Algorithms For Approximation of J-Fixed Points of Nonexpansive - Type Maps, Zeros of Monotone Maps, Solutions of Feasibility and Variational Inequality Problems

Nnakwe, Monday Ogudu (2019-07-05)

Main Thesis


It is well known that many physically significant problems in different areas of research can be transformed at 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 zerosof 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 vi 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 ∗ = 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 gener-alizations.In chapter five and six of the thesis, we present convergence theorems for approximating solu-tions of variational inequalities and a convex feasibility problem; and solutions of split varia-tional inequalities and generalized split feasibility problems.