Positive Solutions for Systems of Coupled Fractional Boundary Value Problems

Affiliation(s)

^{1}
Department of Mathematics, Baylor University, Waco, Texas, USA.

^{2}
Department of Mathematics, Gh. Asachi Technical University, Iasi, Romania.

^{3}
Faculty of Computer Engineering and Automatic Control, Gh. Asachi Technical University, Iasi, Romania.

ABSTRACT

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.

We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann-Liouville fractional differential equations with coupled integral boundary conditions which contain some positive constants.

KEYWORDS

Riemann-Liouville Fractional Differential Equations, Coupled Integral Boundary Conditions, Positive Solutions

Riemann-Liouville Fractional Differential Equations, Coupled Integral Boundary Conditions, Positive Solutions

Cite this paper

Henderson, J. , Luca, R. and Tudorache, A. (2015) Positive Solutions for Systems of Coupled Fractional Boundary Value Problems.*Open Journal of Applied Sciences*, **5**, 600-608. doi: 10.4236/ojapps.2015.510059.

Henderson, J. , Luca, R. and Tudorache, A. (2015) Positive Solutions for Systems of Coupled Fractional Boundary Value Problems.

References

[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.

http://dx.doi.org/10.1016/j.nonrwa.2010.01.002

[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.

http://dx.doi.org/10.14232/ejqtde.2012.1.93

[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.

http://dx.doi.org/10.1016/j.camwa.2010.03.031

[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. http://dx.doi.org/10.1016/j.na.2009.07.012

[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.

http://dx.doi.org/10.1016/j.amc.2014.10.028

[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.

http://dx.doi.org/10.1515/fca-2015-0024

[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.

[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.

http://dx.doi.org/10.1016/j.nonrwa.2010.01.002

[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.

http://dx.doi.org/10.14232/ejqtde.2012.1.93

[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.

http://dx.doi.org/10.1016/j.camwa.2010.03.031

[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. http://dx.doi.org/10.1016/j.na.2009.07.012

[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.

http://dx.doi.org/10.1016/j.amc.2014.10.028

[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.

http://dx.doi.org/10.1515/fca-2015-0024

[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.