Higher Order Functional Differential Equations Research Articles

High-Order Nonlinear Functional Differential Equations: New Monotonic Properties and Their Applications

We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation test requires three independent conditions, but we provide criteria with two-conditions without checking the additional conditions. Lastly, we give examples to highlight the significance of the findings.

Amended Criteria for Testing the Asymptotic and Oscillatory Behavior of Solutions of Higher-Order Functional Differential Equations

Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them.

Long time behavior of higher-order delay differential equation with vanishing proportional delay and its convergence analysis using spectral method

<abstract> <p>Delay differential equations (DDEs) are used to model some realistic systems as they provide some information about the past state of the systems in addition to the current state. These DDEs are used to analyze the long-time behavior of the system at both present and past state of such systems. Due to the oscillatory nature of DDEs their explicit solution is not possible and therefore one need to use some numerical approaches. In this article, we developed a higher-order numerical scheme for the approximate solution of higher-order functional differential equations of pantograph type with vanishing proportional delays. Some linear and functional transformations are used to change the given interval [0, T] into standard interval [-1, 1] in order to fully use the properties of orthogonal polynomials. It is assumed that the solution of the equation is smooth on the entire domain of interval of integration. The proposed scheme is employed to the equivalent integrated form of the given equation. A Legendre spectral collocation method relative to Gauss-Legendre quadrature formula is used to evaluate the integral term efficiently. A detail theoretical convergence analysis in L<sub>∞</sub> norm is provided. Several numerical experiments were performed to confirm the theoretical results.</p> </abstract>

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.

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.

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.

Lasota\u2013Opial type conditions for periodic problem for systems of higher-order functional differential equations

In the paper we study the question of solvability and unique solvability of systems of the higher-order functional differential equations ui(mi)(t)=ℓi(ui+1)(t)+qi(t)(i=1,n‾) for t∈I:=[a,b]\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{i}^{(m_{i})}(t)=\\ell _{i}(u_{i+1}) (t)+ q_{i}(t) \\quad (i= \\overline{1, n}) \\text{ for } t\\in I:=[a, b] $$\\end{document} and ui(mi)(t)=Fi(u)(t)+q0i(t)(i=1,n‾) for t∈I\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{i}^{(m_{i})} (t)=F_{i}(u) (t)+q_{0i}(t) \\quad (i = \\overline{1, n}) \\text{ for } t\\in I $$\\end{document} under the periodic boundary conditions ui(j)(b)−ui(j)(a)=cij(i=1,n‾,j=0,mi−1‾),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{i}^{(j)}(b)-u_{i}^{(j)}(a)=c_{ij} \\quad (i=\\overline{1, n},j= \\overline{0, m_{i}-1}), $$\\end{document} where u_{n+1}=u_{1} , m_{i}geq 1, ngeq 2 , c_{ij}in R, q_{i},q_{0i}in L(I; R), ell _{i}:C^{0}_{1}(I; R)to L(I; R) are monotone operators and F_{i} are the local Caratheodory’s class operators. In the paper in some sense optimal conditions that guarantee the unique solvability of the linear problem are obtained, and on the basis of these results the optimal conditions of the solvability and unique solvability for the nonlinear problem are proved.

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.

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.

Oscillation of higher order functional differential equations with positive and negative coefficients and damping term

The oscillation for a class of even order nonlinear variable delay damped functional differential equation with positive and negative coefficients and nonlinear neutral term was discussed.By introducing parameter function and the generalized Riccati transformation,using Holder inequality and some necessary analytic techniques,some new criteria for the oscillation of the equation were proposed.These criteria improve and generalize some known results.Examples were given to illustrate the main results.

Minimal periods of solutions to higher-order functional differential equations

We obtain sharp bounds for minimal periods of periodic solutions to higher-order functional differential equations.

Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

We investigate a class of higher order functional differential equations with damping. By using a generalized Riccati transformation and integral averaging technique, some oscillation criteria for the differential equations are established.

On forced oscillation of higher order functional differential equations

In this paper, we establish some new oscillation criteria for the following forced functional differential equationx(n)(t)+q(t)|x(τ(t))|λ-1x(τ(t))=e(t),t∈[t0,∞).No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. Two cases have been taken into consideration: (i) q(t)<0,λ>1 and τ(t)⩽t(⩾t); (ii) q(t) changes its sign, 0<λ<1 and τ(t)⩽t(⩾t). The main results generalize some existing results and can be applied to the oscillation problem that are not covered in the literature.

Global asymptotic stability for a class of nonlinear systems which are equivalent to neutral higher order Functional Differential Equations

This paper has a strong motivation in the models occurring in the problem of PIO (Pilot In-the-Loop Oscillations) where any pilot model contains at least one pure delay; at the same time in the so-called PIO II category all dynamics are assumed linear except the position and rate limiters whose models include the saturation nonlinearity - a typical bounded nonlinear function. Interesting enough, an equally motivating application arises from neural networks dynamics where both delays and bounded sigmoids are to be met. In this paper we start from some classical results for integral equations concerning boundedness and asymptotic behavior and extend them to an interesting class of systems whose structure incorporates those described by input/output neutral functional differential equations of higher order.

On solvability conditions for nonlinear operator equations

New sufficient conditions are found for solvability and unique solvability of nonlinear operator equations in the Banach space. In particular, abstract analogues of the Conti–Opial-type theorems are established, which concern the solvability of nonlinear boundary value problems. On the basis of these results, new sufficient conditions are obtained for the solvability of a periodic problem at resonance for nonlinear higher-order functional differential equations.

Multiple solutions for impulsive higher order functional differential equations

Multiple solutions for impulsive higher order functional differential equations

On Periodic Solutions of Higher-Order Functional Differential Equations

For higher-order functional differential equations and, particularly, for nonautonomous differential equations with deviated arguments, new sufficient conditions for the existence and uniqueness of a periodic solution are established.

Oscillations of Higher Order Functional Differential Equations with Impulses

Abstract A kind of higher order sub-and super-linear FDE with impulses is studied in this paper. Several criteria on the oscillations of solutions are given. In particular, in the case where the coefficients of equations are positive and continuous functions, we find some suitable impulse functions such that all solutions of the equation are oscillatory under the impulse control.

Existence of positive solutions for higher-order functional differential equations

Under suitable conditions on f( t, y( t+ θ)), the boundary value problem of higher-order functional differential equation (FDE) of the form (BVP) (FDE) y (n)(t)+f(t,y(t+θ))=0, t∈[0,1], θ∈[−τ,a], (BC) y (i)(0)=0, 0⩽i⩽n−3, αy (n−2)(t)−βy (n−1)(t)=η(t), t∈[−τ,0], γy (n−2)(t)+δy (n−1)(t)=ξ(t), t∈[1,1+a], has at least one positive solution, where θ∈[− τ, a] is a fixed constant. Moreover, we also apply this main result to establish several existence theorems which guarantee (BVP) has multiple positive solutions.

On Higher Order Functional Differential Equations with Property A

where 𝑛 ≥ 2 and 𝐹 : 𝐶(𝑅+; 𝑅) → 𝐿loc(𝑅+; 𝑅) is a continuous mapping. Sufficient conditions for this equation to have the so-called Property A are established. In the case of ordinary differential equation the obtained results lead to an integral generalization of the well-known theorem by Kondrat'ev.