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Existence theory for nonlinear Fredholm and Volterra integral equations on half-open intervals. (English) Zbl 0920.45006

The authors study the existence of solutions of Volterra integral equations of the form \[ y(t) = h(t) + \int_0^t k(t,s)g(s,y(s)) ds, \quad 0\leq t < T, \] where \(0\leq T \leq \infty\) with particular emphasis on the case \(T=\infty\). Some results on the corresponding Fredholm equation \(y(t) = h(t) + \int_0^T k(t,s)g(s,y(s)) ds\) are also given. First the authors derive some existence principles based mainly on a nonlinear alternative of Leray-Schauder type but with complicated assumptions. These principles are then used to derive a number of existence and comparison results, including results on the asymptotic behavior of the solution, with easily verifiable assumptions. In many cases the assumptions on the kernel \(k\) involve some form of ”log-convexity”. A large number of examples are given too.

MSC:

45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
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