Asymptotic periodicity for hyperbolic evolution equations and applications

Andrade, Filipe Moreira de; Cuevas, Claudio; Silva, Clessius; Soto, Herme

doi:10.1016/j.amc.2015.07.046

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…In 2008, Henríquez et al [15] formally introduced the concept of S-asymptotically ω-periodic function, which is a more general approximate period function. Since then, the S-asymptotically periodic functions have been widely studied in fractional evolution equations, and the existence and uniqueness of S-asymptotically ω-periodic solutions have been well studied (see [3,9,10,19,29,32,33,35]). It is not difficult to find that in most of the above work, the Lipschitz-type conditions for nonlinear functions are necessary.…”

Section: Introductionmentioning

confidence: 99%

Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

Liu

Wei

2021

NAMC

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In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

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Section: Introductionmentioning

confidence: 99%

Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

Liu

Wei

2021

NAMC

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“…Remark 1.1. A result similar to Theorem 1.4 was obtained in [13] for obtaining the existence and uniqueness of S-asymptotically !-periodic mild solution to a class of abstract di erential equations.…”

Section: Statement Of Resultsmentioning

confidence: 85%

About a composite fractional relaxation equation via regularized families

2017

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Abstract. This work deals with asymptotic periodicity and compactness for a class of composite fractional relaxation equations. Some di culties arise when the e ect of di erent kinds of nonhomogeneous terms is taken into consideration. To overcome these, we use methods resulting from regularized families and xed point techniques, which are an important tool to study nonlinear phenomena. We can cover a large class of nonlinearities.© 2018 Sharif University of Technology. All rights reserved. Statement of resultsFractional calculus is a eld of mathematical analysis, which deals with the investigation and applications of integration and di erentiation of any order, not necessarily integer. This eld has in recent years become a powerful tool to investigate various concrete problems of mathematical physics. For this reason, there is much interest in developing the qualitative theory of fractional evolution equation, i.e., evolution equations where the integer derivative with respect to time is replaced by a derivative of fractional order (see [1][2][3]). We set up our problem and formulate the obtained results precisely. Let X be an arbitrary Banach space. In this work, we study asymptotic periodicity and compactness properties of solutions for a composite fractional relaxation equation in X. Let us start with the linear case (when = 1 2 corresponds to the Basset problem a classical in uid dynamics)(see and A is a closed linear operator, which is the generator of an (a; k)-regularized family (in this setting, we comment that the notion of (a; k)-regularized families of operators, introduced in [5], includes k-convoluted semigroups, r-times integrated cosine families, and integral resolvent [4,6]) R (t) of bounded linear operators from X into X (see De nition 2.1), with k(t) = e t and a(t) = t E 1;1 ( t), where E ; ( ) denotes the Mittag-Le er function, which is de ned as follows: where H a is a Henkel path, i.e. a contour with starts

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“…Recently, the S-asymptotically ω-periodic functions were formally introduced by Henrquez et al [24] and after that, its wild applications in functional differential equations, integro-differential equations, fractional differential equations and the existence and uniqueness of S-asymptotically ω-periodic solutions to these equations have been well studied. See, for instance, [6,[15][16][17][18]21,23,24,31]. In 2013, Pierri and Rolnik [32] introduced the concept of pseudo S-asymptotically ω-periodic (P SAP ω , in short) functions, which is a natural generalization of S-asymptotically ω-periodic functions.…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader to [14,22,36] and the references therein. In [6] the authors established the sufficient conditions to ensure the existence and uniqueness of pseudo S-asymptotically ω-periodic mild solutions for hyperbolic evolution equations respectively by hyperbolic semigroup and uniformly stable semigroup and considered the P SAP ω -mild solutions in intermediate spaces. Xia [37] discussed the pseudo asymptotically periodic solutions of two-term time fractional differential equations with delay.…”

Section: Introductionmentioning

confidence: 99%