A Sum Operator Method for Unique Positive Solution to a Class of Boundary Value Problem of Nonlinear Fractional Differential Equation

Authors: Fengxia Zheng, Chuanyun Gu

Abstract:

By using a fixed point theorem of a sum operator, the existence and uniqueness of positive solution for a class of boundary value problem of nonlinear fractional differential equation is studied. An iterative scheme is constructed to approximate it. Finally, an example is given to illustrate the main result.

Keywords: Fractional differential equation, Boundary value problem, Positive solution, Existence and uniqueness, Fixed point theorem of a sum operator.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1110630

References:

[1] K. Diethelm, AD. Freed; On the solutions of nonlinear fractional order differential equations used in the modelling of viscoplasticity, Scientific computing in chemical engineering II C computational fluid dynamics,reaction engineering and molecular properties, 1999,217-224.
[2] X. Ding, Y. Feng, R. Bu; Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations, J Appl Math Comput, 40 (2012), 371-381.
[3] CS. Goodrich; Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055.
[4] D. Guo, V. Lakshmikantham; Nonlinear Problems in Abstract Cones., Boston and New York: Academic Press Inc, 1988.
[5] WG. Glockle, TF. Nonnenmacher; A fractional calculus approach of selfsimilar protein dynamics, Biophys J, 68 (1995), 46-53.
[6] D. Jiang, C. Yuan; The positive properties of the Green function for Dirichlet-type boundary value problems of nonlinear fractional differential equations and its application, Nonlinear Analysis, 72 (2010), 710-719.
[7] AA. Kilbas, HM. Srivastava, JJ. Trujillo; Theory and applications of fractional differential equations, North-Holland mathematics studies, 2006,204.
[8] S. Liang, J. Zhang; Existence and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem, Comput Math Appl, 62 (2011), 1333-1340.
[9] F. Mainardi; Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 1997, 291-348.
[10] KS. Miller, B. Ross; An introduction to the fractional calculus and fractional differential equations, John Wiley, New York,1993.
[11] KB. Oldham, J. Spanier; The fractional calculus, Williams and Wilkins, New York: Academic Press, 1974.
[12] I. Podlubny; Fractional differential equations, mathematics in science and engineering, New York: Academic Pres, 1999.
[13] EM. Rabei, KI. Nawaeh, RS. Hijjawi, SI. Muslih, D. Baleanu; The Hamilton formalism with fractional derivatives, J Math Anal Appl, 327 (2007), 891-897.
[14] SG. Samko, AA. Kilbas. OI Marichev; Fractional integral and derivatives: theory and applications, Gordon and Breach, Switzerland, 1993.
[15] X.Yang, Z. Wei, W. Dong; Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations, Commun Nonlinear Sci Numer Simulat, 17 (2012), 85-92.
[16] C. Yang, C. Zhai; Uniquess of positive solutions for a fractional differdential equations via a fixed point theorem of a sum operator, Electronic Journal of Differential Equations, 2012 (70) (2012), 1-8.
[17] C. Zhai, DR. Anderson; A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations,J. Math. Anal. Appl. 375 (2011), 388-400.
[18] C. Zhai, M. Hao; Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems, Boundary Value Problems, 2013 (2013),85.
[19] Y. Zhao, S. Sun, Z. Han; The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations, Commun Nonlinear Sci Numer Simulat, 16 (2011), 2086-2097.
[20] C. Zhai, W. Yan, C. Yang; A sum operator method for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems, Commun Nonlinear Sci Numer Simulat, 18 (2013), 858-866.