A Sum Operator Method for Unique Positive Solution to a Class of Boundary Value Problem of Nonlinear Fractional Differential Equation
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:71683", author = "Fengxia Zheng and Chuanyun Gu", title = "A Sum Operator Method for Unique Positive Solution to a Class of Boundary Value Problem of Nonlinear Fractional Differential Equation", 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.", volume = "9", number = "8", pages = "498-4", }
