# Algorithms for approximation of J-Fixed Points of Nonexpensive-Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems.

Main Thesis

Thesis

It is well known that many physically significant problems in different areas of research can be transformed at an equilibrium state into an inclusion problem of the form 0 ∈ Au, where A is either a multi-valued accretive map from a real Banach space into itself or a multi-valued monotone map from a real Banach space into its dual space. In several applications, the solutions of the inclusion problem, when the map A is monotone, corresponds to minimizers of some convex functions. It is known that the sub-differential of any convex function, say g, and denoted by ∂g is monotone, and for any vector, say v, in the domain of g, 0 ∈ ∂g(v) if and only if v is a minimizer of g. Setting ∂g ≡ A, solving the inclusion problem, is equivalent to finding minimizers of g. The method of approximation of solutions of the inclusion problem 0 ∈ Au, when the map A is monotone in real Banach spaces, was not known until in 2016 when Chidume and Idu [52] introduced J-fixed points technique. They proved that the J-fixed points correspond to zeros of monotone maps which are minimizers of some convex functions. In general, finding closed form solutions of the inclusion problem, where A is monotone is extremely difficult or impossible. Consequently, solutions are sought through the construction of iterative algorithms for approximating J-fixed points of nonlinear maps. In chapter three, four and seven of the thesis, we present a convergence result for approximating zeros of the inclusion problem 0 ∈ Au. Let H1 and H2 be real Hilbert spaces and K1, K2, · · · , KN , and Q1, Q2, · · · , QP , be nonempty, closed and convex subsets of H1 and H2, respectively, with nonempty intersections K and Q, respectively, that is, K = K1 ∩ K2 ∩ · · · ∩ KN ̸= ∅ and Q = Q1 ∩ Q2 ∩ · · · ∩ QP ̸= ∅. Let B : H1 → H2 be a bounded linear map, Gi : H1 → H1, i = 1, · · · , N and Aj : H2 → H2, j = 1, · · · , P be given maps. The common split variational inequality problem introduced by Censor et al. [32] in 2005, and denoted by (CSVIP), is the problem of finding an element u∗ ∈ K for which(⟨u − u∗, Gi(u∗)⟩ ≥ 0, ∀ u ∈ Ki, i = 1, 2, · · · , N, such that v∗ = Bu∗ ∈ Q solves ⟨v − v∗, Aj (v∗)⟩ ≥ 0, ∀ v ∈ Qj , j = 1, 2, · · · , P. The motivation for studying this class of problems with N > 1 stems from a simple observation that if we choose Gi ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki , which is the known convex feasibility problem (CFP) such that Bu∗ ∈ ∩P j=1V I(Qj , Aj ). If the sets Ki are the fixed point sets of maps Si : H1 → H1, then, the convex feasibility problems (CFP) is the common fixed points problem(CFPP) whose image under B is a common solution to variational inequality problems (CSVIP). If we choose Gi ≡ 0 and Aj ≡ 0, the problem reduces to finding u ∗ ∈ ∩N i=1Ki such that the point Bu∗ ∈ ∩P j=1Qj which is the well known multiple-sets split feasibility problem or common split feasibility problem which serves as a model for many inverse problems where the constraints are imposed on the solutions in the domain of a linear operator as well as in the range of the operator. A lot of research interest is now devoted to split variational inequality problem and its generalizations. In chapter five and six of the thesis, we present convergence theorems for approximating solutions of variational inequalities and a convex feasibility problem; and solutions of split variational inequalities and generalized split feasibility problem