The existence and exponential stability of semilinear functional differential equations with random impulses under non-uniqueness

In this paper, the existence and exponential stability of mild solutions of semilinear differential equations with random impulses are studied under non-uniqueness in a real separable Hilbert space. The results are obtained by using the Leray-Schauder alternative fixed point theorem.     

Existence and exponential stability for neutral stochastic fractional differential equations with impulses driven by Poisson jumps

The paper is mainly concerned with a class of neutral stochastic fractional integro-differential equation with Poisson jumps. First, the existence and uniqueness for mild solution of an impulsive stochastic system driven by Poisson jumps is established by using the Banach fixed point theorem and resolvent operator. The exponential stability in the pth moment for mild solution to neutral stochastic fractional integro-differential equations with Poisson jump is obtained by establishing an integral inequality.     

Impulsive differential equations with Gamma distributed moments of impulses and p-moment exponential stability

Differential equations with impulses at random moments are set up and investigated. We study the case of Gamma distributed random moments of impulses. Several properties of solutions are studied based on properties of Gammma distributions. Some sufficient conditions for p-moment exponential stability of the solutions are given.     

Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses

This paper is devoted to establish Bielecki–Ulam–Hyers–Rassias stability, generalized Bielecki–Ulam–Hyers–Rassias stability, and Bielecki–Ulam–Hyers stability on a compact interval [0,T], for a class of higher‐order nonlinear differential equations with fractional integrable impulses. The phrase ‘fractional integrable’ brings one to fractional calculus. Hence, applying usual methods for analysis offers many difficulties in proving the results of existence and uniqueness of solution and stability theorems. Picard operator is applied in showing existence and uniqueness of solution. Stability results are obtained by using the tools of fractional calculus and Hölder's inequality of integration. Along with tools of fractional calculus, Bielecki's normed Banach spaces are considered, which made the results more interesting. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence results for fractional functional differential equations with impulses

In this paper, we discuss some existence results for a class of multi-point boundary value problem for impulsive fractional functional differential equations. Some sufficient conditions are obtained by using suitable fixed point theorems. Examples are also given to illustrate our results.     

A class of nonlinear differential equations with fractional integrable impulses

In this paper, we introduce a new class of impulsive differential equations, which is more suitable to characterize memory processes of the drugs in the bloodstream and the consequent absorption for the body. This fact offers many difficulties in applying the usual methods to analysis and novel techniques in Bielecki’s normed Banach spaces and thus makes the study of existence and uniqueness theorems interesting. Meanwhile, new concepts of Bielecki–Ulam’s type stability are introduced and generalized Ulam–Hyers–Rassias stability results on a compact interval are established. This is another novelty of this paper. Finally, an interesting example is given to illustrate our theory results.     

Improved global exponential stability for delay difference equations with impulses

In this paper, by means of constructing an impulsive delay difference inequality, we study the global exponential stability for a class of delay difference equations with impulses. A new criterion ensuring the global exponential stability of the equilibrium point is obtained, which is less restrictive and conservative than that given in the earlier reference. An illustrative example is given to demonstrate the effectiveness and advantage of the obtained result.     

Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions

In this paper, we consider the properties of Green’s function for a class of nonlinear Caputo fractional differential equations with integral boundary conditions by constructing an available integral operator. By means of well-known fixed point theorems and lower and upper solutions method, some new existence and nonexistence criteria of single or multiple positive solutions for fractional differential equation boundary value problems are established. As applications, some interesting examples are presented to illustrate the main results.     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.     

Ulam‐Hyers stability of Caputo fractional difference equations

We study the Ulam‐Hyers stability of linear and nonlinear nabla fractional Caputo difference equations on finite intervals. Our main tool used is a recently established generalized Gronwall inequality, which allows us to give some Ulam‐Hyers stability results of discrete fractional Caputo equations. We present two examples to illustrate our main results.     

Uniform stability theorems for delay differential equations with impulses

In the paper, we obtain sufficient conditions for the uniform stability of the zero solution of the delay differential equation with impulses

     

A sufficient condition of viability for fractional differential equations with the Caputo derivative

In this paper viability results for nonlinear fractional differential equations with the Caputo derivative are proved. We give the sufficient condition that guarantees fractional viability of a locally closed set with respect to nonlinear function. As an example we discuss positivity of solutions, particularly in linear case.     

Nonlinear differential equations with Caputo fractional derivatives in the class of monotonic functions

A fixed point theorem is established to prove the existence of a global solution to some fractional differential equations involving the Caputo derivatives under a, not necessarily small, growth condition on the nonlinearity.     

A short note on the solvability of impulsive fractional differential equations with Caputo derivatives

In this paper, we prove the existence and non-existence of solutions to two impulsive fractional differential equations with strong or weak Caputo derivatives in Euclidean space, respectively.     

Weak exponential stability of stochastic differential equations

In this paper we introduce weak exponential stability of stochastic differential equations. In particular, we introduce weak exponential stability in mean, weak exponential asymptotical stability in mean and weak uniform asymptotical stability in mean. We also derive some results related to the above concepts     

Periodic nonautonomous differential equations with noninstantaneous impulsive effects

In this paper, we study a new class of periodic nonautonomous differential equations with periodic noninstantaneous impulsive effects. A concept of noninstantaneous impulsive Cauchy matrix is introduced, and some basic properties are considered. We give the representation of solutions to the homogeneous problem and nonhomogeneous problem by using noninstantaneous impulsive Cauchy matrix, and the variation of constants method, adjoint systems, and periodicity of solutions is verified under standard periodicity conditions. Further, we show the existence and uniqueness of solutions of semilinear problem and establish existence result for periodic solutions via Brouwer fixed point theorem and uniqueness and global asymptotic stability via Banach fixed point theorem.     

Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives

We discuss existence, uniqueness and stability of solutions of the system of nonlinear fractional differential equations