Existence of solutions for first-order Hamiltonian random impulsive differential equations with Dirichlet boundary conditions. (English) Zbl 1507.37087
Summary: 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.
MSC:
| 37J51 | Action-minimizing orbits and measures for finite-dimensional Hamiltonian and Lagrangian systems; variational principles; degree-theoretic methods |
| 37H10 | Generation, random and stochastic difference and differential equations |
| 34F05 | Ordinary differential equations and systems with randomness |
| 34K50 | Stochastic functional-differential equations |
Keywords:
random impulsive differential equation; variational method; critical point; mountain pass lemma; Dirichlet boundary conditionReferences:
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