Some coupled fixed point results for set­valued mappings with applications

Document Type : Original Article

Authors

1 LAMDA-RO Laboratory, Department of Mathematics, University of Blida, Algeria.

2 Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

Abstract

This paper deals with the study of coupled fixed point theorems for φ-pseudo-contractive set-valued
mappings without using the mixed g-monotone property on the closed ball of partial metric spaces. Generalizations
of some well-known results concerning existence and location of coupled fixed points are obtained.
These coupled fixed point theorems are applied for obtaining the existence results for an elliptic system.

Keywords

[1] I. Addou, A. Benmeza, Boundary-value problems for the one-dimensional p-laplacian with even superlinearity,
Electron. J. Differential Equations, 1999 (1999), 29 pages. 3
[2] I. Altun, F. Sola, H. Simsek, Generalized contractions on partial metric spaces, Topology Appl., 157 (2010),
2778-2785.
[3] H. Aydi, Some coupled fixed point results on partial metric spaces, Int. J. Math. Math. Sci., 2011 (2011), 11
pages.
[4] H. Aydi, M. Abbas, C. Vetro, Partial Hausdor metric and Nadler's fixed point theorem on partial metric spaces,
Topology Appl., 159 (2012), 3234-3242.
[5] H. Aydi, S. Hadj Amor, E. Karapnar, Berinde-type generalized contractions on partial metric spaces, Abstr.
Appl. Anal., 2013 (2013), 10 pages.
[6] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund.
Math., 3 (1922), 133-181.
[7] I. Beg, A. R. Butt, Coupled fixed points of set-valued mappings in partially ordered metric spaces, J. Nonlinear
Sci. Appl., 3 (2010), 179-185.
[8] A. Benterki, A local fixed point theorem for set-valued mappings on partial metric spaces, Appl. Gen. Topol., 17
(2016), 37-49.
[9] N. Bilgili, I. M. Erhan, E. Karapnar, D. Turkoglu, A note on 'Coupled fixed point theorems for mixed g-monotone
mappings in partially ordered metric spaces', Fixed Point Theory Appl., 2014 (2014), 6 pages.
[10] A. Castro, R. Shivaji, Multiple solutions for a Dirichlet problem with jumping nonlinearities. II, J. Math. Anal.
Appl., 133 (1988), 509-528
[11] X. Cheng, C. Zhong, Existence of three nontrivial solutions for an elliptic system, J. Math. Anal. Appl., 327
(2007), 1420-1430.
[12] M. Chhetri, P. Girg, Existence and nonexistence of positive solutions for a class of superlinear semipositone
systems, Nonlinear Anal., 71 (2009), 4984-4996.
[13] A. L. Dontchev, W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc., 121
(1994), 481-489.
[14] W.-S. Du, E. Karapnar, N. Shahzad, The study of fixed point theory for various multivalued non-self-maps, Abstr.
Appl. Anal., 2013 (2013), 9 pages. 1
[15] T. Gnana Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications,
Nonlinear Anal., 65 (2006), 1379-1393.
[16] D. J. Guo, V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal.,
11 (1987), 623-632.
[17] J. Harjani, B. Lopez, K. Sadarangani, Fixed point theorems for mixed monotone operators and applications to
integral equations, Nonlinear Anal., 74 (2011), 1749-1760.
[18] A. D. Io e, V. M. Tihomirov, Theory of extremal problems. Translated from the Russian by K. Makowski. Studies
in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, (1979). 
[19] H. Isik, M. Imdad, D. Turkoglu, N. Hussain, Generalized Meir-Keeler type -contractive mappings and applications
to common solution of integral equations, Int. J. Anal. Appl., 13 (2017), 185-197.
[20] H. Isik, S. Radenovic, A new version of coupled fixed point results in ordered metric spaces with applications,Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79 (2017), 131-138.
[21] H. Isik, D. Turkoglu, Coupled fixed point theorems for new contractive mixed monotone mappings and applications
to integral equations, Filomat, 28 (2014), 1253-1264.
[22] E. Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, Inc., New York, (1989).
[23] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric
spaces, Nonlinear Anal., 70 (2009), 4341-4349.
[24] H. Lee, A coupled fixed point theorem for mixed monotone mappings on partial ordered G-metric spaces, Kyungpook
Math. J., 54 (2014), 485-500.
[25] P. S. Macansantos, A generalized Nadler-type theorem in partial metric spaces, Int. Journal of Math. Anal., 7
(2013), 343-348.
[26] P. S. Macansantos, A fixed point theorem for multifunctions in partial metric spaces, J. Nonlinear Anal. Appl.,
2013 (2013), 7 pages.
[27] R. T. Marinov, D. K. Nedelcheva, Implicit mapping theorem for extended metric regularity in metric spaces, Ric.
Mat., 62 (2013), 55-66.
[28] S. G. Matthews, Partial metric topology, New York Acad. Sci., New York, (1994).
[29] D. Motreanu, Z. Zhang, Constant sign and sign changing solutions for systems of quasilinear elliptic equations,
Set-Valued Var. Anal., 19 (2011), 255-269.
[30] S. B. Nadler, Multi-valued contraction mappings, Paci c J. Math., 30 (1969), 475-488.
[31] S. Radenovic, Remarks on some coupled fixed point results in partial metric spaces, Nonlinear Funct. Anal. Appl.,
18 (2013), 39-50.
[32] S. Rasouli, M. Bahrampour, A remark on the coupled fixed point theorems for mixed monotone operators in
partially ordered metric spaces, Int. J. Math. Comput. Sci., 3 (2011), 246-261.
[33] S. Romaguera, Fixed point theorems for generalized contractions on partial metric spaces, Topology Appl., 159
(2012), 194-199. 
[34] M. Rouaki, Nodal radial solutions for a superlinear problem, Nonlinear Anal. Real World Appl., 8 (2007), 563-571.
[35] M. Rouaki, Existence and classi ction of radial solutions of a nonlinear nonautonomous Dirichlet problem, arXiv
preprint arXiv:1110.4019, (2011).
[36] B. Ruf, S. Solimini, On a class of superlinear Sturm-Liouville problems with arbitrarily many solutions, SIAM J.
Math. Anal., 17 (1986), 761-771.
[37] M. Sangurlu, A. Ansari, D. Turkoglu, Coupled fixed point theorems for mixed g-monotone mappings in partially
ordered metric spaces via new functions, Gazi Univ. J. Sci., 29 (2016), 149-158.
[38] W. Shatanawi, B. Samet, M. Abbas, Coupled fixed point theorems for mixed monotone mappings in ordered partial
metric spaces, Math. Comput. Modelling, 55 (2012), 680-687.
[39] D. Turkoglu, M. Sangurlu, Coupled fixed point theorems for mixed g-monotone mappings in partially ordered
metric spaces, Fixed Point Theory Appl., 2013 (2013), 11 pages.