Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem

Abstract

We develop a generalized monotone method using coupled lower and upper solutions for Caputo fractional differential equations with periodic boundary conditions of order q, where 0 < q < 1. We develop results which provide natural monotone sequences or intertwined monotone sequences which converge uniformly and monotonically to coupled minimal and maximal periodic solutions. However, these monotone iterates are solutions of linear initial value problems which are easier to compute.

On asymptotic properties of solutions to fractional differential equations

“…Based on arguments as in [25,Theorem 2.3], we obtain the following comparison proposition. Proposition 23.…”

“…Remark 24. Proposition 23 improved [25,Theorem 2.3] in the way that we do not need to require continuous differentiability of m 1 (·), m 2 (·), and Lipschitz property of L(·). This improvement is very useful for our purpose in the next steps.…”

A note on one-dimensional time fractional ODEs

“…In [15,Lemma 2.6], f (t, ·) was assumed to be non-increasing. In [14,Theorem 2.3], there is no assumption on the monotonicity of f (t, ·), but the function v is assumed to be C 1 so that the pointwise value of D γ c v can be defined. Combining these ideas and establishing a crucial lemma (Lemma 2.1), we prove some generalized comparison principles in this section.…”

“…Taking ǫ → 0 then gives the result. [12,14]). Now, we establish a generalized Grönwall inequality (or another version of comparison principle), consistent with the new definition of Caputo derivative.…”

Analysis of Nonlinear Fractional Differential Equations Involving Atangana-Baleanu-Caputo Derivative

“…The following theorem is the comparison result for periodic boundary value problems involving ABC-fractional derivative. The proof of the same one can finish watching the comparable kind of steps of Theorem 2.6 [46].…”