Random Impulsive Differential Equations Research Articles

Existence of solution sets for Φ-Laplacian for random impulsive differential equations

In this article, we provide results on the existence and compactness of the set of solutions for a class of initial-value problem representing a first-order random impulsive differential equation. The key tool to achieve our goal is based on an alternative nonlinear Leary-Schauder type theorem in the settings of generalized Banach spaces.

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.

Existence and uniqueness of solutions for random impulsive differential equation

In this paper, we study the existence and uniqueness of the mild solutions for random impulsive differential equations through fixed point technique. An example is provided to illustrate the theory.

Existence and Multiplicity of Mild Solutions for First-Order Hamilton Random Impulsive Differential Equations with Dirichlet Boundary Conditions

Existence and Multiplicity of Mild Solutions for First-Order Hamilton Random Impulsive Differential Equations with Dirichlet Boundary Conditions

HJB equation for optimal control system with random impulses

This paper studies the optimal control problem of random impulsive differential equations. Based on the influence of random impulse generation, we define a more reasonable performance index by setting the random function and obtain the HJB equation of random impulse. Using the basic analysis method and stochastic process theory, we prove that the value function satisfies the random impulse HJB equation, and the value function is the viscosity solution of the random impulse HJB. As an application, we present an example of optimal feedback control.

Existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions

<abstract><p>In this paper, we study sufficient conditions for the existence of solutions to a class of damped random impulsive differential equations under Dirichlet boundary value conditions. By using variational method we first obtain the corresponding energy functional. Then the existence of critical points are obtained by using Mountain pass lemma and Minimax principle. Finally we assert the critical point of enery functional is the mild solution of damped random impulsive differential equations.</p></abstract>

Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions

<p style='text-indent:20px;'>In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian random impulsive differential equations under Dirichlet boundary value conditions. By using the variational method, we first obtain the corresponding energy functional. And by using Legendre transformation, we obtain the conjugation of the functional. Then the existence of critical point is obtained by mountain pass lemma. Finally, we assert that the critical point of the energy functional is the mild solution of the first order Hamiltonian random impulsive differential equation. Finally, an example is presented to illustrate the feasibility and effectiveness of our results.</p>

The existence of solutions for Sturm\u2013Liouville differential equation with random impulses and boundary value problems

In this article, we consider the existence of solutions to the Sturm–Liouville differential equation with random impulses and boundary value problems. We first study the Green function of the Sturm–Liouville differential equation with random impulses. Then, we get the equivalent integral equation of the random impulsive differential equation. Based on this integral equation, we use Dhage’s fixed point theorem to prove the existence of solutions to the equation, and the theorem is extended to the general second order nonlinear random impulsive differential equations. Then we use the upper and lower solution method to give a monotonic iterative sequence of the generalized random impulsive Sturm–Liouville differential equations and prove that it is convergent. Finally, we give two concrete examples to verify the correctness of the results.

Existence of periodic solutions of second-order nonlinear random impulsive differential equations via topological degree theory

In this paper, we investigate the existence of periodic solutions for a class of second order nonlinear random impulse differential equations. By extending the definitions of continuous function bound set, curvature bound set and Nagumo set in topological degree to PC space, and defining the appropriate operators and conditions, using the theory of topological degree and coincidence degree, we prove that the solution must be bounded if it exists and find the boundary value. Finally, we define appropriate Nagumo set and autonomous curvature bound set, and obtain the existence of periodic solutions of equations and simultaneous equations by using Mawhin’s continuity theorem.

Coexistence of Random Subharmonic Solutions of Random Impulsive Differential Equations and Inclusions on a Circle

The coexistence of random periodic solutions with various periods (i.e. subharmonics) is proved to random differential equations on a circle with random impulses of all integer orders. One of the theorems is also extended to random differential inclusions on a circle with multivalued deterministic impulses. These results can be roughly characterized as a further application of the randomized Sharkovsky type theorems to random impulsive differential equations and inclusions on a circle.

Randomized Sharkovsky-type theorems and their application to random impulsive differential equations and inclusions on tori

Our randomized versions of the Sharkovsky-type cycle coexistence theorems on tori and, in particular, on the circle are applied to random impulsive differential equations and inclusions. The obtained effective coexistence criteria for random subharmonics with various periods are formulated in terms of the Lefschetz numbers (in dimension one, in terms of degrees) of the impulsive maps and their iterates w.r.t. the (deterministic) state variables. Otherwise, the forcing properties of certain periods of the given random subharmonics are employed, provided there exists a random harmonic solution. In the single-valued case, the exhibition of deterministic chaos in the sense of Devaney is detected for random impulsive differential equations on the factor space [Formula: see text]. Several simple illustrative examples are supplied.

Application of the Randomized Sharkovsky-Type Theorems to Random Impulsive Differential Equations and Inclusions

Our randomized Sharkovsky-type theorems are applied to random impulsive differential equations and inclusions in order to establish the coexistence of random periodic solutions with various periods, which are forced according to the Sharkovsky ordering of positive integers. The impulses can be single-valued or multivalued and deterministic or random. The obtained theorems can be rather curiously stronger than their deterministic analogies. The relationship to deterministic chaos is also indicated.

On Existence and Uniqueness of Random Impulsive Differential Equations

This paper studies existence and uniqueness for random impulsive differential equations. The authors first generalize a random fixed point theorem of Schaefer’s type. Then the authors shall rely on the generalized Schaefer’s type random fixed point theorem to discuss the existence of the system. In addition, the authors study the existence and uniqueness of random impulsive differential equations by applying random Banach fixed point theorem and obtain some less conservative results. Finally, an example is given to illustrate the effectiveness of the results.

The Euler scheme for random impulsive differential equations

Random impulsive differential equations (RIDEs) are a kind of mathematical models with extensive applications. In this paper, the Euler scheme for RIDEs is first brought forward, one of whose important applications is to generate the whole approximate trajectories of RIDEs. Thus the proposed Euler scheme allows us to approximate moments, functionals and the distribution for the underlying process and perform Monte-Carlo type analysis. The obtained results show that the Euler scheme is at least 1-order of step h when the right terms of equation satisfy Lipschitz conditions and the waiting times of random impulses follow the mutually independent exponential distribution with the same parameter λ. Thus it is an efficient method for numerical simulation.