On the approximation of solutions to a fixed point problem with inequality constraints in a Banach space partially ordered by a cone

Abstract

Let E be a Banach space with a cone P. Let T, ϕi: E → E (i = 1, 2) be three given operators. We address the following question: Find x ε E such that where ≤P is the partial order on E induced by the cone P, and 0E is the zero vector of E. We obtain sufficient conditions for the existence and uniqueness of solutions to this problem. We present an iterative algorithm to approximate the solution. The error estimates as well as results concerning the data dependence, well-posedness, limit shadowing property, and sequences of operators are provided. Some interesting consequences are deduced from our main results. © Springer Nature Singapore Pte Ltd. 2017. All rights reserved.