Abstract
We consider a discrete fractional nonlinear boundary value problem in which nonlinear term is involved with the fractional order difference. And we transform the fractional boundary value problem into boundary value problem of integer order difference equation. By using a generalization of Leggett-Williams fixed-point theorem due to Avery and Peterson, we provide sufficient conditions for the existence of at least three positive solutions.
1. Introduction
Let and be the sets of real numbers and integers, respectively. For , with , define . Assume that is a given positive integer with . We consider the fractional difference boundary value problem (briefly FBVP) of the forms where , with , and , , , is fractional difference operator. We give the following assumptions: is continuous; is nonnegative on does not hold on , and .
Fixed-point theorems and their applications in nonlinear problems have a long history, some of which are documented in Zeidler’s book [1]. There seems to be increasing interest in multiple fixed-point theorems and their applications to boundary value problems for ordinary differential equations or finite difference equations. The applications can be found in the papers [2–6]. An interest in triple solutions has evolved from the Leggett-Williams multiple fixed-point theorem [7]. And, lately, two triple fixed-point theorems by Avery and Peterson [4] and Avery [8] have been applied to obtain triple solutions of certain boundary value problems for ordinary differential equations as well as for their discrete analogues. On the other hand, fractional differential and difference “operators” are found themselves in concrete applications, and hence attention has to be paid to associated fractional difference and differential equations under various boundary or side conditions. For example, Atici and Eloe [9] explored some of the theories of a discrete conjugate FBVP in [9]. Similarly, in [10], a discrete right-focal FBVP was analyzed. Other recent advances in the theory of the discrete fractional calculus may be found in [11–26]. In particular, an interesting recent paper by Atici and Şengül [14] addressed the use of fractional difference equations in tumor growth modeling. Thus, it seems that there exists some promise in using fractional difference equations as mathematical models for describing physical problems in more accurate manners.
In order to handle the existence problem for FBVP, various methods (among which are some standard fixed-point theorems) can be used. For example, in [10, 12, 27], authors investigated the existence to some boundary value problems by fixed-point theorems on a cone. In [28], we established the existence conditions for a boundary value problem by using the coincidence degree theory. In [29], authors pointed out the existence of multiple solutions for a FBVP with parameter by establishing the corresponding variational framework and using the mountain pass theorem, linking theorem, and Clark theorem in critical point theory. To the best of our knowledge, Leggett-Williams fixed-point theorem has not been used in discrete fractional boundary value problems. The aim of this paper is to establish the existence conditions for boundary value problem (1)-(2). The proof relies on the Leggett-Williams fixed-point theorem.
Throughout this paper, we make the convention that for and denote , .
2. Preliminaries
In this section, we collect some basic definitions and lemmas for manipulating discrete fractional operators. These and other related results can be found in [4, 12, 14, 16].
For any integer , let . We define , for any and for which the right-hand side is defined. We also appeal to the convention that if is a pole of the Gamma function and is not a pole, then .
Definition 1. The th fractional sum of for is defined by for . We also define the th fractional difference for by , where , and is chosen so that .
Definition 2. Let be a real Banach space. A nonempty closed convex set is called a cone of if it satisfies the following conditions:(1), implies ;(2), implies .
Every cone induces a partial ordering “” on defined by if and only if .
Definition 3. Given a cone in a real Banach space , a functional is said to be increasing on , provided that for all with .
Let and be nonnegative continuous convex functionals on a nonnegative continuous concave functional on , and a nonnegative continuous functional on . Then, for positive real numbers , and , we define the following convex sets: , , and a closed set .
The following fixed-point theorem due to Avery and Peterson is fundamental in the proof of our main results.
Lemma 4 (see [4]). Let be a Banach space and let be a cone. Let and be nonnegative continuous convex functionals on , a nonnegative continuous concave functional on , and a nonnegative continuous functional on satisfying for , such that, for some positive numbers and and for all . Suppose is completely continuous and there exist positive numbers with such that and for ; for with ; and for with .Then has at least three fixed points such that for ; , , with .
Lemma 5 (see [16]). Let and with , and then for , where .
Lemma 6 (see [16]). Let be given and suppose and . Then, for ,
Moreover, if with , then, for ,
Lemma 7 (see [16]). Let be given and suppose with . Then, for ,
Lemma 8 (see [16]). Let be given and suppose with . Then, for ,
Lemma 9 (see [16]). Let and be given. Then, for any for which both sides are well-defined. Furthermore, for ,
3. Triple Positive Solutions
In this section, we impose growth conditions on to obtain the triple positive solutions for FBVP (1)-(2).
For the sake of convenience, for , , , , we set the following notations:
Then, from (11), we immediately obtain some properties of as follows:
We will also need the following elementary facts:
Indeed, if , then, by (12) and (14), clearly, .
If , then It follows that .
If , then, since We have .
Now, suppose that is a solution of the problem (1)-(2), and let . Then, from Lemmas 6, 7, 8, and 9, we have , and Similarly, we have Thus, by the transformation , the problem (1)-(2) is equivalent to the following problem (20)–(22): where
Now, suppose that is a solution of (20)–(22). We will show that Firstly, it is easy to see that from .
Secondly, if , then by (21) and (22). If , we have
So we obtain that and hence for . Note that . Thus (24) holds.
Next, clearly if .
On the other hand, if is a solution of (20)–(22), then, from , we have
that is,
This implies that the problem (1)-(2) has positive solutions if and only if the problem (20)–(22) has positive solutions. In the sequel, we will concretely consider the boundary problem (20)–(22).
Summing (20) from to , we find that Thus,
Summing (29) from to , it follows that which, together with (22), implies that
Next, let be endowed with the norm , where . Choose the cone defined by
Let the nonnegative continuous concave functional , the nonnegative continuous convex functional , and the nonnegative continuous functional be defined on the cone by where and is the greatest integer not greater than . Clearly, , , and .
For , define an operator by
We next require a preliminary lemma.
Lemma 10. Let be defined by the above equation. If , then(i) for , for ;(ii), ;(iii) is completely continuous;(iv)finding positive solutions of FBVP (20)–(22) is equivalent to finding fixed points of the operator on ;(v)If ,, for .
The proof is simple and omitted.
By Lemma 10 and (33), for all , the functionals defined above satisfy
Furthermore, since we have
Therefore, and are satisfied, where .
We now put growth condition on such that the boundary value problem (20)–(22) has at least three positive solutions belonging to the cone . Then (1)-(2) has at least three positive solutions.
Theorem 11. Let , , and suppose that - hold. In addition, assume that there exist positive numbers with such that the following conditions are satisfied: for , for , for ,. Then FBVP (1)-(2) has at least three positive solutions , and satisfying
Proof. By the definition of operator and its properties (i)–(v), it suffices to show that the conditions of Lemma 4 hold with respect to .
Firstly, if , we know . From (37), we have
Since is monotonic decreasing in variable , then
Thus, assumption and Lemma 10 imply that
Hence, .
To check condition of Lemma 4, we choose as follows for :
It is easy to verify that and . So .
Therefore, if , then , for .
By (27), we have . According to assumption and Lemma 10, we get
This shows that condition of Lemma 4 is satisfied.
Secondly, from (35), we have
for all with . Thus, condition of Lemma 4 is satisfied.
We finally exhibit that of Lemma 4 is also satisfied. Clearly, as , we have . Suppose that with . Since is monotonic increasing in variable , we have
Next, we consider two cases.
Case i. Consider . By and the inequality for , we can obtain that
Case ii. Consider . By and the inequality , for , we can have
So, condition of Lemma 4 is satisfied. Therefore, Lemma 4 implies that the FBVP (20)–(22) has at least three positive solutions , and satisfying (38); that is, the FBVP (1)-(2) has at least three positive solutions , and satisfying (38). The proof is complete.
Theorem 12. Let and hold. In addition, assume that the following condition is satisfied:.
Then the FBVP (1)-(2) has at least three positive solutions , and satisfying (38).
Proof. If , (21) is equivalent to by (14). Similar to discussion in Theorem 11, we know that (24) holds.
For , define an operator byThe rest of the proof is similar to Theorem 11, so it is omitted. The proof is complete.
Example 13. Consider the boundary value problem: Compared with (1), we have , , , , , , , , , , , and Then, FBVP (49) has at least three positive solutions.
Proof. Choose , , , , , and .
By computation, we know , , , , , and .
It is easy to see that . And satisfies that
for ;
for ;
for . Consider
Thus, the conditions of Theorem 11 are satisfied. Therefore, the FBVP (49) has at least three positive solutions satisfying
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The Project was supported by the National Natural Science Foundation of China (11161049).