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Constant-Sign Lp Solutions for a System of Integral Equations

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Abstract

We consider the following system of integral equations

$$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,1\rbrack,1\leq i\leq n.$$

Our aim is to establish criteria such that the above system has a constant-sign solution (u1, u2, …, u n) ∈ (Lp[0, 1])n, where the integer 1 ≤ p < ∞ is fixed. We shall tackle the case when f is ‘nonnegative’ as well as the case when f is ‘semipositone’. The above problem is also extended to that on the half-line [0, ∞)

$$u_i(t)=\int^1_0g_i(t,s)f(s,u_1(s),u_2(s),\cdots,u_n(s))ds,\quad t\in \lbrack 0,\infty ),1\leq i\leq n.$$

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Authors and Affiliations

  1. Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida, 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

    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 Lp Solutions for a System of Integral Equations. Results. Math. 46, 195–219 (2004). https://doi.org/10.1007/BF03322881

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  • DOI: https://doi.org/10.1007/BF03322881

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