132 Downloads 421 Views
Author(s)
Fengxia Zheng, Sichuan University of Arts and Science, Sichuan Dazhou, China
Abstract
In this paper, by introducing the concept of a generalized concave operator and the properties of cone and monotone iterative technique in ordered Banach spaces, some new existence and uniqueness theorems of fixed points for the operator under more extensive conditions are obtained. Finally, as applications, we apply the results obtained in this paper to study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems.
Keywords
fixed point; generalized concave operator; normal cone; positive solution; fractional differential equation; boundary value problems
Cite this paper
Fengxia Zheng,
Fixed point theorems for generalized concave operators and applications to fractional differential equation boundary value problems, SCIREA Journal of Mathematics. Vol.
2
, No.
3
,
2017
, pp.
41

54
.
References
[ 1 ]  A. Tarski, A latticetheoretical fixed point theorem and its applications, Pacific J. Math. 5(1955) 285–309. 
[ 2 ]  H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (4) (1976) 620–709. 
[ 3 ]  D.J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press Inc., Boston, New York, 1988. 
[ 4 ]  M.A. Krasnosel’skii, Positive Solutions of Operators Equations, Noordoff, Groningen, 1964. 
[ 5 ]  W.X. Wang, Z.D. Liang, Fixed point theorems of a class of nonlinear operators and applications, Acta Math. Sin. 48 (4) (2005) 789–800 (in Chinese). 
[ 6 ]  Z.D. Liang, W.X. Wang, S.J. Li, On concave operators, Acta Math. Sin. (Engl. Ser.) 22 (2) (2006) 577–582. 
[ 7 ]  Z. Zhao, X. Du, Fixed points of generalized econcave (generalized econvex) operators and their applications, J. Math. Anal. Appl. 334 (2007)1426–1438. 
[ 8 ]  Potter A.J.B., Applications of Hilbert’s projective metric to certain classes of nonHomogeneous operators, Quart.J.Math.Oxford, 1977, 28(2):9399. 
[ 9 ]  C. Zhai, W. Wang, L. Zhang, Generalizations for a class of concave and convex operators, ACTA MATHEMATICA SINICA, Chinese Series,2008, 51 (3):529540.(in Chinese) 
[ 10 ]  C.B. Zhai and X.M. Cao, “Fixed point theorems for concave operators and applications,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 532–538, 2010. 
[ 11 ]  C.B. Zhai, C.M. Guo, On αconvex operators, J. Math. Anal. Appl. 316 (2006) 556–565. 
[ 12 ]  F.Y. Li, Z.D. Liang, Fixedpoint theorems of concave( convex) operator and application,J.Sys.Sci.&Math.Scis.14(4)(1994),355360. (in Chinese) 
[ 13 ]  C.B. Zhai, Y.J. Li, Fixed point theorems of concave operators and applications, Acta Mathematica Scientia, 2008, 28A (6):10231028y. (in Chinese) 
[ 14 ]  A.A. Kilbas, O.I. Marichev, S.G. Samko, Fractional Integral and Derivatives (Theory and Applications), Gordon and Breach, Switzerland, 1993. 
[ 15 ]  A.A. Kilbas, J.J. Trujillo, Differential equations of fractional order: methods, results and problems I, Appl.Anal. 78 (2001) 153–192. 
[ 16 ]  K.S. Miller, Fractional differential equations, J. Fract. Calc. 3 (1993) 49–57. 
[ 17 ]  K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. 
[ 18 ]  I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press,New York, 1999. 
[ 19 ]  S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integral and Derivatives, in: Theory and Applications, Gordon and Breach, Switzerland, 1993. 
[ 20 ]  Bai Z.B., Boundary value problem of fractional differential equation theory and Application.China Science and Technology Press, Beijing, 2013. (in Chinese). 
[ 21 ]  Bai Z, Lü H. Positive solutions for boundary value problem of nonlinear fractional differential equation. J Appl Math 2005; 311:495–505. 
[ 22 ]  Jiang D, Yuan C. The positive properties of the Green function for Dirichlettype boundary value problems of nonlinear fractional differential equations and its application. Nonlinear Analysis 2010; 72:710–719. 
[ 23 ]  Liang S, Zhang J. Existence and uniqueness of strictly nondecreasing and positive solution for a fractional threepoint boundary value problem. Comput Math Appl 2011; 62:1333–40. 
[ 24 ]  Yang X, Wei Z, Dong W. Existence of positive solutions for the boundary value problem of nonlinear fractional differential equations. Commun Nonlinear Sci Numer Simulat 2012;17:85–92. 
[ 25 ]  Ding X , Feng Y, Bu R. Existence, nonexistence and multiplicity of positive solutions for nonlinear fractional differential equations. J Appl Math Comput 2012; 40:371–381. 