Document Type : Research Paper

Authors

1 Department of pure Mathematics, Sarab Branch, Islamic Azad University, Sarab, Iran.

2 Department of Mathematics, Faculty of Science, Tabriz Branch, Islamic Azad University Tabriz, Iran.

3 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran.

Abstract

In 2012, Samet, et al. introduced the notion of $\alpha$-$\psi$-contractive type mappings. They have been establish some fixed point theorems for the mappings on complete metric
spaces. In this paper, we introduce the notion of generalized $\alpha_*$-$\psi$-contractive multi-valued mappings and we give some related fixed point results on ordered metric spaces via  application to an initial value problem.

Keywords

[1] I. Altun and V. Rakocevic, Ordered cone metric spaces and fixed point results, CAMWA-D-09-00221,(2009).

[2] A. Amini-Harandi, Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem, Math. Compute. Modelling, (2012).

[3] B.C. Dhage, Condensing mappings and applications to existence theorems for common solution of differential equations, Bull. Korean Math. Soc., 36 (3),(1999), pp. 565-578.

[4] B.C. Dhage, D. ORegan and R.P. Agarwal, Common fixed theorems for a pair of countably condensing mappings in ordered Banach spaces, J. Apple. Math Stoch. Anal., 16 (3),(2003), pp. 243-248.

[5] Y. Feng and S. Liu, Fixed point theorems for multi-valued increasing operators in partially ordered spaces, Soochow J. Math., 30 (4),(2004), pp. 461-469.

[6] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications Nonlinear Analysis. Theory, Methods & Applications., 11 (1987), pp. 623-632.

[7] J. Hasanzadeh Asl, Common fixed point theorems for $alpha$-$psi$-contractive type mappings, Int. J. Anal.,(2013), Article ID 654659, 7 pages.

[8] J. Hasanzadeh Asl, Sh. Rezapour and N. Shahzad, On fixed points of $alpha-psi$-contractive multifunction's, Fixed Point Theory Appl., 212 (2012), 7 pages.

[9] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for $alpha$-$psi$-contractive type mappings, Nonlinear Analysis, 75 (2012), pp. 2154-2165.

[10] Hu Xin-qi and Ma Xiao-yan, Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces, Nonlinear Analysis, 74 (2011), pp. 6451-6458.