International Journal of Differential Equations
Volume 2016 (2016), Article ID 6906049, 10 pages
http://dx.doi.org/10.1155/2016/6906049

Research Article

Multiplicity of Positive Solutions for Fractional Differential Equation with -Laplacian Boundary Value Problems

Sabbavarapu Nageswara Rao

Department of Mathematics, Jazan University, P.O. Box 114, Jazan, Saudi Arabia

Received 15 October 2015; Accepted 6 April 2016

Academic Editor: Kanishka Perera

Copyright © 2016 Sabbavarapu Nageswara Rao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Therefore

Thus, 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 prove

In 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 have

Finally, 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 obtain

Hence, 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 equation

where

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.

References

R. P. Agarwal, D. O'Regan, and P. J. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic, Dordrecht, The Netherlands, 1999.

D. R. Anderson and J. M. Davis, “Multiple solutions and eigenvalues for third-order right focal boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 267, no. 1, pp. 135–157, 2002.

R. W. Leggett and L. R. Williams, “Multiple positive fixed points of nonlinear operators on ordered Banach spaces,” Indiana University Mathematics Journal, vol. 28, no. 4, pp. 673–688, 1979.

Y. Sun, Q. Sun, and X. Zhang, “Existence and nonexistence of positive solutions for a higher-order three-point boundary value problem,” Abstract and Applied Analysis, vol. 2014, Article ID 513051, 7 pages, 2014.

S. Nageswararao, “Solvability of second order delta-nabla p-Laplacian m-point eigenvalue problem on time scales,” TWMS Journal of Applied and Engineering Mathematics, vol. 5, no. 1, pp. 98–109, 2015.

M. Ramaswamy and R. Shivaji, “Multiple positive solutions for classes of p-Laplacian equations,” Differential and Integral Equations, vol. 17, no. 11-12, pp. 1255–1261, 2004.

C. Yang and J. Yan, “Positive solutions for third-order Sturm-Liouville boundary value problems with p-Laplacian,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2059–2066, 2010.

C. Zhai and C. Guo, “Positive solutions for third-order Sturm-Liouville boundary-value problems with p-Laplacian,” Electronic Journal of Differential Equations, vol. 2009, no. 154, pp. 1–9, 2009.

S. Liang and J. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5545–5550, 2009.

S. Nageswararao, “Multiple positive solutions for a system of Riemann-Liouville fractional order two-point boundary value problems,” Panamerican Mathematical Journal, vol. 25, no. 1, pp. 66–81, 2015.

S. Nageswararao, “Existence and multiplicity for a system of fractional higher-order two-pointboundary value problem,” Journal of Applied Mathematics and Computing, vol. 53, pp. 93–107, 2016.

Y. Zhao, S. Sun, Z. Han, and M. Zhang, “Positive solutions for boundary value problems of nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 217, no. 16, pp. 6950–6958, 2011.

G. Chai, “Positive solutions for boundary value problem of fractional Differential equation with p-Laplacian operator,” Boundary Value Problems, vol. 2012, article 18, 2012.

Z. Han, H. Lu, S. Sun, and D. Yang, “Positive solutions to boundary-value problems of p-Laplacian fractional differential equations with a parameter in the boundary conditions,” Electronic Journal of Differential Equations, vol. 2012, no. 213, pp. 1–14, 2012.

H. Lu, Z. Han, and S. Sun, “Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with p-Laplacian,” Boundary Value Problems, vol. 2014, article 26, 2014.

C. T. Ledesma, “Boundary value problem with fractional p-Laplacian operator,” Advances in Nonlinear Analysis, 2015.

Y. Li and A. Qi, “Positive solutions for multi-point boundary value problems of fractional differential equations with p-Laplacian,” Mathematical Methods in the Applied Sciences, vol. 39, no. 6, pp. 1425–1434, 2016.

K. R. Prasad and B. M. B. Krushna, “Existence of multiple positive solutions for p-Laplacian fractional order boundary value problems,” International Journal of Analysis and Applications, vol. 6, no. 1, pp. 63–81, 2014.

B. Zhang, “Solvability of fractional boundary value problems with p-Laplacian operator,” Advances in Difference Equations, vol. 2015, article 352, 2015.

A. Kameswararao and S. Nageswararao, “Eigenvalue problems for systems of nonlinear higher order boundary value problems,” Journal of Applied Mathematics and Informatics, vol. 28, no. 3-4, pp. 711–721, 2010.

D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Orlando, Fla, USA, 1988.

K. Deimling, Nonlinear Functional Analysis, Springer, New York, NY, USA, 1985.