Electron. J. Diff. Equ., Vol. 2011 (2011), No. 104, pp. 1-19.

Existence of solutions of systems of Volterra integral equations via Brezis-Browder arguments

Ravi P. Agarwal, Donal O'Regan, Patricia J. Y. Wong

Abstract:
We consider two systems of Volterra integral equations
$$ u_i(t)=h_i(t) + \int_{0}^{t}g_i(t,s)f_i(s,u_1(s),u_2(s),\dots, u_n(s))ds, \quad 1\leq i\leq n $$
where t is in the closed interval $[0,T]$, or in the half-open interval $[0,T)$. By an argument originated from Brezis and Browder [8], criteria are offered for the existence of solutions of the systems of Volterra integral equations. We further establish the existence of constant-sign solutions, which include positive solutions (the usual consideration) as a special case. Some examples are also presented to illustrate the results obtained.

Submitted April 26, 2011. Published August 16, 2011.
Math Subject Classifications: 45B05, 45G15, 45M20.
Key Words: System of Volterra integral equations; Brezis-Browder argument.

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Ravi P. Agarwal
Department of Mathematics, Texas A&M University - Kingsville
Kingsville, Texas 78363-8202, USA
email: agarwal@tamuk.edu
Donal O'Regan
Department of Mathematics, National University of Ireland
Galway, Ireland
email: donal.oregan@nuigalway.ie
Patricia J. Y. Wong
School of Electrical and Electronic Engineering
Nanyang Technological University
50 Nanyang Avenue, Singapore 639798, Singapore
email: ejywong@ntu.edu.sg

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