Eigenvalues of a system of Fredholm integral equations

We consider the following system of Fredholm integral equations: u"i=@l@!01g"i(t,s)P"i(s,u"1(s),u"2(s),...,u"n(s))ds, t@e[0,1], 1@?i@?n, where @l > 0. Our aim is to determine those values of @l such that the above system has a constant-sign solution. In addition, explicit intervals for @l 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, ~) u"i=@l@!0~g"i(t,s)P"i(s,u"1(s),u"2(s),...,u"n(s))ds, t@e[0,~], 1@?i@?n, Finally, both the systems above are extended to the general case when @l is replaced by @l"i.

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