Order Functional Differential Equations Research Articles

Existence and uniqueness for fractional order functional differential equations with Hilfer derivative

Existence and uniqueness for fractional order functional differential equations with Hilfer derivative

О существовании и единственности положительного решения краевой задачи для одного нелинейного функционально-дифференциального уравнения второго порядка

This work continues a series of author's papers devoted to the problems of the existence and uniqueness of positive solutions of boundary value problems for nonlinear second order functional-differential equations. We study a boundary value problem for a nonlinear second order functional-differential equation with homogeneous boundary conditions. Based on the theory of semi-ordered spaces and with the help of special topological tools we establish the existence of a unique positive solution to the problem.

Periodic solutions of a class of third-order differential equations with two delays depending on time and state

The goal of the present paper is to establish some new results on the existence, uniqueness and stability of periodic solutions for a class of third order functional differential equations with state and time-varying delays. By Krasnoselskii's fixed point theorem, we prove the existence of periodic solutions and under certain sufficient conditions, the Banach contraction principle ensures the uniqueness of this solution. The results obtained in this paper are illustrated by an example.

Solution to a Zero-Sum Differential Game with Fractional Dynamics via Approximations

The paper deals with a zero-sum differential game in which the dynamical system is described by a fractional differential equation with the Caputo derivative of an order $\alpha \in (0, 1).$ The goal of the first (second) player is to minimize (maximize) the value of a given quality index. The main contribution of the paper is the proof of the fact that this differential game has the value, i.e., the lower and upper game values coincide. The proof is based on the appropriate approximation of the game by a zero-sum differential game in which the dynamical system is described by a first order functional differential equation of a retarded type. It is shown that the values of the approximating differential games have a limit, and this limit is the value of the original game. Moreover, the optimal players' feedback control procedures are proposed that use the optimally controlled approximating system as a guide.

Global existence results for second order functional differential equations with delay

Our aim in this work is to study the existence of global mild solutions of a second order functional differential equation with delay. We use the Schauder fixed point theorem combined with the cosine family of linear bounded operators for the existence of solutions.

ON HYERS-ULAM STABILITY OF NONLINEAR SECOND ORDER ORDINARY AND FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, we consider the Hyers-Ulam stability of some nonlinear second order ordinary and functional differential equations. As mathematical technique, we use some nonlinear extension of the Gr{\o}nwall integral inequality.

Realization of Nonlinear Time-Delay Input–Output Equations

In this letter the problem of transforming a nonlinear multi-input multi-output time-delay control system, described by a set of higher order functional differential equations relating system inputs and outputs, into a set of first order functional differential equations, is studied. The globally linearized system equations are described in terms of polynomials in a delay operator and derivative operator. The Euclidean left division of polynomials is used to compute the basis of a certain free submodule of differential 1-forms. The integrability of the submodule is proved to be necessary and sufficient for the solvability of the problem.

Oscillation of a perturbed nonlinear third order functional differential equation

Oscillation of a perturbed nonlinear third order functional differential equation

О РАЗРЕШИМОСТИ СИНГУЛЯРНОЙ ЗАДАЧИ КОШИ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ С ОТКЛОНЕНИЕМ СПЕЦИАЛЬНОГО ВИДА

At this paper unique solvability conditions of singular at independent variable linear first order functional-differential equation with special type argument deviation and singular coefficient were obtained.

Dhage iteration method in the theory of IVPs of nonlinear first order functional differential equations

Dhage iteration method in the theory of IVPs of nonlinear first order functional differential equations

Controllability for a class of second order functional evolution differential equations without uniqueness

Controllability for a class of second order functional evolution differential equations without uniqueness

Oscillatory behavior of the second order functional differential equations

Establishing new properties of nonoscillatory solutions we introduce new oscillatory criteria for the second order advanced differential equation y ′ ′ ( t ) + p ( t ) y ( σ ( t ) ) = 0 . Our oscillatory results essentially extend the earlier ones. The progress is illustrated via Euler differential equation.

A new numerical method for variable order fractional functional differential equations

In this letter, a high order numerical scheme is proposed for solving variable order fractional functional differential equations. Firstly, the problem is approximated by an integer order functional differential equation. The integer order differential equation is then solved by the reproducing kernel method. Numerical examples are given to demonstrate the theoretical analysis and verify the efficiency of the proposed method.

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential equations

Existence and stability of Hadamard p-type fractional functional differential equations

In this paper, we investigate initial value problems for Hadamard p-type fractional order functional and neutral functional differential equations. Sufficient conditions which guarantee the existence and Hyers-Ulam stability are new and obtained by some classical fixed point theory. Examples are also given to illustrate our theoretical results.

On positive periodic solutions of linear second order functional differential equations

Abstract The periodic boundary value problem for linear second order functional differential equations is considered. Sharp sufficient conditions for the positiveness of solutions are obtained.

On nonlinear boundary value problems for higher order functional differential equations

Abstract For higher order nonlinear functional differential equations, sufficient conditions for the solvability and unique solvability of some nonlinear nonlocal boundary value problems are established.

Best Proximity point for Z-contraction and Suzuki type Z-contraction mappings with an application to fractional calculus

In this article, we introduced the best proximity point theorems for $\mathcal{Z}$-contraction and Suzuki type $\mathcal{Z}$-contraction in the setting of complete metric spaces. Also by the help of weak $P$-property and $P$-property, we proved existence and uniqueness of best proximity point. There is a simple example to show the validity of our results. Our results extended and unify many existing results in the literature. Moreover, an application to fractional order functional differential equation is discussed.<br /><br />

Measure of Noncompactness and Partial Functional Differential Equations with State-Dependent Delay

Our aim in this work is to study the existence of solutions of first and second order functional differential equations with state-dependent delay. We use the Monch’s fixed point theorem for the existence of solutions and the concept of measures of noncompactness.

Existence and stability for Hadamard p-type fractional functional differential equations

In this paper, we study the existence and Hyers–Ulam stability of solutions for a Hadamard p-type fractional order functional and neutral functional differential equations involving initial value problems. Sufficient conditions which guarantee the existence and Hyers–Ulam stability are new and obtained.