Graph convergence for the H ( ⋅ , ⋅ ) -mixed mappingwith an application for solving the system of generalized variationalinclusions
Shamshad Husain, Sanjeev Gupta, Vishnu Narayan Mishra · Fixed Point Theory and Applications · 2013
In this paper, we investigate a class of accretive mappings called the H(•, •)-mixed mappings in Banach spaces.We prove that the proximal-point mapping associated with the H(•, •)-mixed mapping is single-valued and Lipschitz continuous.Some examples are given to justify the definition of H(•, •)-mixed mapping.Further, a concept of graph convergence concerned with the H(•, •)-mixed mapping is introduced in Banach spaces and some equivalence theorems between graph-convergence and proximal-point mapping convergence for the H(•, •)-mixed mappings sequence are proved.As an application, we consider a system of generalized variational inclusions involving H(•, •)-mixed mappings in real q-uniformly smooth Banach spaces.Using the proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithm under some suitable conditions.