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Pointwise conditions for perturbed Hammerstein integral equations with monotone nonlinear, nonlocal elements

  • Christopher S. GoodrichEmail author
Original Paper

Abstract

We consider perturbed Hammerstein integral equations of the form
$$\begin{aligned} y(t)= & {} \gamma _1(t)F_1(\varphi _1(y),\varphi _2(y),\dots ,\varphi _{n_1} (y))+\gamma _2(t)F_2(\psi _1(y),\psi _2(y),\dots ,\psi _{n_2}(y))\\&+\,\lambda \int _0^1G(t,s)f(s,y(s))\ ds, \end{aligned}$$
where the \(\varphi _i\)’s and \(\psi _i\)’s are linear functionals realized as Stieltjes integrals with associated signed Stieltjes measures. Positive solutions of the integral equation can be related to positive solutions of boundary value problems, and we demonstrate that such problems have solutions under relatively mild hypotheses on the functions \(F_1\), \(F_2\), and f. We provide examples to illustrate the applicability of the results and their relationship to existing results in the literature, and we mention applications to radially symmetric solutions of PDEs as well as beam deflection modeling. Our results are achieved by using a nonstandard order cone together with an associated nonstandard open set.

Keywords

Hammerstein integral equation Nonlocal boundary condition Nonlinear boundary condition Positive solution Coercivity 

Mathematics Subject Classification

47H30 34B10 34B18 45G10 45M20 47H14 

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUNSW AustraliaSydneyAustralia

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