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Results in Mathematics

, Volume 72, Issue 1–2, pp 369–383 | Cite as

Generalized Hammerstein Equations and Applications

  • John GraefEmail author
  • Lingju Kong
  • Feliz Minhós
Article

Abstract

In this paper the authors study the Hammerstein generalized integral equation
$$\begin{aligned} u(t)=\int _{0}^{1}k(t,s)\text { }g(s)\text { }f(s,u(s),u^{\prime }(s),\dots ,u^{(m)}(s))\,ds, \end{aligned}$$
where \(k:[0,1]^{2}\rightarrow {\mathbb {R}}\) are kernel functions, \(m\ge 1\), \(g:[0,1] \rightarrow [0,\infty )\), and \(f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )\) is a \(L^{\infty }-\)Carathéodory function. The existence of solutions of integral equations has been studied in concrete and abstract cases, by different methods and techniques. However, in the existing literature, the nonlinearity depends only on the unknown function. This paper is one of a very few to consider equations having discontinuous nonlinearities that depend on the derivatives of the unknown function and having discontinuous kernels functions that have discontinuities in the partial derivatives with respect to their first variable. Our approach is based on the Krasnosel’skiĭ–Guo compression/expansion theorem on cones and it can be applied to boundary value problems of arbitrary order \(n>m\). The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.

Keywords

Hammerstein integral equation Krasnosel’skiĭ–Guo theorem Boundary value problems Discontinuous kernels Nonlinearities depending on derivatives 

Mathematics Subject Classification

Primary 45G10 Secondary 34B15 47H30 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Tennessee at ChattanoogaChattanoogaUSA
  2. 2.Departamento de Matemática, Escola de Ciências e Tecnologia Centro de Investigação em Matemática e Aplicações (CIMA), Instituto de Investigação e Formação AvançadaUniversidade de ÉvoraÉvoraPortugal

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