Order Nonlinear Delay Differential Equations Research Articles

Galerkin Approximations for Neutral Delay Differential Equations

In this work, Galerkin approximations are developed for a system of first order nonlinear neutral delay differential equations (NDDEs). The NDDEs are converted into an equivalent system of hyperbolic partial differential equations (PDEs) along with the nonlinear boundary constraints. Lagrange multipliers are introduced to enforce the boundary constraints. The explicit expressions for the Lagrange multipliers are derived by exploiting the equivalence of partial derivatives in space and time at a given point on the domain. To illustrate the method, comparisons are made between numerical solution of NDDEs and its Galerkin approximations for different NDDEs.

Galerkin Approximations for Higher Order Delay Differential Equations

In this work, Galerkin approximations are developed for a system of n first order nonlinear delay differential equations (DDEs) and also for an nth order nonlinear scalar DDE. The DDEs are converted into an equivalent system of partial differential equations of the same order along with the nonlinear boundary constraints. Lagrange multipliers are then introduced and explicit expressions for the Lagrange multipliers are derived to enforce the nonlinear boundary constraints. To illustrate the method, comparisons are made between the numerical solution of nonlinear DDEs and its Galerkin approximations for different parameter values.

An instability theorem for a certain sixth order nonlinear delay differential equation

The aim of this paper is to establish an instability theorem for a certain sixth order nonlinear delay differential equation. The proof of the theorem is based on the use of Lyapunov–Krasovskii functional approach. By this work, we improve an instability result obtained in the literature for a certain sixth order nonlinear differential equation without delay to the instability of the zero solution of a certain sixth order nonlinear delay differential equation.

Oscillation for second order nonlinear delay differential equations with impulse

Oscillation for second order nonlinear delay differential equations with impulse

Global existence and Mann iterative algorithms of positive solutions for first order nonlinear neutral delay differential equations

This paper deals with the first order nonlinear neutral delay differential equation d dt [ x ( t ) + p ( t ) x ( t - τ ) ] + f ( t , x ( σ 1 ( t ) ) , x ( σ 2 ( t ) ) , … , x ( σ n ( t ) ) ) = 0 , t ⩾ t 0 , where τ > 0 , p ∈ C ( [ t 0 , + ∞ ) , R ) , f ∈ C ( [ t 0 , + ∞ ) × R n , R ) and σ l ∈ C ( [ t 0 , + ∞ ) , R ) with lim t→+∞ σ l ( t) = +∞ for l ∈ {1, 2, … , n}. By using the Banach fixed point theorem, we prove the global existence of uncountably many bounded positive solutions for the above equation relative to all ranges of the function p, construct some Mann type iterative algorithms with errors to approximate these positive solutions and discuss several error estimates between the sequences generated by the iterative algorithms and these positive solutions. Seven examples are presented to illuminate the results obtained in this paper.

Existence of oscillatory solutions of forced second order delay differential equations

In this paper, under weaker hypothesis, the global existence of oscillatory solutions is established for a forced second order nonlinear delay differential equation. The results of this paper generalize and improve some known results.

Asymptotic behavior of solutions for third-order half-linear delay dynamic equations on time scales

By means of Riccati transformation technique and integral averaging technique, we establish some new sufficient conditions which guarantee that every solution of the third-order half-linear delay dynamic equations $$(r(t)(x^{\Delta\Delta}(t))^\gamma)^\Delta+p(t)x^\gamma(\tau(t))=0$$ oscillates or converges to zero on a time scale \(\mathbb{T}\), here γ>0 is a quotient of odd positive integers with r and p real-valued positive rd-continuous functions defined on \(\mathbb{T}\). Our results not only extend and improve the results given in literatures but also unify the asymptotic behavior results of the third order nonlinear delay differential equation and the third order nonlinear delay difference equation. Some examples are considered to illustrate the importance of our results.

On the oscillatory behavior for a certain class of third order nonlinear delay difference equations

By employing the generalized Riccati transformation technique, we will establish some new oscillation criteria for a certain class of third order nonlinear delay difference equations. Our results extend and improve some previously obtained ones. An example is worked out to demonstrate the validity of the proposed results.

On some qualitative behaviors of solutions to a kind of third order nonlinear delay differential equations

Sufficiency criteria are established to ensure the asymptotic stability and boundedness of solutions to third-order nonlinear delay differential equations of the form ... x(t) +e(x(t), u x(t), ¨(t))¨x(t) +g(x(t −r), u x(t −r)) + (x(t −r))

Oscillation Criteria for Second-Order Nonlinear Neutral Delay Differential Equations

Some oscillation criteria for the second order nonlinear neutral delay differential equations[a(t)x(t)+∑li=1ci(t)x(t-τi(t))]″+∑mi=1pi(t)fi(x(t-δi(t))) -∑ni=1qi(t)gi(x(t-σi(t)))=0,t0.(E)are established in this paper.we give sufficient conditions for the oscillation of all soiutions.And the results of some references have been extended.

Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation

The paper is concerned with the higher order nonlinear neutral delay differential equation r ( t ) ( x ( t ) + p ( t ) x ( t - τ ) ) ( m ) ( n - m ) + ( - 1 ) n - m + 1 f t , x σ 1 ( t ) , … , x ( σ l ( t ) ) = g ( t ) , t ⩾ t 0 . Using the Banach fixed-point theorem, we establish five existence results of uncountably many bounded nonoscillatory solutions for the above equation, construct five Mann iterative sequences with mixed errors for approximating these nonoscillatory solutions and discuss five error estimates between the approximate solutions and these nonoscillatory solutions. To dwell the importance and applications of our results, five nontrivial examples are constructed.

Necessary and sufficient conditions for oscillation of higher order nonlinear neutral delay differential equations

We first show that the n order superlinear neutral delay differential equation with unstable type $$ [x(t) - px(t - \tau )]^{(n)} = q(t)|x(t - \sigma )|^{\beta - 1} x(t - \sigma ),t \geqslant t_0 $$ always has an unbounded and nonoscillatory solution, where n > 1 is an even integer, σ > 0, τ > 0, β > 1, 0 < p < 1 and q ∈ C([t0, ∞), [0, ∞)). Then a necessary and sufficient condition is obtained for all bounded solution of the above equation to be oscillatory.

Stability criteria for certain third order nonlinear delay differential equations

Stability criteria for certain third order nonlinear delay differential equations

Interval oscillation theorems for second order nonlinear partial delay differential equations

Using the integral averaging method and the generalized Riccati technique, we derive new interval oscillation criteria for second order nonlinear partial delay differential equations. These results are different from most known ones in the sense that they are based on information only on a sequence of subintervals of (0,∞), rather than on the whole (0,∞). Our results are of a high degree of generality and sharper than the existing results in literature.

A note on oscillation criteria of second order nonlinear neutral delay differential equations

In recent years, many results on oscillation criteria of second order nonlinear neutral delay differential equations have been obtained, but some of these criteria are incorrect due to a mistake in a crucial auxiliary result. This note examines several of these results, provides some counterexamples and points out what causes the problems.

Oscillation of second-order delay dynamic equations on time scales

By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations $$\bigl(p(t)\bigl(x^{\Delta}(t)\bigr)^{\gamma}\bigr)^{\Delta}+q(t)f\bigl(x\bigl(\tau(t)\bigr)\bigr)=0$$ on a time scale \(\mathbb{T}\) , here γ≥1 is a quotient of odd positive integers with p and q real-valued positive rd-continuous functions defined on \(\mathbb{T}\) . Our results improve and extend some results established by Saker (J. Comput. Appl. Math. 177:375–387, 2005) but also unify the oscillation of the second order nonlinear delay differential equation and the second order nonlinear delay difference equation.

Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations

In this paper we consider the second order nonlinear neutral delay differential equation [ a ( t ) ( x ( t ) + b ( t ) x ( t − τ ) ) ′ ] ′ + [ h ( t , x ( h 1 ( t ) ) , x ( h 2 ( t ) ) , … , x ( h k ( t ) ) ) ] ′ + f ( t , x ( f 1 ( t ) ) , x ( f 2 ( t ) ) , … , x ( f k ( t ) ) ) = g ( t ) , t ≥ t 0 , where τ > 0 , a , b , g ∈ C ( [ t 0 , + ∞ ) , R ) with a ( t ) > 0 for t ≥ t 0 , h ∈ C 1 ( [ t 0 , + ∞ ) × R k , R ) , f ∈ C ( [ t 0 , + ∞ ) × R k , R ) , h l ∈ C 1 ( [ t 0 , + ∞ ) , R ) and f l ∈ C ( [ t 0 , + ∞ ) , R ) with lim t → + ∞ h l ( t ) = lim t → + ∞ f l ( t ) = + ∞ , l = 1 , … , k . Under suitable conditions, by making use of the Banach fixed point theorem, we show the existence of infinitely many nonoscillatory solutions, which are uncountable, for the above equation, suggest several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. Five nontrivial examples are given to illustrate the advantages of our results.

POSITIVE INCREASING SOLUTIONS ON THE HALF-LINE TO SECOND ORDER NONLINEAR DELAY DIFFERENTIAL EQUATIONS

AbstractSecond order nonlinear delay differential equations with positive delays are considered, and sufficient conditions are given that guarantee the existence of positive increasing solutions on the half-line with first order derivatives tending to zero at infinity. The approach is elementary and is essentially based on an old idea which appeared in the author's paper Arch. Math. (Basel)36 (1981), 168–178. The application of the result obtained to second order Emden-Fowler type differential equations with constant delays and, especially, to second order linear differential equations with constant delays, is also presented. Moreover, some (general or specific) examples demonstrating the applicability of the main result are given.

Existence and iterative approximations of nonoscillatory solutions of higher order nonlinear neutral delay differential equations

This paper studies a few existence results of nonoscillatory solutions for the following higher order nonlinear neutral delay differential equation: dndtnx(t)+cx(t-τ)+(-1)n-i+1didtih(t,x(h1(t)),x(h2(t)),…,x(hk(t)))+(-1)n+1f(t,x(f1(t)),x(f2(t)),…,x(fk(t)))=g(t),t⩾t0, where n is a positive integer, 0⩽i⩽n-1, c∈R⧹{-1}, τ>0, h∈Ci([t0,+∞)×Rk,R), f∈C([t0,+∞)×Rk,R), g∈C([t0,+∞),R), hl∈Ci([t0,+∞),R) and fl∈C([t0,+∞),R) with limt→+∞hl(t)=limt→+∞fl(t)=+∞,l=1,…,k constructs several Mann type iterative approximation sequences with errors for these nonoscillatory solutions and establish some error estimates between the approximate solutions and the nonoscillatory solutions. In addition sufficient conditions for the existence of infinitely many nonoscillatory solutions for the above equation are given. Three nontrivial examples are given to illustrate the advantages of our results.

Global Solutions Approaching Lines at Infinity to Second Order Nonlinear Delay Differential Equations

This article is concerned with second order nonlinear delay, and especially ordinary, differential equations. By the use of the fixed point technique based on the classical Schauder's theorem, for any given line, sufficient conditions are established in order that there exists at least one global solution which is asymptotic at ∞ to this line. In the special case of ordinary differential equations, via the Banach's Contraction Principle, for any given line, conditions are presented which guarantee that there exists a unique global solution that is asymptotic at ∞ to this line. The application of the results obtained to second order delay, and ordinary, differential equations of Emden-Fowler type is presented, and some examples demonstrating the applicability of the results are given. Finally, some supplementary results are obtained, which provide sufficient conditions for all global solutions belonging to a suitable class to be asymptotic at ∞ to lines.