On fractional resolvent operator functions

Chuang, Chi-Tai; Li, Miao

doi:10.1007/s00233-009-9184-7

Cited by 73 publications

(43 citation statements)

References 14 publications

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“…and using the fact that R(t)R(s) = R(s)R(t) for all t, s ≥ 0 (which can be proved similarly as that of α-times resolvent family in [8]), one gets …”

Section: The Algebraic Equationmentioning

confidence: 82%

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On a class of time-fractional differential equations

Kostić

et al. 2012

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“…and using the fact that R(t)R(s) = R(s)R(t) for all t, s ≥ 0 (which can be proved similarly as that of α-times resolvent family in [8]), one gets …”

Section: The Algebraic Equationmentioning

confidence: 82%

“…In particular, we show that if the Cauchy problem (1.2) admits an analytic solution families, then the problem (0.1) is also well-posed. In Section 4 we derive the algebraic equation for such resolvent families, similarly as we did in [8].…”

Section: Introductionmentioning

confidence: 99%

On a class of time-fractional differential equations

Kostić

et al. 2012

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“…We establish the sufficient conditions for generation of global α-times C-regularized resolvent families (α > 1) in SCLCSs. In contrast to the recent papers of C. Chen, M. Li [3] and M. Li, C.-C. Li, F.-B. Li [13], the constructed resolvent families need not be exponentially equicontinuous, so that there is a certain novelty value in our approach.…”

Section: Introductionmentioning

confidence: 93%

Abstract time-fractional equations: Existence and growth of solutions

Kostić

2011

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We contribute to the existence theory of abstract time-fractional equations by stating the sufficient conditions for generation of not exponentially bounded α-times C-regularized resolvent families (α > 1) in sequentially complete locally convex spaces. We also consider the growth order of constructed solutions.MSC 2010 : Primary 47D06: Secondary 47D09, 47D60, 47D62, 47D99 Key Words and Phrases: abstract time-fractional equations, α-times C-regularized resolvent families IntroductionThroughout this paper, we assume that E is a Hausdorff sequentially complete locally convex space, SCLCS for short, and that the abbreviation stands for the fundamental system of seminorms which defines the topology of E. By L(E) is denoted the space of all continuous linear mappings from E into E. Henceforth A is a closed linear operator acting on E, L(E) C is an injective operator which satisfies CA ⊆ AC, and the convolution like mapping * is given by f * g(t) : Then (S α (t)) t≥0 is said to be locally equicontinuous, resp. exponentially equicontinuous, if for each T > 0, the family {S α (t) : t ∈ [0, T ]} is equicontinuous, resp. if there exists ω ≥ 0 such that the family {e −ωt S α (t) : t ≥ 0} is equicontinuous; if C = I, then it is also said that (S α (t)) t≥0 is an α-times regularized resolvent family having A as a subgenerator.It is worthwhile to mention that the local equicontinuity of (S α (t)) t≥0 automatically holds if E is a barrelled space ([9]).Let α > 0, let β > 0 and let the Mittag-Leffler functionThen it is well known ([2], [5]) that, for every α > 1, there exists c α ≥ 1 such that:

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“…Peng and Li [10] proved that the α-order fractional semigroup is closely related to the solution operator of the Caputo fractional abstract Cauchy problem (1.1) (for the definition of solution operator, we refer to [2]). Moreover, the equality (1.7) is equivalent to the following equality In the special case when T (·) is exponentially bounded (hence it is Laplace transformable), taking Laplace transform on both sides of (1.5) with respect to s and t, we derive that…”

Section: Theorem 11 Assume That a Is The Generator Of The A-order Fmentioning

confidence: 99%