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Continuity, compactness, fixed points, and integral equations. (English) Zbl 1022.45001

The integral equation \[ x(t)= a(t)- \int^t_{-\infty} D(t,s)g(x(s)) ds \] is studied by means of a Lyapunov functional, introduced by the first author [Proceedings of the first world congress, Tampa, FL, USA, August 19-26, 1992. 4 volumes. Berlin: de Gruyter, 1203-1214 (1996; Zbl 0843.45007)]. The existence of a solution is proved using a fixed point theorem. This equation generalizes the equation in which \(g(x)= x\) and \(D(t,s)= \exp(-(t-s))\), reducible to a trivial, ordinary, differential equation.
The fixed point theorem used in the paper is a change of H. Schaefer’s theorem [Math. Ann. 129, 415-416 (1955; Zbl 0064.35703)], modified by the authors according to the circumstances imposed by the study of the problem. The result which motivates the paper is the following: “There is a constant \(L\) such that if \(\phi\) is a solution and if for \(t< 0\) it is bounded, then \(|\phi(t)|< L\) for \(-\infty< t< \infty\).”
The solutions of the studied equation are determined on chosen by him to illustrate the obtained results. The second kind equation is studied with the same method and all the results are summarised in three theorems giving the conditions in which the put problems are soluble.

MSC:

45G10 Other nonlinear integral equations
47H10 Fixed-point theorems
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