\lambda \theta -\beta )}\parallel x\oplus y\parallel ,for\lambda \theta >\beta .\end{array}$$

This completes the proof.

In support of Proposition 3–5, we have the following example.  □

Example 1.

Let ${\mathcal{H}}_p=[0,\infty )$ with the usual inner product and norm, and let $C=[0,1]$ be a normal cone in $[0,\infty )$. Let $H:{\mathcal{H}}_p\to {\mathcal{H}}_p$ and $F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ be the mappings defined by:

$$H(x)=\frac{x}{3}+1,andF(x)=\frac{x}{2},\forall x\in [0.\infty ).$$

Let $x,y\in {\mathcal{H}}_p,x\propto y,$ then we calculate:

$$\begin{array}{ccc}\hfill (H\circ F)(x)\oplus (H\circ F)(y)& =& H(F(x))\oplus H(F(y))\hfill \\ & =& \left(\frac{F(x)}{3}+1\right)\oplus \left(\frac{F(y)}{3}+1\right)\hfill \\ & =& \left(\frac{x}{6}+1\right)\oplus \left(\frac{y}{6}+1\right)\hfill \\ & =& \left(\left(\frac{x}{6}+1\right)-\left(\frac{y}{6}+1\right)\right)\vee \left(\left(\frac{y}{6}+1\right)-\left(\frac{x}{6}+1\right)\right)\hfill \\ & =& \left(\frac{x}{6}-\frac{y}{6}\right)\vee \left(\frac{y}{6}-\frac{x}{6}\right)\hfill \\ & =& \frac{1}{6}\left((x-y)\vee (y-x)\right)\hfill \\ & =& \frac{1}{6}(x\oplus y)\hfill \\ & \le & \frac{1}{5}(x\oplus y),\hfill \end{array}$$

i.e.,

$$(H\circ F)(x)\oplus (H\circ F)(y)\le \frac{1}{5}(x\oplus y),\forall x,y\in [0,\infty ).$$

Hence, $H\circ F$ is $\frac{1}{5}$-ordered compression mapping.

Suppose that $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is a the set-valued mapping defined by:

$$M(x)=\{x+1\},\forall x\in [0,\infty ).$$

It can be easily verified that M is a comparison mapping, 1-XOR-ordered strongly monotone comparison mapping, and 1-non-ordinary difference mapping.

Let $v_x=x+1\in M(x)$ and $v_y=y+1\in M(y),$ then we evaluate:

$$\begin{array}{ccc}\hfill ⟨v_x\odot v_y,-(x\oplus y)⟩& =& ⟨v_x\oplus v_y,x\oplus y⟩\hfill \\ & =& ⟨(x+1)\oplus (y+1),x\oplus y⟩\hfill \\ & =& ⟨x\oplus y,x\oplus y⟩\hfill \\ & =& {\parallel x\oplus y\parallel }^2\hfill \\ & \ge & \frac{1}{2}{\parallel x\oplus y\parallel }^2,\hfill \end{array}$$

i.e.,

$$⟨v_x\odot v_y,-(x\oplus y)⟩\ge \frac{1}{2}{\parallel x\oplus y\parallel }^2,\forall x,y\in [0,\infty ).$$

Thus, M is a $\frac{1}{2}$-ordered rectangular comparison mapping. Further, it is clear that for $\lambda =1,$ $[(H\circ F)\oplus \lambda M][0,\infty )=[0,\infty )$. Hence, M is an $(1,1)$-XOR-NODSM set-valued mapping.

The resolvent operator defined by (1) is given by:

$$\begin{array}{c}\hfill {\mathcal{J}}_{\lambda ,M}^{H,F}(x)=\frac{6x}{5},\forall x\in [0,\infty ).\end{array}$$

(8)

It is easy to check that the resolvent operator defined above is a comparison and single-valued mapping.

Further:

$$\begin{array}{ccc}\hfill \parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel & =& \parallel \frac{6x}{5}\oplus \frac{6y}{5}\parallel \hfill \\ & =& \frac{6}{5}\parallel x\oplus y\parallel \hfill \\ & \le & \frac{10}{3}\parallel x\oplus y\parallel ,\hfill \end{array}$$

i.e.,

$$\parallel {\mathcal{J}}_{\lambda ,M}^{H,F}(x)\oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(y)\parallel \le \frac{10}{3}\parallel x\oplus y\parallel ,\forall x,y\in [0,\infty ).$$

That is, the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}$ is $\frac{10}{3}$-Lipschitz-type-continuous.

3. Formulation of The Problem and Existence of Solution

Let ${\mathcal{H}}_p$ be a real positive Hilbert space. Let $A,B,C:{\mathcal{H}}_p\to C^{*}({\mathcal{H}}_p)$ and $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ be the set-valued mappings and let $G:{\mathcal{H}}_p\times {\mathcal{H}}_p\times {\mathcal{H}}_p\to {\mathcal{H}}_p$ be a single-valued mapping. Then, we consider the following problem:

Find $x\in {\mathcal{H}}_p$, $w\in A(x)$, $u\in B(x)$ and $v\in C(x)$ such that:

$$0\in G(w,u,v)\oplus M(x).$$

(9)

We call the problem in Equation (9) a generalized implicit set-valued variational inclusion problem with ⊕ operation.

(i) 

If $C\equiv 0$ and $G(w,u,v)=G(w,v)$, then problem (9) coincides with the problem studied by Ahmad et al. [1].

(ii) 

If $B,C\equiv 0$ and A is single-valued such that $G(w,u,v)=A(x)$, then problem (9) reduces to the problem studied by Ahmad et al. [7].

(iii) 

If $G\equiv 0$, then problem (9) becomes the problem studied by Li [22].

It is clear that for suitable choices of operators involved in the formulation of problem (9), one can obtain many related problems.

The following Lemma is a fixed point formulation of the problem in Equation (9).

Lemma 3.

The generalized implicit set-valued variational inclusion problem involving ⊕ operation (9) has a solution $x\in {\mathcal{H}}_p$, $w\in A(x)$, $u\in B(x)$, $v\in C(x)$ if and only if it satisfies the following equation:

$$x={\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w,u,v)\oplus (H\circ F)(x)],$$

(10)

where $\lambda >0$ is a constant.

Proof. 

Using the definition of the resolvent operator ${\mathcal{J}}_{\lambda ,M}^{H,F}$, and Equation (10), we get:

$$\begin{array}{ccc}\hfill x& =& {\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w,u,v)\oplus (H\circ F)(x)]\hfill \\ & =& {[(H\circ F)\oplus \lambda M]}^{-1}[\lambda G(w,u,v)\oplus (H\circ F)(x)],\hfill \\ \hfill (H\circ F)\oplus \lambda M(x)& =& \lambda G(w,u,v)\oplus (H\circ F)(x),\hfill \end{array}$$

which implies that $0\in G(w,u,v)\oplus M(x)$, the required generalized implicit set-valued variational inclusion problem with ⊕ operation (9).

Conversely, suppose that generalized implicit set-valued variational inclusion problem with ⊕ operation (9) is satisfied, that is, $x\in {\mathcal{H}}_p$, $w\in A(x)$, $u\in B(x)$ and $v\in C(x)$ such that:

$$0\in G(w,u,v)\oplus M(x),$$

which shows that:

$$\begin{array}{ccc}\hfill G(w,u,v)& =& M(x),\hfill \\ \hfill \lambda G(w,u,v)& =& \lambda M(x),\hfill \\ \hfill \lambda G(w,u,v)\oplus (H\circ F)(x)& =& (H\circ F)(x)\oplus \lambda M(x),\hfill \\ \hfill \lambda G(w,u,v)\oplus (H\circ F)(x)& =& [(H\circ F)\oplus \lambda M](x),\hfill \\ \hfill x& =& {[(H\circ F)\oplus \lambda M]}^{-1}[\lambda G(w,u,v)\oplus (H\circ F)(x)]\hfill \\ \hfill x& =& {\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w,u,v)\oplus (H\circ F)(x)]\hfill \end{array}$$

Thus, Equation (10) is satisfied.  □

Based on Lemma 3, we establish the following iterative algorithm to obtain the solution of the problem in Equation (9).

Iterative Algorithm 1.

For any given $x_0\in {\mathcal{H}}_p$, choose $w_0\in A(x_0)$, $u_0\in B(x_0)$, $v_0\in C(x_0)$ and using (10), let:

$$x_1=(1-\alpha )x_0+\alpha {\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w_0,u_0,v_0)\oplus (H\circ F)(x_0)].$$

Since $w_0\in A(x_0)$, $u_0\in B(x_0)$, $v_0\in C(x_0)$, by the Nadler’s theorem [28], there exists $w_1\in A(x_1)$, $u_1\in B(x_1)$, $v_1\in C(x_1)$, and using Proposition 2, we have:

$$\begin{array}{ccc}\hfill \parallel w_0\oplus w_1\parallel & \le & \parallel w_0-w_1\parallel \le (1+1)\mathcal{D}(A(x_0),A(x_1))\hfill \\ \hfill \parallel u_0\oplus u_1\parallel & \le & \parallel u_0-u_1\parallel \le (1+1)\mathcal{D}(B(x_0),B(x_1))\hfill \\ \hfill \parallel v_0\oplus v_1\parallel & \le & \parallel v_0-v_1\parallel \le (1+1)\mathcal{D}(C(x_0),C(x_1)),\hfill \end{array}$$

where $\mathcal{D}$ is the Hausdorff metric on $C^{*}({\mathcal{H}}_p)$. Let:

$$x_2=(1-\alpha )x_1+\alpha {\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w_1,u_1,v_1)\oplus (H\circ F)(x_1)].$$

Again by Nadler’s theorem [28], there exist $w_2\in F(x_2)$, $u_2\in B(x_2)$, $v_2\in C(x_2)$ such that:

$$\begin{array}{ccc}\hfill \parallel w_1\oplus w_2\parallel & \le & \parallel w_1-w_2\parallel \le (1+2^{-1})\mathcal{D}(A(x_1),A(x_2)),\hfill \\ \hfill \parallel u_1\oplus u_2\parallel & \le & \parallel u_1-u_2\parallel \le (1+2^{-1})\mathcal{D}(B(x_1),B(x_2)),\hfill \\ \hfill \parallel v_1\oplus v_2\parallel & \le & \parallel v_1-v_2\parallel \le (1+2^{-1})\mathcal{D}(C(x_1),C(x_2)).\hfill \end{array}$$

Continuing the above procedure inductively, we have the following scheme:

$$x_{n+1}=(1-\alpha )x_n+\alpha {\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w_n,u_n,v_n)\oplus (H\circ F)(x_n)].$$

Since $w_{n+1}\in A(x_{n+1})$, $u_{n+1}\in B(x_{n+1})$, $v_{n+1}\in C(x_{n+1})$, such that:

$$\begin{array}{ccc}\hfill \parallel w_n\oplus w_{n+1}\parallel & \le & \parallel w_n-w_{n+1}\parallel \le (1+{(n+1)}^{-1})\mathcal{D}(A(x_n),A(x_{n+1})),\hfill \\ \hfill \parallel u_n\oplus u_{n+1}\parallel & \le & \parallel u_n-u_{n+1}\parallel \le (1+{(n+1)}^{-1})\mathcal{D}(B(x_n),B(x_{n+1})),\hfill \\ \hfill \parallel v_n\oplus v_{n+1}\parallel & \le & \parallel v_n-v_{n+1}\parallel \le (1+{(n+1)}^{-1})\mathcal{D}(C(x_n),C(x_{n+1})).\hfill \end{array}$$

where $\alpha \in [0,1],n=0,1,2,\cdots$.

Theorem 1.

Let $C\subset {\mathcal{H}}_p$ be a normal cone with constant ${\lambda }_N,H,F:{\mathcal{H}}_p\to {\mathcal{H}}_p$ and $G:{\mathcal{H}}_p\times {\mathcal{H}}_p\times {\mathcal{H}}_p\to {\mathcal{H}}_p$ be the single-valued mappings such that $(H\circ F)$ be strongly comparison, β-ordered compression mapping, G is ${\beta }_1$-ordered compression mapping in the first argument, ${\beta }_2$-ordered compression mapping in the second argument, and ${\beta }_3$-ordered compression mapping in the third argument. Let $A,B,C:{\mathcal{H}}_p\to C^{*}({\mathcal{H}}_p)$ be the set-valued mappings such that A is ${\lambda }_A$-$\mathcal{D}$-Lipschitz-continuous, B is ${\lambda }_B$-$\mathcal{D}$-Lipschitz-continuous, and C is ${\lambda }_C$-$\mathcal{D}$-Lipschitz-continuous. Suppose that $M:{\mathcal{H}}_p\to 2^{{\mathcal{H}}_p}$ is $(\alpha ,\lambda )$- XOR-NODSM set-valued mapping with respect to ${\mathcal{J}}_{\lambda ,M}^{H,F}$ and θ-ordered rectangular mapping with $\lambda \theta >\beta$. If $x_{n+1}\propto x_n$, $n=0,1,2,\dots .$ and the following condition is satisfied:

$$\mid \lambda \mid [{\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C]+\beta <\frac{1-{\lambda }_N(1-\alpha )}{{\lambda }_N\alpha {\theta }^{{}^{\prime }}},$$

(11)

where ${\theta }^{{}^{\prime }}=\frac{1}{\lambda \theta -\beta }$ and $\lambda \theta >\beta$; ${\beta }_1,{\beta }_2,{\beta }_3,{\lambda }_A,{\lambda }_B,{\lambda }_C,{\lambda }_N,\alpha ,\theta ,\beta$ all are positive constants.Then, the generalized implicit set-valued variational inclusion problem with ⊕ operation (9) has a solution $x\in {\mathcal{H}}_p,w\in A(x),u\in B(x),andv\in C(x)$. Moreover, the iterative sequences $\{x_n\},\{w_n\}\{u_n\},and\{v_n\}$ generated by Algorithm 1 converge strongly to $x,w,u,andv$, the solution of generalized implicit set-valued variational inclusions problem with ⊕ operation (9).

Proof. 

By Algorithm 1 and Proposition 1, we have:

$$\begin{array}{ccc}\hfill 0& \le & x_{n+1}\oplus x_n\hfill \\ & =& \left[(1-\alpha )x_n+\alpha \left({\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w_n,u_n,v_n)\oplus (H\circ F)(x_n)]\right)\right]\hfill \\ & & \oplus \left[(1-\alpha )x_{n-1}+\alpha \left({\mathcal{J}}_{\lambda ,M}^{H,F}[\lambda G(w_{n-1},u_{n-1},v_{n-1})\oplus (H\circ F)(x_{n-1})]\right)\right]\hfill \\ & =& (1-\alpha )(x_n\oplus x_{n-1})+\alpha ({\mathcal{J}}_{\lambda ,M}^{H,F}(\lambda G(w_n,u_n,v_n)\oplus (H\circ F)(x_n))\hfill \\ & & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(\lambda G(w_{n-1},u_{n-1},v_{n-1})\oplus (H\circ F)(x_{n-1}))).\hfill \end{array}$$

(12)

Using Proposition 2 and Lipschitz-type-continuity of the resolvent operators (1) and (12), we have:

$$\begin{array}{ccc}\parallel x_{n+1}\oplus x_n\parallel \hfill & \le \hfill & {\lambda }_N\parallel (1-\alpha )(x_n\oplus x_{n-1})+\alpha [{\mathcal{J}}_{\lambda ,M}^{H,F}(\lambda G(w_n,u_n,v_n)\oplus (H\circ F)(x_n))\hfill \\ & & \oplus {\mathcal{J}}_{\lambda ,M}^{H,F}(\lambda G(w_{n-1},u_{n-1},v_{n-1})\oplus (H\circ F)(x_{n-1}))]\parallel \hfill \\ & \le \hfill & {\lambda }_N\parallel (1-\alpha )(x_n\oplus x_{n-1})\parallel +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\parallel (\lambda G(w_n,u_n,v_n)\oplus (H\circ F)(x_n))\hfill \\ & & \oplus (\lambda G(w_n-1,u_n-1,v_n-1)\oplus (H\circ F)(x_{n-1}))\parallel \hfill \\ & \le \hfill & {\lambda }_N(1-\alpha )\parallel x_n\oplus x_{n-1}\parallel +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\parallel (\lambda G(w_n,u_n,v_n)\oplus \lambda G(w_{n-1},u_{n-1},v_{n-1})-((H\circ F)(x_n)\hfill \\ & & \oplus (H\circ F)(x_{n-1}))\parallel \hfill \\ & \le \hfill & {\lambda }_N(1-\alpha )\parallel x_n\oplus x_{n-1}\parallel +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\left|\lambda \right|\parallel G(w_n,u_n,v_n)\oplus \lambda G(w_{n-1},u_{n-1},v_{n-1})\parallel \hfill \\ & & +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\parallel (H\circ F)(x_n)\oplus (H\circ F)(x_{n-1})\parallel ,\hfill \end{array}$$

(13)

where ${\theta }^{{}^{\prime }}=\frac{1}{\lambda \theta -\beta }$.

Since G is ${\beta }_1$-compression mapping in the first argument, ${\beta }_2$-compression mapping in the second argument, and ${\beta }_3$-compression mapping in third argument, A is ${\lambda }_A$-$\mathcal{D}$-Lipschitz-continuous, B is ${\lambda }_B$-$\mathcal{D}$-Lipschitz-continuous, and C is ${\lambda }_C$-$\mathcal{D}$-Lipschitz-continuous, using Algorithm 1, we have:

$$\begin{array}{ccc}\parallel G(w_n,u_n,v_n)\oplus G(w_{n-1},u_{n-1},v_{n-1})\parallel \hfill & =\hfill & \parallel G(w_n,u_n,v_n)\oplus G(w_{n-1},u_n,v_n)\oplus G(w_{n-1},u_n,v_n)\hfill \\ & & \oplus G(w_{n-1},u_{n-1},v_n)\oplus G(w_{n-1},u_{n-1},v_n)\hfill \\ & & \oplus G(w_{n-1},u_{n-1},v_{n-1})\parallel \hfill \\ & \le \hfill & {\beta }_1\parallel w_n\oplus w_{n-1}\parallel +{\beta }_2\parallel u_n\oplus u_{n-1}\parallel +{\beta }_3\parallel v_n\oplus v_{n-1}\parallel \hfill \\ & \le \hfill & {\beta }_1\parallel w_n-w_{n-1}\parallel +{\beta }_2\parallel u_n-u_{n-1}\parallel +{\beta }_3\parallel v_n-v_{n-1}\parallel \hfill \\ & \le \hfill & {\beta }_1(1+n^{-1})\mathcal{D}(A(x_n),A(x_{n-1}))+{\beta }_2(1+n^{-1})\mathcal{D}(B(x_n),B(x_{n-1}))\hfill \\ & & +{\beta }_3(1+n^{-1})\mathcal{D}(C(x_n),C(x_{n-1}))\hfill \\ & \le \hfill & {\beta }_1(1+n^{-1}){\lambda }_A\parallel x_n-x_{n-1}\parallel +{\beta }_2(1+n^{-1}){\lambda }_B\parallel x_n-x_{n-1}\parallel \hfill \\ & & +{\beta }_3(1+n^{-1}){\lambda }_C\parallel x_n-x_n-1\parallel \hfill \\ & =\hfill & ({\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C)(1+n^{-1})\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(14)

As $(H\circ F)$ is $\beta$-ordered compression mapping, we have:

$$\parallel (H\circ F)(x_n)\oplus (H\circ F)(x_{n-1})\parallel \le \beta \parallel x_n\oplus x_{n-1}\parallel .$$

(15)

Using Equations (14) and (15), (13) becomes:

$$\begin{array}{ccc}\parallel x_{n+1}\oplus x_n\parallel \hfill & \le \hfill & {\lambda }_N(1-\alpha )\parallel x_n\oplus x_{n-1}\parallel +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\left|\lambda \right|[({\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C)(1+1/n)]\parallel x_n-x_{n-1}\parallel \hfill \\ & & +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\beta \parallel x_n\oplus x_{n-1}\parallel \hfill \\ & =\hfill & {\lambda }_N(1-\alpha )\parallel x_n-x_{n-1}\parallel +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\left|\lambda \right|[({\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C)(1+1/n)]\parallel x_n-x_{n-1}\parallel \hfill \\ & & +{\lambda }_N\alpha {\theta }^{{}^{\prime }}\beta \parallel x_n-x_{n-1}\parallel \hfill \\ & =\hfill & \partial (P_n)\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(16)

As $x_{n+1}\propto x_n$, $n=0,1,2,\cdots ,$ we have:

$$\parallel x_{n+1}-x_n\parallel \le \partial (P_n)\parallel x_n-x_{n-1}\parallel ,$$

where $\partial (P_n)={\lambda }_N(1-\alpha )+{\lambda }_N\alpha {\theta }^{{}^{\prime }}\left|\lambda \right|[({\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C)(1+1/n)]+{\lambda }_N\alpha {\theta }^{{}^{\prime }}\beta .$

Let $\partial (P)={\lambda }_N(1-\alpha )+{\lambda }_N\alpha {\theta }^{{}^{\prime }}\left|\lambda \right|[{\beta }_1{\lambda }_A+{\beta }_2{\lambda }_B+{\beta }_3{\lambda }_C]+{\lambda }_N\alpha {\theta }^{{}^{\prime }}\beta .$

We know that $\partial (P_n)\to \partial (P)$ as $n\to \infty$. It follows from condition (11) that $0<\partial (p)<1$, and consequently, $\{x_n\}$ is a cauchy sequence in ${\mathcal{H}}_p$ and since ${\mathcal{H}}_p$ is complete, there exists an $x\in {\mathcal{H}}_p$ such that $x_n\to x$, as $n\to \infty$. From Algorithm 1, we have:

$$\begin{array}{ccc}\hfill \parallel w_n\oplus w_{n+1}\parallel & \le & \parallel w_n-w_{n+1}\parallel \hfill \\ & \le & (1+{(n+1)}^{-1})\mathcal{D}(A(x_n),A(x_{n+1}))\hfill \\ & \le & (1+{(n+1)}^{-1}){\lambda }_F\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill \parallel u_n\oplus u_{n+1}\parallel & \le & \parallel u_n-u_{n+1}\parallel \hfill \\ & \le & (1+{(n+1)}^{-1})\mathcal{D}(B(x_n),B(x_{n+1}))\hfill \\ & \le & (1+{(n+1)}^{-1}){\lambda }_B\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill and\parallel v_n\oplus v_{n+1}\parallel & \le & \parallel v_n-v_{n+1}\parallel \hfill \\ & \le & (1+{(n+1)}^{-1})\mathcal{D}(C(x_n),C(x_{n+1}))\hfill \\ & \le & (1+{(n+1)}^{-1}){\lambda }_C\parallel x_n-x_{n-1}\parallel .\hfill \end{array}$$

(19)

It is clear from Euqations (17)–(19) that $\{w_n\}$, $\{u_n\}$, and $\{v_n\}$ are also cauchy sequences in ${\mathcal{H}}_p$. Let $w_n\to w$, $u_n\to u$ and $v_n\to v$, as $n\to \infty$. In view of Lemma 3, we conclude that $(x,w,u,v),$ such that $x\in {\mathcal{H}}_p$, $w\in A(x)$, $u\in B(x)$ and $v\in C(x)$ is a solution of a generalized implicit set-valued variational inclusion problem with ⊕ operation (9). Now, we show that with $w\in A(x)$, we have:

$$\begin{array}{ccc}\hfill d(w,A(x))& \le & \parallel w\oplus w_n\parallel +d(w_n,A(x))\hfill \\ & \le & \parallel w-w_n\parallel +\parallel w_n\oplus A(x)\parallel \hfill \\ & \le & \parallel w-w_n\parallel +\mathcal{D}(A(x_n),A(x))\hfill \\ & \le & \parallel w-w_n\parallel +{\lambda }_A\parallel x_n-x\parallel \to 0,asn\to \infty ,\hfill \end{array}$$

which implies that $d(w,A(x))=0$, and since $A(x)\in C^{*}({\mathcal{H}}_p)$, it follows that $w\in A(x)$. Similarly, we can show that $u\in B(x)$ and $v\in C(x)$, respectively. This complete the proof.  □

4. Conclusions

In this paper, we considered a generalized implicit set-valued variational inclusion problem with ⊕ operation, which includes many previously studied problems in ordered spaces as special cases. A resolvent operator which involves composition of two mappings was considered, and we proved some properties of it. An existence and convergence result was proven for our problem in real ordered positive Hilbert spaces.

We remark that our results may be generalized further in higher dimensional spaces.