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Existence and uniqueness of integrable solutions to fractional Langevin equations involving two fractional orders with initial value problems

Choukri Derbazi, Hadda Hammouche (2021)

Mathematica Bohemica

We study the existence and uniqueness of integrable solutions to fractional Langevin equations involving two fractional orders with initial value problems. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem. Examples are provided to illustrate the main results.

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Małgorzata Klimek, Marek Błasik (2012)

Open Mathematics

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary...

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In the present paper the author discusses certain multiple integrals I ( u ) of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals ( u ) , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals I ( u ) and ( u ) are reduced to simpler form H ( v ) and ( v ) to which the existence theorems above apply. Thus, we derive that I ( u ) ( u ) , H ( v ) ( v ) , we obtain the existence of the absolute minimum for the Serrin forms ( u ) and ( v ) , and...

Existence of positive solution of a singular partial differential equation

Shu Qin Zhang (2008)

Mathematica Bohemica

Motivated by Vityuk and Golushkov (2004), using the Schauder Fixed Point Theorem and the Contraction Principle, we consider existence and uniqueness of positive solution of a singular partial fractional differential equation in a Banach space concerning with fractional derivative.

Existence of solutions for impulsive fractional partial neutral integro-differential inclusions with state-dependent delay in Banach spaces

Zuomao Yan, Hongwu Zhang (2014)

Annales Polonici Mathematici

We study the existence of mild solutions for a class of impulsive fractional partial neutral integro-differential inclusions with state-dependent delay. We assume that the undelayed part generates an α-resolvent operator and transform it into an integral equation. Sufficient conditions for the existence of solutions are derived by means of the fixed point theorem for discontinuous multi-valued operators due to Dhage and properties of the α-resolvent operator. An example is given to illustrate the...

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