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Abstract: Two new stability results, Hyers–Ulam–Rassias stability and Hyers–Ulam stability, of a certain fractional integral equation with delay are presented by using a fixed point theorem in a generalized complete metric space. Some applications of our results are also provided. [Copyright &y& Elsevier]

Abstract: In this paper, we propose the cubic spline collocation method with two parameters for solving delay differential equations (DDEs). Some results of the local truncation error and the convergence of the spline collocation method are given. We also obtain some results of the linear stability and the nonlinear stability of the method for DDEs. In particular, we design an algorithm to obtain the ranges of the two parameters which are necessary for the P-stability of the collocation method. Some illustrative examples successfully verify our theoretical results. [Copyright &y& Elsevier]

Abstract: The purpose of this note is twofold: First we highlight the importance of an implicit assumption in [Murat Adıvar, Youssef N. Raffoul, Stability and periodicity in dynamic delay equations, Computers and Mathematics with Applications 58 (2009) 264–272]. Second we emphasize one consequence of the bijectivity assumption which enables ruling out the commutativity condition on the delay function. [Copyright &y& Elsevier]

Abstract: Let be an arbitrary time scale that is unbounded above. By means of a variation of Lyapunov’s method and contraction mapping principle this paper handles asymptotic stability of the zero solution of the completely delayed dynamic equations Moreover, if is a periodic time scale, then necessary conditions are given for the existence of a unique periodic solution of the above mentioned equation. [Copyright &y& Elsevier]

Abstract: As an alternative to the classical splitting methods, two new splitting schemes have been developed recently: the additive and the iterative splitting. In this paper we discuss the most important properties, the advantages and disadvantages of these schemes, and investigate their performance on simple examples as well as on more complex physical models. [Copyright &y& Elsevier]

In the paper, employing the variational Lyapunov method, stability and instability properties in terms of two measures for impulsive delay differential equations with fixed moments of impulsive effects are discussed. Some stability and instability criteria are obtained. These results much generalize the known ones. Some examples are given to illustrate the advantages of them as well. [Copyright &y& Elsevier]