Fixed point theorems for generalized contractive multi-valued maps

✍ Scribed by Quanita Kiran; Tayyab Kamran

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 347 KB
• Volume: 59
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/j.camwa.2010.04.015

✦ Synopsis

In [N. Mizoguchi, W. Takahashi, Fixed point theorems for multi-valued mappings on complete metric spaces, J. Math. Anal. Appl. 141 (1989) 177-188] the authors gave a positive answer to the conjecture of S. Reich concerning the existence of fixed points of multi-valued mappings that satisfy certain contractive conditions. In this paper, we establish some results for multi-valued mappings that satisfy a generalized contractive condition in a way that it contains Mizoguchi's result as one of its special cases. In addition, our results not only improve the results of Kiran and Kamran [Q. Kiran, T. Kamran, Nadler's type principle with high order of convergence, Nonlinear Anal. TMA 69 (2008) 4106-4120] and some results of Agarwal et al. [R.P. Agarwal, Jewgeni Dshalalow, Donal O'Regan, Fixed point and homotopy results for generalized contractive maps of Reich type, Appl. Anal. 82 (4) (2003) 329-350] but also provide the high order of convergence of the iterative scheme and error bounds. As an application of our results, we obtain an existence result for a class of integral inclusions.