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Generalized Hammerstein Equations and Applications
- Source :
- Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos), Agência para a Sociedade do Conhecimento (UMIC)-FCT-Sociedade da Informação, instacron:RCAAP
- Publication Year :
- 2017
- Publisher :
- Springer, 2017.
-
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 $$L^{\infty }-$$ Caratheodory 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.
- Subjects :
- Applied Mathematics
Hammerstein integral equation
010102 general mathematics
Mathematical analysis
Order (ring theory)
Function (mathematics)
Krasnosel’skiĭ–Guo theorem
01 natural sciences
Integral equation
Prime (order theory)
010101 applied mathematics
Combinatorics
Nonlinear system
Mathematics (miscellaneous)
Discontinuous kernels
Partial derivative
Boundary value problem
0101 mathematics
Mathematics
Variable (mathematics)
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Journal :
- Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos), Agência para a Sociedade do Conhecimento (UMIC)-FCT-Sociedade da Informação, instacron:RCAAP
- Accession number :
- edsair.doi.dedup.....3bc3d81d84a1d17d8bf4b4f4842121b3