On mappings of higher order and their applications to nonlinear equations
Dariusz Bugajewski Piotr Kasprzak
Communications on Pure & Applied Analysis 2012, 11(2): 627-647 doi: 10.3934/cpaa.2012.11.627
One of the main goals of this paper is to investigate mappings of higher order which possess ``good'' properties, in particular, when we treat them as perturbations of nonlinear differential as well as integral equations. We draw a particular attention to nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan or in the sense of Young. We provide sufficient conditions which guarantee that nonlinear Hammerstein operators are of higher order in such spaces. We also prove a few extensions of Lovelady's fixed point theorem in Archimedean as well as non-Archimedean setting. Finally, we apply our results to prove the existence and uniqueness results to some commonly known nonlinear equations with perturbations.
keywords: non-Archimedean normed space Lipschitz condition superposition operator. Banach contraction principle mapping of higher order function of bounded variation in the sense of Jordan Hammerstein integral operator functional-integral equation function of bounded $\phi$-variation fixed point theorem mixed Volterra-Fredholm integral equation

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