Positive solutions for singular coupled integral boundary value problems of nonlinear Hadamard fractional differential equations
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.
References
[1] R.P. Agarwal, V. Lakshmikantham and 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 and A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher-order nonlinear fractional differential equations, Fixed Point Theory Appl. 2010 (2010), Art. ID 364560, 17 pp.
[4] B. Ahmad and J. Nieto, Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations, Abstr. Appl. Anal. 2009 (2009), Art. ID 494720, 9pp.
[5] B. Ahmad and 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 and J.J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl. 2011 (2011) 36.
[7] B. Ahmad and S.K. Ntouyas, On Hadamard fractional integro-differential boundary value problems, J. Appl. Math. Comput. (2014), doi: 10.1007/s12190-014-0765-6.
[8] B. Ahmad and 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 and 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 and 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 and 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 and 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 and 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 and X. Zhang, Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems, Abstr. Appl. Anal. 2013 (2013), Art. ID 340487, 9 pages.
[16] A. Debbouche and 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 and R. Khaldi, Solvability of a two-point fractional boundary value problem, J. Nonlinear Sci. Appl. 5 (2012), 64-73.
[20] M. Hao and 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, and 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. J. 31 (1982), 223-241.
[27] Y. Li, and H. Zhang, Positive solutions for a nonlinear higher order differential system with coupled integral boundary conditions, J. Appl. Math. 2014 (2014), Art. ID 901094, 7 pages.
[28] Y. Liu and 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 and 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 and 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, and J. Tariboon, Existence and uniqueness results for Hadamard-type fractional differential equations with nonlocal fractional integral boundary conditions, Abstr. Appl. Anal. 2014 (2014), Art. ID 902054, 9 pages.
[35] Y. Wang, L. Liu and 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. (2014), doi: 10.1007/s12190-014-0843-9.
[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 and 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, and Y. Cui, The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance, Abstr. Appl. Anal. 2014 (2014), Art. ID 314083, 8 pages.