Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

Abstract

In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain fixed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel ′ slkiȋ's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann-Stieltjes type. Some examples are presented in order to illustrate the obtained results.2010 Mathematics Subject Classification. Primary 45G99; Secondary 34B10, 47H10, 47H30.

Existence of positive solutions of a Hammerstein integral equation using the layered compression-expansion fixed point theorem

“…In this paper, we show that under the same assumptions on the kernel G, the Layered Expansion-Compression Fixed Point Theorem can be applied to show the existence of positive and positive symmetric solutions of the Hammerstein integral equation. More examples of recent work on Hammerstein integral equations, can be found in [14,15,19,21,26].…”

On Λ p B V -Solutions of Some Nonlinear Integral Equations on the Plane

“…One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose rst derivative exists almost everywhere and are frequently used to de ne generalized solutions to nonlinear problems involving functionals, and partial di erential equations in mathematics, physics, and engineering. In recent decades, the solutions of this type of integral equations have been studied by various authors in various spaces of bounded variation, for example, in the space of the functions of bounded variation in the Jordan sense and in the Waterman sense, see [1,2], in addition to other generalized spaces of bounded variation, some of these have been studied in [3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”