Existence and multiplicity results for some generalized Hammerstein equations with a parameter

Abstract

This paper considers the existence and multiplicity of fixed points for the integral operatorwhere λ > 0 is a positive parameter, k : I × I → R is a kernel function such that k ∈ W m,1 (I × I), m is a positive integer with m ≥ 1, and f :The existence of solutions for these Hammerstein equations is obtained by fixed point index theory on new type of cones. Therefore some assumptions must hold only for, at least, one of the derivatives of the kernel or, even, for the kernel, on a subset of the domain. Assuming some asymptotic conditions on the nonlinearity f , we get sufficient conditions for multiplicity of solutions.Two examples will illustrate the potentialities of the main results, namely the fact that the kernel function and/or some derivatives may only be positive on some subintervals, which can degenerate to a point. Moreover, an application of our method to general Lidstone problems improves the existent results on the literature in this field.

Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

“…We also highlight recent works, not necessarily in real line or half-line, on Hammerstein-type integral equations, with several approaches and applications in References [13,[15][16][17][18][19][20][21][22] and the references therein.…”

Positive Solution for a Singular Fourth-Order Differential System

“…By constructing a new type of cone and using fixed point index theory, López-Somoza and Minhós [15] investigated existence of solutions for the Hammerstein equations.…”