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Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. (English) Zbl 1253.47037
The authors study the existence and uniqueness of positive fixed points for mixed monotone operators with perturbation. They consider the existence and uniqueness of positive solutions for the operator equation A(x,x)+B(x)=x in ordered Banach spaces, where A is a mixed monotone operator and B is an increasing sub-homogeneous or α-concave operator. In fact, using the properties of cones and a fixed point theorem for mixed monotone operators, the authors obtain existence and uniqueness results for the above equation without assuming the operators to be continuous or compact. They also give some applications to boundary value problems for nonlinear fractional differential equations.
MSC:
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H07Monotone and positive operators on ordered topological linear spaces
34A08Fractional differential equations
47N20Applications of operator theory to differential and integral equations
References:
[1]Guo, D.; Lakskmikantham, V.: Coupled fixed points of nonlinear operators with applications, Nonlinear anal. 11, No. 5, 623-632 (1987) · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0
[2]Guo, D.: Fixed points of mixed monotone operators with application, Appl. anal. 34, 215-224 (1988) · Zbl 0688.47019 · doi:10.1080/00036818808839825
[3]Chen, Y.: Thompson’s metric and mixed monotone operators, J. math. Anal. appl. 177, 31-37 (1993) · Zbl 0804.47054 · doi:10.1006/jmaa.1993.1241
[4]Zhang, Z.: New fixed point theorems of mixed monotone operators and applications, J. math. Anal. appl. 204, 307-319 (1996) · Zbl 0880.47036 · doi:10.1006/jmaa.1996.0439
[5]Zhang, S.; Ma, Y.: Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solution for a class of functional equations arising in dynamic programming, J. math. Anal. appl. 160, 468-479 (1991) · Zbl 0753.47029 · doi:10.1016/0022-247X(91)90319-U
[6]Sun, Y.: A fixed point theorem for mixed monotone operator with applications, J. math. Anal. appl. 156, 240-252 (1991) · Zbl 0761.47040 · doi:10.1016/0022-247X(91)90394-F
[7]Liang, Z.; Zhang, L.; Li, S.: Fixed point theorems for a class of mixed monotone operators, Z. anal. Anwend. 22, No. 3, 529-542 (2003) · Zbl 1065.47060 · doi:10.4171/ZAA/1160
[8]Wu, Y.; Li, G.: On the fixed point existence and uniqueness theorems of mixed monotone operators and applications, Acta math. Sinica 46, No. 1, 161-166 (2003) · Zbl 1027.47062
[9]Li, K.; Liang, J.; Xiao, T.: New existence and uniqueness theorems of positive fixed points for mixed monotone operators with perturbation, J. math. Anal. appl. 328, 753-766 (2007) · Zbl 1115.47044 · doi:10.1016/j.jmaa.2006.04.069
[10]Lian, X.; Li, Y.: Fixed point theorems for a class of mixed monotone operators with applications, Nonlinear anal. 67, 2752-2762 (2007) · Zbl 1144.47043 · doi:10.1016/j.na.2006.09.040
[11]Wu, Y.; Liang, Z.: Existence and uniqueness of fixed points for mixed monotone operators with applications, Nonlinear anal. 65, 1913-1924 (2006) · Zbl 1111.47049 · doi:10.1016/j.na.2005.10.045
[12]Wang, W.; Liu, X.; Cheng, S.: Fixed points for mixed monotone operators and applications, Nonlinear stud. 14, No. 2, 189-204 (2007) · Zbl 1142.47034
[13]Zhang, Z.; Wang, K.: On fixed point theorems of mixed monotone operators and applications, Nonlinear anal. 70, 3279-3284 (2009) · Zbl 1219.47072 · doi:10.1016/j.na.2008.04.032
[14]Lakshmikantham, V.; Ciric, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear anal. 70, 4341-4349 (2009) · Zbl 1176.54032 · doi:10.1016/j.na.2008.09.020
[15]Zhao, Z.: Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear anal. 73, 1481-1490 (2010) · Zbl 1229.47082 · doi:10.1016/j.na.2010.04.008
[16]Harjani, J.; López, B.; Sadarangani, K.: Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear anal. 74, 1749-1760 (2011) · Zbl 1218.54040 · doi:10.1016/j.na.2010.10.047
[17]Zhai, C. B.; Zhang, L. L.: New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems, J. math. Anal. appl. 382, 594-614 (2011) · Zbl 1225.47074 · doi:10.1016/j.jmaa.2011.04.066
[18]Lin, X.; Jiang, D.; Li, X.: Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. comput. Appl. math. 196, 155-161 (2006) · Zbl 1107.34307 · doi:10.1016/j.cam.2005.08.016
[19]Yuan, C.; Jiang, D.; Zhang, Y.: Existence and uniqueness of solutions for singular higher order continuous and discrete boundary value problems, Bound. value probl. (2008) · Zbl 1154.34315 · doi:10.1155/2008/123823
[20]Lei, P.; Lin, X.; Jiang, D.: Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems, Nonlinear anal. 69, 2773-2779 (2008) · Zbl 1155.35375 · doi:10.1016/j.na.2007.08.049
[21]Deimling, K.: Nonlinear functional analysis, (1985) · Zbl 0559.47040
[22]Guo, D.; Lakshmikantham, V.: Nonlinear problems in abstract cones, (1988)
[23]Nussbaum, R.: Iterated nonlinear maps and Hilbert’s projective metric, Mem. amer. Math. soc. 75, 391 (1988)
[24]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integral and derivatives, Theory and applications (1993) · Zbl 0818.26003
[25]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[26]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[27]Podlubny, I.: Fractional differential equations, mathematics in science and engineering, (1999)
[28]El-Sayed, A. M. A.: Nonlinear functional differential equations of arbitrary orders, Nonlinear anal. 33, 181-186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[29]Bakakhani, A.; Gejji, V. D.: Existence of positive solutions of nonlinear fractional differential equations, J. math. Anal. appl. 278, 434-442 (2003) · Zbl 1027.34003 · doi:10.1016/S0022-247X(02)00716-3
[30]Zhang, S. Q.: Existence of positive solution for some class of nonlinear fractional differential equations, J. math. Anal. appl. 278, 136-148 (2003) · Zbl 1026.34008 · doi:10.1016/S0022-247X(02)00583-8
[31]Bai, Z. B.; Lu, H. S.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[32]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[33]Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. syst. Differ. equ. 1, No. 4, 239-244 (2008) · Zbl 1175.34081 · doi:10.1504/IJDSDE.2008.022988
[34]Kaufmann, E. R.; Mboumi, E.: Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. theory differ. Equ. 2008, No. 3, 1-11 (2008) · Zbl 1183.34007 · doi:emis:journals/EJQTDE/2008/200803.html
[35]Bai, C. Z.: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. Qual. theory differ. Equ. 2008, No. 24, 1-10 (2008) · Zbl 1183.34005 · doi:emis:journals/EJQTDE/2008/200824.html
[36]Zhou, Y.: Existence and uniqueness of solutions for a system of fractional differential equations, J. fract. Calc. appl. Anal. 12, 195-204 (2009)
[37]Kosmatov, N.: A singular boundary value problem for nonlinear differential equations of fractional order, J. appl. Math. comput. 29, 125-135 (2009) · Zbl 1191.34006 · doi:10.1007/s12190-008-0104-x
[38]Xu, X.; Jiang, D.; Yuan, C.: Multiple positive solutions for boundary value problem of nonlinear fractional differential equation, Nonlinear anal. 71, 4676-4688 (2009) · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030
[39]Wang, Y. Q.; Liu, L. S.; Wu, Y. H.: Positive solutions for a nonlocal fractional differential equation, Nonlinear anal. 74, 3599-3605 (2011) · Zbl 1220.34006 · doi:10.1016/j.na.2011.02.043
[40]Lizama, C.: An operator theoretical approach to a class of fractional order differential equations, Appl. math. Lett. 24, 184-190 (2011) · Zbl 1226.47048 · doi:10.1016/j.aml.2010.08.042