Results of well-posedness for mixed equilibrium problems

This thesis is an attempt to study in a rigorous manner some specific cases of well posedness (w-p) in applied sciences. More precisely, the thesis looks at mixed hemi-equilibrium in Banach space, all with the aim of obtaining results on the well posed-ness of equilibrium problems in the case of convex subsets in B.S, compactness, and convex function. All this is done through three tasks. Task one presents a host of essential definitions along with some theorems. The aim of this task is to help the reader to get some understanding of main results grounded in this thesis. Task two is designed to establish a new kind of Mixed-hemi-equilibrium problem (abbreviation for (MHEP)) involving a novel type of monotonicity called ψ- monotone in the setting of B.S. This will demonstrate that the idea of w-p for(MHEP) is equivalent to well posedness for ψ- monotone. Task three deals with the optimization of solution for Mixed-hemiequilibrium with parametric characters. In so aiming, the results in this thesis help us develop and improve some corresponding solutions of several authors. Consider that Π:K×K⊸K be a closed, convex set-valued mapping. Then (MHEP) is s-w-p if, and only if the solution set π≠∅ of (MHEP) and (lim)┬(ϵ→0)⁡dim Γ(ϵ)=0. Assume that the hypothesis H_1-H_4 hold. Then (MHEP) is s-w-p in the generalized sense if, only if the solution set π≠∅ compact of (MHEP) and (lim)┬(ϵ→0^+ )⁡〖e(Γ(ϵ),π)=0.〗 Assume that S is a B.S and K is a nonempty convex closed subset of S if (OPMHEPC) is strongly w-p . Then ψ(ϵ,c)≠φ∀c,ϵ>0 , diam ψ(ϵ,c)→0.where(ϵ,c)→(0,0). Also, if the suppositions (H_1-H_4) and H_5 (ii) hold. Then the converse hold.