Abstract
We investigate the existence of multiple positive solutions of fractional differential equations with -Laplacian operator , , , , where , , , , , is a fixed integer, and , by applying Leggett–Williams fixed point theorems and fixed point index theory.
1. Introduction
The goal of differential equations is to understand the phenomena of nature by developing mathematical models. Fractional calculus is the field of mathematical analysis, which deals with investigation and applications of derivatives and integrals of an arbitrary order. Among all, a class of differential equations governed by nonlinear differential operators appears frequently and generated a great deal of interest in studying such problems. In this theory, the most applicable operator is the classical -Laplacian, given by .
The positive solutions of boundary value problems associated with ordinary differential equations were studied by many authors [1–4] and extended to -Laplacian boundary value problems [5–8]. Later, these results are further extended to fractional order boundary value problems [9–12] by applying various fixed point theorems on cones. Recently, researchers are concentrating on the theory of fractional order boundary value problems associated with -Laplacian operator [13–19]. The above few papers motivated this work.
In this paper, we are concerned with the existence of multiple positive solutions for the fractional differential equation with -Laplacian operator with the boundary conditions where , , , , , , , , is a fixed integer, and , are constants. The function is continuous and , , are the standard Riemann-Liouville fractional order derivatives.
The rest of this paper is organized as follows. In Section 2, the Green functions for the homogeneous BVPs corresponding to (1)-(2) are constructed and the bounds for the Green functions are estimated. In Section 3, sufficient conditions for the existence of at least two or at least three positive solutions are established, by using fixed point index theory and Leggett-Williams fixed point theorems. In Section 4, as an application, an example is presented to illustrate our main result.
2. Green’s Function and Bounds
In this section, we construct Green’s function for the homogeneous boundary value problem and estimate bounds for Green’s function that will be used to prove our main theorems.
Let be Green’s function for the homogeneous BVP
Lemma 1. Let . If , then the fractional order BVP has a unique solution, , where here
Proof. Assume that is a solution of fractional order BVP (4)-(5) and is uniquely expressed as , so that From , , we have . Then From , we have ThereforeThus, the unique solution of (4)-(5) is where is given in (6).
Lemma 2. If , then the fractional order differential equation satisfying (5) and has a unique solution, where Here is Green’s function for
Proof. An equivalent integral equation for (13) is given by By (14), one can determine and .
Thus, the unique solution of (13), (2) is Therefore, . Consequently, .
Hence,
Lemma 3. Green’s function satisfies the following inequalities:(i),(ii).
Proof. Consider Green’s function given by (6).
Let . Then, we have Let . Then, we have Therefore is increasing in , which implies .
Now we proveIn fact, if , obviously, (23) holds. If , one has That implies that (23) is also true. Therefore, by (6), (21), and (23), we find Therefore is increasing with respect to . Hence the inequality (i) is proved. Now, we establish the inequality (ii).
On the other hand, if , we have If , we have Therefore From (6) and (28) we have Therefore Hence the inequality (ii) is proved.
Lemma 4. Green’s function satisfies the following inequalities:(i),(ii),where and .
The method of proof is similar to that [20], and we omit it here.
Theorem 5 (Leggett-Williams [3]). Let be completely continuous and let be a nonnegative continuous concave functional on such that for all . Suppose that there exist , , , and with such that() and for ,() for ,() for with . Then has at least three fixed points , , and in satisfying , and .
Theorem 6 (see [3]). Let be a completely continuous operator and let be a nonnegative continuous concave functional on such that for all . Suppose that there exist , , and with such that() and for ,() for ,() for with . Then has at least two fixed points and in satisfying , and .
Theorem 7 (see [21]). Let be a closed convex set in a Banach space and let be a bounded open set such that . Let be a compact map. Suppose that for all :Existence: if , then has a fixed point in .Normalization: if , then , where for .Homotopy: let be a compact map such that for and . Then .Additivity: if are disjoint relatively open subsets of such that for , then , where .
Theorem 8 (see [22]). Let be a cone in a Banach space . For , define . Assume that is a compact map such that for . Thus, one has the following conclusions:If for , then .If for , then .
3. Main Results
In this section, the existence of at least two or at least three positive solutions for fractional differential equation with -Laplacian operator BVP (1)-(2) is established by using fixed point index theory and Leggett-Williams fixed point theorems.
Let be the real Banach space equipped with the norm . Define the cone by Let be the operator defined by If is a fixed point of , then satisfies (32) and hence is a positive solution of -Laplacian fractional order BVP (1)-(2).
Lemma 9. The operator defined by (32) is a self-map on .
Proof. Let . Clearly, , for all and so that On the other hand, by Lemma 3, we have Hence and so . Standard argument involving the Arzela-Ascoli theorem shows that is completely continuous.
For convenience of the reader, we denote
Theorem 10. Let be nonnegative continuous on . Assume that there exist constants , with such that the following conditions are satisfied: () for all .() for all .Then fractional order BVP (1)-(2) has at least two positive solutions and satisfying , , and .
Proof. Let be the nonnegative continuous concave functional defined by . Evidently, for each , we have .
It is easy to see that is completely continuous and . We choose ; then So . Hence, if , then for . Thus for , from assumption , we have Consequently, for . That is, Therefore, condition of Theorem 6 is satisfied. Now if , then . By assumption , we have which shows that , that is, for . This shows that condition of Theorem 6 is satisfied. Finally, we show that of Theorem 6 also holds. Assume that with ; then by the definition of cone , we have So condition of Theorem 6 is satisfied. Thus using Theorem 6, has at least two fixed points. Consequently, boundary value problem (1)-(2) has at least two positive solutions and in satisfying
Theorem 11. Let be nonnegative continuous on . Assume that there exist constants , , with such that() for all ,() for all ,() for all .Then fractional order BVP (1)-(2) has at least three positive solutions , , and with , , , and .
Proof. If , then . By assumption , we have This shows that . Using the same arguments as in the proof of Theorem 10, we can show that is a completely continuous operator. It follows from conditions and in Theorem 11 that . Similarly to the proof of Theorem 10, we have and for all .
Moreover, for and , we have So all the conditions of Theorem 5 are satisfied. Thus using Theorem 5, has at least three fixed points. So, th boundary value problem (1)-(2) has at least three positive solutions , , and with , , , and .
Theorem 12. Let be nonnegative continuous on . If the following assumptions are satisfied:();()there exists a constant such that , for , then boundary value problem (1)-(2) has at least two positive solutions and such that .
Proof. From Lemma 1, we obtain being completely continuous. In view of , there exists such that , for , , where . Let . Then, for any , we have which implies for . Hence, Theorem 8 implies On the other hand, since , there exists such that , for , where . Let and . Then , for any . By using the method to get (48), we obtain , which implies for . Thus, from Theorem 8, we haveFinally, let . Then, for any , by , we then get which implies for . Using Theorem 8 again, we get Note that , by the additivity of fixed point index and (48)–(51); we obtain Hence, has a fixed point in , and it has a fixed point in . Clearly, and are positive solutions of boundary value problem (1)-(2) and .
Theorem 13. Let be nonnegative continuous on . If the following assumptions are satisfied:();()there exists a constant such that , for , then boundary value problem (1)-(2) has at least two positive solutions and such that .
Proof. From Lemma 9, we obtain being completely continuous. In view of , there exists such that , for , , where . Let . Then, for any , we have which implies for . Hence, Theorem 8 implies Next, since , there exists such that , for , where . We consider two cases.
Case (i). Suppose that is bounded, which implies that there exists such that for all and . Take . Then, for with , we get Case (ii). Suppose that is unbounded. In view of being continuous, there exist and such that , for , . Then, for with , we obtain So, in either case, if we always choose , then we have , for . Thus, from Theorem 8, we have Finally, let . Then, for any , , by , and we then obtain which implies for . An application of Theorem 8 again shows that Note that by the additivity of fixed point index and (54)–(59); we obtainHence, has a fixed point in , and it has a fixed point in . Consequently, and are positive solutions of boundary value problem (1)-(2) and .
4. Example
In this section, we consider boundary value problem of the fractional differential equationwhere
Let We note that , , , , , , , , and . By a simple calculation, we obtain , , , and . Choosing , , and , then and satisfies(i) for all ,(ii) for all ,(iii) for all .Consequently, all of the conditions of Theorem 11 are satisfied. With the use of Theorem 11, boundary value problem (61) has at least three positive solutions , , and with
Competing Interests
The author declares that he has no competing interests regarding the publication of this paper.
Acknowledgments
The author expresses his gratitude to his guide Professor K. Rajendra Prsasd.