Multiple positive fixed points for the sum of expansive mappings and k‐set contractions

Abstract

In this work, we are concerned with the existence of multiple positive fixed points for the sum of an expansive mapping with constant h > 1 and a k-set contraction when 0 ≤ k < h − 1. In particular, the case of the sum of an expansive mapping with constant h > 1 and an e-concave operator and an e-convex operator is considered. Two examples of application illustrate some of the theoretical results. KEYWORDSexpansive mapping, sum of operators, e-concave operator, e-convex operator, k-set contraction INTRODUCTIONMany problems in science, when modeled under the mathematical point of view, lead to the nonlinear equation Tx + Fx = x, posed in some closed convex subset of a Banach space. In particular, ordinary and partial differential equations and integral equations can be formulated like abstract equations. It is the reason for which it becomes desirable to develop fixed point theorems for such situations. When T is compact continuous and F is a contraction, a classical tool generally used to deal with such problems is the well-known Krasnosel'skii fixed point theorem. 1 For contributions which develop this capturing result, see previous studies. [2][3][4] The study of positive solutions to nonlinear equations, especially ordinary, partial differential equations, and integral equations, draws great attention. Positivity can be developed by arbitrary cones, that is, nonempty closed convex subsets  in some Banach space E satisfying  ⊂  for all real positive number and  ∩ (−) = {0}.On the other hand, many researchers have been interested in the extension of the fixed point theory for positive nonlinear operators with convexity and concavity, which is applied to the study of various nonlinear differential equations defined on a cone in a Banach space. Note that the definitions and properties of e-concave operators and e-convex operators were first given by Krasnosel'skii 5 and the concepts of -concavity and -convexity for nonlinear operators were introduced by Potter. 6 Zhao 7 discussed the existence of multiple positive fixed points for the sum of e-concave and e-convex operators by using the cone expansion fixed point theorem for strict k-set contraction. Djebali and Mebarki 8 developed a generalized fixed point index for the sum of an expansive mapping and a k-set contraction.Motivated by the results obtained in Zhao 7 and Djebali and Mebarki, 8 the main goal of this work is the study of the existence of multiple positive fixed points for some operators that are of the form T + F, where T is an expansive operator and F is a k-set contraction. For goal, we use the generalized fixed point index introduced in Djebali and Mebarki. 8 As corollaries of the main theorem, we obtain some fixed point results in the case where the k-set contraction F is a sum of e-concave and e-convex operators, or the sum of -concave and -convex operators. Our results generalize and improve the corresponding ones in Zhao 7 and Sang et al. 9 4412

Multiple Solutions for Some Classes Integro-Dynamic Equations on Time Scales

“…The following result will be used to prove our main result. We refer the reader to [16,17] for more details.…”

Multiple nonnegative solutions for a class IVPs for second order ODEs

“…The following result will be used to prove our main result. We refer the reader to [8] and [11] for more details.…”

An extension of Krasnoselskii's cone fixed point theorem for a sum of two operators and applications to nonlinear boundary value problems

“…By this result, a solution is localized in a conical shell of a normed linear space. This theorem has been recently deeply improved in various directions; see [1,2,3,6,9,12,13,14] and references therein. A vector version of Krasnoselskii's fixed point theorem in cones has been given in [4,15,16].…”