scholarly journals Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Valentin Keyantuo ◽  
Carlos Lizama ◽  
Mahamadi Warma
Keyword(s):  
Existence Of Solutions ◽  
Mild Solutions ◽  
Explicit Representation ◽  
First Order ◽  
Order Case ◽  
Unique Mild Solution ◽  
Positivity Of Solutions ◽  
Fractional Differential ◽  
Time Fractional Differential Equations ◽  
Semilinear Problem

We investigate mild solutions of the fractional order nonhomogeneous Cauchy problemDtαu(t)=Au(t)+f(t),  t>0, where0<α<1.WhenAis the generator of aC0-semigroup(T(t))t≥0on a Banach spaceX, we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary conditionu(0)=u(1)admits a unique mild solution for eachf∈C([0,1];X)if and only if the operatorI-Sα(1)is invertible. Here, we use the representationSα(t)x=∫0∞‍Φα(s)T(stα)x ds,  t>0in whichΦαis a Wright type function. For the first order case, that is,α=1, the corresponding result was proved by Prüss in 1984. In caseXis a Banach lattice and the semigroup(T(t))t≥0is positive, we obtain existence of solutions of the semilinear problemDtαu(t)=Au(t)+f(t,u(t)),t>0,0<α<1.

scholarly journals On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Ahmed Alsaedi ◽  
Dumitru Baleanu ◽  
Sina Etemad ◽  
Shahram Rezapour
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Continuous Functions ◽  
Coupled Systems ◽  
Fractional Differential Inclusions ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

The existence of solutions for a coupled system of time-fractional differential equations including continuous functions and the Caputo-Fabrizio fractional derivative is examined. After that we investigated a coupled system of time-fractional differential inclusions including compact- and convex-valuedL1-Caratheodory multifunctions and the Caputo-Fabrizio fractional derivative.

scholarly journals Existence of solutions for some nonlinear delay integrodifferential equations with nonlocal initial conditions

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
Zuomao Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence Of Solutions ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Integrodifferential Equations ◽  
Resolvent Operators ◽  
First Order ◽  
Nonlocal Initial Conditions ◽  
Nonlinear Delay

AbstractThe main purpose of this paper is the existence of mild solutions for a class of first-order nonlinear delay integrodifferential equations with nonlocal initial conditions in Banach spaces. We show that the solutions are given by the application of the theory of resolvent operators and the Sadovskii’s fixed point theorem. An example is presented in the end to show the applications of the obtained results.

On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 451-460 ◽  
Author(s):  
Mohammed Belmekki ◽  
Kheira Mekhalfi
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Kuratowski Measure Of Noncompactness ◽  
State Dependent ◽  
State Dependent Delay ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals Positive Solvability for Conjugate Fractional Differential Inclusion of (k,n-k) Type without Continuity and Compactness

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 170
Author(s):  
Ahmed Salem ◽  
Aeshah Al-Dosari
Keyword(s):  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Inclusion Type ◽  
Fractional Differential Inclusion ◽  
Fractional Differential

The monotonicity of multi-valued operators serves as a guideline to prove the existence of the results in this article. This theory focuses on the existence of solutions without continuity and compactness conditions. We study these results for the (k,n−k) conjugate fractional differential inclusion type with λ>0,1≤k≤n−1.

scholarly journals On resonant mixed Caputo fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Assia Guezane-Lakoud ◽  
Adem Kılıçman
Keyword(s):  
Differential Equation ◽  
Fractional Derivatives ◽  
Existence Of Solutions ◽  
Coincidence Degree ◽  
Degree Theory ◽  
Caputo Fractional Derivatives ◽  
At Resonance ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Problem At Resonance

Abstract The purpose of this study is to discuss the existence of solutions for a boundary value problem at resonance generated by a nonlinear differential equation involving both right and left Caputo fractional derivatives. The proofs of the existence of solutions are mainly based on Mawhin’s coincidence degree theory. We provide an example to illustrate the main result.

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