Periodic boundary value problems for higher‐order fractional differential systems

Abstract

Approximation of solutions of fractional differential systems (FDS) of higher orders is studied for periodic boundary value problem (PBVP). We propose a numerical‐analytic technique to construct a sequence of functions convergent to the limit function, which is a solution of the given PBVP, if the corresponding determined equation has a root. We also study scalar fractional differential equations (FDE) with asymptotically constant nonlinearities leading to Landesman‐Lazer–type conditions.

On One Interpolation Type Fractional Boundary-Value Problem

“…Since the approach of the numerical-analytic method [32] was appied to the fractional differential systems for the first time in [25][26][27][28], it is resonable to give an overview of the results that will allow the reader to follow and will open up possible perspectives for future research in this direction.…”

“…As a generalization of the aforementioned in Sections 3.1 and 3.2 problems, in [27] we studied the generalized fractional differential system…”

“…i.e., instead of the original three-point restrictions (25), we study a problem with some parametrized two-point constraints (27).…”

“…Remark 5. Note, that under additional conditions (26) the parametrized FBVP (24), (27) is equivalent to the original BVP of the interpolation type (24), (25).…”

“…Recently, in the series of papers (see discussions [25][26][27][28]), a completely new approach for study of a class of fractional periodic and anti-periodic boundary-value problems was developed. To construct the approximate solutions of the studied problems, the numerical-analytic method, based on the successive iterations, was used.…”