Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions

Abstract

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin’s continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin’s continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results.

New Results Achieved for Fractional Differential Equations with Riemann–Liouville Derivatives of Nonlinear Variable Order

“…Remark 3. Since the operators involved in the description of fractional differential equations of variable nonlinear fractional order have complex properties, the research results in this direction are still limited [23,34,[36][37][38][39][40][41][42][43]. With the proposed new criteria in Theorems 1 and 2, we complement and extend the existence of theoretical results for such initial value problems.…”

“…Recently, Souid et al have intensively contributed to this field [34][35][36][37][38][39][40][41][42]. A series of papers is concerned with the questions of the existence and uniqueness of solutions of different classes of differential systems with fractional derivatives of variable order [34,[36][37][38][39][40][41][42]. Some of our research papers are devoted to the qualitative analysis of such problems.…”

Advances in Boundary Value Problems for Fractional Differential Equations

“…Paper [3] is concerned with the nonlinear boundary value problem for a fractional differential equation of variable order at resonance…”

Problem Involving the Riemann–Liouville Derivative and Implicit Variable-Order Nonlinear Fractional Differential Equations

“…Te existence of the solutions variable-order problems is rarely discussed in literature [27][28][29][30][31][32][33][34][35][36][37]; specifcally, Souid et al [38,39] presented the existence, uniqueness, and stability of solutions to many diferent problems (implicit, thermostat, and resonance).…”