Existence Theorems on Solvability of Constrained Inclusion Problems and Applications

Teffera M Asfaw

doi:10.1155/2018/6953649

Abstract

LetXbe a real locally uniformly convex reflexive Banach space with locally uniformly convex dual spaceX⁎. LetT:X⊇D(T)→2X⁎be a maximal monotone operator andC:X⊇D(C)→X⁎be bounded and continuous withD(T)⊆D(C). The paper provides new existence theorems concerning solvability of inclusion problems involving operators of the typeT+Cprovided thatCis compact orTis of compact resolvents under weak boundary condition. The Nagumo degree mapping and homotopy invariance results are employed. The paper presents existence results under the weakest coercivity condition onT+C. The operatorCis neither required to be defined everywhere nor required to be pseudomonotone type. The results are applied to prove existence of solution for nonlinear variational inequality problems.

Highlights

PreliminariesIn what follows, the norm of the spaces X and X∗ will be denoted by ‖ ⋅ ‖
An operator T : X ⊃ D(T) 󳨀→ Y is “demicontinuous” if it is continuous from the strong topology of D(T) to the weak topology of Y
Existence results concerning pseudomonotone perturbations of maximal monotone operators under coercivity condition can be found in the papers due to Kenmochi [4, 7, 8], Asfaw and Kartsatos [9], Le [10], Asfaw [11], and the references therein

Summary

Introduction

PreliminariesIn what follows, the norm of the spaces X and X∗ will be denoted by ‖ ⋅ ‖. Let T : X ⊇ D(T) 󳨀→ 2X∗ be a maximal monotone operator and C : X ⊇ D(C) 󳨀→ X∗ be bounded and continuous with D(T) ⊆ D(C). The main objective of this paper is to establish sufficient conditions which guarantee existence of solution for inclusions of the type (T + C)(D(T) ∩ BR(0)) ∋ f∗ (where T : X ⊇ D(T) 󳨀→ 2X∗ is maximal monotone of compact resolvents and C : X ⊇ D(C) 󳨀→ X∗ is bounded and continuous with D(T) ⊆ D(C)) provided that there exists R > 0 such that (T + C + εJ)(D(T) ∩ ∂BR(0)) ∌ f∗ for all ε > 0.

Objectives

Results

Conclusion