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

TL;DR

This paper establishes conditions for the existence of solutions to first-order Hamiltonian stochastic impulsive differential equations with Dirichlet boundary conditions using variational methods and critical point theory.

Contribution

It introduces a novel approach combining variational methods, Legendre transformation, and mountain pass lemma to prove existence of solutions for these complex stochastic impulsive equations.

Findings

Existence of solutions is proven under certain conditions.

Energy functional and its conjugation are constructed for analysis.

An example demonstrates the practical applicability of the theoretical results.

Abstract

In this paper, we study the sufficient conditions for the existence of solutions of first-order Hamiltonian stochastic 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 stochastic impulsive differential equation.Finally, an example are presented to illustrate the feasibility and effectiveness of our results.