ON THE NONLINEAR GENERALIZED LANGEVIN EQUATION INVOLVING ψ-CAPUTO FRACTIONAL DERIVATIVES

Nguyen Minh Dien and Dang Duc Trong
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Nguyen Minh Dien: Faculty of Education, Thu Dau Mot University, Binh Duong Province, Vietnam
Dang Duc Trong: ��Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam‡Vietnam National University, Ho Chi Minh City, Vietnam

FRACTALS (fractals), 2021, vol. 29, issue 06, 1-20

Abstract: This paper considers the generalized Langevin equation involving ψ-Caputo fractional derivatives in a Banach space. The fractional derivative is generalized from the Caputo derivative (ψ(t) = t), the Caputo–Katugampola (ψ(t) = ÏˆÏ (t) = (tÏ âˆ’ 1)/Ï , the Hadamard derivative (ψ(t) = ψH(t) =ln t). We investigate the existence of mild solutions uψ of the problem, in which the source function is assumed to satisfy some weakly singular conditions. Before proceeding to the main results, we transform the problem into an integral equation. Based on the obtained integral equation, the main results are proved via the nonlinear Leray–Schauder alternatives and Banach fixed point theorems. To prove this end, a new generalized weakly Gronwall-type inequality is established. Further, we prove that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders and the friction constant. As a consequence, we deduce that the solution uÏˆÏ of the equation involving the Caputo–Katugampola derivative tends to the solution uψH of the equation involving the Hadamard derivative as Ï â†’ 0+.

Keywords: Langevin Equation; ψ-Caputo Fractional Derivatives; Weakly Singular Source; Existence and Unique Solution; Continuity of Solution (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1142/S0218348X21501280

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