Positive Green’s function and triple positive solutions of a second-order impulsive differential equation with integral boundary conditions and a delayed argument. (English) Zbl 1381.34045
Summary: In this paper, we first establish the expression of positive Green’s function for a second-order impulsive differential equation with integral boundary conditions and a delayed argument. Furthermore, applying Legget-William’s fixed point theorem and Hölder’s inequality, we obtain the existence results of at least three positive solutions under three cases: \(p=1\), \(1< p<+\infty\), and \(p=+\infty\). We discuss our problem with impulsive effects and a delayed argument. In this case, our results cover second-order boundary value problems without impulsive effects and delayed arguments and are compared with some recent results. Finally, we give an example to illustrate our main results.
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
| 34B27 | Green’s functions for ordinary differential equations |
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
| 34K10 | Boundary value problems for functional-differential equations |
Keywords:
differential equations with impulsive effects and delayed arguments; positive Green’s function; three positive solutions; Legget-William’s fixed point theorem; Hölder’s inequalityReferences:
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