New generalized systems of nonlinear ordered variational inclusions involving ⊕ operator in real ordered Hilbert spaces.

This manuscript deals with two general systems of nonlinear ordered variational inclusion problems. We also construct some new iterative algorithms for finding approximation solutions to the general systems of nonlinear ordered variational inclusions and prove the convergence of the sequences obtained by the schemes. The results presented in the manuscript are new and improve some well-known results in the literature.

Introduction

A lot of work has been added into the theory of variational inequalities since its seed was planted by Lions et al. [24]. On account of its wide applications in physics and applied sciences etc., the classical variational inequalities have been extensively studied by many researchers in different ways [1, 4, 5, 7–10].

A useful and important generalization of variational inequality problem is variational inclusion problem which was introduced and studied by Hasounni et al. [16]. Furthermore, they proposed a perturbed iterative algorithm for solving the variational inclusion problem.

Fang et al. [12] introduced and studied H-monotone operators, which was used to design a resolvent operator and to prove its Lipschitz continuity. Furthermore, they also introduced a class of variational inclusions in Hilbert space. Fang et al. [13] additionally presented another class of generalized monotone operators, called -monotone operators, which generalize different classes of maximal monotone, maximal η-monotone and H-monotone operators.

Recently, Lan et al. [17] presented another idea of -accretive mappings, which generalized the current monotone or accretive operators, and concentrated a few properties of mappings. They examined a class of variational inclusions using the resolvent operator related with -accretive mappings.

Amann [6] studied the number of fixed points for a continuous operator on a bounded order interval , an ordered Banach space. The nonlinear mapping fixed point theory and applications have been widely studied in ordered Banach spaces [4, 14, 15]. In this manner, it is essential that summed up nonlinear ordered variational inclusions (ordered equation) are contemplated.

Plenty of research concerned with the ordered equations and ordered variational inequalities in ordered Banach spaces has been done by Li et al.; see [21, 23]. Many problems concerning ordered variational inclusions are answered by the resolvent technique linked with RME set-valued mappings [19], -NODM set-valued mapping [20], -weak RRD mapping [2] and -weak ANODD set-valued map with strongly comparison mapping A [21] and many more see; e.g., [3, 22, 25, 26, 29] and the references therein.

In this work, we make use of the resolvent operator approach for the approximation solvability of solutions of implicit system of generalized nonlinear ordered variational inclusions in real ordered Hilbert spaces.

Preliminaries

In this part, we present some basic notions and results for the building up the manuscript.

Allow to be a real ordered Hilbert space endowed with a norm , and an inner product , d be a metric induced by the norm , be a collection of all closed and bounded subsets of and be a Hausdorff metric on defined as where , and .

Definition 2.1

Let be a nonvoid closed, convex subset of . Then is called a cone if

and , ;

x and , then .

Definition 2.2

([11])

A cone is said to be normal iff there exists with implying , where is called a normal constant of .

Definition 2.3

A relation ≤ defined as iff for is known as a partial order relation expounded by in ; then () is called a real ordered Hilbert space.

Definition 2.4

([27])

Members having the relation (or ) are called comparable with each other.

Definition 2.5

([27])

For arbitrary elements , and mean the least upper bound and the greatest upper bound of the set . Suppose and exist; some binary relations are defined as follows: The operations ∨, ∧, ⊕ and ⊙ are called OR, AND, XOR and XNOR operations, respectively.

;

;

;

.

Proposition 1

([11])

For any positive integer n, if and (), then .

Proposition 2

([11, 20])

Let be two operations on . Then the following hold:

;

;

, if ;

, if ;

if , then if and only if ;

;

;

, if , and .

Proposition 3

([11])

Let be a normal cone in with normal constant , then, for each , the following hold:

;

;

;

if , then .

Definition 2.6

([20])

Let to be a single-valued map.

A is called a δ-order non-extended map, if there is a positive constant such that

A is called a strongly comparison map, if it is a comparison map and iff , for all .

Definition 2.7

([2])

A single-valued map is termed a β-ordered compression, if it is comparison map and

Definition 2.8

([18])

A map is called -restricted-accretive map, if it is a comparison and ∃ constants such that where I is the identity map on .

Lemma 2.1

([28])

Let be a constant. Then the function for is nonnegative and strictly decreases and . Furthermore, if , then .

Lemma 2.2

([30])

Assume that and be two sequences of nonnegative real numbers such that where and . Then .

Ordered weak-ARD mapping in ordered Hilbert spaces

Definition 3.1

Let be a strong comparison and β-ordered compression mapping and be a set-valued mapping. Then

M is said to be a comparison mapping, if for any , and if , then, for any and any , , for all ;

a comparison mapping M is said to be ordered rectangular, if for each , and such that

a comparison mapping M is said to be a γ-ordered rectangular with respect to A, if there exists a constant for any , there exist and such that holds, where and are said to be -elements, respectively;

M is said to be a weak comparison mapping with respect to A, if, for any , , there exist and such that , , where and are said to be weak comparison elements, respectively;

M is said to be a λ-weak ordered different comparison mapping with respect to A, if there exists a constant such that, for any , there exist and , holds, where and are said to be λ-elements, respectively;

a weak comparison mapping M is said to be a -weak ARD mapping with respect to A, if M is a -ordered rectangular and λ-weak ordered different comparison mapping with respect to A and , for and there exist and such that and are -elements, respectively.

Definition 3.2

A set-valued mapping is said to be -Lipschitz continuous, if for each , there exists a constant such that

Definition 3.3

Let be a set-valued mapping, be a single-valued mapping and be an identity mapping. Then a weak comparison mapping M is said to be a -weak-ARD mapping with respect to , if M is a -ordered rectangular and λ-weak ordered different comparison mapping with respect to and , for and there exist and such that and are called -elements, respectively.

Definition 3.4

Let be a normal cone with normal constant and be a weak-ARD set-valued mapping. Let be the identity mapping and be a set-valued mapping and be a single-valued mapping. The relaxed resolvent operator associated with I, A and M is defined by

The relaxed resolvent operator defined by (1) is single-valued, a comparison mapping and Lipschitz continuous.

Proposition 4

([2])

Let be a β-ordered compression mapping and be the set-valued ordered rectangular mapping. Then the resolvent is single-valued, for all .

Proposition 5

([2])

Let be a -weak-ARD set-valued mapping with respect to . Let be a strongly comparison mapping with respect to and be the identity mapping. Then the resolvent operator is a comparison mapping.

Proposition 6

([2])

Let be a -weak-ARD set-valued mapping with respect to . Let be a strongly comparison and β-ordered compression mapping with respect to with condition . Then the following condition survives:

Formulation of the problems

Let , and to be single-valued mappings, for . Let be a set-valued map and be set-valued weak-ARD mapping. Then we have the problem:

Find and , for , such that where and are given positive constants. Problem (2) is called a generalized set-valued system of nonlinear ordered variational inclusions problem for weak-ARD mappings.

If is a single-valued mapping, then problem (2) becomes:

Find , such that This problem is known as a generalized system of nonlinear ordered variational inclusions problem involving weak-ARD mappings.

Remark

Here, we discuss special cases for our problem (2), which was encountered by Li et al.

For , , and , then problem (2) is reduced to finding such that This problem was considered by Li et al. [23] and coined a general nonlinear mixed-order quasi-variational inclusion (GNMOQVI) involving the ⊕ operator in an ordered Banach space.

If (zero mapping), then problem (4) is reduced to finding such that This problem were considered by Li for ordered RME set-valued mappings [19] and -NODM set-valued mappings [20].

Lemma 4.1

Let and for . Then is a solution of problem (2) if and only if it satisfies where and for .

Proof

The proof follows from the definition of the relaxed resolvent operator. □

Design of the algorithms

If we choose and for , is single-valued operator, then Algorithm 1 reduces to Algorithm 2 for problem (3).

for the problem ( ):

for the problem ( ):

Main results

Theorem 6.1

Let , and be the single-valued mappings such that be -ordered compression mapping, be -ordered compression, -ordered restricted-accretive mapping and be -ordered compression mapping with respect to the jth argument. Let be a --ordered Lipschitz continuous set-valued mapping. Let be a -weak rectangular different compression mapping with respect to and if , and for all , , then the following condition holds: for all , which in turn, implies that problem (2) admits a solution , where and . Moreover, iterative sequences and generated by Algorithm 1, converge strongly to and , for , respectively.

Using Algorithm 1 and Proposition 2, for , we have Using Definition 2.2, Proposition 6 and Eq. (10), we get Now, from Eq. (11), we compute By the definition of as a -ordered compression map with respect to the jth argument, we have Using Proposition 6 and Eq. (13) in Eq. (11), we obtain which implies that where and From Eq. (14), we know that the sequence is monotonic decreasing and as . Thus, . Since for . We get . By Lemma 2.1, we have , from Eq. (14), it follows that is a Cauchy sequence and there exists such that as for . Next, we show that as for . It follows from Eq. (13) that the are also Cauchy sequences. Hence, there exists such that as for . Furthermore, Since is closed for , we have for . By using continuity and for satisfy Eq. (6) and so by Lemma 4.1, problem (2) has a solution , where for and . This completes the proof. □

Theorem 6.2

Suppose that and are the same as in Theorem 6.1 for . Let be -Lipschitz continuous and be -ordered compression mapping with respect to the jth argument. Let there be constants , for such that Then problem (3) has a unique solution . Moreover, the iterative sequence generated by Algorithm 2 converges strongly to for .

Let us define a norm on the product space by Then it can easily be seen that is a Banach space.

Setting Define a mapping as For any , we have First of all, we have to calculate as follows: From Definition 2.2 and Proposition 3, we have Further, we calculate Now we calculate the inner part estimate of the above expression with the help of the properties of the -operator for . We have By using the Lipschitz continuity of -operator in Eq. (19), we have Using Eq. (20) in Eq. (18) and then use it in Eq. (17), we have Now, Eq. (16) can be rewritten as where . Finally, from Eq. (22), Eq. (16) can be written as It follows from the condition (9) that . This implies that is a contraction which in turn implies that there exists a unique such that . Thus, is the unique solution of problem (3). Now, we prove that as for . In fact, it follows from Eq. (8) and the Lipschitz continuity of the relaxed resolvent operator that From the previous calculations, we have where , . Algorithm 2 yields . Now, Lemma 2.2 implies that , and so as for . This completes the proof. □

Conclusion

Two of the most troublesome and imperative issues identified with inclusions are the foundation of generalized inclusions and the improvement of an iterative calculation. In this article, two systems of variational inclusions were presented and contemplated, which is a broader aim than the numerous current systems of generalized ordered variational inclusions in the literature. An iterative calculation is performed with a weak ARD mapping to an inexact solution of our systems, and the convergence criterion is likewise addressed.

We comment that our outcomes are new and valuable for additionally investigations. Considerably more work is required in every one of these regions to address utilizations of the system of general ordered variational inclusions in engineering and physical sciences.