Generalized Set-valued Nonlinear Variational-like Inequalities and Fixed Point Problems: Existence and Approximation Solvability Results

The paper is devoted to the introduction of a new class of generalized set-valued nonlinear variational-like inequality problems in the setting of Banach spaces. By means of the notion of P-eta-proximal mapping, we prove its equivalence with a class of generalized implicit Wiener-Hopf equations and employ the obtained equivalence relationship and Nadler's technique to suggest a new iterative algorithm for finding an approximate solution of the considered problem. The existence of solution and the strong convergence of the sequences generated by our proposed iterative algorithm to the solution of our considered problem are verified. The problem of finding a common element of the set of solutions of a generalized nonlinear variational-like inequality problem and the set of fixed points of a total asymptotically nonexpansive mapping is also investigated. The final section deals with the investigation and analysis of the main results appeared in Kazmi and Bhat (Appl Math Comput 166:164-180, 2005) and some comments relating to them are given. The results presented in this article extend and improve some known results in the literature.