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We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in

Fractional derivatives and integrals have been vastly used in different fields, facing a huge development especially during the last few decades (see, e.g., [

In particular, fractional differential equations as an important research branch of fractional calculus attracted much more attention (see, e.g., [

Having all the aforementioned facts in mind, in this paper we study the existence and uniqueness of solutions for a class of delayed fractional differential equations, namely,

Here

The notion of the phase space

Our approach is based on the Banach fixed-point theorem and on the nonlinear alternative of Leray-Schauder type [

In Section

In this section, we present some basic notations and properties which are used throughout this paper. First of all, we will explain the phase space

If

There exist functions

There exists a positive constant

The quotient space

See [

Let

The most common notation for

For a function

Let

Let

In this paper we use the alternative Leray-Schauder’s theorem and Banach’s contraction principle for getting the main results. These theorems can be found in [

In this section, we prove the existence results for (

For the existence results on (

Equation (

The proof is an immediate consequence of Proposition

To study the existence and uniqueness of solutions for (

Assume that

It is enough to show that the operator

Let

For any

Let

It is an immediate consequence from (i)–(iii), together with the Arzela-Ascoli theorem.

We show in the following that there exists an open set

Let

The solution of (

Let

There exist

The function

Consider the operator

Let

The unique solution of (

Let

The proof is a similar process Theorem

In this paper, the existence and the uniqueness of solutions for the nonlinear fractional differential equations with infinite delay comprising standard Riemann-Liouville derivatives have been discussed in the phase space. Leray-Schauder’s alternative theorem and the Banach contraction principle were used to prove the obtained results. Further generalizations can be developed to some other class of fractional differential equations such as

The authors would like to thank the referee for helpful comments and suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.