Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

Feliz Minhós,Robert De Sousa

doi:10.3390/math8010111

Feliz Minhós, Robert De Sousa

Open Access

https://doi.org/10.3390/math8010111

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Journal: Mathematics	Publication Date: Jan 10, 2020
Citations: 4	License type: CC BY 4.0

Affiliation: University of Évora, University of Cape Verde

Abstract

In this work, we consider a generalized coupled system of integral equations of Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is Schauder’s fixed point theorem in the space of bounded and continuous functions with bounded and continuous derivatives on R , combined with the equiconvergence at ± ∞ to recover the compactness of the correspondent operators. To the best of our knowledge, it is the first time where coupled Hammerstein-type integral equations in real line are considered with nonlinearities depending on several derivatives of both variables and, moreover, the derivatives can be of different order on each variable and each equation. On the other hand, we emphasize that the kernel functions can change sign and their derivatives in order to the first variable may be discontinuous. The last section contains an application to a model to study the deflection of a coupled system of infinite beams.

Highlights

Integral equations are of many types and Hammerstein-type is a particular case of them.These equations appear naturally in inverse problems, fluid dynamics, potential theory, spread of interdependent epidemics, elasticity,
Hammerstein-type integral equations in real line play an important role in physical problems and are often used to reformulate or rewrite mathematical problems
Along with the proof of theorems on the existence of solutions, profound constructive solvability theorems were proposed with analysis of branching solutions of nonlinear Hammerstein integral equation presented in Reference [6]

Summary

Introduction

Integral equations are of many types and Hammerstein-type is a particular case of them.These equations appear naturally in inverse problems, fluid dynamics, potential theory, spread of interdependent epidemics, elasticity, . . . (see References [1,2,3]). The authors have shown that the above problem is equivalent to Hammerstein-type integral equations in the real line u( x ) −

Results

Conclusion