Existence and Uniqueness of Solution to Two-Point Boundary Value for Two-Sided Fractional Differential Equations

Affiliation(s)

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China.

School of Science, Beijing University of Civil Engineering and Architecture, Beijing, China.

ABSTRACT

In this paper, existence and uniqueness of solution to two-point boundary value for two-sided fractional differential equations involving Caputo fractional derivative is discussed, by means of the Min-Max Theorem.

In this paper, existence and uniqueness of solution to two-point boundary value for two-sided fractional differential equations involving Caputo fractional derivative is discussed, by means of the Min-Max Theorem.

Cite this paper

A. Shi and Y. Bai, "Existence and Uniqueness of Solution to Two-Point Boundary Value for Two-Sided Fractional Differential Equations,"*Applied Mathematics*, Vol. 4 No. 6, 2013, pp. 914-918. doi: 10.4236/am.2013.46127.

A. Shi and Y. Bai, "Existence and Uniqueness of Solution to Two-Point Boundary Value for Two-Sided Fractional Differential Equations,"

References

[1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.

[2] I. Podlubny, “Fractional Differential Equations,” In: Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

[3] D. Delbosco and L. Rodino, “Existence and Uniqueness for a Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 2, 1996, pp. 609-625. doi:10.1006/jmaa.1996.0456

[4] V. Lakshmikantham and A. S. Vatsala, “Theory of Fractional Differential Inequalities and Applications,” Communications on Pure and Applied Analysis, Vol. 11, No. 3-4, 2007, pp. 395-402.

[5] V. Lakshmikantham and A. S. Vatsala, “General Uniqueness and Monotone Iterative Technique for Fractional Differential Equations,” Applied Mathematics Letters, Vol. 21, No. 8, 2008, pp. 828-834. doi:10.1016/j.aml.2007.09.006

[6] V. Lakshmikantham and A. S. Vatsala, “Basic Theory of Fractional Differential Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 8, 2008, pp. 2677-2682. doi:10.1016/j.na.2007.08.042

[7] H. Liang and J. H. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equation,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 11, 2009, pp. 5545-5550. doi:10.1016/j.na.2009.04.045

[8] S. Q. Zhang, “Positive Solutions to Singular Boundary Value Problem for Nonlinear Fractional Differential Equation,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1300-1309. doi:10.1016/j.camwa.2009.06.034

[9] Z. Bai and H. Lu, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 2, 2005, pp. 495-505. doi:10.1016/j.jmaa.2005.02.052

[10] R. P. Agarwal, Y. Zhou, J. R. Wang and X. N. Luo, “Fractional Functional Differential Equations with Causal Operators in Banach Spaces,” Mathematical and Computer Modelling, Vol. 54, No. 5-6, 2011, pp. 1440-1452. doi:10.1016/j.mcm.2011.04.016

[11] F. L. Chen, J. J. Nieto and Y. Zhou, “Global Attractivity for Nonlinear Fractional Differential Equations,” Nonlinear Analysis: Real World Applications, Vol. 13, No. 1, 2012, pp. 287-298. doi:10.1016/j.nonrwa.2011.07.034

[12] F. Jiao and Y. Zhou, “Existence of Solutions for a Class of Fractional Boundary Value Problems via Critical Point theory,” Computers & Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 1181-1199. doi:10.1016/j.camwa.2011.03.086

[13] F. Jiao and Y. Zhou, “Existence of Solutions for a Class of Fractional Boundary Value Problems via Critical Point Theory,” International Journal of Bifurcation and Chaos, Special Issue, to Appear.

[14] R. F. Manasevich, “A Min-Max Theorem,” Journal of Mathematical Analysis and Applications, Vol. 90, No. 1, 1982, pp. 64-71. doi:10.1016/0022-247X(82)90044-0

[1] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, “Theory and Applications of Fractional Differential Equations,” Elsevier, Amsterdam, 2006.

[2] I. Podlubny, “Fractional Differential Equations,” In: Mathematics in Science and Engineering, Academic Press, San Diego, 1999.

[3] D. Delbosco and L. Rodino, “Existence and Uniqueness for a Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 204, No. 2, 1996, pp. 609-625. doi:10.1006/jmaa.1996.0456

[4] V. Lakshmikantham and A. S. Vatsala, “Theory of Fractional Differential Inequalities and Applications,” Communications on Pure and Applied Analysis, Vol. 11, No. 3-4, 2007, pp. 395-402.

[5] V. Lakshmikantham and A. S. Vatsala, “General Uniqueness and Monotone Iterative Technique for Fractional Differential Equations,” Applied Mathematics Letters, Vol. 21, No. 8, 2008, pp. 828-834. doi:10.1016/j.aml.2007.09.006

[6] V. Lakshmikantham and A. S. Vatsala, “Basic Theory of Fractional Differential Equations,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 69, No. 8, 2008, pp. 2677-2682. doi:10.1016/j.na.2007.08.042

[7] H. Liang and J. H. Zhang, “Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equation,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 11, 2009, pp. 5545-5550. doi:10.1016/j.na.2009.04.045

[8] S. Q. Zhang, “Positive Solutions to Singular Boundary Value Problem for Nonlinear Fractional Differential Equation,” Computers & Mathematics with Applications, Vol. 59, No. 3, 2010, pp. 1300-1309. doi:10.1016/j.camwa.2009.06.034

[9] Z. Bai and H. Lu, “Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation,” Journal of Mathematical Analysis and Applications, Vol. 311, No. 2, 2005, pp. 495-505. doi:10.1016/j.jmaa.2005.02.052

[10] R. P. Agarwal, Y. Zhou, J. R. Wang and X. N. Luo, “Fractional Functional Differential Equations with Causal Operators in Banach Spaces,” Mathematical and Computer Modelling, Vol. 54, No. 5-6, 2011, pp. 1440-1452. doi:10.1016/j.mcm.2011.04.016

[11] F. L. Chen, J. J. Nieto and Y. Zhou, “Global Attractivity for Nonlinear Fractional Differential Equations,” Nonlinear Analysis: Real World Applications, Vol. 13, No. 1, 2012, pp. 287-298. doi:10.1016/j.nonrwa.2011.07.034

[12] F. Jiao and Y. Zhou, “Existence of Solutions for a Class of Fractional Boundary Value Problems via Critical Point theory,” Computers & Mathematics with Applications, Vol. 62, No. 3, 2011, pp. 1181-1199. doi:10.1016/j.camwa.2011.03.086

[13] F. Jiao and Y. Zhou, “Existence of Solutions for a Class of Fractional Boundary Value Problems via Critical Point Theory,” International Journal of Bifurcation and Chaos, Special Issue, to Appear.

[14] R. F. Manasevich, “A Min-Max Theorem,” Journal of Mathematical Analysis and Applications, Vol. 90, No. 1, 1982, pp. 64-71. doi:10.1016/0022-247X(82)90044-0