Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case |
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| Authors: | Ravi P Agarwal Donal O’Regan Patricia J Y Wong |
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| Institution: | (1) Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA;(2) Department of Mathematics, National University of Ireland, Galway, Ireland;(3) School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore |
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| Abstract: | We consider the system of Fredholm integral equations and also the system of Volterra integral equations where T>0 is fixed and the nonlinearities h
i
(t,u
1,u
2,…,u
n
) can be singular at t=0 and u
j
=0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ
i
u
i
(t)≥0 for t∈0,1] and 1≤i≤n, where θ
i
∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.
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| Keywords: | Constant-sign solutions System of Fredholm integral equations System of Volterra integral equations Singular equations |
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![$$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\8pt]\quad t\in0,T],~1\leq i\leq n\end{array}$$](/content/9v875r8303h65437/10440_2008_9234_Article_Equa.gif)
![$$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\8pt]\quad t\in0,T],~1\leq i\leq n\end{array}$$](/content/9v875r8303h65437/10440_2008_9234_Article_Equb.gif)