In this paper we focus on three fixed point theorems and an integral equation. Schaefer's fixed point theorem will yield a T -periodic solution of z(t) = a(t) + D(t, s)g(s, z(s)) ds 6, if D and g satisfy certain sign conditions independent of their magnitude. A combination of the contraction mapping theorem and Schauder's theorem (known as Krasnoselskii's theorem) will yield a T -periodic solution of z(t) = f(t, 4 t ) ) + D ( t , s)g(., 4 3 ) ) d.9 (0.2) L, if j defines a contraction and if D and g are small enough.We prove a fixed point theorem which is a combination of the contraction mapping theorem and Schaefer's theorem which yields a T-periodic solution of (0.2) when f defines a contraction mapping, while D and g satisfy the aforementioned sign conditions.

No abstract

Abstract. The problem is to show that (1) V (t, x) = S(t,t 0 H (t, s, x(s)) ds) has a solution, where V defines a contraction,Ṽ , and S defines a compact map,S. A fixed point of P ϕ =Sϕ + (I −Ṽ )ϕ would solve the problem. Such equations arise naturally in the search for a solution of f (t, x) = 0 where f(0, 0) = 0, but ∂f(0, 0)/∂x = 0 so that the standard conditions of the implicit function theorem fail. Now P ϕ =Sϕ + (I −Ṽ )ϕ would be in the form for a classical fixed point theorem of Krasnoselskii if I −Ṽ were a contraction. But I −Ṽ fails to be a contraction for precisely the same reasons that the implicit function theorem fails. We verify that I −Ṽ has enough properties that an extension of Krasnoselskii's theorem still holds and, hence, (1) has a solution. This substantially improves the classical implicit function theorem and proves that a general class of integral equations has a solution.

No abstract

The scalar equation (1) x (t) = − t t−r(t) a(t, s)g(x(s)) ds with variable delay r(t) ≥ 0 is investigated, where t − r(t) is increasing and xg(x) > 0 (x = 0) in a neighborhood of x = 0. We find conditions for r, a, and g so that for a given continuous initial function ψ a mapping P for (1) can be defined on a complete metric space C ψ and in which P has a unique fixed point. The end result is not only conditions for the existence and uniqueness of solutions of (1) but also for the stability of the zero solution. We also find conditions ensuring the zero solution is asymptotically stable by changing to an exponentially weighted metric on a closed subset of C ψ . Finally, we parlay the methods for (1) into results for (2) x (t) = − t t−r(t) a(t, s)g(s, x(s)) ds and (3) x (t) = −a(t)g(x(t − r(t))).

No abstract

No abstract

No abstract