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A note on perturbed Hammerstein equations with applications to nonlocal boundary value problems. (English) Zbl 1346.45006

Summary: We consider the perturbed Hammerstein equation \[ y(t)=\gamma_{1}(t)H_{1}(\phi(y))+\gamma_{2}(t)H_{2}(\psi(y))+\lambda\int_{0}^{1}G(t,s)f(s,y(s))\,ds, \] where \(\phi\) and \(\psi\) are linear functionals, which can be represented as Stieltjes integrals with signed measures, and \(\lambda>0\) is a parameter. We demonstrate that by imposing some mild conditions on \(H_{1}\) and \(H_{2}\), together with an assumption that the functionals decompose in a particular way, it follows that this problem can have multiple positive solutions even if no conditions at all are imposed on \(f\) except for a positivity assumption and continuity. Applications are, in particular, given to boundary value problems with nonlocal boundary conditions, and one of these existence results allows \(H_{1}\) to be affine and \(H_{2}\) to be linear.

MSC:

45G10 Other nonlinear integral equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
45M20 Positive solutions of integral equations
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