Journal of Advances in Applied Mathematics

Download PDF (709.8 KB) PP. 241 - 259 Pub. Date: October 1, 2016

DOI: 10.22606/jaam.2016.14005

• Bhuvaneswari Sambandham
  Department of Mathematics, Southern Utah University, Cedar city, United States
• Aghalaya S Vatsala^*
  Department of Mathematics, University of Louisiana at Lafayette, Lafayette, United States

Generalized monotone method together with coupled lower and upper solutions yield monotone sequences which converge uniformly and monotonically to coupled minimal and maximal solutions of the nonlinear problem under consideration. In this work, we have developed generalized monotone method for sequential Caputo fractional boundary value problem with mixed boundary conditions which are in terms of Caputo fractional derivative. For that purpose, we have obtained a representation form for the corresponding linear Caputo sequential boundary value problem in terms of the Green’s function. In addition, we have obtained a linear comparison result for the linear sequential differential inequalities with linear mixed boundary conditions. The comparison result is useful in proving the monotonicity of the iterates as well as the uniqueness of the solution of the nonlinear sequential boundary value problem. Our method yields the integer results as a special case. Some numerical examples for the linear sequential Caputo fractional boundary value problems have also been presented.

Sequential Caputo fractional derivative, fractional boundary value problem, Green’s function, coupled lower and upper solutions.

[1] I. Podlubny, Fractional Differential Equations, vol. 198, Academics Press, San Diego, 1999.

[2] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.

[3] V. Lakshmikantham, S. Leela, and D.J. Vasundhara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.

[4] G.S. Ladde, V. Lakshmikantham and A. S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman publishing Inc, 1985.

[5] V. Lakshmikantham, A. S. Vatsala, General Uniqueness and Monotone Iterative Technique for Fractional Differential Equations, Applied Mathematics Letters, vol. 21, no. 8, pp. 828aAS834, 2008.

[6] Alberto Cabada, Juan J. Nieto, A Generalization of Monotone Iterative Techniques for Nonlinear Second Order Periodic Boundary Value Problems, Journal of Mathematical Analysis and Applications, 151, 181-189(1990).

[7] I. H.West, A. S. Vatsala, Generalized Monotone Iterative Method for Initial Value Problems, Applied Mathematics Letter, 17, 1231-1237(2004).

[8] Z. Denton, A. S. Vatsala, Monotone Iterative Technique for Finite System of Nonlinear Riemann–Liouville Differential Equations, Opuscula Mathematica, 31(3):327-339, (2011).

[9] J. D. Ramirez, A.S.Vatsala Generalized Monotone Iterative Technique for Caputo Fractional Differential Equation with Periodic Boundary Condition via Initial Value Problem, International Journal of Differential Equations, 17, 842813(2012).

[10] J. D. Ramirez, A. S. Vatsala, Generalized Monotone Method for Caputo Fractional Differential Equation with Periodic Boundary Conditions. In Proceedings of Neural, Parallel, and Scientific Computations, vol. 4, pp. 332-337, Dynamic, Atlanta, Ga, USA, 2010.

[11] Wenli Wanga, Jingfeng Tianb Generalized Monotone Iterative Method for Integral Boundary Value Problems with Causal Operators, J. Nonlinear Sci, Appl. 8 (2015), 600-609.

[12] Youzheng Ding, Zhongli Wei, Qingli Zhao Solutions for a Nonlinear Fractional Boundary Value Problem with Sign-Changing Green’s Function, J. Nonlinear Sci, Appl. 8 (2015), 650-659.

[13] Sutthirut Charoenphon, Green’s Functions of Discrete Fractional Calculus Boundary Value Problems and an Application of Discrete Fractional Calculus to a Pharmacokinetic Model, Masters Thesis and Specialist Projects, Paper 1328,(2014).

[14] Yujun Cui, Yumei Zou, Existence Results and the Monotone Iterative Technique for Nonlinear Fractional Differential Systems with Coupled Four-Point Boundary Value Problems, Abstract and Applied Analysis, Volume 2014, Article ID 242-591, 6 pages.

[15] Rahmat Darzi, Bahar Mohammadzadeh, New Existence Results to Solution of Fractional Boundary Value Problems, Applications and Applied Mathematics, Vol. 8, Issue 2 (December 2013), pp. 535-552.

[16] Jie Zhou, Meiqiang Feng, GreenaAZs Function for Sturm-Liouville-type Boundary Value Problems of Fractional Order Impulsive Differential Equations and its Applications, Boundary Value Problems 2014, 2014:69.

[17] F. Mainardi, The Fundamental Solutions for the Fractional Diffusion-Wave Equation, Appl. Math. Lett, 9(6),(1996),(23-28).

[18] Wenquan Feng, Shurong Sun, Ying Sun, Existence of Positive Solutions for a Generalized and Fractional Ordered Thomas-Fermi Theory of Neutral Atoms, Advances in Difference Equations 2015, 2015:350.

[19] Shuqin Zhang, Positive Solutions for Boundary Value Problems of Nonlinear Fractional Differential Equations, Electronic Journal of Differential Equations, 2006, 36, pp(1-12), Vol. 2006(2006).

[20] Nanware J.A, D.B Dhaigude, Existence of Uniqueness of Solutions of Riemann-Liouville Fractional Differential Equations with Integral Boundary Conditions, International Journal of Nonlinear Science Vol 14 (2012) 4, pp.410-415.

[21] Mouffak Benchohra, Benaouda Heida, Positive Solutions for Boundary Value Problems with Fractional Order, International Journal of Advanced Mathematical Science, (1) (2013), 12-22.

[22] Fuquan Jiang, Xiaojie Xu, Zhongwei Cao, The Positive Solutions of Green’s Functions for Fractional Differential Equations and its Applications, Abstract and Applied Analysis, vol 2103, 531038, 12.

[23] Ma, Ruyun, Positive solutions of a nonlinear three-point boundary-value problem, Electron. J. Differential Equations, vol 34, (1999).

[24] Webb, JRL, Positive solutions of some three point boundary value problems via fixed point index theory, Nonlinear Analysis: Theory, Methods & Applications, vol 47,(7) (2001).

[25] Liang, Sihua and Zhang, Jihui, Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear Analysis: Theory, Methods & Applications, vol 71,(11) (2009).