Dipankar DAS,Nilakshi GOSWAMI.Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition[J].数学研究及应用,2016,36(1):70~78
Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition
Fixed Points of Mappings Satisfying a Weakly Contractive Type Condition

DOI：10.3770/j.issn:2095-2651.2016.01.009

 作者 单位 Dipankar DAS Mathematics Department, Gauhati University, Assam, India Nilakshi GOSWAMI Mathematics Department, Gauhati University, Assam, India

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.

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.