On uniformly generalized Lipschitzian mappings
In [\textit{J. S. Jung} and \textit{B. S. Thakur}, Int. J. Math. Math. Sci. 28, No.~1, 41--50 (2001; Zbl 1017.47042)] the notion of \(G_1\)-generalized mapping was introduced. Definition 1: A mapping \(T: X\to X\) is said to be generalized \(G_1\)-Lipschitzian if \[ d(T^n x,T^n y)\leq a_nd(x, y)+ b_n(d(x, T^n x)+ d(y, T^n y))+ c_n(d(x, T^n y)+ d(y, T^n x)), \] for each \(x,y\in X\) and \(n\geq 1\), where \(a_n\), \(b_n\), \(c_n\) are nonnegative constants such that there exists an integer \(n_0\) such that \(b_n+ c_n< 1\) for all \(n> n_0\). Recently \textit{A. H. Soliman} [Sci. Math. Jpn. 69, No.~3, 323--328 (2009; Zbl 1178.54027)] defined another class of generalized Lipschitzian mappings on metric spaces. Definition 2: A mapping \(T: X\to X\) is said to be a generalized \(G_2\)-Lipschitzian mapping if for each integer \(n\geq 1\) there exists a constant \(k_n> 0\) (depending on \(n\)) such that \[ d(T^n x, T^n y)\leq k_n\max\left\{d(x,y),\tfrac12\,d(x, T^n x),\tfrac12 d(y, T^n y)\right\} \] for every \(x,y\in X\). If \(k_n= k\) for all \(n\geq 1\), then \(T\) is called uniformly \(G_2\)-Lipschitzian. In the present paper another class of generalized Lipschitzian mappings on metric spaces is introduced. Definition 3: A mapping \(T: X\to X\) is said to be a generalized \(G_3\)-Lipschitzian mapping if for each integer \(n\geq 1\) there exists a constant \(k_n> 0\) (depending on \(n\)) such that \[ d(T^nx, T^ny)\leq k_n\max\left\{d(x,y),\tfrac12\, d(x, T^n x),\tfrac12\, d(y, T^ny),\tfrac12\, d(x, T^ny),\tfrac12\, d(y, T^nx)\right\} \] for every \(x,y\in X\). If \(k_n= k\) for all \(n\geq 1\), then \(T\) is called uniformly \(G_2\)-Lipschitzian. Definition 5: A semigroup \({\mathcal I}= \{T(t): t\in G\}\) of self-mappings defined on \(X\) is called a uniformly \(G_1\)-Lipschitzian semigroup if \[ \begin{multlined} d(T(t)x, T(t)y)\leq a(t) d(x,y)+ b(t)(d(x, T(t) x)+ d(y, T(t)y))+\\ c(t)(d(x, T(t)y)+ d(y, T(t)x)),\end{multlined} \] for each \(x,y\in X\), where \(a(t),b(t)\) and \(c(t)\) are nonnegative constants, \(b(t)+ c(t)< 1\), \(\sup\{a(t): t\in G\}= a<\infty\), \(\sup\{b(t): t\in G\}= b<\infty\), and \(\sup\{c(t): t\in G\}= c<\infty\) with \(b+ c< 1\). Definition 6: A semigroup \({\mathcal I}= \{T(t): t\in G\}\) of self-mappings defined on \(X\) is called a uniformly \(G_3\)-Lipschitzian semigroup if \(\sup\{k(t): t\in G\}= k<\infty\) whenever \(d(T(t) x,T(t)y)\leq k(t) M(x,y)\) for each \(x,y\in X\) and \[ M(x,y)= \max\left\{d(x,y), \tfrac12\, d(x,T(t) x),\tfrac12\,d(y,T(t)y),\tfrac12\,d(x,T(t)y),\tfrac12\, d(y, T(t)y)\right\}\neq 0. \] The first result of this paper (Theorem 3.1) is a fixed point theorem for uniform \(G_1\)-Lipschitzian semigroups of self-mappings defined on bounded metric spaces with uniform normal structure. A second result (Theorem 3.2) is a fixed point theorem for uniformly \(G_2\)-Lipschitzian semigroups of self-mappings defined on bounded metric spaces equipped with uniform normal structure. These results are generalizations of certain results from: \textit{J. C. Yao} and \textit{L. C. Zheng} [J. Nonlinear Convex Anal. 8, No.~1, 153--163 (2007; Zbl 1121.47045)] and from a paper of \textit{A. Soliman} [loc. cit.]
\(G_1\), \(G_2\), \(G_3\) Lipschitzian mappings
uniformly \(G_1\), \(G_2\), \(G_3\) Lipschitzian semigroup
scientific article; zbMATH DE number 5792456
scientific article