Solutions of Hammerstein equations in the spaceBV()

Abstract

Mathematics Subject Classification (2010): Primary 93B05; Secondary 93C25.

Integral Representation of Functions of Bounded Variation

“…As recent publications show, finding solutions of integral equations in the spaces of bounded variations in various fields is coming to the focus of attention of international research in mathematics like solution of Volterra-Hammerstein integral equations. This research was supported among others by the Venezuelan Central Bank [2][3][4].…”

On Λ p B V -Solutions of Some Nonlinear Integral Equations on the Plane

“…One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose rst derivative exists almost everywhere and are frequently used to de ne generalized solutions to nonlinear problems involving functionals, and partial di erential equations in mathematics, physics, and engineering. In recent decades, the solutions of this type of integral equations have been studied by various authors in various spaces of bounded variation, for example, in the space of the functions of bounded variation in the Jordan sense and in the Waterman sense, see [1,2], in addition to other generalized spaces of bounded variation, some of these have been studied in [3][4][5][6][7][8][9][10][11][12][13][14][15][16].…”

“…In [16] Aziz, Leiva, and Merentes studied the solutions for the nonlinear Hammerstein equation and for the Volterra-Hammerstein integral equation in the space of functions of bounded variation, BV(I b a ), on the plane, which are de ned by the following:…”

“…a , moreover, together with some additional hypotheses, they showed that there exists a number ρ > 0 such that for every λ, with |λ| < ρ, the (1) has a unique solution in BV(I b a , R), defined on I b a . is research work is motivated by the paper [16]. Here, we consider the Hammerstein equation ( 1) and the Volterra-Hammerstein integral (2); under certain hypotheses, the solutions of these equations are studied in the classes of functions of bounded variation in the sense of Shiba on the plane (Λ p BV(I b a , R)), with p ≥ 1. is paper is structured as follows: Section 2 focuses on preliminaries, which gives a round-up of the necessary results for proofs of the main theorems.…”