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

Yu Guo; Xiao-Bao Shu; Qianbao Yin

doi:10.3934/dcdsb.2021236

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2022, Volume 27, Issue 8: 4455-4471. Doi: 10.3934/dcdsb.2021236

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

College of Mathematics, Hunan University, Changsha 410012, China

^* Corresponding author: Xiao-Bao Shu
^* Corresponding author: Xiao-Bao Shu

Yu Guo and Qianbao Yin contributed equally to this paper

Received: April 30, 2021

Revised: May 31, 2021

Early access: September 2021

Published: August 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 34B37, 37H10; Secondary: 37J51.

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