scholarly journals On Fractional Order Hybrid Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Mohamed A. E. Herzallah ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Nonlinear Perturbations ◽  
Linear And Nonlinear ◽  
Hybrid Differential Equations ◽  
Hybrid Equations

We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2019 ◽  
Vol 2 (2) ◽  
pp. 18 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
Chokkalingam Ravichandran ◽  
Shanmugam Dhanalakshmi ◽  
Rangasamy Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Fractional Order ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Order System ◽  
Semi Group ◽  
Non Local

In this paper, we establish the existence of piece wise (PC)-mild solutions (defined in Section 2) for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained using techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we used the distributed characteristic operators to define a mild solution of the system. We also discussed the controversy related to the solution operator for the fractional order system using weak and strong Caputo derivatives. Examples are given to illustrate the theory.

scholarly journals Abstract Semilinear Evolution Equations with Convex-Power Condensing Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Lanping Zhu ◽  
Qianglian Huang ◽  
Gang Li
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Point Theorems ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Condensing Operators ◽  
Semilinear Evolution Equations

By using the techniques of convex-power condensing operators and fixed point theorems, we investigate the existence of mild solutions to nonlocal impulsive semilinear differential equations. Two examples are also given to illustrate our main results.

scholarly journals L -Simulation Functions over b -Metric-Like Spaces and Fractional Hybrid Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shimaa I. Moustafa ◽  
Ayman Shehata
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Unique Solution ◽  
Caputo Fractional Derivative ◽  
Multipoint Boundary Conditions ◽  
Simulation Functions ◽  
Hybrid Differential Equation ◽  
Hybrid Differential Equations

In this paper, we establish some fixed point results for α q s p -admissible mappings embedded in L -simulation functions in the context of b -metric-like spaces. As an application, we discuss the existence of a unique solution for fractional hybrid differential equation with multipoint boundary conditions via Caputo fractional derivative of order 1 < α ≤ 2 . Some examples and corollaries are also considered to illustrate the obtained results.

scholarly journals Impulsive partial hyperbolic differential inclusions of fractional order

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra
Keyword(s):  
Fixed Point ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Caputo Fractional Derivative ◽  
Differential Inclusions ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Hyperbolic Differential Inclusions

AbstractIn this paper we investigate the existence of solutions of a class of partial impulsive hyperbolic differential inclusions involving the Caputo fractional derivative. Our main tools are appropriate fixed point theorems from multivalued analysis.

scholarly journals Applications of Krasnoselskii-Dhage Type Fixed-Point Theorems to Fractional Hybrid Differential Equations

2021 ◽  
Vol 52 ◽  
Author(s):  
Habibulla Akhadkulov ◽  
Fahad Alsharari ◽  
Teh Yuan Ying
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Integral Operators ◽  
Existence Of A Solution ◽  
Hybrid Differential Equation ◽  
Hybrid Differential Equations

In this paper, we prove the existence of a solution of a fractional hybrid differential equation involving the Riemann-Liouville differential and integral operators by utilizing a new version of Kransoselskii-Dhage type fixed-point theorem obtained in [13]. Moreover, we provide an example to support our result.

scholarly journals Stability of the mixed Caputo fractional integro-differential equation by means of weighted space method

2021 ◽  
Vol 7 (2) ◽  
pp. 2498-2511
Author(s):  
Qun Dai ◽  
◽  
Shidong Liu
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Weighted Space ◽  
Research Work ◽  
Banach Fixed Point Theorem ◽  
Space Method

<abstract><p>In this research work, we consider a class of nonlinear fractional integro-differential equations containing Caputo fractional derivative and integral derivative. We discuss the stabilities of Ulam-Hyers, Ulam-Hyers-Rassias, semi-Ulam-Hyers-Rassias for the nonlinear fractional integro-differential equations in terms of weighted space method and Banach fixed-point theorem. After the demonstration of our results, an example is given to illustrate the results we obtained.</p></abstract>

scholarly journals On the existence of mild solutions for nonlocal differential equations of the second order with conformable fractional derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mustapha Atraoui ◽  
Mohamed Bouaouid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Conformable Fractional Derivative ◽  
The Family ◽  
Krasnoselskii Fixed Point Theorem

AbstractIn the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019), the authors have used the Krasnoselskii fixed point theorem for showing the existence of mild solutions of an abstract class of conformable fractional differential equations of the form: $\frac{d^{\alpha }}{dt^{\alpha }}[\frac{d^{\alpha }x(t)}{dt^{\alpha }}]=Ax(t)+f(t,x(t))$ d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , $t\in [0,\tau ]$ t ∈ [ 0 , τ ] subject to the nonlocal conditions $x(0)=x_{0}+g(x)$ x ( 0 ) = x 0 + g ( x ) and $\frac{d^{\alpha }x(0)}{dt^{\alpha }}=x_{1}+h(x)$ d α x ( 0 ) d t α = x 1 + h ( x ) , where $\frac{d^{\alpha }(\cdot)}{dt^{\alpha }}$ d α ( ⋅ ) d t α is the conformable fractional derivative of order $\alpha \in\, ]0,1]$ α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family $(\{C(t),S(t)\})_{t\in \mathbb{R}}$ ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X. The elements $x_{0}$ x 0 and $x_{1}$ x 1 are two fixed vectors in X, and f, g, h are given functions. The present paper is a continuation of the work (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) in order to use the Darbo–Sadovskii fixed point theorem for proving the same existence result given in (Bouaouid et al. in Adv. Differ. Equ. 2019:21, 2019) [Theorem 3.1] without assuming the compactness of the family $(S(t))_{t>0}$ ( S ( t ) ) t > 0 and any Lipschitz conditions on the functions g and h.

scholarly journals Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

Symmetry ◽  
2019 ◽  
Vol 11 (6) ◽  
pp. 829 ◽  
Author(s):  
Piyachat Borisut ◽  
Poom Kumam ◽  
Idris Ahmed ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Integral Condition ◽  
Supremum Norm ◽  
Caputo Fractional Derivatives ◽  
Integral Boundary

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n - 1 < q < n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , ⋯ , n - 2 , i = 1 , 2 , ⋯ , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. R L I 0 + p i is the Riemann–Liouville fractional integral of order p i > 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n - 1 Γ ( n ) Γ ( n + p i ) ≠ T n - 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

scholarly journals Existence of Fractional Impulsive Functional Integro-Differential Equations in Banach Spaces

2018 ◽  
Author(s):  
Dimplekumar Chalishajar ◽  
C. Ravichandran ◽  
S. Dhanalakshmi ◽  
R. Murugesu
Keyword(s):  
Fixed Point ◽  
Group Theory ◽  
Differential Equations ◽  
Banach Spaces ◽  
Mild Solution ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Semi Group ◽  
Finite Delay ◽  
Non Local

In this paper, we established the existence of PC-mild solutions for non local fractional impulsive functional integro-differential equations with finite delay. The proofs are obtained by using the techniques of fixed point theorems, semi-group theory and generalized Bellman inequality. In this paper, we have used the distributed characteristic operators to define the mild solution of the system. Results obtained here improve and extend some known results.

scholarly journals Analysis of boundary value problem with multi-point conditions involving Caputo-Hadamard fractional derivative

2020 ◽  
Vol 39 (6) ◽  
pp. 1555-1575
Author(s):  
Muthaiah Subramanian ◽  
Thangaraj Nandha Gopal
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Boundary Value Problems ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Fractional Differential

We study the boundary value problems (BVPs) of the Caputo-Hadamard type fractional differential equations (FDEs) supplemented by multi-point conditions. Many new results of existence and uniqueness are obtained with the use of fixed point theorems for single-valued maps. With the help of examples, the results are well illustrated.

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