^{1}

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We introduce boundary value conditions involving antiperiodic and nonlocal three-point boundary conditions. We solve a nonlinear fractional differential equation supplemented with those conditions. We obtain some existence results for the given problem by applying some standard tools of fixed point theory. These results are well illustrated with the aid of examples.

In recent years, several kinds of boundary value problems of nonlinear fractional differential equations, supplemented with a variety of boundary conditions (including Dirichlet, Neumann, mixed, periodic, antiperiodic, multipoint, integral type, and nonlocal), have been investigated by several researchers. This investigation includes a wide collection of results ranging from theoretical to analytic and numerical methods. For details and examples, see [

In this paper, we study a new class of problems of fractional differential equations supplemented with antiperiodic and three-point nonlocal boundary conditions. Precisely, we consider the following fractional problem:

We emphasize that the second boundary condition in (

In order to define the solutions for the given problem, we consider the following lemma.

Let

It is well known that the solution of equation

Let

In view of Lemma

In this section, we present our main results. The first result relies on classical Banach’s contraction mapping principle.

Let

Then problem (

In the first step, it will be shown that

The next existence result is based on the following Schaefer’s fixed point theorem [

Let

Assume that there exists a positive constant

We first show that the operator

Next, we consider the set

Now we show the existence of solutions for problem (

Suppose that there exist constants

Define a ball

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

Let

For

In view of condition (

Finally, we make use of Leray-Schauder nonlinear alternative to show the existence of solutions for problem (

Let

there is

Let

there exist a function

there exists a constant

where

Then problem (

As a first step, we show that the operator

Next, it will be shown that

Let

Notice that the operator

Consider a three-point boundary value problem of nonlinear fractional differential equations given by

In (

Clearly

Taking

in (

Choosing

in (

The authors declare that there is no conflict of interests regarding the publication of this paper.

Jorge Losada acknowledges financial support by Xunta de Galicia under grant Plan I2C ED481A-2015/272.