Abstract
In this paper, we consider and study a system of multi-valued mixed variational inclusions with XOR-operation ⊕ in real ordered uniformly smooth Banach spaces. This system consists of bimappings, multi-valued mappings and Cayley operators. An iterative algorithm is suggested to find the solution to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and consequently an existence and convergence result is proved. In support of our main result, an example is constructed.
Highlights
In 1964, Stampacchia [1] investigated the theory of variational inequality which provides us a lenient way for solving perplexities occurring in industry, finance, economics, operation research, optimization, decision sciences and several other branches of pure and applied sciences, and so forth, see, for example, [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
Hassouni and Moudafi [18] studied a mixed type variational inequality which involves a nonlinear term called variational inclusion. They used the resolvent operator technique in order to find the solution to their problem as the projection method does not work due to the nonlinear term
Yao [20] and many more researchers considered various system of variational inequalities, see [21,22,23,24,25,26,27,28,29]. It has been shown by Pang [30] that the Nash equilibrium problem and various equilibrium type problems, like the traffic equilibrium problem, spatial equilibrium problem and the general equilibrium programming problems from operation research, game theory, mathematical physics, and so forth, can be formulated as a variational inequality problem defined over a product of sets, which is equivalent to a system of variational inequalities
Summary
In 1964, Stampacchia [1] investigated the theory of variational inequality which provides us a lenient way for solving perplexities occurring in industry, finance, economics, operation research, optimization, decision sciences and several other branches of pure and applied sciences, and so forth, see, for example, [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Hassouni and Moudafi [18] studied a mixed type variational inequality which involves a nonlinear term called variational inclusion. They used the resolvent operator technique in order to find the solution to their problem as the projection method does not work due to the nonlinear term. We consider and study a system of multi-valued mixed variational inclusions with. We prove the existence of solutions to a system of multi-valued mixed variational inclusions with XOR-operation ⊕ and we discuss the convergence of the iterative sequences generated by the proposed algorithm.
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