Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013View this Special Issue
Research Article | Open Access
Positive Solutions of Nonlocal Boundary Value Problem for High-Order Nonlinear Fractional -Difference Equations
We study the nonlinear -difference equations of fractional order , , , , , where is the fractional -derivative of the Riemann-Liouville type of order , , , , and . We obtain the existence and multiplicity results of positive solutions by using some fixed point theorems. Finally, we give examples to illustrate the results.
The -difference calculus or quantum calculus is an old subject that was initially developed by Jackson [1, 2]. It is rich in history and in applications as the reader can find in the work by Ernst . For some recent existence results on -difference equations, see [4–7] and the references therein.
The fractional -difference calculus had its origin in the works by Al-Salam and Agarwal. Henceforth, fractional -difference equations have gained considerable importance due to their application in various sciences, such as physics, chemistry, aerodynamics, biology, economics, control theory, mechanics, electricity, signal and image processing, biophysics, blood flow phenomena, and fitting of experimental data. It has been a significant development in difference equations involving fractional -derivatives; see [8–11] and references therein. As well known, fractional differential equations boundary value problems is currently under strong research, see [12–21] and references therein. In particular, in recent years, fractional -difference boundary value problem (BVP) was in its infancy, and many people begin to study the existence of positive solutions for this kind of BVP; see [22–27] and references therein. However, there are few related results available. Lots of work and development should be done in the future.
Recently, in , Li et al. considered the BVP of nonlinear fractional difference equation where , , , , and satisfies Caratheodory type conditions.
More recently, in , Ferreira considered the BVP of fractional -difference equation where and is a nonnegative continuous function.
Motivated by the work above, in this paper, we will discuss the following BVP: where , , , , and satisfies Caratheodory type conditions. We discuss the existence of positive solutions for BVP(3) and obtain multiplicity results which extend and improve the known results by using some fixed point theorems.
2. Preliminary Results
In this section, we introduce definitions and preliminary facts which are used throughout this paper.
Let and define
The -analogue of the power function with is More generally, if , then Note that, if , then . The -gamma function is defined by and satisfies .
Then, 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
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 defined by
Similarly as done for derivatives, an operator can be defined; 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 let be a function defined on . The fractional -integral of the Riemann-Liouville type is and
Definition 6 (see ). The fractional -derivative of the Riemann-Liouville type of is defined by and where is the smallest integer greater than or equal to .
Remark 8. Assume that and are two constants such that . Then
Lemma 9 (see ). Let and let be a positive integer. Then, the following equality holds:
Lemma 10. Let ; then the unique solution of is where where .
Proof. Let be a solution of (21); in view of Lemma 7 and Lemma 9, (21) is equivalent to the integral equation where are some constants to be determined. The boundary conditions , , imply that . Thus, By Remark 8, we have For , Hence, The proof is complete.
Remark 11. For the special case where , it is easy to see that can be written as
Lemma 12. Green function in Lemma 10 satisfies the following conditions:(i) for ;(ii) for ;(iii) for .
We first prove part (i). For , from Remark 4, for , Since , it is easy to know , , and . Therefore, .
Next, we prove part (ii). Fix , and That is, is increasing function of . By the same way, we can conclude that , , and are increasing functions of for fixed . Thus, for .
Finally, we prove part (iii). Suppose that ; then
For other circumstances, we also get and this completes the proof.
Remark 13. Let ; then and
Lemma 14 (see ). Let be a Banach space with being closed and convex. Assume that is a relatively open subset of with and is complete continuous. Then either(i) has a fixed point in , or(ii)there exist and with .
Lemma 15 (see Krasnoselskii’s ). Let be a Banach space, and is a cone in . Assume that and are open subsets of with and . Let be a completely continuous operator. In addition, suppose that either, for all and , for all or, for all and , for all holds. Then has a fixed point in .
Lemma 16 (see ). Let be a cone in a real Banach space , , is a nonnegative continuous concave functional on such that , for all , and . Suppose that is completely continuous and there exist positive constants such that and for , for , for with .Then has at least three fixed points , and with
Remark 17. If , then () implies ().
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.
Let be the Banach space endowed with the norm . Let for a given , and define the cone by
Let the nonnegative continuous concave functional on the cone be defined by
In this paper, we assume that satisfies the following conditions of Caratheodory type: is Lebesgue measurable with respect to on ; is continuous with respect to on .
Theorem 18. Assume that the conditions and hold. Suppose further that there exists a real-valued function such that for almost every and all . If then there exist unique positive solutions of BVP (3) on .
Proof. Consider the operator defined by For any , we have This implies that is a contraction mapping. By the Banach contraction mapping principle, we deduce that has a unique fixed point which is obviously a solution of BVP (3). The proof is complete.
Corollary 19. Assume that the conditions and hold. Suppose further that there exists a positive constant with , where then there exists a unique positive solution of BVP (3) on .
Corollary 20. Assume that the conditions and hold. Suppose further that there exists a real-valued function such that , , . If then there exists a unique positive solution of BVP (3) on .
Corollary 21. Assume that the conditions and hold. Suppose further that there exists a positive constant with , , ; then there exists a unique positive solution of BVP (3) on .
Next, we discuss multiple solutions of BVP (3).
Lemma 22. Assume that the conditions and hold. Suppose further that there exist two nonnegative real-valued functions such that for almost every and all . Then the operation defined by (39) is completely continuous.
Proof. We will divide the proof into three parts.(I) We show that is continuous.
For any , , with , we have Thus So, we can obtain that On the other hand, we have It implies that Therefore, as . This means that is continuous.(II) We will prove that maps bounded sets into bounded sets in .
For any , there exists a positive constant such that for each , we have . By the definition of , for each , we get That is, .(III) We will show that maps bounded sets into equicontinuous sets of .
Let be a bounded set, and . For any , we have Since is continuous on , then is uniformly continuous in . Hence, for any , there exists , whenever , and we have So ; that is, is equicontinuous.
By Arzela-Ascoli theorem, we can conclude that is completely continuous. This completes the proof.
Remark 23. If is continuous, is also completely continuous.
Proof. Let , where
and . From Lemma 22, is completely continuous.
Assume that there exist and such that ; we claim that : and then That is, . By Lemma 14, has a fixed point . Therefore, BVP (3) has at least a positive solution. The proof is complete.
In the following, we set
Proof. Because is completely continuous, we just only show that has a solution for .
Let . For any , we know on . Using (56) and (34), we have which implies that , .
Let . For any , we get on . Using (57), we have that is, , .
In view of Lemma 15, has a fixed point which is the solution of BVP (3).
Proof. First, if , then . So , . By , we have
which implies that . Hence, . In view of Lemma 22, is completely continuous.
By using the analogous argument, from , we can get that if , then .
Set , , so , . Therefore, .
On the other hand, if , then . By , we have which implies that , for .
By Lemma 16, BVP(3) has at least three positive solutions , , and with The proof is complete.
Example 29. Consider the following BVP:
Let , . By calculation, we get