Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition
Received:February 01, 2015  Revised:May 27, 2015
Key Words: tensor product   weakly contractive map   projective tensor norm  
Fund Project:
Author NameAffiliation
Dipankar DAS Mathematics Department, Gauhati University, Assam, India 
Nilakshi GOSWAMI Mathematics Department, Gauhati University, Assam, India 
Hits: 2570
Download times: 1998
Abstract:
      In this paper, we discuss a fixed point theorem for mappings derived by a pair of mappings satisfying weak $(k,k^/)$ contractive type condition on the tensor product spaces. Let $X$ and $Y$ be Banach spaces and $T_1:X\otimes_\gamma Y \to X$ and $T_2:X\otimes_\gamma Y\to Y$ be two operators which satisfy weak $(k,k^/)$ contractive type condition. Using $T_1$ and $T_2$, we construct an operator $T$ on $X \otimes_\gamma Y$ and show that $T$ has a unique fixed point in a closed and bounded subset of $X\otimes_\gamma Y$. We derive an iteration scheme converging to this unique fixed point of $T$. Conversely, using a weakly contractive mapping $T$, we construct a pair of mappings $(T_1,T_2)$ satisfying weak $(k,k^/)$ contractive type condition on $X\otimes_\gamma Y$ and from this pair, we also obtain two self mappings $S_1$ and $S_2$ on $X$ and $Y$ respectively with unique fixed points.
Citation:
DOI:10.3770/j.issn:2095-2651.2016.01.009
View Full Text  View/Add Comment