Krasnoselskii‐type fixed point theorems with applications to Hammerstein integral equations in spaces

Abstract

In this paper, we study fixed point theorems and new variants of some nonlinear altenatives of Krasnoselskii type in Banach spaces by using measures of weak noncompactness. Then we give an application to solve a nonlinear Hammerstein integral equation in L1 spaces.

Fixed points of Hammerstein-type equations on general cones

“…An equation of the form ( 7) is usually known as a Hammerstein-type equation, and there are many papers in the literature which deal with this type of equations, see for instance [7][8][9]. Typically, as we said in Section 1, these equations appear when looking for solutions of certain type of boundary value problems.…”

“…On the other hand, Therefore, condition (I 1 ρ 2 ) is satisfied. This means that condition (S 1 ) in Theorem 3.6 holds and, hence, there exists a solution u of problem (8)…”

“…Moreover, this cone is contained in the cone of nonnegative continuous functions. Observe that we can rewrite problem (8) in terms of a fixed-point problem for the operator…”

Application of measure of noncompactness to $\ell_{1}$-solvability of infinite systems of second order differential equations

“…Intuitively, such applications are characterized by some "loss of compactness" which arises in many fields: imbedding theorems between Sobolev spaces with critical exponent, imbedding over domains with irregular boundary, linear composition operators over the complex unit disc, nonlinear integral equations (also with delay), differential equations over unbounded domains, fractional differential equations, infinite systems, etc. Recently, the measure of noncompactness has been applied in several papers (see [1][2][3][4], [8][9][10], [13,27]). The aim of this paper is to present a new generalization of Darbo type fixed point theorem which also improves the corresponding results given by the first author et.…”

Fixed Points for Sum or Product of Operators in Weak Topology With Applications (A Survey) H. H. G. Hashem

“…In 2014, Gang Cai and Shangquan Bu [31] study fixed point theorems and new variants of some nonlinear alternatives of Krasnosel'skii type in Banach spaces by using measures of weak noncompactness. Then they give an application to solve a nonlinear Hammerstein integral equation in L 1 spaces.…”