scholarly journals Periodic Solutions andS-Asymptotically Periodic Solutions to Fractional Evolution Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jia Mu ◽  
Yong Zhou ◽  
Li Peng
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic ◽  
Liouville Fractional Derivative ◽  
Asymptotically Periodic Solutions

This paper deals with the existence and uniqueness of periodic solutions,S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

Asymptotically periodic solutions for Caputo type fractional evolution equations

2018 ◽  
Vol 21 (5) ◽  
pp. 1294-1312 ◽  
Author(s):  
Lulu Ren ◽  
JinRong Wang ◽  
Michal Fečkan
Keyword(s):  
Periodic Solutions ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic ◽  
Theoretical Results ◽  
Asymptotically Periodic Solutions

Abstract In this paper, we prove that Caputo type linear fractional evolution equations do not have nonconstant periodic solutions. Then, we study asymptotically periodic solutions of semilinear fractional evolution equations and establish existence and uniqueness results by using theory of semigroup and fixed point theorems. Finally, two examples are given to illustrate the theoretical results.

Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2

2019 ◽  
Vol 69 (3) ◽  
pp. 599-610 ◽  
Author(s):  
Lulu Ren ◽  
Jinrong Wang ◽  
Donal O’Regan
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Fractional Evolution Equations ◽  
Periodic Behavior ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Asymptotically Periodic

Abstract In this paper we investigate the asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2 and in particular existence and uniqueness results are established. Two examples are given to illustrate our results.

scholarly journals Positive Mild Solutions of Periodic Boundary Value Problems for Fractional Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Jia Mu ◽  
Hongxia Fan
Keyword(s):  
Existence And Uniqueness ◽  
Periodic Boundary ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Boundary Value ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Uniqueness Results ◽  
Periodic Boundary Value Problems

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

scholarly journals Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 792-805
Author(s):  
Junfei Cao ◽  
Zaitang Huang
Keyword(s):  
Periodic Solutions ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Nonlocal Initial Conditions ◽  
Asymptotically Periodic ◽  
Semilinear Evolution Equations ◽  
Asymptotically Periodic Solutions

AbstractIn this paper we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of asymptotically periodic mild solutions. As one would expect, the results presented here would generalize and improve some results in this area.

scholarly journals On the Existence and Uniqueness Results for Fuzzy Linear and Semilinear Fractional Evolution Equations Involving Caputo Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Ali El Mfadel ◽  
Said Melliani ◽  
M’hamed Elomari
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Existence Theorems ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Banach Contraction

In this manuscript, we establish new existence and uniqueness results for fuzzy linear and semilinear fractional evolution equations involving Caputo fractional derivative. The existence theorems are proved by using fuzzy fractional calculus, Picard’s iteration method, and Banach contraction principle. As application, we conclude this paper by giving an illustrative example to demonstrate the applicability of the obtained results.

scholarly journals Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
Jia Mu
Keyword(s):  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Monotone Iterative Technique ◽  
Iterative Technique ◽  
Ordered Banach Space ◽  
Fractional Evolution Equations ◽  
Existence And Uniqueness Results ◽  
Monotone Iterative ◽  
Fractional Evolution Equation ◽  
Uniqueness Results

We mainly study the fractional evolution equation in an ordered Banach space , , , . Using the monotone iterative technique based on lower and upper solutions, the existence and uniqueness results are obtained. The necessary perturbation results for accomplishing this approach are also developed.

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

2021 ◽  
Vol 26 (5) ◽  
pp. 928-946
Author(s):  
Qiang Li ◽  
Lishan Liu ◽  
Mei Wei
Keyword(s):  
Evolution Equations ◽  
Mild Solutions ◽  
Upper And Lower Solutions ◽  
Periodic Problem ◽  
Growth Conditions ◽  
Monotone Iterative Method ◽  
Monotone Iterative ◽  
Asymptotically Periodic ◽  
Asymptotically Periodic Solutions ◽  
Ordered Banach Spaces

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.

Semilinear evolution equations and fractional powers of a closed pair of operators

1987 ◽  
Vol 105 (1) ◽  
pp. 101-115 ◽  
Author(s):  
Marié Grobbelaar-Van Dalsen
Keyword(s):  
Existence And Uniqueness ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dynamic Boundary Conditions ◽  
Fractional Powers ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Nonlinear Evolution Problem ◽  
Semilinear Evolution Equations

SynopsisThe nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

scholarly journals Existence and Attractivity for Fractional Evolution Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Yong Zhou ◽  
Jia Wei He ◽  
Bashir Ahmad ◽  
Ahmed Alsaedi
Keyword(s):  
Fractional Derivative ◽  
Evolution Equations ◽  
Global Attractivity ◽  
Mild Solutions ◽  
Sufficient Conditions ◽  
Cauchy Problems ◽  
Fractional Evolution Equations ◽  
Liouville Fractional Derivative

We study the existence and attractivity of solutions for fractional evolution equations with Riemann-Liouville fractional derivative. We establish sufficient conditions for the global attractivity of mild solutions for the Cauchy problems in the case that semigroup is compact.

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