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
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.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
