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Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case

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Abstract

We consider the system of Fredholm integral equations

$$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$

and also the system of Volterra integral equations

$$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$

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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Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL, 32901-6975, USA

    Ravi P. Agarwal

  2. Department of Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  3. School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

    Patricia J. Y. Wong

Authors
  1. Ravi P. Agarwal
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  2. Donal O’Regan
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  3. Patricia J. Y. Wong
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Correspondence to Ravi P. Agarwal.

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Agarwal, R.P., O’Regan, D. & Wong, P.J.Y. Constant-Sign Solutions for Systems of Fredholm and Volterra Integral Equations: The Singular Case. Acta Appl Math 103, 253–276 (2008). https://doi.org/10.1007/s10440-008-9234-2

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