Order Functional Differential Equations Research Articles

Some sufficient conditions of existence and trajectory controllability for impulsive and initial value fractional order functional differential equations

For a class of impulsive fractional differential equation with fractional order r∈(1,2) by means of solution operator Sr(t), we develop the mild solution and deal with the existence results of mild solution for the considered equation with state-dependent delay through three fixed point theorems, namely Banach, Nonlinear Leray–Schauder Alternative and Krasnoselskii’s fixed point theorem. Next, we define the mild solution via an operator function Wr,β(τ) for abstract version of the considered equation. Further, we study the trajectory controllability (which is a stronger notion than other controllability) of the equation. In the last, we present some examples to illustrate the established results through partial fractional differential equations.

Periodic perturbations of reducible scalar second order functional differential equations

Using a topological approach we investigate the structure of the set of forced oscillations of a class of reducible second order functional retarded differential equations subject to periodic forcing. More precisely, we consider a delay-type functional dependence involving a gamma probability distribution and, using a linear chain trick, we formulate a first order system of ODEs whose T -periodic solutions correspond to those of the functional equation.

New monotonicity properties and oscillation of $n$-order functional differential equations with deviating argument

In this paper, we offer new technique for investigation of the even order linear differential equations of the form ( E ) y ( n ) ( t ) = p ( t ) y ( τ ( t ) ) . We establish new criteria for bounded and unbounded oscillation of ( E ) which improve a number of related ones in the literature. Our approach essentially involves establishing stronger monotonicities for the positive solutions of ( E ) than those presented in known works. We illustrate the improvement over known results by applying and comparing our technique with the other known methods on the particular examples.

Cattmul-Rom spline approach and the order of convergence of Green’s functional method for functional differential equations

The purpose of this work is to investigate the convergence properties of Green’s function method applied to boundary value problems for functional differential equations. Recently, involving Picard and Mann iterations, a Green’s function technique was developed (in Int. J. Computer Math. 95, no. 10 (2018) 1937-1949) for third order functional differential equations, but without specifying the order of convergence of the proposed method. In order to improve this aspect, here we establish the maximal order of convergence of Green’s function method applied to two-point boundary value problems associated to second and third order functional differential equations. In this context, by using suitable quadrature rule and appropriate spline interpolation procedure, the Picard iterations are approximated by a sequence of cubic splines on uniform mesh. Some numerical experiments are presented in order to test the theoretical results and to illustrate the accuracy of the method.

The convergence properties of the Green’s function method for third order functional differential equations

The convergence properties of the Green’s function method for third order functional differential equations

Existence and uniqueness of monotonic positive solution of a functional differential equation with nonlocal conditions

In this paper we study the existence and uniqueness of at least one monotonic positive solution of a nonlocal two-point, with parameters, boundary value problem of the second order functional differential equation.

Comparison theorems on the oscillation of even order nonlinear mixed neutral differential equations

This study purposes to present some new comparison theorems that guarantee the oscillation of all solutions of even order functional differential equations with a neutral term including both advanced and delayed arguments. The obtained results are based on comparisons with associated first‐order delay differential inequalities and first‐order delay differential equations, and they are applicable to the both cases where the neutral coefficients are unbounded and/or bounded. In the last section, several examples showing the applicability of the results are provided.

Controllability of Second Order Functional Random Differential Equations with Delay

In this article, we study some existence and controllability results for two classes of second order functional differential equations with delay and random effects. To begin, we employ a random fixed point theorem with a stochastic domain to demonstrate the existence of mild random solutions. Next, we prove that our problems are controllable. Finally, an example is given to validate the theory part.

Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for a Second Order Functional-Differential Equation

Existence and Uniqueness of a Positive Solution to a Boundary Value Problem for a Second Order Functional-Differential Equation

Oscillation of Second Order Functional Difference Equations of Non-canonical Type

Oscillation of Second Order Functional Difference Equations of Non-canonical Type

Oscillation and Nonoscillation of First Order Functional Differential Equations with Multi-Deviating Arguments

The oscillation and nonoscillation are studied in this paper for a class of first order functional differential equations with multi-deviating arguments. The obtained results extend the corresponding results in the literature.

Some two-point boundary value problems for systems of higher order functional differential equations

In the paper we study the question of the solvability and unique solvability of systems of the higher order differential equations with the argument deviations \begin{equation*} u_i^{(m_i)}(t)=p_i(t)u_{i+1}(\tau _{i}(t))+ q_i(t), (i=\overline {1, n}), \text {for $t\in I:=[a, b]$}, \end{equation*} and \begin{equation*}u_i^{(m_i)} (t)=F_{i}(u)(t)+q_{0i}(t), (i = \overline {1, n}), \text {for $ t\in I$}, \end{equation*} under the conjugate $u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {1, m_i-k_i}$, $i=\overline {1, n}$, and the right-focal $u_i^{(j_1-1)}(a)=a_{i j_1}$, $u_i^{(j_2-1)}(b)=b_{i j_2}$, $j_1=\overline {1, k_i}$, $j_2=\overline {k_i+1,m_i}$, $i=\overline {1, n}$, boundary conditions, where $u_{n+1}=u_1, $ $n\geq 2, $ $m_i\geq 2, $ $p_i \in L_{\infty }(I; R), $ $q_i, q_{0i}\in L(I; R), $ $\tau _i\colon I\to I$ are the measurable functions, $F_i$ are the local Caratheodory's class operators, and $k_i$ is the integer part of the number $m_i/2$.In the paper are obtained the efficient sufficient conditions that guarantee the unique solvability of the linear problems and take into the account explicitly the effect of argument deviations, and on the basis of these results are proved new conditions of the solvability and unique solvability for the nonlinear problems.

Classification of Second Order Functional Differential Equations with Constant Coefficients to Solvable Lie Algebras

In this paper, we shall apply symmetry analysis to second order functional differential equations with constant coefficients. The determining equations of the admitted Lie group are constructed in a manner different from that of the existing literature for delay differential equations. We define the standard Lie bracket and make a complete classification of the second order linear functional differential equations with constant coefficients, to solvable Lie algebras. We also classify some second order non-linear functional differential equations with constant coefficients, to solvable Lie algebras.

Stability, boundedness and existence of unique periodic solutions to a class of fourth order functional differential equations

In this paper a novel class of fourth order functional differential equations is discussed. By reducing the fourth order functional differential equation to system of first order, a suitable complete Lyapunov functional is constructed and employed to obtain sufficient conditions that guarantee existence of a unique periodic solution, asymptotic and uniform asymptotic stability of the zero solutions, uniform boundedness and uniform ultimate boundedness of solutions. The obtained results are new and include many prominent results in literature. Finally, two examples are given to show the feasibility and reliability of the theoretical results.

On solvability of focal boundary value problems for higher order functional differential equations with integral restrictions

Sharp conditions are obtained for the unique solvability of focal boundary value problems for higher-order functional differential equations under integral restrictions on functional operators. In terms of the norm of the functional operator, unimprovable conditions for the unique solvability of the boundary value problem are established in the explicit form. If these conditions are not fulfilled, then there exists a positive bounded operator with a given norm such that the focal boundary value problem with this operator is not uniquely solvable. In the symmetric case, some estimates of the best constants in the solvability conditions are given. Comparison with existing results is also performed.

Existence and Uniqueness of Positive Solution to a Boundary Value Problem for a Nonlinear Second Order Functional-Differential Equation

Existence and Uniqueness of Positive Solution to a Boundary Value Problem for a Nonlinear Second Order Functional-Differential Equation

Quasi-synchronization of heterogenous fractional-order dynamical networks with time-varying delay via distributed impulsive control

Abstract This paper investigates the quasi-synchronization problem of a heterogeneous dynamical network. All nodes have fractional order dynamical behavior with time-varying delay. The distributed impulsive control strategy is applied to drive all the nodes to approximately synchronize with the target orbit within a nonzero error bound. A new comparison principle of impulsive fractional order functional differential equation has been built at first. Then, based on the Lyapunov stability theory, some basic theories of fractional order functional differential equation, and the definition of an average impulsive interval, some quasi-synchronization criteria are derived with explicit expressions of the error bound. Both synchronizing impulses and desynchronizing impulses are discussed in this paper. Finally, two numerical examples are presented to illustrate the validity of the theoretical analysis.

Unbounded oscillation of fourth order functional differential equations

Abstract In this paper, sufficient conditions for oscillation of unbounded solutions of a class of fourth order neutral delay differential equations of the form ( r ( t ) ( y ( t ) + p ( t ) y ( t − τ ) ) ″ ) ″ + q ( t ) G ( y ( t − α ) ) − h ( t ) H ( y ( t − σ ) ) = 0 $$\begin{array}{} \displaystyle (r(t)(y(t)+p(t)y(t-\tau))'')''+q(t)G(y(t-\alpha))-h(t)H(y(t-\sigma))=0 \end{array}$$ are discussed under the assumption ∫ 0 ∞ t r ( t ) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}\text{d}~~ t=\infty \end{array}$$

Oscillation of higher order functional differential equations with an advanced argument

This paper is concerned with the oscillatory behavior of solutions of the nth-order nonlinear differential equations with an advanced argument x(n)(t)+δq(t)xλ(g(t))=0,t≥t0>0, where n≥2 is even or odd and δ=±1. Here q(t)>0, g(t)≥t, and λ is the ratio of odd positive integers. The results are illustrated with an example.

On unbounded oscillation of fourth order functional difference equations

On unbounded oscillation of fourth order functional difference equations