The author obtain results on the asymptotic behavior of the nonoscillatory solutions of first order nonlinear neutral differential equations. Keywords. Neutral differential equations, Oscillatory and Nonoscillatory solutions.

     In this paper, we conduct some qualitative analysis that involves the global asymptotic stability (GAS) of the Neutral Differential Equation (NDE) with variable delay, by using  Banach contraction mapping theorem, to give some necessary conditions to achieve the GAS of the zero solution.

In this article, the backstepping control scheme is proposed to stabilize the fractional order Riccati matrix differential equation with retarded arguments in which the fractional derivative is presented using Caputo's definition of fractional derivative. The results are established using Mittag-Leffler stability. The fractional Lyapunov function is defined at each stage and the negativity of an overall fractional Lyapunov function is ensured by the proper selection of the control law. Numerical simulation has been used to demonstrate the effectiveness of the proposed control scheme for stabilizing such type of Riccati matrix differential equations.

This article addresses a new numerical method to find a numerical solution of the linear delay differential equation of fractional order , the fractional derivatives described in the Caputo sense. The new approach is to approximating second and third derivatives. A backward finite difference method is used. Besides, the composite Trapezoidal rule is used in the Caputo definition to match the integral term. The accuracy and convergence of the prescribed technique are explained. The results  are shown through numerical examples.

    In this paper, we introduce new conditions to prove that the existence and boundedness of the solution by convergent sequences and convergent series. The theorem of Krasnoselskii, Lebesgue’s dominated convergence theorem and fixed point theorem are used to get some sufficient conditions for the existence of solutions. Furthermore, we get sufficient conditions to guarantee the oscillatory property for all solutions in this class of equations. An illustrative example is included as an application to the main results.

This study focuses on studying an oscillation of a second-order delay differential equation. Start work, the equation is introduced here with adequate provisions. All the previous is braced by theorems and examplesthat interpret the applicability and the firmness of the acquired provisions

In this paper, we present an approximate analytical and numerical solutions for the differential equations with multiple delay using the extend differential transform method (DTM). This method is used to solve many linear and non linear problems.

An efficient modification and a novel technique combining the homotopy concept with  Adomian decomposition method (ADM) to obtain an accurate analytical solution for Riccati matrix delay differential equation (RMDDE) is introduced  in this paper  . Both methods are very efficient and effective. The whole integral part of ADM is used instead of the integral part of homotopy technique. The major feature in current technique gives us a large convergence region of iterative approximate solutions .The results acquired by this technique give better approximations for a larger region as well as previously. Finally, the results conducted via suggesting an efficient and easy technique, and may be addressed to other non-linear problems.

     The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations.  Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decompos