Cauchy problems involving a Hadamard-type fractional derivative

2018 ◽  
Vol 68 (6) ◽  
pp. 1353-1366
Author(s):  
Rafał Kamocki
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Unique Solution ◽  
Nonlinear Problem ◽  
Point Theorem ◽  
Linear Problem ◽  
Cauchy Formula ◽  
Cauchy Problems

Abstract In this paper, we investigate some Cauchy problems involving a left-sided Hadamard-type fractional derivative. A theorem on the existence of a unique solution to a nonlinear problem is proved. The main result is obtained using a fixed point theorem due to Banach, as well as the Bielecki norm. A Cauchy formula for the solution of the linear problem is derived.

scholarly journals Existence and Uniqueness of Mild Solutions to Impulsive Nonlocal Cauchy Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Mohamed Hannabou ◽  
Khalid Hilal ◽  
Ahmed Kajouni
Keyword(s):  
Fixed Point ◽  
Partial Differential Equations ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Existence And Uniqueness ◽  
Cauchy Problems ◽  
Partial Differential ◽  
Krasnoselskii’S Fixed Point Theorem

In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced. We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.

scholarly journals Existence of solutions for hybrid fractional pantograph equations

2015 ◽  
Vol 9 (1) ◽  
pp. 150-167 ◽  
Author(s):  
Mohamed Darwish ◽  
Kishin Sadarangani
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Existence Of Solutions ◽  
Banach Algebras ◽  
Type Condition ◽  
Measures Of Noncompactness ◽  
Pantograph Equation ◽  
Liouville Fractional Derivative

In this paper, we study the existence of the hybrid fractional pantograph equation {D?0+[x(t)/f(t,x(t),x(?t))= g(t,x(t), x(?t)), 0 < t < 1, x(0) = 0, where ?,?,? ?((0,1) and D?0+ denotes the Riemann-Liouville fractional derivative. The results are obtained using the technique of measures of noncompactness in the Banach algebras and a fixed point theorem for the product of two operators verifying a Darbo type condition. Some examples are provided to illustrate our results.

scholarly journals Controllability Analysis for Impulsive Integro-differential Equation via AB Fractional Derivative

2021 ◽  
Author(s):  
KALIMUTHU KALIRAJ ◽  
E. Thilakraj ◽  
Ravichandran C ◽  
Kottakkaran Nisar
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Integro Differential Equation

In this work, we analyse the controllability for certain classes of impulsive integro - differential equations(IIDE) of fractional order via Atangana Baleanu derivative involving finite delay with initial and nonlocal conditions using Banach fixed point theorem.

scholarly journals Existence of a unique solution for a third-order boundary value problem with nonlocal conditions of integral type

2021 ◽  
Vol 26 (5) ◽  
pp. 914-927
Author(s):  
Sergey Smirnov
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Unique Solution ◽  
Point Theorem ◽  
Boundary Value ◽  
Nonlocal Conditions ◽  
Integral Condition ◽  
Banach Fixed Point Theorem ◽  
Third Order

The existence of a unique solution for a third-order boundary value problem with integral condition is proved in several ways. The main tools in the proofs are the Banach fixed point theorem and the Rus’s fixed point theorem. To compare the applicability of the obtained results, some examples are considered.

scholarly journals FIXED POINT RESULTS IN COMPLEX VALUED METRIC SPACES WITH AN APPLICATION

2021 ◽  
pp. 237
Author(s):  
Rashwan A. Rashwan ◽  
Hasanen A. Hammad ◽  
Liliana Guran
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unique Solution ◽  
Integral Equations ◽  
Point Theorem ◽  
Metric Spaces ◽  
Contractive Condition ◽  
Urysohn Integral Equations ◽  
Complex Valued

In this paper, we introduce fixed point theorem for a general contractive condition in complex valued metric spaces. Also, some important corollaries under this contractive condition areobtained. As an application, we find a unique solution for Urysohn integral equations and some illustrative examples are given to support our obtaining results. Our results extend and generalize the results of Azam et al. [2] and some other known results in the literature.

scholarly journals Existence of Mild Solutions for a Class of Impulsive Hilfer Fractional Coupled Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Karim Guida ◽  
Khalid Hilal ◽  
Lahcen Ibnelazyz ◽  
Ming Mei
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Banach Spaces ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Existence Results ◽  
Coupled Systems ◽  
Hilfer Fractional Derivative

The aim of this paper is to give existence results for a class of coupled systems of fractional integrodifferential equations with Hilfer fractional derivative in Banach spaces. We first give some definitions, namely the Hilfer fractional derivative and the Hausdorff’s measure of noncompactness and the Sadovskii’s fixed point theorem.

scholarly journals On isolated singularities of Kirchhoff equations

2020 ◽  
Vol 10 (1) ◽  
pp. 102-120
Author(s):  
Huyuan Chen ◽  
Mouhamed Moustapha Fall ◽  
Binling Zhang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Unique Solution ◽  
Positive Solutions ◽  
Point Theorem ◽  
Singular Solutions ◽  
Kirchhoff Equations ◽  
Subcritical Case ◽  
Bounded Smooth Domain ◽  
Bounded Smooth

Abstract In this note, we study isolated singular positive solutions of Kirchhoff equation $$\begin{array}{} \displaystyle M_\theta(u)(-{\it\Delta}) u =u^p \quad{\rm in}\quad {\it\Omega}\setminus \{0\},\qquad u=0\quad {\rm on}\quad \partial {\it\Omega}, \end{array}$$ where p > 1, θ ∈ ℝ, Mθ(u) = θ + ∫Ω |∇ u| dx, Ω is a bounded smooth domain containing the origin in ℝN with N ≥ 2. In the subcritical case: 1 < p < $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ if N ≥ 3, 1 < p < + ∞ if N = 2, we employee the Schauder fixed point theorem to derive a sequence of positive isolated singular solutions for the above equation such that Mθ(u) > 0. To estimate Mθ(u), we make use of the rearrangement argument. Furthermore, we obtain a sequence of isolated singular solutions such that Mθ(u) < 0, by analyzing relationship between the parameter λ and the unique solution uλ of $$\begin{array}{} \displaystyle -{\it\Delta} u+\lambda u^p=k\delta_0\quad{\rm in}\quad B_1(0),\qquad u=0\quad {\rm on}\quad \partial B_1(0). \end{array}$$ In the supercritical case: $\begin{array}{} \displaystyle \frac{N}{N-2} \end{array}$ ≤ p < $\begin{array}{} \displaystyle \frac{N+2}{N-2} \end{array}$ with N ≥ 3, we obtain two isolated singular solutions ui with i = 1, 2 such that Mθ(ui) > 0 under other assumptions.

scholarly journals FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS WITH NONLOCAL CONDITIONS AND Ψ–HILFER FRACTIONAL DERIVATIVE

2019 ◽  
Vol 24 (4) ◽  
pp. 564-584 ◽  
Author(s):  
Mohammed S. Abdo ◽  
Satish K. Panchal ◽  
Hussien Shafei Hussien
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Nonlocal Conditions ◽  
Banach Fixed Point Theorem ◽  
Cauchy Type ◽  
Uniqueness Of Solutions ◽  
Hilfer Fractional Derivative ◽  
Krasnoselskii’S Fixed Point Theorem

Considering a fractional integro-differential equation with nonlocal conditions involving a general form of Hilfer fractional derivative with respect to another function. We show that weighted Cauchy-type problem is equivalent to a Volterra integral equation, we also prove the existence, uniqueness of solutions and Ulam-Hyers stability of this problem by employing a variety of tools of fractional calculus including Banach fixed point theorem and Krasnoselskii's fixed point theorem. An example is provided to illustrate our main results.

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.

L 1-solutions of the initial value problems for implicit differential equations with Hadamard fractional derivative

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Benoumran Telli ◽  
Mohammed Said Souid
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Point Theorem ◽  
Initial Value Problems ◽  
Initial Value ◽  
Hadamard Fractional Derivative ◽  
Implicit Differential Equations ◽  
Banach Contraction

Abstract In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

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