Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations

Volume 8, Issue 2, pp 110--129 http://dx.doi.org/10.22436/jnsa.008.02.04

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1883 Downloads
• 3256 Views

Authors

Wengui Yang - Ministry of Public Education, Sanmenxia Polytechnic, Sanmenxia, Henan 472000, China.

Abstract

In this paper, we study the existence of positive solutions for a class of coupled integral boundary value problems of nonlinear semipositone Hadamard fractional differential equations \[D^\alpha u(t) + \lambda f(t, u(t), v(t)) = 0,\quad D^\alpha v(t) + \lambda g(t, u(t), v(t)) = 0,\quad t \in (1, e),\quad \lambda > 0\] \[u^{(j)}(1) = v^{(j)}(1) = 0, 0 \leq j \leq n - 2; u(e) = \mu\int^e_1 v(s) \frac{ds}{ s} , v(e) = \nu\int^e_1 u(s) \frac{ds}{ s},\] where \(\lambda,\mu,\nu\) are three parameters with \(0<\mu<\beta\) and \(0<\nu<\alpha,\quad \alpha,\beta\in (n - 1; n]\) are two real numbers and \(n\geq 3, D^\alpha, D^\beta\) are the Hadamard fractional derivative of fractional order, and \(f; g\) are sign-changing continuous functions and may be singular at \(t = 1\) or/and \(t = e\). First of all, we obtain the corresponding Green's function for the boundary value problem and some of its properties. Furthermore, by means of the nonlinear alternative of Leray-Schauder type and Krasnoselskii's fixed point theorems, we derive an interval of \(\lambda\) such that the semipositone boundary value problem has one or multiple positive solutions for any \(\lambda\) lying in this interval. At last, several illustrative examples were given to illustrate the main results.

Share and Cite

Keywords

• Hadamard fractional differential equations
• coupled integral boundary conditions
• positive solutions
• Green's function
• fixed point theorems.

MSC

•  34A08
•  34B16
•  34B18

References

• [1] R. P. Agarwal, V. Lakshmikantham, J. J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonliear Anal., 72 (2010), 2859-2862.


• [2] R. P. Agarwal, M. Meehan, D. ORegan, Fixed Point Theory and Applications, Cambridge University Press, (2001)


• [3] B. Ahmad, A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations, Fixed Point Theory Appl., Art. ID 364560, 2010 (2010), 17 pages.


• [4] B. Ahmad, J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal., Art. ID 494720, 2009 (2009), 9 pages.


• [5] B. Ahmad, J. J. Nieto, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions , Comput. Math. Appl., 58 (2009), 1838-1843.


• [6] B. Ahmad, J. J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011 (2011), 36


• [7] B. Ahmad, S. K. Ntouyas, On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput., doi: 10.1007/s12190-014-0765-6. (2014)


• [8] B. Ahmad, S. K. Ntouyas, A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations, Fract. Calc. Appl. Anal., 17 (2014), 348-360.


• [9] B. Ahmad, S. K. Ntouyas , On three-point Hadamard-type fractional boundary value problems, Int. Electron. J. Pure Appl. Math., 8 (4) (2014), 31-42.


• [10] B. Ahmad, S. K. Ntouyas, A. Alsaedi, New results for boundary value problems of Hadamard-type fractional differential inclusions and integral boundary conditions, Bound. Value Probl., 2013 (2013), 275


• [11] A. Anguraj, M. L. Maheswari, Existence of solutions for fractional impulsive neutral functional infinite delay integrodifferential equations with nonlocal conditions, J. Nonlinear Sci. Appl. , 5 (2012), 271-280.


• [12] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl., 269 (2002), 387-400.


• [13] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl., 269 (2002), 1-27.


• [14] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270 (2002), 1-15.


• [15] Y. Cui, L. Liu, X. Zhang, Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems, Abstr. Appl. Anal., Art. ID 340487, 2013 (2013), 9 pages.


• [16] A. Debbouche, J. J. Nieto, Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls, Appl. Math. Comput., 245 (2014), 74-85.


• [17] Y. Gambo et al., On Caputo modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2014 (2014), 10


• [18] C. S. Goodrich, Existence of a positive solution to systems of differential equations of fractioanl order, Comput. Math. Appl., 62 (2011), 1251-1268.


• [19] A. Guezane-Lakoud, R. Khaldi, Solvability of a two-point fractional boundary value problem, J. Nonlinear Sci. Appl., 5 (2012), 64-73.


• [20] M. Hao, C. Zhai , Application of Schauder fixed point theorem to a coupled system of differential equations of fractional order, J. Nonlinear Sci. Appl., 7 (2) (2014), 131-137.


• [21] J. Henderson, R. Luca, Positive solutions for a system of nonlocal fractional boundary value problems, Fract. Calc. Appl. Anal., 16 (2013), 985-1008.


• [22] F. Jarad, T. Abdeljawad, D. Baleanu, Caputo-type modification of the Hadamard fractional derivatives, Adv. Difference Equ., 2012 (2012), 142


• [23] J. Jiang, L. Liu, Y. Wu, Positive solutions to singular fractional differential system with coupled boundary conditions, Commun. Nonlinear Sci. Numer. Simulat., 18 (2013), 3061-3074.


• [24] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Boston (2006)


• [25] M. A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff Gronigen, Netherland (1964)


• [26] A. Leung, A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability, Indiana Univ. Math., 31 (1982), 223-241.


• [27] Y. Li, H. Zhang, Positive solutions for a nonlinear higher order differential system with coupled integral boundary conditions, J. Appl. Math., Art. ID 901094, 2014 (2014), 7 pages.


• [28] Y. Liu, H. Shi, Existence of unbounded positive solutions for BVPs of singular fractional differential equations, J. Nonlinear Sci. Appl., 5 (2012), 281-293.


• [29] S. Liu, G. Wang, L. Zhang, Existence results for a coupled system of nonlinear neutral fractional differential equations, Appl. Math. Lett., 26 (2013), 1120-1124.


• [30] J. A. Nanware, D. B. Dhaigude, Existence and uniqueness of solutions of differential equations of fractional order with integral boundary conditions, J. Nonlinear Sci. Appl., 7 (4) (2014), 246-254.


• [31] T. Qiu, Z. Bai, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Nonlinear Sci. Appl., 1 (3) (2008), 123-131.


• [32] C. V. Pao, Finite difference reaction-diffusion systems with coupled boundary conditions and time delays, J. Math. Anal. Appl., 272 (2002), 407-434.


• [33] M. Rehman, R. Khan, A note on boundary value problems for a coupled system of fractional differential equations, Comput. Math. Appl., 61 (2011), 2630-2637.


• [34] P. Thiramanus, S. K. Ntouyas, J. Tariboon, Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal., Art. ID 902054, 2014 (2014), 9 pages.


• [35] Y. Wang, L. Liu, Y. Wu, Positive solutions for a class of higher-order singular semipositone fractional differential systems with coupled integral boundary conditions and parameters, Adv. Differ. Equ, 2014 (2014), 268


• [36] W. Yang, Positive solutions for a coupled system of nonlinear fractional differential equations with integral boundary conditions, Comput. Math. Appl., 63 (2012), 288-297.


• [37] W. Yang, Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions, J. Appl. Math. Comput., 44 (2014), 39-59.


• [38] W. Yang, Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions, Appl. Math. Comput., 244 (2014), 702-725.


• [39] W. Yang, Positive solutions for singular Hadamard fractional differential system with four-point coupled boundary conditions, J. Appl. Math. Comput., doi: 10.1007/s12190-014-0843-9. (2014)


• [40] C. Yuan, Two positive solutions for (n - 1; 1)-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 930- 942.


• [41] C. Yuan et al., Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions, Electron. J. Qual. Theory Differ. Equ., 13 (2012), 1-17.


• [42] Y. Zou, Y. Cui, Monotone iterative method for differential systems with coupled integral boundary value problems, Bound. Value Probl., 2013 (2013), 245


• [43] Y. Zou, L. Liu, Y. Cui , The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance, Abstr. Appl. Anal., Art. ID 314083, 2014 (2014), 8 pages.