Existence and multiplicity results for some generalized Hammerstein equations with a parameter
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Título: | Existence and multiplicity results for some generalized Hammerstein equations with a parameter |
Autor/a: | López Somoza, Lucía Minhós, Feliz |
Centro/Departamento: | Universidade de Santiago de Compostela. Departamento de Estatística, Análise Matemática e Optimización Universidade de Santiago de Compostela. Instituto de Matemáticas |
Palabras chave: | Hammerstein equations | Nonlinear boundary value problems | Parameter dependence | Degree theory | Fixed points in cones | |
Data: | 2019 |
Editor: | Springer |
Cita bibliográfica: | López-Somoza, L., Minhós, F. Existence and multiplicity results for some generalized Hammerstein equations with a parameter. Adv Differ Equ 2019, 423 (2019). https://doi.org/10.1186/s13662-019-2359-y |
Resumo: | This paper considers the existence and multiplicity of fixed points for the integral operator Tu(t)=λ∫T0k(t,s)f(s,u(s),u′(s),…,u(m)(s))ds,t∈[0,T]≡I, where λ>0 is a positive parameter, k:I×I→R is a kernel function such that k∈Wm,1(I×I), m is a positive integer with m≥1, and f:I×Rm+1→[0,+∞[ is an L1-Carathéodory function. 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 in the literature in this field |
Versión do editor: | https://doi.org/10.1186/s13662-019-2359-y |
URI: | http://hdl.handle.net/10347/21100 |
DOI: | 10.1186/s13662-019-2359-y |
E-ISSN: | 1687-1847 |
Dereitos: | © The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made |
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© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made
© The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in anymedium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made