Some probability densities and fundamental solutions of fractional evolution equations

El-Borai, Mahmoud M.

doi:10.1016/s0960-0779(01)00208-9

Cited by 329 publications

(151 citation statements)

References 9 publications

Supporting

Mentioning

149

Contrasting

Unclassified

Order By: Relevance

“…Many physical processes appear to exhibit fractional order behavior that may vary with time or space. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; we only enumerate here the monographs of Kilbas et al [26,27], Diethelm [28], Hilfer [29], Podlubny [30], Miller [31], and Zhou [32] and the papers of Agarwal et al [33,34], Benchohra et al [35,36], El-Borai [37], Lakshmikantham et al [38][39][40][41], Mophou et al [42][43][44][45], N'Guérékata [46], and Zhou et al [47][48][49][50] and the reference therein.…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

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

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

View full text Add to dashboard Cite

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.

show abstract

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

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

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…El-Borai [5][6][7] used a probability density function to obtain the solutions to Cauchy problems for different fractional evolution equations. In 2010, Hernandez et al [8] proved that the concepts of mild solutions of fractional evolution equations considered in some previous papers were not appropriate.…”

Section: Q X(t) = Ax(t) + F T X(t)mentioning

confidence: 99%

Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations

Sin¹,

In²,

Kim

2018

Adv Differ Equ

View full text Add to dashboard Cite

show abstract

“…The following lemma follows from the results in [16][17][18][19][20] and will be used throughout this paper. [21]) The operatorsS q (t) andT q (t) have the following properties:…”

Section: Definition 23 the Caputo Derivative Of Ordermentioning

confidence: 99%

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

2017

View full text Add to dashboard Cite

This paper investigates the approximate controllability for Sobolev type stochastic perturbed control systems of fractional order with fractional Brownian motion and Sobolev fractional stochastic nonlocal conditions in a Hilbert space, A new set of sufficient conditions are established by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is also given to illustrate the obtained theory.

show abstract

Some probability densities and fundamental solutions of fractional evolution equations

Cited by 329 publications

References 9 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

Existence and uniqueness of mild solutions to initial value problems for fractional evolution equations

Stochastic fractional perturbed control systems with fractional Brownian motion and Sobolev stochastic non local conditions

Contact Info

Product

Resources

About