Dynamics and stability for Katugampola random fractional differential equations

1 Laboratory of Mathematics, University of Saı̈da–Dr. Moulay Tahar, P. O. Box 138, EN-Nasr, 20000 Saı̈da, Algeria 2 Department of Mathematics, University of Saı̈da–Dr. Moulay Tahar, P. O. Box 138, EN-Nasr, 20000 Saı̈da, Algeria 3 Laboratory of Mathematics, Djillali Liabes University of Sidi Bel-Abbès, P. O. Box 89, Sidi Bel-Abbès 22000, Algeria 4 Departamento de Estatistica, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain


Introduction
The history of fractional calculus dates back to the 17th century. So many mathematicians define the most used fractional derivatives, Riemann-Liouville in 1832, Hadamard in 1891 and Caputo in 1997 [24,28,34]. Fractional calculus plays a very important role in several fields such as physics, chemical technology, economics, biology; see [2,24] and the references therein. In 2011, Katugampola introduced a derivative that is a generalization of the Riemann-Liouville fractional operators and the fractional integral of Hadamard in a single form [21,22].

Preliminaries
By C(I) := C(I, E) we denote the Banach space of all continuous functions x : and L 1 (I, E) denotes the Banach space of measurable function x : I → E with are Bochner integrable, equipped with the norm Let C ς,ρ (I) be the weighted space of continuous functions defined by with the norm Definition 2.1. [2]. The Riemann-Liouville fractional integral operator of the function h ∈ L 1 (I, E) of order ς ∈ R + is defined by Definition 2.2. [2]. The Riemann-Liouville fractional operator of order ς ∈ R + is defined by Definition 2.3. (Hadamard fractional integral) [4]. The Hadamard fractional integral of order r is defined as provided that the left-hand side is well defined for almost every ξ ∈ (0, T ).
Definition 2.4. (Hadamard fractional derivative ) [4]. The Hadamard fractional derivative of order r is defined as provided that the left-hand side is well defined for almost every ξ ∈ (0, T ).
We present in the following theorem some properties of Katugampola fractional integrals and derivatives.
[18] Let X be a nonempty, closed convex bounded subset of the separable Banach space E and let G : Ω × X → X be a compact and continuous random operator. Then the random equation G(w)u = u has a random solution.

Existence and Ulam stability results
We shall make use of the following hypotheses: (H 1 ) f is a random Carathéodory function.
(H 2 ) There exist measurable and essentially bounded functions l i : Ω → C(I); i = 1, 2 such that for all x ∈ E and t ∈ I with Proof. Let N : Ω × C ς,ρ (I) → C ς,ρ (I) be the operator defined by and set and define the ball For any w ∈ Ω and each t ∈ I, we have Hence N(w)(B R ) ⊂ B R . We shall prove that N : Ω × B R → B R satisfies the assumptions of Theorem 2.18.
Step 1. N(w) is a random operator. From (H 1 ), the map w −→ f (t, x, w) is measurable and further the integral is a limit of a finite sum of measurable functions therefore the map Step 2. N(w) is continuous.

Conclusions
In this paper, we provided some sufficient conditions ensuring the existence of random solutions and the Ulam stability for a class of fractional differential equations involving the Katugampola fractional derivative in Banach spaces. The techniques used are the random fixed point theory and the notion of Ulam-Hyers-Rassias stability. 34. Y. Zhou, Basic theory of fractional differential equations, World Scientific, Singapore, 2014.