Some Classes of Function Spaces, Their Properties, and Their Applications 2014View this Special Issue
Positive Solutions for a Class of Singular Boundary Value Problems with Fractional -Difference Equations
We discuss a class of singular boundary value problems of fractional -difference equations. Some existence and uniqueness results are obtained by a fixed point theorem in partially ordered sets. Finally, we give an example to illustrate the results.
In recent years, many papers on fractional differential equations have appeared, because of their demonstrated applications in various fields of science and engineering; see [1–11] and the references therein. Based on the increasingly extensive application of discrete fractional calculus and the development of -difference calculus or quantum calculus (see [12–19] and the references therein), fractional -difference equations have attracted the attention of researchers for the numerous applications in a number of fields such as physics, chemistry, aerodynamics, biology, economics, control theory, mechanics, electricity, signal and image processing, biophysics, blood flow phenomena, aerodynamics, and fitting of experimental data; see [20–23]. Some recent work on the existence theory of fractional -difference equations can be found in [24–29]. However, the study of singular boundary value problems (BVPs) with fractional -difference equations is at its infancy and lots of work on the topic should be done.
Recently, in , Ferreira has investigated the existence of positive solution for the following fractional -difference equations BVP by applying a fixed point theorem in cones.
More recently, in , Caballero et al. have studied positive solutions for the following BVP: by fixed point theorem in partially ordered sets.
Motivated by the work above, in this paper, we discuss the existence and uniqueness of solutions for the singular BVPs of factional -difference equations given by where and is continuous with (i.e., is singular at ).
2. Preliminary Results
For convenience, we present some definitions and lemmas which will be used in the proofs of our results.
Let and define
The -analogue of the power function with is More generally, if , then Note that if then . The -gamma function is define by and it satisfies .
Following, let us recall some basic concepts of -calculus .
Definition 1. For , we define the -derivative of a real-value function as Note that .
Definition 2. The higher order -derivatives are defined inductively as For example, , where is a positive integer and the bracket . In particular, .
Definition 3. The -integral of a function in the interval is given by If and is defined in the interval , its integral from to is define by Similarly as done for derivatives, an operator can be define, namely, Observe that and if is continuous at , then .
We now point out three formulas ( denotes the derivative with respect to variable )
Remark 4. We note that if and , then .
Definition 5 (see ). Let and be a function defined on . The fractional -integral of the Riemann-Liuville type is and
Definition 6 (see ). The fractional -derivative of the Riemann-Liuville type of is defined by and where is the smallest integer greater than or equal to .
Lemma 8 (see ). Let and let be a positive integer. Then, the following equality holds:
Lemma 9. Let and ; then the BVP has a unique solutionwhere
Proof. By Lemmas 7 and 8, we see thatwhere , and are some constants to be determined. Since , we must have . Now, differentiating both sides of (22) and using (15), we obtain Using and , we must set , and Finally, we obtain The proof is complete.
Lemma 10. Function defined above satisfies the following conditions:(i) is a continuous function on ;(ii) for .
Proof. (i) Obviously, is continuous on .
(ii) Let It is clear that , for . Now, in view of Remark 4, for Therefore, . This proof is complete.
By we denote the class of those functions satisfying the following condition; implies .
Theorem 11 (see ). Let be a partially ordered set and suppose that there exists a metric in such that is a complete metric space. Let be a nondecreasing mapping such that there exists an element with . Suppose that there exists such thatAssume that either is continuous or is such that Besides if then has a unique fixed point.
Let be the Banach space with the classic metric given by .
3. Main Result
In this section, we will consider the question of positive solutions for BVP (3). At first, we prove some lemmas required for the main result.
Lemma 12. Let , and is a continuous function with . Suppose that is a continuous function on . Then the function defined by is continuous on , where is Green function be given in Lemma 9.
Proof. We will divide the proof into three parts.
Case 1 (). First, . Since is continuous on , we can find a positive constant such that for any . Thus, For , let ; then we obtainHence,where denotes the -beta function.
When , we see that ; that is is continuous at .
Case 2 (). We should prove when . Without loss of generality, we consider (it is the same argument for ). In fact, where When , we verify .
As , then , when . Moreover, We have converges pointwise to the zero function and is bounded by a function belonging to , by Lebesgue’s dominated convergence theorem when .
Now, we prove when .
In fact, as and taking into account that , we get when .
Above all, we obtain , when .
Case 3 (). It is easy to check that and is continuous at . The proof is the same as the proof of Case 1.
Lemma 13. Suppose that . Then,where .
Proof. Let , .
Since , let ; we can get has a maximum at the point .
For the convenience, we denote by .
Next, we denote the class of functions by satisfying(i) is nondecreasing;(ii) for any ;(iii), where is the class of functions appearing in Theorem 11.
We give our main result as follows.
Theorem 14. Let , is continuous and , and is a continuous function on . Assume that there exists such that for with and , where . Then the BVP (3) has a unique positive solution (i.e., for ).
Proof. We define the cone by It is clear that is a complete metric space as is a closed set of . It is also easy to check that satisfies conditions (29) and (30) of Theorem 11.
We define the operator by In view of Lemma 12, . Moreover, it follows from the nonnegativeness of and that for . Thus, .
Next, we will prove that assumptions in Theorem 11 are satisfied.
First, for , we have Hence, the operator is nondecreasing. Besides, for and , Since is nondecreasing and , Moreover, when , we get Obviously, the last inequality is satisfied for .
Taking into account that the zero function satisfies , in view of Theorem 11, the operator has a unique fixed point in .
At last, we will prove is a positive solution. We assume that there exists such that . Since of problem (3) is a fixed point of the operator , we have For the nonnegative character of and , the last relation gives is continuous and ; then for , we can find , and, for , we have . It is clear that and , where is the Lebesgue measure on . That is to say, a.e. . This is a contradiction because is a rational function in .
Therefore, for .
The proof is complete.
Consider the following singular BVP: Here, , , , and for . Notice that is continuous in and .
At first, we define by It is clear that is a nondecreasing function; for , we can get
Moreover, for , also satisfies In fact, when , equivalently Above all, for .
Second, for and , we have that is, satisfies assumptions of Theorem 14.
Third, we should prove belongs to . By elemental calculus, it is easy to check that is nondecreasing and , for .
In order to prove , we notice that if , then the sequence is a bounded sequence because in contrary case, that is, , we get Now, we assume that , and then we find such that for each there exists with .
Since is a bounded sequence, we can find a subsequence of with , for certain . When , it implies that and, as the unique solution of is , we deduce that . Therefore, and this contradicts the fact that for every .
Thus, and this proves that .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is supported by the Natural Science Foundation of China (11201112), the Natural Science Foundation of Hebei Province (A2013208147), (A2014208152), and (A2015208114), and the Foundation of Hebei Education Department (Z2014062).
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