POSITIVE SOLUTIONS OF INTEGRODIFFERENTIAL AND DIFFERENCE-EQUATIONS WITH UNBOUNDED DELAY

We establish a necessary and sufficient condition for the existence of a positive solution of the integrodifferential equation x'(t) + integral-infinity/0 x(t - s) dn(s) = 0, where n is an increasing real-valued function on the interval [0, infinity); that is, if and only if the characteristic equation -lambda + integra-infinity/0 e(lambdas) dn(s) = 0 admits a positive root. Consider the difference equation x(n+1) - x(n) + SIGMA(k=0)infinity c(k)x(n-k) = 0, where (c(k))k greater-than-or-equal-to 0 is a sequence of non-negative numbers. We prove this has positive solution if and only if the characteristic equation -lambda + SIGMA(k=0)infinity lambda(-k)c(k) = 0 admits a root in (0, 1). For general results on integrodifferential equations we refer to the book by Burton [1] and the survey article by Corduneanu and Lakshmikantham [2]. Existence of a positive solution and oscillations in integrodifferential equations or in systems of integrodifferential equations recently have been investigated by Ladas, Philos and Sficas [5], Gyori and Ladas [4], Philos and Sficas [12], Philos [9], [10], [11]. Recently, there has been some interest in the existence or the non-existence of positive solutions or the oscillation behavior of some difference equations. See Ladas, Philos and Sficas [6], [7]. The purpose of this paper is to investigate the positive solutions of integrodifferential equations (Section 1) and difference equations (Section 2) with unbounded delay. We obtain also some results for integrodifferential and difference inequalities.