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

• Published in: Advances in difference equations Vol. 2021; no. 1; pp. 1 - 11
• Main Authors: Atraoui, Mustapha, Bouaouid, Mohamed
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 10.10.2021; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Banach spaces; Cauchy problems; Conformable fractional derivative; Cosine family of linear operators; Derivatives; Difference and Functional Equations; Differential equations; Existence theorems; Fixed points (mathematics); Fractional differential equations; Functional Analysis; Mathematical analysis; Mathematics; Mathematics and Statistics; Measure of noncompactness; Nonlocal conditions; Ordinary Differential Equations; Partial Differential Equations; Nonlocal conditions; Measure of noncompactness; Fractional differential equations; Cosine family of linear operators; Conformable fractional derivative; 47D09; 34A08
• ISSN: 1687-1847; 1687-1839; 1687-1847
• DOI: 10.1186/s13662-021-03593-5

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: d α d t α [ d α x ( t ) d t α ] = A x ( t ) + f ( t , x ( t ) ) , t ∈ [ 0 , τ ] subject to the nonlocal conditions x ( 0 ) = x 0 + g ( x ) and d α x ( 0 ) d t α = x 1 + h ( x ) , where d α ( ⋅ ) d t α is the conformable fractional derivative of order α ∈ ] 0 , 1 ] and A is the infinitesimal generator of a cosine family ( { C ( t ) , S ( t ) } ) t ∈ R on a Banach space X . The elements x 0 and 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 and any Lipschitz conditions on the functions g and h .