Abstract
By virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem, we analytically establish several sufficient criteria for the existence of at least two or three positive solutions in the -Laplacian dynamic equations on time scales with a particular kind of -Laplacian and -point boundary value condition. It is this kind of boundary value condition that leads the established criteria to be dependent on the time scales. Also we provide a representative and nontrivial example to illustrate a possible application of the analytical results established. We believe that the established analytical results and the example together guarantee the reliability of numerical computation of those -Laplacian and -point boundary value problems on time scales.
Highlights
The investigation of dynamic equations on time scales, originally attributed to Stefan Hilger’s seminal work 1, 2 two decades ago, is undergoing a rapid development
Some analytical criteria have been established for the existence of positive solutions in some specific boundary value problems for the p-Laplacian dynamic equations on time scales 22, 33
Some novel and time-scale-dependent sufficient conditions are established for the existence of multiple positive solutions in a specific kind of boundary value problems on time scales
Summary
The investigation of dynamic equations on time scales, originally attributed to Stefan Hilger’s seminal work 1, 2 two decades ago, is undergoing a rapid development. Φp u is supposed to be the p-Laplacian operator, that is, φp u |u|p−2u and φp −1 φq, in which p > 1 and 1/p 1/q 1 With these configurations and with the aid of the Avery-Henderson fixed point theorem 34 , He established the criteria for the existence of at least two positive solutions in 1.1 fulfilling the boundary value conditions 1.2. This paper analytically establishes some new and time-scale-dependent criteria for the existence of at least double or triple positive solutions in the boundary value problems 1.9 and 1.10 by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. These obtained criteria significantly extend the results existing in 26–28.
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