Systems of Fredholm and Volterra Integral Equations: Integrable Singularities

Ravi P. Agarwal, Donal O’Regan and Patricia J. Y. Wong
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Ravi P. Agarwal: Texas A&M University – Kingsville, Department of Mathematics
Donal O’Regan: National University of Ireland, Galway, School of Mathematics, Statistics and Applied Mathematics
Patricia J. Y. Wong: Nanyang Technological University, School of Electrical & Electronic Engineering

Chapter Chapter 7 in Constant-Sign Solutions of Systems of Integral Equations, 2013, pp 209-230 from Springer

Abstract: Abstract In this chapter we consider three systems of singular integral equations. Specifically we are interested in the following systems of Fredholm integral equations 7.1.1 $$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n }$$ 7.1.2 $$\displaystyle{ u_{i}(t) =\int _{ 0}^{\infty }g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,\infty ),\ 1 \leq i \leq n }$$ and the system of Volterra integral equations 7.1.3 $$\displaystyle{ u_{i}(t) =\int _{ 0}^{t}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,T],\ 1 \leq i \leq n }$$ where T > 0 is fixed. The nonlinearities $$f_{i},\ 1 \leq i \leq n$$ in the above systems may be singular in the independent variable and may also be singular at $$u_{j} = 0,\ j \in \{ 1,2,\cdots \,,n\}.$$

Keywords: Volterra Integral Equation; Constant Sign Solutions; Schauder-Tychonoff Fixed Point Theorem; Respective Domains; Similar Argument (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/978-3-319-01255-1_7

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