We study existence and uniqueness results for Caputo fractional sum-difference equations with fractional sum boundary value conditions, by using the Banach contraction principle and Schaefer’s fixed point theorem. Our problem contains different numbers of order in fractional difference and fractional sums. Finally, we present some examples to show the importance of these results.

In this paper we consider a Caputo fractional sum-difference equation with nonlocal fractional sum boundary value conditions of the form

Mathematicians have employed this fractional calculus in recent years to model and solve various applied problems. In particular, fractional calculus is a powerful tool for the processes which appear in nature, for example, biology, ecology, and other areas, and can be found in [

At present, there is a development of boundary value problems for fractional difference equations which shows an operation of the investigative function. The study may also have another function which is related to the one we are interested in. These creations are incorporating with nonlocal conditions which are both extensive and more complex, for instance.

Agarwal et al. [

Kang et al. [

Sitthiwirattham [

The plan of this paper is as follows. In Section

In the following, there are notations, definitions, and lemmas which are used in the main results.

One defines the generalized falling function by

Assume that the following factorial functions are well defined:

If

For

For

Assume that

The following lemma deals with linear variant of boundary value problem (

Let

Using Lemma

Substituting

Now we are in a position to establish the main results. First, we transform boundary value problem (

For

It is easy to see that problem (

Assume that

where

We will show that

Consequently,

The following result is based on Schaefer’s fixed point theorem.

A set of function in

If a set is closed and relatively compact then it is compact.

Assume that X is a Banach space and that

Assume that

We will use Schaefer’s fixed point theorem to prove this result. Let

Let

In this section, in order to illustrate our results, we consider some examples.

Consider the following fractional sum boundary value problem:

Here

Since

Also, we have

We can show that

Consider the following fractional sum boundary value problem:

Here

Clearly for

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

The authors would like to thank the editor and the referees for their useful comments. This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-GOV-58-50).