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Perturbed Integral Operator Equations of Volterra Type with Applications to \({\varvec{p}}\)-Laplacian Equations

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Abstract

We consider perturbed Volterra integral operator equations of the form

$$\begin{aligned} y(t)=\gamma (t)H(\psi (y))+\lambda \int _0^tf(\tau ,y(\tau ))\,\mathrm{d}\tau , \end{aligned}$$

where \(\psi :{\mathscr {C}}([0,1])\rightarrow {\mathbb {R}}\) is a linear functional. By utilizing a nonstandard order cone and an associated nonstandard open set, we demonstrate the existence of at least one positive solution to this problem under some more general conditions previously seen in the literature. As a related problem we consider the one-dimensional p-Laplacian equation

$$\begin{aligned} (\varphi _{p}(y'(t)))'=-\lambda f(t,y(t)),\quad t\in (0,1) \end{aligned}$$

subject to the nonlocal boundary conditions

$$\begin{aligned}&\varphi _p(y'(0))=0\\&y(1)=H(\psi (y)), \end{aligned}$$

where \(\varphi _p(z):=|z|^{p-2}z\), for \(p>2\), is the one-dimensional p-Laplacian.

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Authors and Affiliations

  1. Department of Mathematics, Creighton Preparatory School, Omaha, NE,  68114, USA

    Christopher S. Goodrich

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Goodrich, C.S. Perturbed Integral Operator Equations of Volterra Type with Applications to \({\varvec{p}}\)-Laplacian Equations. Mediterr. J. Math. 15, 47 (2018). https://doi.org/10.1007/s00009-018-1090-3

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  • DOI: https://doi.org/10.1007/s00009-018-1090-3

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