System of Urysohn Integral Equations: Existence of a Constant-Sign Solution

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 18 in Constant-Sign Solutions of Systems of Integral Equations, 2013, pp 539-570 from Springer

Abstract: Abstract In this chapter we shall consider the following system of Urysohn integral equations $$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n. }$$ A solution $$u = (u_{1},u_{2},\cdots \,,u_{n})$$ of (18.1.1) will be sought in $${(C[0,1])}^{n} = C[0,1] \times \cdots \times C[0,1]$$ (n times). We are particularly interested in achieving a constant-sign solution u, i.e., for each 1 ≤ i ≤ n, we have $$\theta _{i}u_{i}(t) \geq 0$$ for t ∈ [0,1], where $$\theta _{i} \in \{ 1,-1\}$$ is fixed.

Keywords: Constant Sign Solutions; Urysohn Integral Equation; Skii Fixed Point Theorem; Lower Solutions Method; Nontrivial Eigenfunction (search for similar items in EconPapers)
Date: 2013
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-01255-1_18

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DOI: 10.1007/978-3-319-01255-1_18

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