Abstract and Applied Analysis

Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions

J. Caballero, I. Cabrera, and K. Sadarangani

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Abstract

We investigate the existence and uniqueness of positive solutions of the following nonlinear fractional differential equation with integral boundary value conditions    C D α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ′′ ( 0 ) = 0 ,    u ( 1 ) = λ 0 1 u ( s ) d s , where 2 < α < 3 , 0 < λ < 2 and    C D α is the Caputo fractional derivative and f : [ 0,1 ] × [ 0 , ) [ 0 , ) is a continuous function. Our analysis relies on a fixed point theorem in partially ordered sets. Moreover, we compare our results with others that appear in the literature.

Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 303545, 11 pages.

Dates
First available in Project Euclid: 4 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1365099928

Digital Object Identifier
doi:10.1155/2012/303545

Mathematical Reviews number (MathSciNet)
MR2984028

Zentralblatt MATH identifier
1253.35197

Citation

Caballero, J.; Cabrera, I.; Sadarangani, K. Positive Solutions of Nonlinear Fractional Differential Equations with Integral Boundary Value Conditions. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 303545, 11 pages. doi:10.1155/2012/303545. https://projecteuclid.org/euclid.aaa/1365099928


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References

  • Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.
  • J. Caballero, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems,” Boundary Value Problems, vol. 2009, Article ID 421310, 10 pages, 2009.
  • L. M. B. C. Campos, “On the solution of some simple fractional differential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 13, no. 3, pp. 481–496, 1990.
  • D. Delbosco and L. Rodino, “Existence and uniqueness for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 204, no. 2, pp. 609–625, 1996.
  • A. A. Kilbas and J. J. Trujillo, “Differential equations of fractional order: methods, results and problems. I,” Applicable Analysis, vol. 78, no. 1-2, pp. 153–192, 2001.
  • J. Jiang, L. Liu, and Y. H. Wu, “Multiple positive solutions of singular fractional differential system involving Stieltjes integral conditions,” Electronic Journal Qualitative Theory of Differential Equations, vol. 43, pp. 1–18, 2012.
  • C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1363–1375, 2010.
  • Y. Ling and S. Ding, “A class of analytic functions defined by fractional derivation,” Journal of Mathematical Analysis and Applications, vol. 186, no. 2, pp. 504–513, 1994.
  • X. Liu and M. Jia, “Multiple solutions of nonlocal boundary value problems for fractional differential equations on the half-line,” Electronic Journal of Qualitative Theory of Differential Equations, no. 56, pp. 1–14, 2011.
  • K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.
  • T. Qiu and Z. Bai, “Existence of positive solutions for singular fractional differential equations,” Electronic Journal of Differential Equations, vol. 2008, no. 146, pp. 1–9, 2008.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
  • Y. Wang, L. Liu, and Y. Wu, “Positive solutions of a fractional boundary value problem with changing sign nonlinearity,” Abstract and Applied Analysis, vol. 2012, Article ID 149849, 12 pages, 2012.
  • Y. Wang, L. Liu, and Y. Wu, “Positive solutions for a nonlocal fractional differential equation,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 74, no. 11, pp. 3599–3605, 2011.
  • J. Wu, X. Zhang, L. Liu, and Y. H. Wu, “Positive solutions of higher-order nonlinear fractional differential equations with changing-sign measure,” Advances in Difference Equations, vol. 2012, article 71, 2012.
  • S. Zhang, “The existence of a positive solution for a nonlinear fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 804–812, 2000.
  • S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1300–1309, 2010.
  • X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a sigular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.
  • X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.
  • X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012.
  • X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.
  • Y. Zhou and F. Jiao, “Nonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 4465–4475, 2010.
  • M. Feng, X. Liu, and H. Feng, “The existence of positive solution to a nonlinear fractional differential equation with integral boundary conditions,” Advances in Difference Equations, vol. 2011, Article ID 546038, 14 pages, 2011.
  • M. Feng, X. Zhang, and W. Ge, “New existence results for higher-order nonlinear fractional differential equation with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 720702, 20 pages, 2011.
  • J. Jiang, L. Liu, and Y. Wu, “Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions,” Applied Mathematics and Computation, vol. 215, no. 4, pp. 1573–1582, 2009.
  • X. Zhang, M. Feng, and W. Ge, “Existence result of second-order differential equations with integral boundary conditions at resonance,” Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 311–319, 2009.
  • T. Jankowski, “Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 73, no. 5, pp. 1289–1299, 2010.
  • M. Benchohra, J. J. Nieto, and A. Ouahab, “Second-order boundary value problem with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 260309, 9 pages, 2011.
  • H. A. H. Salem, “Fractional order boundary value problem with integral boundary conditions involving Pettis integral,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 661–672, 2011.
  • B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis. Real World Applications, vol. 9, no. 4, pp. 1727–1740, 2008.
  • A. Cabada and G. Wang, “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 403–411, 2012.
  • J. Harjani and K. Sadarangani, “Fixed point theorems for weakly contractive mappings in partially ordered sets,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 71, no. 7-8, pp. 3403–3410, 2009.
  • J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005.
  • D. O'Regan and A. Petruşel, “Fixed point theorems for generalized contractions in ordered metric spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 1241–1252, 2008.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  • G. A. Anastassiou, Fractional Differentiation Inequalities, Springer, Dordrecht, The Netherlands, 2009.
  • J. Caballero, J. Harjani, and K. Sadarangani, “Uniqueness of positive solutions for a class of fourth-order boundary value problems,” Abstract and Applied Analysis, vol. 2011, Article ID 543035, 13 pages, 2011.