Marasi, H. R.; Afshari, H.; Daneshbastam, M.; Zhai, C. B. Fixed points of mixed monotone operators for existence and uniqueness of nonlinear fractional differential equations. (English) Zbl 1370.34017 J. Contemp. Math. Anal., Armen. Acad. Sci. 52, No. 1, 8-13 (2017) and Izv. Nats. Akad. Nauk Armen., Mat. 52, No. 1, 78-84 (2017). Summary: In this paper we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems by using new fixed point results of mixed monotone operators on cones. Cited in 11 Documents MSC: 34A08 Fractional ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations 47H10 Fixed-point theorems Keywords:fractional differential equation; normal cone; boundary value problem; mixed monotone operator PDFBibTeX XMLCite \textit{H. R. Marasi} et al., J. Contemp. Math. Anal., Armen. Acad. Sci. 52, No. 1, 8--13 (2017; Zbl 1370.34017) Full Text: DOI References: [1] I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999. · Zbl 0924.34008 [2] A. A. Kilbas, H.M. Srivastava, J.J. Trujillo, “Theory and Applications of Fractiona differential Equations”, North-Holland Mathematics studies, 204, 7-10, 2006). · Zbl 1092.45003 [3] J. Sabatier, O.P. Agrawal, J.A.T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007). · Zbl 1116.00014 · doi:10.1007/978-1-4020-6042-7 [4] D. Guo, V. Lakskmikantham, “Coupled fixed points of nonlinear operators with applications”, Nonlinear Anal., 11 (5), 623-632, 1987). · Zbl 0635.47045 · doi:10.1016/0362-546X(87)90077-0 [5] D. Guo, “Fixed points of mixed monotone operators with application”, Appl. Anal., 34, 215-224, 1988). · Zbl 0688.47019 · doi:10.1080/00036818808839825 [6] D. Guo, V. Lakskmikantham, Nonlinear Problems in Abstract Cones (Academic Press, New York, 1988). · Zbl 0661.47045 [7] D. Guo, Partial Order Methods in Nonlinear Analysis (Shandong Science and Technology Press, Jinan, 2000). [8] S. S. Zhang, “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 0751.30016 · doi:10.1016/0022-247X(91)90330-3 [9] T. G. Bhaskar, V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications”, Nonlinear Anal. TMA, 65, 1379-1393, 2006). · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 [10] Dz. Burgic, S. Kalabusic, M.R. Kulenovic, “Global attractivity results for mixed monotone mappings in partially ordered complete metric spaces”, Fixed Point Theory Appl., Article ID 762478, 2009. · Zbl 1168.54327 [11] Z. Drici, F.A. McRae, J. Vasundhara Devi, “Fixed point theorems for mixed monotone operators with PPF dependence”, Nonlinear Anal. TMA, 69, 632-636, 2008). · Zbl 1162.47042 · doi:10.1016/j.na.2007.05.044 [12] J. Harjani, B. Lopez, K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations”, Nonlinear Anal. TMA, 74, 1749-1760, 2011). · Zbl 1218.54040 · doi:10.1016/j.na.2010.10.047 [13] Y. Sang, “Existence and uniqueness of fixed points for mixed monotone operators with perturbations”, Electronic Journal of Differential Equations, 233, 1-16, 2013). · Zbl 1302.47078 [14] X. J. Xu, D.Q. Jiang, C. J. Yuan, “Multiple positive solutions for the boundary value problems of a nonlinear fractional differential equation”, Nonlinear Anal., 71, 4676-4688, 2009). · Zbl 1178.34006 · doi:10.1016/j.na.2009.03.030 [15] M. Benchohra, F. Ouaar, “Existence results for nonlinear Fractional Differential Equations with integral boundary conditions”, Bulletin of Math. Anal. and Appl., 4, 7-15, 2010). · Zbl 1312.34047 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.