Abstract
We analytically establish the conditions for the existence of at least two or three positive solutions in the generalized -point boundary value problem for the -Laplacian dynamic equations on time scales by means of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem. Furthermore, we illustrate the possible application of our analytical results with a concrete and nontrivial dynamic equation on specific time scales.
Highlights
Since the seminal work by Stefan Hilger in 1988, there has been a rapid development in the research of dynamic equations on time scales
The gradually maturing theory of dynamic equations includes the majority of the existing analytical results on both differential equations and difference equations with uniform time-steps and establishes a solid foundation for the research of hybrid equations on different kinds of time scales
Among the topics in the research of dynamic equations on time scales, the investigation of the boundary value problems for some specific dynamic equations on time scales has become a focal one that attained a great deal of attention from many mathematicians
Summary
Since the seminal work by Stefan Hilger in 1988, there has been a rapid development in the research of dynamic equations on time scales. In this paper, inspired by the aforementioned results and methods in dealing with those boundary value problems on time scales, we intend to analytically discuss the possible existence of multiple positive solutions for the following one-dimensional p-Laplacian dynamic equation: 1.4 with m-point boundary value conditions:. A question naturally appears: “can we still establish some criteria for the existence of at least double or triple positive solutions in the generalized boundary value problems 1.4 and 1.5 ?” In this paper, we will give a positive answer to this question by virtue of the Avery-Henderson fixed point theorem and the five functionals fixed point theorem Those obtained criteria will significantly extend the results in literature 19, 21, 23.
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