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The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems

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Abstract

We consider perturbed Hammerstein integral equations of the form:

$$\begin{aligned} y(t)=\gamma _1(t)H_1\big (\varphi _1(y)\big )+\gamma _2(t)H_2\big (\varphi _2(y)\big )+\lambda \int _0^1G(t,s)f\big (s,y(s)\big )\ \mathrm{d}s \end{aligned}$$

in the case, where \(H_1\) and \(H_2\) are continuous functions, which can be either linear or nonlinear subject to some restrictions, and \(\varphi _1\) and \(\varphi _2\) are linear functionals. We demonstrate that by introducing a specially constructed order cone, one can equip \(\varphi _1\) and \(\varphi _2\) with coercivity conditions that are useful in improving existence results for both the integral equation and associated boundary value problems for ODEs and elliptic PDEs on annuli with nonlocal boundary conditions. We illustrate this theory with some specific examples.

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Acknowledgements

The author would like to thank the anonymous referee for his or her very helpful and thoughtful comments, which led to a considerable improvement of the original manuscript.

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

    Christopher S. Goodrich

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  1. Christopher S. Goodrich
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Correspondence to Christopher S. Goodrich.

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Goodrich, C.S. The effect of a nonstandard cone on existence theorem applicability in nonlocal boundary value problems. J. Fixed Point Theory Appl. 19, 2629–2646 (2017). https://doi.org/10.1007/s11784-017-0448-7

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