New existence results for positive solutions of boundary value problems for coupled systems of multi-term fractional differential equations

Yıl 2016, Cilt: 45 Sayı: 2, 391 - 416, 01.04.2016

Yuji Liu

Öz

In this article, we establish some new existence results on positive solutions of a boundary value problem of coupled systems of nonlinear
multi-term fractional differential equations. Our analysis rely on the
well known fixed point theorems. Numerical examples are given to
illustrate the main theorems.

Anahtar Kelimeler

Four-point boundary value problem, multi-term fractional differential system, non-Caratheodory function, fixed-point theorem

Kaynakça

• Ahmad, A. and Nieto, J.J. Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. Math. Appl. 58, 1838- 1843, 2009.
• Ahmad, B. and Sivasundaram, S. On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Appl. Math. Comput. 217, 480-487, 2010.
• Avery R.I., and Peterson, A.C. Three Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces, Comput. Math. Appl. 42, 313-322, 2001.
• Basset, A.B. On the descent of a sphere in a vicous liquid, Q. J. Pure Appl. Math. 41, 369-381, 1910.
• Bai, C. and Fang, J. The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150, (3), 611-621, 2004.
• Caponetto, R., Dongona, G. and Fortuna, L. Fractional Differential Systems, Modeling and control applications, World Scientific Series on Nonlinear Science, Ser. A, 72, World Scientific Publishing Co. Pte. Ltd. 2010.
• Duan, J. and Temuer, C. Solution for system of linear fractional differential equations with constant coefficients, Journal of Mathematics, 29, 599-603, 2009.
• Gaber, M. and Brikaa, M.G. Existence results for a coupled system of nonlinear fractional differential equation with four-point boundary conditions, ISRN Mathematical Analysis, 2011, Article ID 468346, 14 pages, 2011.
• Goodrich, C.S. Existence of a positive solution to systems of differential equations of fractional order, Comput. Math. Appl. 62, 1251-1268, 2011.
• Liu, B. Positive solutions of a nonlinear four-point boundary value problems in Banach spaces, J. Math. Anal. Appl. 305, 253-276, 2005.
• Liu, L., Zhang, X. and Wu, Y. On existence of positive solutions of a two-point boundary value problem for a nonlinear singular semipositone system, Appl. Math. Comput. 192, 223-232, 2007.
• Mainardi, F. Fraction Calculus: Some basic problems in continuum and statistical machanics, In: A. Carpinteri, F. Mainardi (eds.) Fratals and Fractional Calculus in Continuum Machanics, (Springer, Vien, 1997), 291-348.
• Miller, K.S. and Ross, B. An Introduction to the Fractional Calculus and Fractional Differential Equation, (Wiley, New York, 1993.)
• Mamchuev, M.O. Boundary value problem for a system of fractional partial differential equations, Partial Differential Equations, 44, 1737-1749, 2008.
• Rehman, M. and Khan, R. A note on boundaryvalueproblems for a coupled system of fractionaldifferential equations, Comput. Math. Appl. 61, 2630-2637, 2011.
• Wang, J., Xiang, H. and Liu, Z. Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations, International Journal of Differential Equations, 2010, Article ID 186928, 12 pages, 2010.
• Su, X. Boundary value problem for a coupled system of nonlinear fractional differential equations, Applied Mathematics Letters 22 (1), 64-69, 2009.
• Torvik, P.J., and Bagley, R.L. On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mech. 51, 294-298, 1984.
• Trujillo, J.J., Rivero, M. and Bonilla, B. On a Riemann-Liouville generalized Taylor’s formula, J. Math. Anal. Appl. 231, 255-265, 1999.
• Yuan, C. Multiple positive solutions for (n-1, 1)-type semipositone conjugate boundary value problems for coupled systems of nonlinear fractional differential equations, E. J. Qualitative Theory of Diff. Equ. 13 1-12, 2012.
• Yang, A. and Ge, W. Positive solutions for boundary value problems of N-dimension nonlinear fractional differential systems, Boundary Value Problems 2008, article ID 437453, 2008.
• Yuan, C., Jiang, D., O0 regan, D. and Agarwal, R.P. Existence and uniqueness of positive solutions of boundary value problems for coupled systems of singular second-order threepoint non-linear differential and difference equations, Appl. Anal. 87 921-932, 2008.
• Yuan, C., Jiang, D., O0 regan, D. and Agarwal, R.P. Multiple positive solutions to systems of nonlinear semipositone fractional differential equations with coupled boundary conditions, E. J. Qualitative Theory of Diff. Equ. 13 (2012), 1-17, 2012.
• Zou, Y., Liu, L. and Cui, Y. The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance, Abstract and Applied Analysis 2014, Article ID 314083, 8 pages, 2014.
• Ji, D. and Ge, W. Positive solution for four-point nonlocal boundary value problems of fractional order, Mathematical Methods in the Applied Sciences 37 (8), 1232-1239, 2014.
• Zhao, J., Geng, F. and Ge, W. Positive solutions to a new kind Sturm-Liouville-like fourpoint boundary value problem, Applied Mathematics and Computation 217 (2), 811-819, 2010.
• Zhao, X., Chai, C. and Ge, W. Positive solutions for fractional four-point boundary value problems, Comm. Nonl. Sci. Numer. Simul. 16 (9), 3665-3672, 2011.