Abstract
In this paper, we propose the study of an integral equation, with deviating arguments, of the typey(t)=ω(t)-∫0∞f(t,s,y(γ1(s)),…,y(γN(s)))ds,t≥0,in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at∞asω(t). A similar equation, but requiring a little less restrictive hypotheses, isy(t)=ω(t)-∫0∞q(t,s)F(s,y(γ1(s)),…,y(γN(s)))ds,t≥0.In the case ofq(t,s)=(t-s)+, its solutions with asymptotic behavior given byω(t)yield solutions of the second order nonlinear abstract differential equationy''(t)-ω''(t)+F(t,y(γ1(t)),…,y(γN(t)))=0,with the same asymptotic behavior at∞asω(t).
Highlights
From the pioneering work of Atkinson [1], and subsequent works found in the literature, we consider the following differential problem, with deviating arguments: y (t) − ω (t) + F (t, y (γ1 (t)), . . . , y (γN (t))) = 0, t ≥ 0, (1)with the task of finding solutions y with the same behavior at ∞ as ω
We propose the study of an integral equation, with deviating arguments, of the type y(t) = ω(t) − ∫0∞ f(t, s, y(γ1(s)), . . . , y(γN(s)))ds, t ≥ 0, in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at ∞ as ω(t)
The purpose of this note is to provide conditions that ensure the existence of solutions to the above integral equation, whose asymptotic behavior at ∞ is the same as that of ω, giving a procedure to show existence of solutions with prescribed asymptotic behavior of differential equation of the type (1)
Summary
‖∞, (i.e., The Schauder fixed point theorem states that any continuous operator T defined on a nonempty, bounded, closed and convex subset C of a Banach space has necessarily a fixed point, provided that T(C) is a relatively compact subset of C. The result on existence of solutions to the integral equation is the following.
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