Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

Abstract

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

On a Non-Convex Lagrange Optimal Control Problem

“…It is recognized that iterative differential inclusions are a generalization of iterative differential equations which proved to be valuable in many fields such as electrodynamics (as in problems related to motions of charged particles with retarded interactions), biology (as in infection models problems), etc. Numerous results were found in the theory of iterative differential equations, for example: smoothness, equivariance, analycity, convexity and numerical solutions (see [11,18,23,27,29,30,32] and references therein). Several approaches have been used to prove existence results of initial value problems; we mention Schauder's fixed point [1,17,19,28,31], contraction principle [6,13,16], Picard's successive method [25,26] and non-expansive operators [3,22].…”