Solutions to Riemann–Liouville fractional integrodifferential equations via fractional resolvents

Abstract

This paper is concerned with the semilinear fractional integrodifferential system with Riemann-Liouville fractional derivative. Firstly, we introduce the suitable C 1-α-solution to Riemann-Liouville fractional integrodifferential equations in the new frame of fractional resolvents. Some properties of fractional resolvents are given. Then we discuss the sufficient conditions for the existence of solutions without the Lipschitz assumptions to nonlinear item. Finally, an example on fractional partial differential equations is presented to illustrate our abstract results.

Controllability of nonlinear fractional integrodifferential systems involving multiple delays in control

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Existence and controllability of higher‐order nonlinear fractional integrodifferential systems via fractional resolvent

“…The existence and controllability results for various types of systems are proved by many researchers 21–52,53–61 . In recent years, the controllability properties of Caputo fractional systems have been extensively studied.…”

“…Therefore, to incorporate the memory effect in these systems, a term of integration is added in the differential system, which turns to integrodifferential system. The integrodifferential systems have been broadly applied in viscoelastic mechanics, fluid dynamics, thermoelastic contact, control theory, heat conduction, industrial mathematics, financial mathematics, biological models, chemical kinetics and aerospace systems, and so on (see previous works [16][17][18][19][20][21][22] ).…”

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

“…The most important concepts in fractional calculus are the derivatives and integrals of fractional order, which have been defined by more than one mathematician. [6][7][8][9][10][11][12][13][14][15][16][17] These definitions enable us to construct the partial differential equations of fractional order (FPDEs), which are used to modulate the non-linear phenomena in different fields supported by explaining their behaviors and physical properties. Furthermore, there are many methods to construct solutions for FPDEs.…”

“…Therefore, many researchers have turned their consideration towards studying the applications of fractional calculus, 1–5 which include all areas of sciences and engineering. The most important concepts in fractional calculus are the derivatives and integrals of fractional order, which have been defined by more than one mathematician 6–17 . These definitions enable us to construct the partial differential equations of fractional order (FPDEs), which are used to modulate the non‐linear phenomena in different fields supported by explaining their behaviors and physical properties.…”