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Perturbed Hammerstein integral equations with sign-changing kernels and applications to nonlocal boundary value problems and elliptic PDEs. (English) Zbl 1352.45006

Summary: We demonstrate the existence of at least one positive solution to the perturbed Hammerstein integral equation \[ y(t)=\gamma_1(t)H_1(\phi_1(y))+\gamma_2(t)H_2(\phi_2(y)+\lambda\int_0^1G(t,s)f(s,y(s)) ds, \] where certain asymptotic growth properties are imposed on the functions \(f\), \(H_1\) and \(H_2\). Moreover, the functionals \(\phi_1\) and \(\phi_2\) 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.

MSC:

45G10 Other nonlinear integral equations
45M20 Positive solutions of integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
35B09 Positive solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
47H14 Perturbations of nonlinear operators
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References:

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