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

Volume 8, Issue 2, pp 110--129 Publication Date: March 28, 2015
• 454 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.

### Keywords

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

•  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.