First-order Impulsive Differential Equations Research Articles

Existence Result for Coupled Random First-Order Impulsive Differential Equations with Infinite Delay

In this paper, we consider a system of random impulsive differential equations with infinite delay. When utilizing the nonlinear variation of Leray–Schauder’s fixed-point principles together with a technique based on separable vector-valued metrics to establish sufficient conditions for the existence of solutions, under suitable assumptions on Y1, Y2 and ϖ1, ϖ2, which greatly enriched the existence literature on this system, there is, however, no hope to discuss the uniqueness result in a convex case. In the present study, we analyzed the influence of the impulsive and infinite delay on the solutions to our system. In addition, to the best of our acknowledge, there is no result concerning coupled random system in the presence of impulsive and infinite delay.

Controllability Results for First Order Impulsive Fuzzy Differential Systems

In this paper, we investigate the controllability of first-order impulsive fuzzy differential equations. Using the direct construction method, the controllability of first-order linear impulsive fuzzy differential equations is considered with a<0, the (c1) solution, and a<0, the (c2) solution, respectively.In addition, by employing the Banach fixed-point theorem, the controllability of first-order nonlinear impulsive fuzzy differential equations is studied. Finally, examples are presented to illustrate our theoretical results.

Ulam’s-Type Stability of First-Order Impulsive Differential Equations with Variable Delay in Quasi–Banach Spaces

Abstract In this paper, Ulam’s-type stabilities are studied for a class of first-order impulsive differential equations with bounded variable delays on compact interval with finite number of impulses. Results of stability are proved via newly established integral inequality of Bellman–Grönwall–Bihari type with delay for discontinuous functions. Using this inequality for the first time and assumption of α $\alpha$ -H o ¨ $\ddot{o}$ lder’s condition instead of common Lipschitz condition is novelty of this paper. Moreover, solution is obtained in quasi–Banach spaces which is best suited for obtaining results under the assumptions of α $\alpha$ -H o ¨ $\ddot{o}$ lder’s condition.

On Anti-Periodic Solutions for a Class of Impulsive Retarded Functional Differential Equations

In this paper, some new results on the existence and uniqueness of anti-periodic solution for a first-order impulsive functional differential equation with delay are obtained. The main tool to be employed here is the coincidence degree theory.

On the Hyers-Ulam Stability of First-Order Impulsive Delay Differential Equations

This paper proves the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability of nonlinear first-order ordinary differential equation with single constant delay and finite impulses on a compact interval. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions.

Periodic Boundary Value Problems for a Class of Impulsive Functional Differential Equations of Hybrid Type

This paper is concerned with the existence of extreme solutions of periodic boundary value problems for a class of first-order impulsive functional differential equations of hybrid type. We obtain the sufficient conditions for existence of extreme solutions by using upper and lower solutions method coupled with monotone iterative technique.

Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

Bifurcation of positive periodic solutions of first-order impulsive differential equations

We give a global description of the branches of positive solutions of first-order impulsive boundary value problem: which is not necessarily linearizable. Where is a parameter, are given impulsive points. Our approach is based on the Krein-Rutman theorem, topological degree, and global bifurcation techniques. MSC:34B10, 34B15, 34K15, 34K10, 34C25, 92D25.

Green’s Function for the Periodic Boundary Value Problem Related to a First-order Impulsive Differential Equation and Applications to Functional Problems

We calculate the explicit expression of the Green’s function for the periodic boundary value problem related to an impulsive linear differential equation and obtain the expression of the unique solution to this problem. This results are useful to obtain sufficient conditions for the existence of nonnegative solutions (resp. non positive) to the above-mentioned boundary value problem. This study is also useful to prove the existence of a unique solution to the periodic boundary value problem for an impulsive linear differential equation subject to functional dependence.

Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form

This paper is concerned with an impulsive neutral differential equation of Euler form with unbounded delays (∗) { d d t [ x ( t ) − C ( t ) x ( α t ) ] + P ( t ) t x ( β t ) = 0 , t ≥ t 0 > 0 , t ≠ t k , x ( t k ) = b k x ( t k − ) + ( 1 − b k ) ∫ β t k t k P ( s / β ) s x ( s ) d s , k = 1 , 2 , … . Sufficient conditions are obtained for every solution of (∗) to tend to a constant as t → ∞ .

Existence of solutions for impulsive partial neutral functional differential equation with infinite delay

Abstract In this paper, the existence of mild solutions for first-order impulsive semilinear neutral functional differential equations with infinite delay in Banach spaces is investigated. We derive conditions in respect of the Hausdorff measure of noncompactness under which the mild solutions exist in Banach spaces. Our results improve and generalize some previous results.

Nonlinear Boundary Value Problem of First-Order Impulsive Functional Differential Equations

This paper investigates the nonlinear boundary value problem for a class of first-order impulsive functional differential equations. By establishing a comparison result and utilizing the method of upper and lower solutions, some criteria on the existence of extremal solutions as well as the unique solution are obtained. Examples are discussed to illustrate the validity of the obtained results.

On Exact Controllability of First-Order Impulsive Differential Equations

On Exact Controllability of First-Order Impulsive Differential Equations

On Exact Controllability of First-Order Impulsive Differential Equations

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work, we present some new results concerning the exact controllability of a nonlinear ordinary differential equation with impulses.

Existence of Solutions of Boundary Value Problems for Impulsive Delay Differential Equations

his paper concerns first-order impulsive delay differential equations with boundary conditions. The existence results are presented by apply a monotone iterative method, and a uniqueness criterion is obtained.

Solution of impulsive boundary value problems on infinite intervals

In this paper, the upper and lower solution method and monotone iterative technique are employed to study the boundary value problem for a class of first-order impulsive ordinary differential equations with Carathéodory condition on infinite intervals. Several existence theorems are obtained.

Oscillation criteria for a first-order impulsive neutral differential equation of Euler form

In this paper, we are concerned with the oscillation of a first-order impulsive neutral differential equation of Euler form with variable delays. Our results reveal the fact that the oscillatory behavior of all solutions of differential equations without impulses can be inherited by impulsive differential equations under certain impulsive perturbations. It is also seen that the oscillatory properties of all solutions of impulsive differential equations may be caused by the impulsive perturbations, though the corresponding differential equations without impulses admit a nonoscillatory solution. Some examples are also given to illustrate the applicability of the results obtained.

FURTHER RESULTS ON POSITIVE PERIODIC SOLUTIONS OF IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS AND APPLICATIONS

AbstractA class of first-order impulsive functional differential equations with forcing terms is considered. It is shown that, under certain assumptions, there exist positive T-periodic solutions, and under some other assumptions, there exists no positive T-periodic solution. Applications and examples are given to illustrate the main results.

Existence of solutions to first-order impulsive periodic boundary value problems

This paper is concerned with the solvability of periodic boundary value problems for first-order impulsive ordinary differential equations. By using Schaeffer’s theorem, a sufficient condition for existence of solution of the problem is obtained, which extends and improves some known results in the literature.

Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations

Motivated by Liu [Y. Liu, Further results on periodic boundary value problems for nonlinear first order impulsive functional differential equations, J. Math. Anal. Appl. 327 (2007) 435–452], Li et al. [X. Li, X. Lin, D. Jiang, X. Zhang, Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects, Nonlinear Anal. 62 (2005) 683–701] and Liu and Ge [Y. Liu, W. Ge, Stability theorems and existence results for periodic solutions of nonlinear impulsive delay differential equations with variable coefficients, Nonlinear Anal. 57 (2004) 363–399], this paper is concerned with the periodic boundary value problem (PBVP for short) for the nonlinear impulsive differential equation { x ′ ( t ) + a ( t ) x ( t ) = f ( t , x ( t ) ) , a.e. t ∈ [ 0 , T ] ∖ { t 1 , … , t p } , Δ x ( t k ) = x ( t k + ) − x ( t k ) = I k ( x ( t k ) ) , k = 1 , … , p , x ( T ) = x ( 0 ) . We obtain new sufficient conditions for the existence of at least one, two or three positive solutions of the above PBVP, respectively. Examples are presented to illustrate the main results.