Lyapunov-type inequalities of multi-layer fractional half-linear $\nabla$-difference boundary value problems

Document Type : Regular research papers

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Department of Applied Mathematics, Sahand University of Technology

Abstract

‎The main problem of interest in this paper is to investigate the nontrivial solutions of $3$ leyer nested half linear higher order fractional order $\nabla$ difference equations subject to the fully left sided boundary conditions. In this way, two classes of discrete fractional half linear boundary value problems are considered‎. ‎In our investigation the discrete Holder inequality and integration by parts play fundamental roles‎.Thanks to these discrete mathematical tools, Lyapunov type inequalities of the related boundary value problems are obtained‎. ‎Relying on the obtained Lyapunov inequalities, we can study three classes of important qualitative dynamics of related fractional-order difference boundary value problems. The first class is investigating on the nontrivial solutions of the corresponding multi layer half linear fractional $\nabla$ difference problems and the second class is devoted to study on the eigenvalue regions of related eigenvalue problems. At the third step, making use of the obtained Lyapunov type inequalities some nonexistence results are presented‎.

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