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

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.

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

Mathematica Slovaca ◽

10.1515/ms-2017-0250 ◽

2019 ◽

Vol 69 (3) ◽

pp. 599-610 ◽

Cited By ~ 1

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.

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Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

Abstract and Applied Analysis ◽

10.1155/2011/968735 ◽

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.

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Periodic Solutions andS-Asymptotically Periodic Solutions to Fractional Evolution Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2017/1364532 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12 ◽

Cited By ~ 3

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.

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

Journal of Applied Mathematics ◽

10.1155/2012/691651 ◽

2012 ◽

Vol 2012 ◽

pp. 1-13 ◽

Cited By ~ 3

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.

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Asymptotically periodic solutions for Caputo type fractional evolution equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2018-0068 ◽

2018 ◽

Vol 21 (5) ◽

pp. 1294-1312 ◽

Cited By ~ 5

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.

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On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

Mathematics ◽

10.3390/math8040476 ◽

2020 ◽

Vol 8 (4) ◽

pp. 476

Author(s):

Jiraporn Reunsumrit ◽

Thanin Sitthiwirattham

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Boundary Conditions ◽

Difference Equations ◽

Existence And Uniqueness ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach Contraction ◽

The Banach Contraction Principle

In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s fixed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

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Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0228 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Choukri Derbazi ◽

Zidane Baitiche ◽

Akbar Zada

Keyword(s):

Fractional Derivative ◽

Positive Solutions ◽

Caputo Fractional Derivative ◽

Existence And Uniqueness ◽

Relaxation Equation ◽

Monotone Iterative ◽

Banach Contraction ◽

Uniqueness Of The Solution ◽

Fractional Relaxation ◽

The Banach Contraction Principle

Abstract This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

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On Separate Fractional Sum-Difference Equations with n-Point Fractional Sum-Difference Boundary Conditions via Arbitrary Different Fractional Orders

Mathematics ◽

10.3390/math7050471 ◽

2019 ◽

Vol 7 (5) ◽

pp. 471 ◽

Cited By ~ 3

Author(s):

Saowaluck Chasreechai ◽

Thanin Sitthiwirattham

Keyword(s):

Fixed Point ◽

Boundary Conditions ◽

Existence And Uniqueness ◽

Nonlinear Functions ◽

Fractional Difference ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach Contraction ◽

Fractional Orders ◽

The Banach Contraction Principle

In this article, we study the existence and uniqueness results for a separate nonlinear Caputo fractional sum-difference equation with fractional difference boundary conditions by using the Banach contraction principle and the Schauder’s fixed point theorem. Our problem contains two nonlinear functions involving fractional difference and fractional sum. Moreover, our problem contains different orders in n + 1 fractional differences and m + 1 fractional sums. Finally, we present an illustrative example.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500021946 ◽

1987 ◽

Vol 105 (1) ◽

pp. 101-115 ◽

Cited By ~ 9

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.

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Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

Open Mathematics ◽

10.2478/s11533-012-0141-4 ◽

2013 ◽

Vol 11 (3) ◽

Cited By ~ 2

Author(s):

Svatoslav Staněk

Keyword(s):

Differential Equation ◽

Fractional Derivative ◽

Fractional Differential Equation ◽

Existence And Uniqueness ◽

Existence Results ◽

Point Boundary ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Carathéodory Function ◽

Fractional Differential

AbstractWe investigate the fractional differential equation u″ + A c D α u = f(t, u, c D μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.

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