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.


45G10 Other nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
Full Text: DOI


[1] Corduneanu, C., Integral Equations and Stability of Feedback Systems (1973), Academic Press: Academic Press New York · Zbl 0268.34070
[2] Corduneanu, C., Integral Equations and Applications (1990), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0824.45013
[3] Friedman, A., On integral equations of Volterra type, J. Anal. Math., 11, 381-413 (1963) · Zbl 0134.31502
[4] Gripenberg, G.; Londen, S. O.; Staffans, O., Volterra Integral and Functional Equations (1990), Cambridge Univ. Press: Cambridge Univ. Press New York · Zbl 0695.45002
[5] Hochstadt, H., Integral Equations (1973), Wiley: Wiley New York · Zbl 0137.08601
[6] Krasnoselskii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Pergamon Press: Pergamon Press Oxford
[8] Miller, R. K., On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl., 22, 319-340 (1968) · Zbl 0167.40901
[9] O’Regan, D., Existence results for nonlinear integral equations, J. Math. Anal. Appl., 192, 705-726 (1995) · Zbl 0851.45003
[11] Wheeden, R. L.; Zygmund, A., Measure and Integral, Monographs and Textbooks in Pure and Applied Mathematics (1977), Marcel Dekker: Marcel Dekker New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.