Smooth solutions of a nonhomogeneous iterative functional differential equation

Abstract

This paper is concerned with an iterative functional differential equation x(t) = c1x(t) + c2x[2](t) + … cmχ[m](t) + F(t), where x[i](t) is the i-th iterate of the function x(t). By means of Schauder's Fixed Point Theorem, we establish a local existence theorem for smooth solutions which also depend continuously on the forcing function F(t).

Nondecreasing and convex C 2-solutions of an iterative functional-differential equation

“…It is also relatively compact. Indeed, for any x = x(t) in Ξ(ξ, η 1 , η 2 ; M, M * ; I), Finally, by Schauder's fixed point theorem, we may assert that there is a function x = x(t) in Ξ(ξ, η 1 , η 2 ; M, M * ; I) which satisfies (12). By differentiating both sides of (12), we see that x is the desired solution of (1).…”

“…Note that this solution is smooth. This motivates the studies [10,11] which consider the existence of analytic solutions of (1), and the studies [12,13], which consider the existence of C n -solutions. Note that when ξ, c ≥ 0, this solution is also smooth, nondecreasing and convex.…”

“…Functional-differential equations involving iterates of the unknown function have attracted the attentions of several authors in recent years (see e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). More specifically, Cooke [1] points out that it is highly desirable to establish the existence and stability properties of periodic solutions for equations of the form and shows that every solution either vanishes identically or is strictly monotonic.…”

Local invertible analytic solution of a functional differential equation with deviating arguments depending on the state derivative

“…However, such equations, when the delay function τ (z) depends not only on the argument of the unknown function, but also state or state derivative, τ (z) = τ (z, x(z), x (z)), have been relatively little researched. In [3][4][5][6][7][8][9][10][11], analytic solutions of the state dependent functional differential equations are found. In [12,13], the authors studied the existence of analytic solutions of the equations with state derivative dependent delay αz + βx (z) = x az + bx (z) and…”

Analytic solutions of a second-order nonautonomous iterative functional differential equation

“…Differential equation with iterates of the unknown function is an important class of functional differential equations with state-dependent delays, modeled extensively in many fields such as classical electrodynamics [5][6][7][8], populations [1], commodity price fluctuations [2], and blood cell productions [10]. Many results [11][12][13][14] are given for first-order equations. Concerning second-order equations, Petahov [15] proves the existence of solutions of the second-order equation…”