Almost automorphic mild solutions to fractional differential equations

Araya, Daniela; Lizama, Carlos

doi:10.1016/j.na.2007.10.004

Cited by 175 publications

(69 citation statements)

References 13 publications

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“…The study of almost periodic and almost automorphic type solutions to fractional differential equations was initiated by Araya and Lizama [11]. In their work, the authors investigated the existence and uniqueness of an almost automorphic mild solution of the semilinear fractional differential equation…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…However, in many papers (for instance, [11,[51][52][53][54][55][56][57][58][59][60][61][62][63][64]) on almost periodic type and almost automorphic type solutions to fractional differential equations, to be able to apply the well-known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearity of corresponding fractional differential equations is needed. As can be seen, our results generalize those as well as related research and have more broad applications.…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…Since then, this pioneer work has attracted more and more attention and has been substantially extended in several different directions. Many authors have made important contributions to this theory (see, for instance, [5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein). Especially, in [5,6], the authors gave an important overview about the theory of almost automorphic functions and their applications to differential equations.…”

Section: Introductionmentioning

confidence: 99%

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Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

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Section: International Journal Of Differential Equationsmentioning

confidence: 99%

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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“…In their turn, mathematical aspects of studies on FDC were discussed by several authors. For those, can be also seen the works in [3,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29].…”

Section: Introduction Definitions Notations and Motivationmentioning

confidence: 99%

A Few Complex Equations Constituted by an Operator Consisting of Fractional Calculus and Their Consequences

Irmak

Frasin

2014

Chinese Journal of Mathematics

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“…[1], [4], [9], [11], [8], [19], [21] the survey paper [7] and references therein. Our motivation to study equation (1.1) comes from recent investigations where a related class appears in connection with partial differential equations and Cauchy-time processes, a type of iterated stochastic process (see [3]).…”

Section: Introductionmentioning

confidence: 99%

An operator theoretical approach to a class of fractional order differential equations

Lizama

2011

Applied Mathematics Letters

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Abstract. We propose a general method to obtain the representation of solutions for linear fractional order differential equations based on the theory of (a, k)-regularized families of operators. We illustrate the method in case of the fractional order differential equationwhere A is an unbounded closed operator defined on a Banach space X and f is a X-valued function. IntroductionWe study in this paper existence of solutions for fractional order differential equations of the form Fractional order differential equations is a subject of increasing interest in different contexts and areas of research, see e. Note that when A = 2∆− 2 ∆ 2 (where ∆ is the Laplace operator) α = 1 and µ = 1/ 2 the above equation was recently considered by Nane [18, Theorem 2.2]. In particular, in case µ = 0, α = 1, A = −∆ 2 and u (0) = − 2 π ∆u 0 with u 0 ∈ D(∆), the equation, t > 0, has been studied in [18, Theorem 2.1] in connection with PDE's and iterated processes. A precise interplay between entire and fractional order differential equations was investigated in reference [10].Observe that one cannot apply semigroup theory directly to solve problem (1.1) in terms of a variation of constant formula. However, our methods based on the theory of (a, k)-regularized families allows us to construct a solution. In fact, we will show that it is possible to give an abstract operator approach to equation (1.1) by defining an ad-hoc family of strongly continuous operators. Then, we are able to show that the solution of equation (1.1) can be written in terms of a kind of variation of constants formula (cf. Theorem 3.1 below). We believe that the method indicated in this paper can be used to handle many classes of linear fractional order differential equations. Our method can be viewed as an extension of the ideas in reference [4] to state the existence of solutions for the abstract fractional order Cauchy problem.Our plan is as follows: In section 2, we introduce some preliminaries on fractional order derivatives, the Mittag-Leffler function and the concept of (α, µ)-regularized families, which give us the necessary 2000 Mathematics Subject Classification. 26A33, 47D06, 45N05.

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Almost automorphic mild solutions to fractional differential equations

Cited by 175 publications

References 13 publications

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

A Few Complex Equations Constituted by an Operator Consisting of Fractional Calculus and Their Consequences

An operator theoretical approach to a class of fractional order differential equations

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