Abstract
The aim of this paper is to study the existence and uniqueness of periodic solutions for a certain type of nonlinear fractional pantograph differential equation with a psi -Caputo derivative. The proofs are based on the coincidence degree theory of Mawhin. To show the efficiency of the results, some illustrative examples are included.
Highlights
IntroductionNonlinear fractional differential equations (NFDEs) have been the focus of many studies due to the intensive development of the theory of fractional calculus and to their frequent applications in many areas such as mechanics, physics, chemistry, engineering, and many other scientific disciplines [15,16]
In last few decades, nonlinear fractional differential equations (NFDEs) have been the focus of many studies due to the intensive development of the theory of fractional calculus and to their frequent applications in many areas such as mechanics, physics, chemistry, engineering, and many other scientific disciplines [15,16].Recently, many definitions and results about fractional derivatives and integrals operators have been generalized [1,2,4,17,22]
We study the nonlinear pantograph fractional equation with ψ-Caputo fractional derivative cDα0+;ψ u(t) = h (t, u(t), u(εt)), t ∈ J := [0, b], (1)
Summary
Nonlinear fractional differential equations (NFDEs) have been the focus of many studies due to the intensive development of the theory of fractional calculus and to their frequent applications in many areas such as mechanics, physics, chemistry, engineering, and many other scientific disciplines [15,16]. Several researchers have investigated some new existence and uniqueness results for NFDE pantograph models and others by applying fixed point theorems, the nonlinear alternative on cones, or coincidence degree theory [3,8,9,10,11,12]. In [23], Shah et al studied a class of ψ-Caputo fractional pantograph equations with nonlocal boundary conditions nu(0) + mu(b) = c, where n, m, and c are real constants with n + m = 0, and obtained some existence and uniqueness results by using the Banach contraction theorem and Schaefer’s fixed point theorem. We construct a suitable operator and use the coincidence degree theory of Mawhin [14] to study the existence of solutions for NFDEs (1) with periodic boundary conditions (2).
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