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Zeitschrift für Analysis und ihre Anwendungen

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Volume 36, Issue 4, 2017, pp. 393–417
DOI: 10.4171/ZAA/1594

Published online: 2017-10-09

Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems

Daria Bugajewska[1], Gennaro Infante[2] and Piotr Kasprzak[3]

(1) Adam Mickiewicz University, Poznan, Poland
(2) Universita della Calabria, Cosenza, Italy
(3) Adam Mickiewicz University, Poznan, Poland

In this paper we present some new results regarding the solvability of nonlinear Hammerstein integral equations in a special cone of continuous functions. The proofs are based on a certain xed point theorem of Leggett and Williams type. We give an application of the abstract result to prove the existence of nontrivial solutions of a periodic boundary value problem. We also investigate, via a version of Krasnosel0ski's theorem for the sum of two operators, the solvability of perturbed Hammerstein integral equations in the space of continuous functions of bounded variation in the sense of Jordan. As an application of these results, we study the solvability of a boundary value problem subject to integral boundary conditions of Riemann–Stieltjes type. Some examples are presented in order to illustrate the obtained results.

Keywords: Boundary value problem, cone, Hammerstein integral equation, functions of bounded variation

Bugajewska Daria, Infante Gennaro, Kasprzak Piotr: Solvability of Hammerstein Integral Equations with Applications to Boundary Value Problems. Z. Anal. Anwend. 36 (2017), 393-417. doi: 10.4171/ZAA/1594