# Generalized Hammerstein Equations and Applications

Generalized Hammerstein Equations and Applications 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} u ( t ) = ∫ 0 1 k ( t , s ) g ( s ) f ( s , u ( s ) , u ′ ( s ) , ⋯ , u ( m ) ( s ) ) d s , where $$k:[0,1]^{2}\rightarrow {\mathbb {R}}$$ k : [ 0 , 1 ] 2 → R are kernel functions, $$m\ge 1$$ m ≥ 1 , $$g:[0,1] \rightarrow [0,\infty )$$ g : [ 0 , 1 ] → [ 0 , ∞ ) , and $$f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )$$ f : [ 0 , 1 ] × R m + 1 → [ 0 , ∞ ) is a $$L^{\infty }-$$ L ∞ - 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 $$n>m$$ n > m . The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Results in Mathematics Springer Journals

# Generalized Hammerstein Equations and Applications

, Volume 72 (2) – Oct 24, 2016
15 pages

/lp/springer_journal/generalized-hammerstein-equations-and-applications-EexxpvEiXD
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1422-6383
eISSN
1420-9012
D.O.I.
10.1007/s00025-016-0615-y
Publisher site
See Article on Publisher Site

### 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} u ( t ) = ∫ 0 1 k ( t , s ) g ( s ) f ( s , u ( s ) , u ′ ( s ) , ⋯ , u ( m ) ( s ) ) d s , where $$k:[0,1]^{2}\rightarrow {\mathbb {R}}$$ k : [ 0 , 1 ] 2 → R are kernel functions, $$m\ge 1$$ m ≥ 1 , $$g:[0,1] \rightarrow [0,\infty )$$ g : [ 0 , 1 ] → [ 0 , ∞ ) , and $$f:[0,1]\times {\mathbb {R}}^{m+1} \rightarrow [0,\infty )$$ f : [ 0 , 1 ] × R m + 1 → [ 0 , ∞ ) is a $$L^{\infty }-$$ L ∞ - 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 $$n>m$$ n > m . The last two sections of the paper contain an application to a third order nonlinear boundary value problem and a concrete example.

### Journal

Results in MathematicsSpringer Journals

Published: Oct 24, 2016

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