Eigenvalues of a system of Fredholm integral equations

We consider the following system of Fredholm integral equations:u(i) (t) = lambda integral(0)(1) g(i) (t, s) P-i (s, u(1) (s), u(2) (s),...,u(n) (s)) ds, t is an element of [0, 1], 1 less than or equal to i less than or equal to n,where lambda 0. Our aim is to determine those values of lambda such that the above system has a constant-sign solution. In addition, explicit intervals for A will be presented. The generality of the results obtained is illustrated through applications to several well-known boundary value problems. We also extend the above problem to that on the half-line [0,infinity)u(i) (t) = lambda integral(0)(infinity) g(i)(t, S) P-i(s, u(1) (S), u(2) (S),..., u(n) (s)) ds, t is an element of [0,infinity], 1 less than or equal to i less than or equal to n.Finally, both the systems above are extended to the general case when lambda is replaced by lambda(i). (C) 2004 Elsevier Ltd. All rights reserved.