SOME SUZUKI TYPE FIXED POINT THEOREMS FOR MULTI-VALUED MAPPINGS AND APPLICATIONS

R.K Bose

doi:10.12732/ijpam.v92i4.4

Abstract

Recently Suzuki (2008) and then Kikkawa and Suzuki (2008) gave a new generalization of the Banach contraction principle. Then, Mot and Petrusel (2009), Dhompapangasa and Yingtaweesttikulue (2009), Bose and Roychowd- hury (2011), Singh and Mishra (2010), and Doric and Lazovic (2011) further extended their work. Recently Bose (2012) obtained some Suzuki-type common fixed point theorems for generalized contractive multivalued mappings using a result of Bose and Mukherjee(1977) which extend the previously obtained re- sults. Recently Damjanovic and Doric(2011) obtained a multivalued generaliza- tion of theorems Kikkawa and Suzuki (2008) concerning Kannan mappings. Also Singh and Mishra (2010) considered coincidence and fixed point theorems for a class of hybrid pair of single-valued and multi-valued maps in metric space setting and Singh et al (2012) prsented a common fixed point theorem for a pair of multi-valued maps in a complete metric space extending a recent theorem of Doric and Lazovic. First we generalize the theorem of Singh et al which also extend the theorem of Singh and Mishra(2010). Then we extend the theorem of Damjanovic and Doric to a common fixed point theorem of a pair of mul- tivalued mappings. As an application, we consider the existence of a common solution for a class of functional equations arising in dynamic programming.

Highlights

The Banach contraction principle plays a very impotant role in nonlinear analysis and has several generalizations([12] and the references there in)
We extend the theorem of Damjanovic and Doric to a common fixed point theorem of a pair of multivalued mappings
Suzuki [19] gave a new type of generalization of the Banach contraction principle

Summary

Introduction

The Banach contraction principle plays a very impotant role in nonlinear analysis and has several generalizations([12] and the references there in). In Bose [4], some Kikkawa-Suzuku type theorems concerning the generalized multivalued contractions were presented which extends the work of Singh and Mishra [16], Bose and Roychowdhury [3], Mot and Petrusel [14], Kikkawa and Suzuki [12], and others, a new coincidence point theorem concerning a hybrid pair of mappings f : X → X and T : X → CB(X) was discussed and an application concerning dynamic programming was discussed. [18]) prsented a common fixed pont theorem for a pair of multi-valued maps in a complete metric space extending a recent theorem of Doric and Lazovic[10] We generalize the theorem of Singh et al[18] which extend the theorem of Singh and Mishra [17]

Peliminaries

Main Theorems

Application