# Positive solutions for a system of nonlinear semipositone fractional $q$-difference equations with $q$-integral boundary conditions

Volume 10, Issue 8, pp 4430--4440 Publication Date: August 27, 2017       Article History
• 611 Views

### Authors

Wei Cheng - School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China. Jiafa Xu - School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, China. Yujun Cui - State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province, China. - Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China.

### Abstract

In this paper, by virtue of fixed point index on cones, we obtain two existence theorems of positive solutions for a system of nonlinear semipositone fractional $q$-difference equations with $q$-integral boundary conditions. Concave functions and nonnegative matrices are used to characterize the coupling behavior of our nonlinearities.

### Keywords

• q-difference equation
• q-integral boundary conditions
• fixed point index
• positive solution
• concave function.

•  34B10
•  34B18
•  34A34
•  45G15
•  45M20

### References

• [1] R. P. Agarwal, Certain fractional q-integrals and q-derivatives, Proc. Cambridge Philos. Soc., 66 (1969), 365–370.

• [2] Y. Cui, Y. Zou, An existence and uniqueness theorem for a second order nonlinear system with coupled integral boundary value conditions, Appl. Math. Comput., 256 (2015), 438–444.

• [3] R. Dahal, D. Duncan, C. S. Goodrich, Systems of semipositone discrete fractional boundary value problems, J. Difference Equ. Appl., 20 (2014), 473–491.

• [4] R. A. C. Ferreira, Nontrivial solutions for fractional q-difference boundary-value problems, Electron. J. Qual. Theory Differ. Equ., 2010 (2010), 10 pages.

• [5] R. A. C. Ferreira, Positive solutions for a class of boundary value problems with fractional q-differences, Comput. Math. Appl., 61 (2011), 367–373.

• [6] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston (1988)

• [7] J. Henderson, R. Luca, Existence of positive solutions for a system of semipositone fractional boundary value problems, Electron. J. Qual. Theory Differ. Equ., 2016 (2016), 28 pages.

• [8] 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.

• [9] V. Kac, P. Cheung, Quantum Calculus, Springer, New York (2002)

• [10] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam (2006)

• [11] B. Li, S. Sun, Z. Han , Positive solutions for singular fractional differential equations with three-point boundary conditions, J. Appl. Math. Comput., 52 (2016), 477–488.

• [12] Y. Li, Z. Wei, Positive solutions for a coupled systems of mixed higher-order nonlinear singular fractional differential equations, Fixed Point Theory, 15 (2014), 167–178.

• [13] Y. Liu, New existence results for positive solutions of boundary value problems for coupled systems of multi-term fractional differential equations, Hacet. J. Math. Stat., 45 (2016), 391–416.

• [14] R. Luca, A. Tudorache, Positive solutions to a system of semipositone fractional boundary value problems, Adv. Difference Equ., 2014 (2014), 11 pages.

• [15] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego (1999)

• [16] P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in q-calculus, Appl. Anal. Discrete Math., 1 (2007), 311–323.

• [17] S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, USA (1993)

• [18] X. Su, Boundary value problem for a coupled systemof nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64–69.

• [19] Y. Wang, L. Liu, X. Zhang, Y. Wu, Positive solutions of an abstract fractional semipositone differential system model for bioprocesses of HIV infection, Appl. Math. Comput., 258 (2015), 312–324.

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

• [21] 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. Simul., 17 (2012), 930–942.

• [22] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a singular fractional differential system involving derivatives, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 1400–1409.

• [23] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26–33.

• [24] X. Zhang, L. Liu, Y. Wu, B. Wiwatanapataphee, The spectral analysis for a singular fractional differential equation with a signed measure, Appl. Math. Comput., 257 (2015), 252–263.

• [25] K. Zhang, J. Xu, D. O’Regan, Positive solutions for a coupled system of nonlinear fractional differential equations, Math. Meth. Appl. Sci., 38 (2015), 1662–1672.