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In this paper, we discuss the existence and uniqueness of solutions for new classes of separated boundary value problems of Caputo-Hadamard and Hadamard-Caputo sequential fractional differential equations by using standard fixed point theorems. We demonstrate the application of the obtained results with the aid of examples.

Fractional differential equations have been of increasing importance for the past decades due to their diverse applications in science and engineering such as biophysics, bioengineering, virology, control theory, signal and image processing, blood flow phenomena, etc.; see [

Sequential fractional differential equations are also found to be of much interest [

In this paper, we discuss existence and uniqueness of solutions for two sequential Caputo-Hadamard and Hadamard-Caputo fractional differential equations subject to separated boundary conditions as

It can be observed that the sequential Caputo-Hadamard and Hadamard-Caputo fractional differential equations in (

The rest of the paper is arranged as follows. In Section

In this section, we introduce some notations and definitions of fractional calculus [

For an at least

The Riemann-Liouville fractional integral of order

For an at least

The Hadamard fractional integral of order

For

In view of Lemma

Let

In order to define the solution of the boundary value problem (

Let

Taking the Riemann-Liouville fractional integral of order

In the same way, we can prove the following lemma, which concerns a linear variant of problem (

Let

We set some abbreviate notations for sequential Riemann-Liouville and Hadamard fractional integrals of a function with two variables as

In this section, we will use fixed point theorems to prove the existence and uniqueness of solution for problems (

For computational convenience we put

Now, we prove the existence and uniqueness result for problem (

Suppose that

Firstly, we define a ball

Let

Our second existence result is based on Krasnoselskii’s fixed point theorem.

Let

Then there exists

Let

(

If

Let

Now, we will show that

Assume that

The above theorem can be proved by applying Krasnoselskii’s fixed point theorem to the operator

If the operators

If

If

The third existence result will be proved by applying Leray-Schauder nonlinear alternative.

Let

there is a

Let us state and prove the existence theorem.

Suppose that

(

(

Let the operator

The result will be followed from the Leray-Schauder nonlinear alternative if we prove the boundedness of the set of the solutions to equation

Assume that the condition

The next two special cases can be obtained by setting

Let

if

if

In this section, we present some examples to illustrate our results.

Consider the following sequential Caputo-Hadamard fractional differential equations with separated boundary conditions

Here

(i) Let

(ii) Given

Consider the following sequential Hadamard-Caputo fractional differential equations with separated boundary conditions

Here

(i) The function

(ii) Let

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was funded by King Mongkut’s University of Technology North Bangkok, Contract no. KMUTNB-60-ART-105.