Abstract
In this article, we employ the lower regularized incomplete gamma functions to demonstrate the existence and uniqueness of solutions for fractional differential equations involving nonlocal fractional derivatives (GPF derivatives) generated by proportional derivatives of the form 1Dρ=(1−ρ)+ρD,ρ∈[0,1],\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ D^{\\rho }= (1-\\rho )+ \\rho D, \\quad \\rho \\in [0,1], $$\\end{document} where D is the ordinary differential operator.
Highlights
Over the last decades there has been an extensive use of fractional dynamic equations in modeling and describing complex and chaotic systems [1,2,3,4,5,6]
Since the appearance of the concept of conformable derivatives, which allow the derivation up to arbitrary order and resemble ordinary derivatives, in [7] and their modifications in [8, 9], several researchers realized that conformable type derivatives can be used to produce nonlocal more generalized fractional derivatives
In [10], the authors used the conformable derivatives presented in [8] to present a class of generalized nonlocal fractional derivatives, called conformable fractional derivatives, slightly different from the so-called Katugampola [12, 13]
Summary
Over the last decades there has been an extensive use of fractional dynamic equations in modeling and describing complex and chaotic systems [1,2,3,4,5,6]. The left fractional derivative of Caputo type of the function ∈ C(n)[a, b] is defined by cD0a (t) = (t) and cDαa (t) = Jan–α (n)(t). The left generalized proportional integral of Riemann–Liouville type of the function ∈ L1[a, b] is defined by (Ja0,ρ )(t) = (t) and. The left generalized proportional derivative of Caputo type of the function ∈ C(n)[a, b] is defined by cD0a,ρ (t) = (t) and cDaα,ρ (t) = Jan–α,ρ Dn,ρ (t). The left generalized proportional derivative of Riemann–Liouville type of the function is defined by RD0a,ρ (t) = (t) and RDaα,ρ (t) = Dn,ρ Jan–α,ρ (t). By using Lemma 2.7, we can write the definition of the left generalized proportional derivative of Caputo type of the function ∈ C(n)[a, b] as follows: cDaα,ρ (t) = ρnJan–α,1 n–1.
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