_{1}

In this paper, we discussed the problem of nonlocal value for nonlinear fractional <i>q</i>-difference equation. The classical tools of fixed point theorems such as Krasnoselskii’s theorem and Banach’s contraction principle are used. At the end of the manuscript, we have an example that illustrates the key findings.

Importance of fractional differential equations appears in many of the physical and engineering phenomena in the last two decades [

In this paper, we obtain the results of the existence and uniqueness of solutions for the Cauchy problem with nonlocal conditions for some fractional q-difference equations given by

Here,

are given continuous functions. It is worth mentioning that the nonlocal condition

Several authors have studied the semi-linear differential equations with nonlocal conditions in Banach space, [

(H_{1})

(H_{3})

(H_{4})

The problem (1) is then devolved to the following formula

See reference [

Let

The q-analogue of the Pochhammer symbol was presented as follows

In general, if

The q-gamma function is defined by

and satisfies

The q-derivative of a function

and

The q-integral of a function f defined in the interval

Now, it can be defined an operator

We can point to the basic formula which will be used at a later time,

where

See reference [

Definition 2.1. [

Definition 2.3. [

where

Theorem 2.1. [

Theorem 2.2. [

Let M be a closed convex non-empty subset of a Banach space

1)

2) A is continuous and AM is contained in a compact set;

3) B is a contraction mapping.

Then there exists

Now, the obtained results are presented.

Theorem 3.1.

Let (H_{1})- (H_{3}) hold, if

Proof. Define

Choose

This shows that

Now, for

Thus

where

Thus, by the Banach’s contraction mapping principle, we find that the problem (1) has a unique solution.

Our next results are based on Krasnoselskii’s fixed-point theorem.

Theorem 3.2.

Let (H_{1}), (H_{2}), (H_{3}) with _{4}) hold, then the problem (1) has at least one solution on I.

Proof. Take

Let A and B the two operators defined on P by

and

respectively. Note that if

Thus

By (H_{2}), it is also clear that B is a contraction mapping.

Produced from Continuity of u, the operator _{1}). Also we observe that

Then A is uniformly bounded on P.

Now, let

which is autonomous of u and head for zero as

Example 4.1 Consider the following nonlocal problem

where

Set

and

Let

and

It is obviously that our assumptions in Theorem 3.1 holds with

Therefore the problem (3) has a unique solution on

・ If

・ If

Maryam Al-Yami, (2016) A Cauchy Problem for Some Fractional q-Difference Equations with Nonlocal Conditions. American Journal of Computational Mathematics,06,159-165. doi: 10.4236/ajcm.2016.62017