A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

Mikhail M Turov,Bui Trong Kien,Vladimir E Fedorov

doi:10.3390/axioms11030096

Mikhail M Turov, Bui Trong Kien + Show 1 more

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https://doi.org/10.3390/axioms11030096

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Journal: Axioms	Publication Date: Feb 24, 2022
Citations: 5	License type: CC BY 4.0

Affiliation: Chelyabinsk State University, South Ural State University

Abstract

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

Highlights

Axioms 2022, 11, 96. https://doi.org/In recent decades, problems with fractional derivatives have been studied by many authors [1,2,3,4,5]
The purpose of this paper is to study the local unique solvability of initial value problems for multi-term equations in Banach spaces with fractional Riemann–Liouville β β derivatives Dt z, β > 0, fractional Riemann–Liouville integrals Jt z, β ≥ 0, and with nonlinearity, which depends on fractional derivatives of lower orders
Under the condition of Lipschitzian continuity of the nonlinear operator F, using the theorem of contraction mapping for Equation (3), we prove the unique solvability of problem (1), (2) on a small enough interval

Summary

Introduction

Axioms 2022, 11, 96. https://doi.org/In recent decades, problems with fractional derivatives have been studied by many authors [1,2,3,4,5]. Problems with fractional derivatives have been studied by many authors [1,2,3,4,5]. Fractional integro-differential calculus is an important tool in modeling various phenomena that arise in physics, chemistry, mathematical biology, engineering, etc. The purpose of this paper is to study the local unique solvability of initial value problems for multi-term equations in Banach spaces with fractional Riemann–Liouville β β derivatives Dt z, β > 0, fractional Riemann–Liouville integrals Jt z, β ≥ 0, and with nonlinearity, which depends on fractional derivatives of lower orders. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in m −1 published maps and institutional affiliations.

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