Order Nonlinear Functional Differential Equations Research Articles

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Abstract The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

A unified approach to study the existence and numerical solution of functional differential equation

In this paper we consider a class of boundary value problems for third order nonlinear functional differential equation. By the reduction of the problem to operator equation we establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method. The approach used for the third order nonlinear functional differential equation can be applied to functional differential equations of any orders.

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.

Three positive periodic solutions of second order nonlinear neutral functional differential equations with delayed derivative

This paper deals with the existence of three positive periodic solutions for a class of second order neutral functional differential equations involving the delayed derivative term in nonlinearity (x(t)-cx(t-delta)){''}+a(t)g(x(t))x(t)=lambda b(t)f(t,x(t),x(t-tau_{1}(t)),x'(t-tau_{2}(t))). By utilizing the perturbation method of positive operator and Leggett–Williams fixed point theorem, a group of sufficient conditions are established.

Some two-point problems for second order integro-differential equations with argument deviations

In the paper we describe the classes of unique solvability of the Dirichlet and mixed two point boundary value problems for the second order linear integro-differential equation $$ u''(t)=p_0(t)u(t)+p_1(t)u(\tau_1(t))+\int_{a}^{b}p(t,s)u(\tau(s))ds+ q(t). $$% On the basis of the obtained and, in some sense, optimal results for the linear problems, by the a priori boundedness principle we prove the theorems of solvability and unique solvability for the second order nonlinear functional differential equations under the mentioned boundary conditions.

Stochastic numerical approach for solving second order nonlinear singular functional differential equation

A new computational intelligence numerical scheme is presented for the solution of second order nonlinear singular functional differential equations (FDEs) using artificial neural networks (ANNs), global operator genetic algorithms (GAs), efficient local operator interior-point algorithm (IPA), and the hybrid combination of GA-IPA. An unsupervised error function is assembled for the DDE optimized by ANNs using the hybrid combination of GA-IPA. Three kinds of the second order nonlinear singular DDEs have been solved numerically and compared their results with the exact solutions to authenticate the performance and exactness of the present designed scheme. Moreover, statistical analysis based on Mean absolute deviation, Theil's inequality coefficient and Nash Sutcliffe efficiency is also performed to validate the convergence and accuracy of the present scheme.

Oscillation of Even Order Impulsive Neutral Partial Differential Equations with Distributed Deviating Arguments

In this paper, we will consider a class of boundary value problems associated with even order nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. Sufficient conditions are obtained for the oscillation of solutions using impulsive differential inequalities and integral averaging scheme with pair of boundary conditions. Examples are specified to point up our important results.

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 criteria for third order nonlinear delay differential equations with damping

This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation \((\ast)\) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation \((\ast)\) oscillates if equation \((\ast\ast)\) is nonoscillatory. We also establish results for the oscillation of equation \((\ast)\) when equation \((\ast\ast)\) is oscillatory.

Oscillation Criteria for Higher order Nonlinear Functional Differential Equations with Advanced Argument

We consider a differential equation $$ {u^{(n) }}(t)+p(t)\left| {u\left( {\sigma (t)} \right)\left| {^{{\mu (t)}}} \right.\mathrm{sign}\,u\left( {\sigma (t)} \right)=0,} \right. $$ where p ∈ L loc (R +; R +), μ ∈ C(R +; (0, +∞)), σ ∈ C(R +; R +), and σ(t) ≥ t for t ∈ R +. We say that the equation is almost linear if the condition lim t→+∞ μ(t) = 1 is satisfied. At the same time, if lim sup t→+∞ μ(t) ≠ 1 or lim inf t→+∞ μ(t) ≠ 1, then Eq. is called an essentially nonlinear differential equation. The oscillatory properties of almost linear differential equations have been extensively studied. In the paper, new sufficient (necessary and sufficient) conditions are established for a general class of essentially nonlinear functional differential equations to have Property A.

Asymptotic Behavior for Nonoscillatory Solutions of Second Order Nonlinear Functional Differential Equation

This paper studied the asymptotic behavior of a class of nonlinear functional differential equations by using the Bellman-Bihari inequality. We obtain results which extend and complement those in references. The results indicate that all non-oscillatory continuable solutions of equation are asymptotic to at+b as under some sufficient conditions, where a,b are real constants. An example is provided to illustrate the application of the results.

Oscillatory and Asymptotic Behavior of a Second-Order Nonlinear Functional Differential Equations

This paper is concerned with oscillatory and asymptotic behavior of solutions of a class of second order nonlinear functional differential equations. By using the generalized Riccati transformation and the integral averaging technique, new oscillation criteria and asymptotic behavior are obtained for all solutions of the equation. Our results generalize and improve some known theorems.

Oscillation and nonoscillation criteria for even order nonlinear functional differential equations

This paper is devoted to the study of the oscillatory and nonoscillatory behavior of even order nonlinear functional differential equations with deviating argument of the type ( p(t)|x(n)(t)|α sgnx(n)(t)(n) +q(t)|x(g(t))|β sgnx(g(t)) = 0. (Ag) Mathematics subject classification (2010): 34C10, 26A12.

Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations

In this paper, we consider the existence of solutions for first order nonlinear impulsive functional integro-differential equations with integral boundary value conditions. We firstly build a new comparison theorem. Then by utilizing the monotone iterative technique and the method of lower and upper solutions, we obtain the existence of extremal solutions or quasi-solutions.

Regularly Varying Solutions of Second Order Nonlinear Functional Differential Equations with Retarded Argument

The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2]) - consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for $t\rightarrow \infty$ of the slowly varying solutions for a subclass of considered equations is presented.

Existence results on positive periodic solutions for impulsive functional differential equations

A class of first order nonlinear functional differential equations with impulses is studied. It is shown that there exist one or two positive T-periodic solutions under certain assumptions, and no positive T-periodic solution under some other assumptions. Applications to some impulsive biological models and an example, which can not be covered by known results, are given to illustrate the main results.

Oscillation results for higher even order nonlinear partial functional differential equations of neutral type

For some higher even order nonlinear partial functional differential equations of neutral type with continuous distributed delay, we obtain new oscillation criteria of solutions for boundary value problem of the equations, in our paper we omit the assumption which has been required for related results given before. The main tool used is the standard Philos integral averaging technique.

An Existence Result for Neumann Type Boundary Value Problems for Second Order Nonlinear Functional Differential Equation

Abstract. New sufficient conditions for the existence of at least one solution of Neumanntype boundary value problems for second order nonlinear differential equationsˆ(p(t)φ(x 0 (t))) 0 = f(t,x(t),x(τ 1 (t)),··· ,x(τ m (t))), t ∈ [0,T],x 0 (0) = 0, x 0 (T) = 0,are established. 1. IntroductionRecently, there have been many papers discussed the solvability of two-point,or multi-point boundary value problems for second order or higher order ordinaryor functional differential equations, we refer the readers to the text books [1], [2]and papers [6]-[9], [13]-[15] and the references therein.Xuan and Chen in [3] studied the solvability of singular one dimensionalp−Laplacian-like equation with Neumann boundary conditions(1)ˆ(A(|u 0 |)u 0 ) 0 −g(u(t)) = h(t), 0 0, h(t) ∈ L 1 [0,1], g is a continuous function definedon (−∞,0) ∪ (0,+∞) such that g(ξ) → 0 as |ξ| → 0, g(ξ) → +∞ as ξ → 0 + ,g(ξ) → −∞ as ξ → 0 − , and g(ξ)ξ > 0 for ξ 6= 0. Suppose(i) H(ξ) = ξA(|ξ|) is strictly increasing homeomorphism of (0,+∞) withH(0) = 0.(ii) lim

On the oscillation of certain third order nonlinear functional differential equations

We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the form d d t a ( t ) d 2 d t 2 x ( t ) α + q ( t ) f ( x [ g ( t ) ] ) = 0 and d d t a ( t ) d 2 d t 2 x ( t ) α = q ( t ) f ( x [ g ( t ) ] ) + p ( t ) h ( x [ σ ( t ) ] ) , when ∫ ∞ a - 1 / α ( s ) d s < ∞ . The case when ∫ ∞ a - 1 / α ( s ) d s = ∞ is also included.

Oscillation theorems for certain higher order nonlinear functional differential equations

Some new oscillation theorems for higher-order nonlinear functional differential equations of the form >0, are established.