Positive almost periodic solutions of some delay integral equations

AbstractThe delay integral equation x(t) = ∝tt − τ f(s, x(s)) ds which arises in models for the spread of epidemics, is studied with the aim of establishing the existence of positive almost periodic solutions for large values of τ when f(t, x) is uniformly almost periodic in t for x in compact subsets of R+. Under reasonable assumptions on f it is shown that there exist two positive numbers τ∗ < τ0 such that if 0 < τ < τ∗ there are no positive almost periodic solutions while for τ > τ0 they do exist. A priori bounds on the set of positive solutions and uniqueness results are also obtained