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This paper is devoted to the study of the existence and uniqueness of solutions
for

Boundary value problems with nonclassical boundary conditions are often used to take into account some peculiarities of physical, chemical or other processes, which are impossible by applying classical boundary conditions. Nonlocal conditions appear when values of the function on the boundary are connected to values inside the domain. Integral nonlocal boundary conditions can be used when it is impossible to directly determine the values of the sought quantity on the boundary while the total amount or integral average on space domain is known.

Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary value problems occur in the mathematical modeling of a variety of physics processes and have recently received considerable attention. For some recent work on boundary value problems with integral boundary conditions we refer to [

In this paper, we discuss some existence and uniqueness results for boundary value problems of

Next, we extend our discussion to the fractional case by considering the problem consisting of the boundary conditions in (

The paper is organized as follows. In Section

Let

It is well known that the solution of the differential equation in (

Substituting the values of

Let

In view of Lemma

Now we are in a position to present several existence results for the problem (

Let

Let

First we show that the operator

Hence, for

Next, we consider the set

Thus,

Our next existence result is based on Leray-Schauder Nonlinear Alternative [

Let

there is a

Let

there exist a function

there exists a constant

Then, the boundary value problem (

Consider the operator

Thus,

Next, we show that

Obviously, the right-hand side of the above inequality tends to zero independently of

Let

In consequence, we have

In view of

Note that the operator

To prove the next existence result, we need the following fixed point theorem.

Let

Let

Define

Our next existence result is based on Krasnoselskii’s fixed point theorem [

Let

Suppose that

Then, the boundary value problem (

Letting

For

Now, we prove the compactness of the operator

Next, we discuss the uniqueness of solutions for the problem (

Assume that

If

Fixing

Thus, we get

Since

We give another uniqueness result for the problem (

Let

Then, the boundary value problem (

For

By the given condition (

Finally, we discuss the uniqueness of solutions for the problem (

Let

Let

Assume that

Then, the boundary value problem (

We consider the operator

Let

For

Consider the boundary value problem

We find that

(

Since

therefore, by Theorem

(

Choose

then,

Thus, by Theorem

(

Clearly,

Choosing

which implies that

(

We choose

Clearly,

Hence, by Theorem

In this section, we consider a Caputo type fractional analogue of problem (

For an at least

The Riemann-Liouville fractional integral of order

It is well known [

Integrating the second term in (

Replacing

In relation to the problem (

By taking

Assume that

If

The authors are grateful to the referees for a careful reading of the paper and suggesting some useful remarks. This paper was funded by King Abdulaziz University under Grant no. (9/34/Gr). The authors, therefore, acknowledge the technical and financial support of KAU. Bashir Ahmad, Sotiris K. Ntouyas, and Hamed H. Alsulami are Members of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

^{p}-solutions for a class of sequential fractional differential equations