Open Access
2016 Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs
Christopher S. Goodrich
J. Integral Equations Applications 28(4): 509-549 (2016). DOI: 10.1216/JIE-2016-28-4-509

Abstract

We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )\] \[\qquad \qquad \qquad \quad +\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\, ds,\] where certain asymptotic growth properties are imposed on the functions $f$, $H_1$ and $H_2$. Moreover, the functionals $\varphi _1$ and $\varphi _2$ are realizable as Stieltjes integrals with signed measures, which means that the nonlocal elements in the Hammerstein equation are possibly of a very general, sign-changing form. We focus here on the case where the kernel $(t,s)\mapsto G(t,s)$ is allowed to change sign and demonstrate the existence of at least one positive solution to the integral equation. As applications, we demonstrate that, by choosing $\gamma _1$ and $\gamma _2$ in particular ways, we obtain positive solutions to boundary value problems, both in the ODEs and elliptic PDEs setting, even when the Green's function is sign-changing, and, moreover, we are able to localize the range of admissible values of the parameter~$\lambda $. Finally, we also provide a result that for each $\lambda >0$ yields the existence of at least one positive solution.

Citation

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Christopher S. Goodrich. "Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs." J. Integral Equations Applications 28 (4) 509 - 549, 2016. https://doi.org/10.1216/JIE-2016-28-4-509

Information

Published: 2016
First available in Project Euclid: 15 December 2016

zbMATH: 1352.45006
MathSciNet: MR3582800
Digital Object Identifier: 10.1216/JIE-2016-28-4-509

Subjects:
Primary: 45G10 , 45M20 , 47H30
Secondary: 34B10 , 34B18 , 35B09 , 35J25 , 47H14

Keywords: asymptotically linear operator , Hammerstein integral equation , nonlinear boundary condition , nonlocal , positive solution , radially symmetric solution

Rights: Copyright © 2016 Rocky Mountain Mathematics Consortium

Vol.28 • No. 4 • 2016
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