as at least one positive solution for and sufficiently small.
Proof. We consider the system of ordinary fractional differential equations
with the coupled integral boundary conditions
The above problem (6)-(7) has the solution
where is defined in (J1). By assumption (J1) we obtain and for all.
We define the functions and, by
where is a solution of (S)-(BC). Then (S)-(BC) can be equivalently written as
with the boundary conditions
Using the Green’s functions, from Lemma 1, a pair is a solution of problem (9)-(10) if and only if is a solution for the nonlinear integral equations
where and, are given in (8).
We consider the Banach space with the supremum norm, the space with the norm, and we define the set
We also define the operators and by
for all, and.
For sufficiently small and, by (J3), we deduce
Then, by using Lemma 3, we obtain, for all and. By Lemma 4, for all, we have
Using standard arguments, we deduce that S is completely continuous. By Theorem 1, we conclude that S has a fixed point, which represents a solution for problem (9)-(10). This shows that our problem (S)-(BC) has a positive solution with for sufficiently small and.
In what follows, we present sufficient conditions for the nonexistence of positive solutions of (S)-(BC).
Theorem 3. Assume that assumptions (J1), (J2) and (J4) hold. Then problem (S)-(BC) has no positive solution for and sufficiently large.
Proof. We suppose that is a positive solution of (S)-(BC). Then with, is a solution for problem (9)-(10), where is the solution of problem (6)-(7) (given by (8)). By (J2) there exists such that, and then, , ,. Now by using Lemma 3, we have, for all, and by Lemma 5 we obtain and.
Using now (8), we deduce that and. Therefore, we obtain and.
We now consider. By using (J4), for R defined above, we conclude that there exists such that, for all. We consider and sufficiently large such that and. By (J2), (9), (10) and the above inequalities, we deduce that and.
Now by using Lemma 4 and the above considerations, we have
Therefore, we obtain, which is a contradiction, because. Then, for and sufficiently large, our problem (S)-(BC) has no positive solution.
4. An Example
We consider, for all, , , , for all, then and. We also consider the functions, , , for all, with. We have.
Therefore, we consider the system of fractional differential equations
with the boundary conditions
Then we obtain
We also deduce
, for all. For the functions, , we obtain
Then we deduce that assumptions (J1), (J2) and (J4) are satisfied. In addition, by using the above functions, , we obtain, ,
, , and then. We choose and if we select, then we conclude that, for all. For example, if and, then the above conditions for f and g are satisfied. So,
assumption (J3) is also satisfied. By Theorems 2 and 3 we deduce that problem (S0)-(BC0) has at least one positive solution for sufficiently small and, and no positive solution for sufficiently large and.
The work of R. Luca and A. Tudorache was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0557.
Cite this paper
JohnnyHenderson,RodicaLuca,AlexandruTudorache, (2015) Positive Solutions for Systems of Coupled Fractional Boundary Value Problems. Open Journal of Applied Sciences,05,600-608. doi: 10.4236/ojapps.2015.510059
- 1. Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J.J. (2012) Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. World Scientific, Boston.
- 2. Das, S. (2008) Functional Fractional Calculus for System Identification and Control. Springer, New York.
- 3. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science, Amsterdam.
- 4. Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
- 5. Sabatier, J., Agrawal, O.P. and Machado, J.A.T., Eds. (2007) Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Springer, Dordrecht.
- 6. Samko, S.G., Kilbas, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon.
- 7. Agarwal, R.P., Andrade, B. and Cuevas, C. (2010) Weighted Pseudo-Almost Periodic Solutions of a Class of Semilinear Fractional Differential Equations. Nonlinear Analysis, Real World Applications, 11, 3532-3554.
- 8. Agarwal, R.P., Zhou, Y. and He, Y. (2010) Existence of Fractional Neutral Functional Differential Equations. Computers and Mathematics with Applications, 59, 1095-1100. http://dx.doi.org/10.1016/j.camwa.2009.05.010
- 9. Aghajani, A., Jalilian, Y. and Trujillo, J.J. (2012) On the Existence of Solutions of Fractional Integro-Differential Equations. Fractional Calculus and Applied Analysis, 15, 44-69. http://dx.doi.org/10.2478/s13540-012-0005-4
- 10. Ahmad, B. and Ntouyas, S.K. (2012) Nonlinear Fractional Differential Equations and Inclusions of Arbitrary Order and Multi-Strip Boundary Conditions. Electronic Journal of Differential Equations, 2012, 1-22.
- 11. Ahmad, B. and Ntouyas, S.K. (2012) A Note on Fractional Differential Equations with Fractional Separated Boundary Conditions. Abstract and Applied Analysis, 2012, Article ID: 818703.
- 12. Bai, Z. (2010) On Positive Solutions of a Nonlocal Fractional Boundary Value Problem. Nonlinear Analysis, 72, 916-924. http://dx.doi.org/10.1016/j.na.2009.07.033
- 13. Balachandran, K. and Trujillo, J.J. (2010) The Nonlocal Cauchy Problem for Nonlinear Fractional Integrodifferential Equations in Banach Spaces. Nonlinear Analysis, 72, 4587-4593. http://dx.doi.org/10.1016/j.na.2010.02.035
- 14. El-Shahed, M. and Nieto, J.J. (2010) Nontrivial Solutions for a Nonlinear Multi-Point Boundary Value Problem of Fractional Order. Computers and Mathematics with Applications, 59, 3438-3443.
- 15. Graef, J.R., Kong, L., Kong, Q. and Wang, M. (2012) Uniqueness of Positive Solutions of Fractional Boundary Value Problems with Non-Homogeneous Integral Boundary Conditions. Fractional Calculus and Applied Analysis, 15, 509- 528. http://dx.doi.org/10.2478/s13540-012-0036-x
- 16. Jiang, D. and Yuan, C. (2010) The Positive Properties of the Green Function for Dirichlet-Type Boundary Value Problems of Nonlinear Fractional Differential Equations and Its Application. Nonlinear Analysis, 72, 710-719.
- 17. Liang, S. and Zhang, J. (2009) Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equation. Nonlinear Analysis, 71, 5545-5550. http://dx.doi.org/10.1016/j.na.2009.04.045
- 18. Yuan, C. (2010) Multiple Positive Solutions for -Type Semipositone Conjugate Boundary Value Problems of Nonlinear Fractional Differential Equations. Electronic Journal of Qualitative Theory of Differential Equations, 2010, 1-12. http://dx.doi.org/10.14232/ejqtde.2010.1.36
- 19. Yuan, C., Jiang, D., O’Regan, D. and Agarwal, R.P. (2012) Multiple Positive Solutions to Systems of Nonlinear Semipositone Fractional Differential Equations with Coupled Boundary Conditions. Electronic Journal of Qualitative Theory of Differential Equations, 2012, 1-17. http://dx.doi.org/10.14232/ejqtde.2012.1.13
- 20. Henderson, J. and Luca, R. (2014) Positive Solutions for a System of Fractional Differential Equations with Coupled integral Boundary Conditions. Applied Mathematics and Computation, 249, 182-197.
- 21. Henderson, J., Luca, R. and Tudorache, A. (2015) On a System of Fractional Differential Equations with Coupled integral Boundary Conditions. Fractional Calculus and Applied Analysis, 18, 361-386.
- 22. Henderson, J. and Luca, R. (2013) Positive Solutions for a System of Nonlocal Fractional Boundary Value Problems. Fractional Calculus and Applied Analysis, 16, 985-1008. http://dx.doi.org/10.2478/s13540-013-0061-4
- 23. Henderson, J. and Luca, R. (2014) Existence and Multiplicity of Positive Solutions for a System of Fractional Boundary Value Problems. Boundary Value Problems, 2014, 60. http://dx.doi.org/10.1186/1687-2770-2014-60
- 24. Henderson, J., Luca, R. and Tudorache, A. (2015) Positive Solutions for a Fractional Boundary Value Problem. Nonlinear Studies, 22, 1-13.
- 25. Luca, R. and Tudorache, A. (2014) Positive Solutions to a System of Semipositone Fractional Boundary Value Problems. Advances in Difference Equations, 2014, 179. http://dx.doi.org/10.1186/1687-1847-2014-179
- 26. Henderson, J. and Luca, R. (2015) Boundary Value Problems for Systems of Differential, Difference and Fractional Equations: Positive Solutions. Elsevier, Amsterdam.