ABSTRACT

In this paper, we investigate the existence of positive solutions for a class of nonlinear q-fractional boundary value problem. By using some fixed point theorems on cone, some existence results of positive solutions are obtained.

1. Introduction

Considering the following boundary value problem of nonlinear fractional q-difference equation:

(P)

where is a nonnegative continuous function and is the fractional q-derivative of the Riemann-Liouville type.

Fractional differential calculus is a discipline to which many researchers are dedicating their time, perhaps because of its demonstrated applications in various fields of science and engineering [1]. Recently, there are many papers dealing with the boundary value problem of fractional differential equations, see [2-5] and references therein.

The q-difference calculus or quantum calculus is an old subject that was initially developed by Jackson [6,7], and basic definitions and properties of q-difference calculus can be found in [8]. The fractional q-difference calculus had its origin in the works by Al-Salam [9] and Agarwal [10]. More recently, maybe due to the explosion in research within the fractional differential calculus setting, new developments in this theory of fractional qdifference calculus were made, see [11,12].

The question of the existence of positive solutions for fractional q-difference boundary value problems is in its infancy, see [13-16]. No contributions exist, as far as we know, concerning the existence of positive solutions for problem (P).

This paper is organized as follows. In Section 2, some preliminaries are presented. In Section 3, we discuss the existence of positive solutions for problem (P).

2. Preliminaries

Let and define

(2.1)

The q-analogue of the power function with is

(2.2)

More generally, if, then

(2.3)

Note that, if then. The q-gamma function is defined by

(2.4)

and satisfies The q-derivative of a function is here defined by

(2.5)

and q-derivative of higher order by

(2.6)

The q-integral of a function defined in the interval is given by

(2.7)

If and defined in the interval, its integral from to is defined by

(2.8)

Remark 2.1. (see [17]) If and on, then

Similarly as done for derivatives, an operator can be defined, namely,

(2.9)

The fundamental theorem of calculus applies to these operators and, i.e.,

(2.10)

and if is continuous at, then

(2.11)

Basic properties of the two operators can be found in [14]. We now point out three formulas that will be used later (denotes the derivative with respect to variable)

Remark 2.2. (see [14]) We note that if and, then

(2.12)

Definition 2.3. (see [10]) Let and be a function defined on. The fractional q-integral of the Riemann-Liouville type is and

(2.13)

Definition 2.4. (see [14-16]) The fractional q-derivative of the Riemann-Liouville type of order is defined by and

(2.14)

where is the smallest integer greater than or equal to.

Next, we list some properties that are already known in the literature.

Lemma 2.5. (see [14-16]) Let and be a function defined on, Then, the next formulas hold:

1)

2)

Lemma 2.6. (see [14-16]) Let and be a positive integer. Then, the following equality holds:

(2.15)

Let, in view of Lemma 2.5 and Lemma 2.6, we see that

for some constants Using the boundary condition we have Differentiating both side of the above equality, one gets

(2.16)

Using the boundary condition, we have similarly, we have From

(2.17)

and boundary value problem, one can obtain

(2.18)

Putting all things together we finally have

(2.19)

If we define a function by

(2.20)

Hence, in order to solve the problem (P), it is sufficient to find positive solutions of the following integral equation

(2.21)

Some properties of the function needed in the sequel are now stated and proved.

Lemma 2.7. Function defined above satisfies the following conditions:

(2.22)

(2.23)

Proof. Let

(2.24)

and

(2.25)

It is clear that. Now,. For, in view of (2.3) and Remark 2.2, we have

(2.26)

Therefore,. Moreover,

(2.27)

i.e., is an increasing function of x. Obviously, is increasing in x, therefore, is an increasing function of x for fixed This concludes the proof of (2.22).

Suppose now that then

(2.28)

If, then

(2.29)

and this finishes the proof of (2.23).

Let be the Banach space endowed with norm Define the cone by

It follows from the non-negativeness and continuity of and that the operator defined by

(2.30)

is completely continuous [18]. Moreover, for in view of (2.22) and (2.23), we have on and

(2.31)

that is

Lemma 2.8. (see [19]) Let be a Banach space, a cone, and two bounded open balls of centered at the origin with. Suppose that is a completely continuous operator such that either 1)

or 2) holds. Then has a fixed point in

3. Main Results

Let where will be defined later.

Theorem 3.1. Suppose that is a nonnegative continuous function on. In addition, suppose that one of the following two conditions holds:

(H1)

(H2)

Then problem (P) has at least one positive solution.

Proof. Note that the operator is completely continuous. Now, assume that condition (H1) holds. Since

, there exists an such that

(3.1)

where the constant such that

(3.2)

Thus,

This, together with the definitions of and Lemma 2.7, implies that for any

(3.3)

That is, for any

On the other hand, fromit follows that there exists a such that

(3.4)

where the constant satisfies

(3.5)

Let. Then we have

(3.6)

Let and

, then

(3.7)

Thus, the operator satisfies condition of Lemma 2.8. Consequently, the operator has at least one fixed point, which is one positive solution of the problem (P).

Next, we suppose that condition (H2) holds. The proof is similar to that of the case in which (H1) holds and will only be sketched here. Let with. Select two positive constants with and, respectively. Then, there exist two positive constants and such that

(3.8)

(3.9)

It follows from that for,

(3.10)

In addition, let. If, then

(3.11)

and

(3.12)

In view of Remark 2.1, for, we have

(3.13)

Thus, for. Consequently, the operator has at least one fixed point , which is one positive solution of the problem (P).

Example 3.2

where Obviously,

Thus, by the first part of Theorem 3.1, we can get that the problem above has at least one positive solution.

REFERENCES

NOTES

^*This work was supported by the Natural Science Foundation of Hunan Province (12JJ9001), Hunan Provincial Science and Technology Department of Science and Technology Project (2012SK3117) and Construct program of the key discipline in Hunan Province.