Sequential evolution conformable differential equations of second order with nonlocal condition

Advances in Mathematical Physics

Hilal

2022

Self Cite

In many research works Bouaouid et al. have proved the existence of mild solutions of an abstract class of nonlocal conformable fractional Cauchy problem of the form: d α x t / d t α = A x t + f t , x t , x 0 = x 0 + g x , t ∈ 0 , τ . The present paper is a continuation of these works in order to study the controllability of mild solution of the above Cauchy problem. Precisely, we shall be concerned with the controllability of mild solution of the following Cauchy problem d α x t / d t α = A x t + f t , x t + B u t , x 0 = x 0 + g x , t ∈ 0 , τ , where d α . / d t α is the vectorial conformable fractional derivative of order α ∈ 0 , 1 in a Banach space X and A is the infinitesimal generator of a semigroup T t t ≥ 0 on X . The element x 0 is a fixed vector in X and f , g are given functions. The control function u is an element of L 2 0 , τ , U with U is a Banach space and B is a bounded linear operator from U into X .

Section: Introductionmentioning

confidence: 99%

Controllability of Mild Solution of Nonlocal Conformable Fractional Differential Equations

Hannabou

Advances in Mathematical Physics

Hilal

2022

Self Cite

“…In consequence, many researchers have paid attention to propose new fractional derivatives in order to deal better with modeling of evolutionary phenomena [23,24]. In the work [23], the authors proposed the so-called fractional conformable derivative, which quickly became the subject of many research papers [1,. For example, in [1] the authors proved the existence of mild solutions for the following nonlocal conformable fractional Cauchy problem:…”

Section: Introductionmentioning

confidence: 99%

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

Atraoui

2021

Adv Differ Equ

Self Cite

In 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.

“…is new fractional derivative quickly becomes the subject of many contributions in several areas of science [11][12][13][14][15][16][17][18][19][20][21][22]. Motivated by the better effect of the fractional derivative and simple properties of the conformable fractional derivative, we consider model (1) in the framework of conformable fractional calculus.…”

Section: Introductionmentioning

confidence: 99%

Nonlocal Conformable-Fractional Differential Equations with a Measure of Noncompactness in Banach Spaces

Hannabou

Hilal

2020

Journal of Mathematics

Self Cite

is paper deals with the existence of mild solutions for the following Cauchy problem:where d α (.)/dt α is the so-called conformable fractional derivative. e linear part A is the infinitesimal generator of a uniformly continuous semigroup (T(t)) t≥0 on a Banach space X, f and g are given functions. e main result is proved by using the Darbo-Sadovskii fixed point theorem without assuming the compactness of the family (T(t)) t>0 and the Lipshitz condition on the nonlocal part g.