Fixed point theorems for nonlinear non-self mappings in Hilbert spaces and applications

Ngai-Ching Wong,Wataru Takahashi,Jen-Chih Yao

doi:10.1186/1687-1812-2013-116

Abstract

Abstract Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem. MSC:Primary 47H10; secondary 47H05.

Highlights

LetR be the real line and let [, π ]be a bounded, closed and convex subset of R.Consider a mapping T : → defined by Tx =
In this paper, motivated by such a problem, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo, Takahashi and Yao [ ] and Kawasaki and Takahashi fixed point theorems [ ]
Using these fixed point theorems for non-self mappings, we prove the Browder and Petryshyn fixed point theorem [ ] for strict pseudo-contractive non-self mappings and the Kocourek, Takahashi and Yao fixed point theorem [ ] for super hybrid non-self mappings