The existence of solutions for Sturm-Liouville differential equation with random impulses and boundary value problems. (English) Zbl 1496.34053
Summary: 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.
MSC:
| 34B24 | Sturm-Liouville theory |
| 34A37 | Ordinary differential equations with impulses |
| 34F05 | Ordinary differential equations and systems with randomness |
| 34B27 | Green’s functions for ordinary differential equations |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34A45 | Theoretical approximation of solutions to ordinary differential equations |
Keywords:
random impulsive differential equation; Green function; upper and lower solution; fixed point theorem; boundary value problemsReferences:
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