Approximations with weak contractions in Hadamard manifolds (Q2787031)

The article deals with a nonexpansive mapping \(T\) in a closed convex subset of a Hadamard manifold \(M\), with \(\mathrm{Fix}\, T \neq \emptyset\). The sequence \(x_n\) (Halpern iterative scheme) is considered defined by NEWLINE\[NEWLINEy_n = \exp_u(1 - \alpha_n) \, \exp_u^{-1} Tx_n, \quad x_{n+1} = \exp_{x_n}(1 - \beta_n) \, \exp_{x_n}^{-1}y_n, \quad n = 1,2,\dots,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE\lim_{n \to \infty} \;\alpha_n = 0, \quad \sum_{n=1}^\infty \alpha_n = \infty, \quad 0 < \liminf_{n \to \infty} \;\beta_n \leq \limsup_{n \to infty} \;\beta_n < 1.NEWLINE\]NEWLINE It is proved, for any \(u, x_1 \in C\), that the sequence \((x_n)\) converges strongly to \(q = P_{\mathrm{Fix}\, T} u\). The similar results hold provided that NEWLINE\[NEWLINE\lim_{n \to \infty} \;\alpha_n = 0, \quad \sum_{n=1}^\infty \alpha_n = \infty,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{either} \quad \sum_{n=1}^\infty |\alpha_{n+1} - _n| < \infty \quad \text{or} \quad \lim_{n \to \infty} \;\frac{\alpha_n}{\alpha_{n+1}} = 1,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\lim_{n \to \infty} \;\beta_n = 0, \quad \sum_{n=1}^\infty |\beta_{n+1} - \beta_n| < \infty.NEWLINE\]NEWLINE Also, the approximations \(x_n\) (Moudafi iterative scheme) are considered defined by NEWLINE\[NEWLINEy_n = \exp_{f(x_n)}(1 - \alpha_n) \, \exp_{f(x_n)}^{-1} Tx_n, \quad x_{n+1} = \exp_{x_n}(1 - \beta_n) \, \exp_{x_n}^{-1}y_n, \quad n = 1,2,\dots,NEWLINE\]NEWLINE where \(f\) is a \(\varphi\)-weak contraction on \(C\): NEWLINE\[NEWLINEd(f(x),f(y)) \leq d(x,y) - \varphi(d(x,y)), \quad x, y \in X.NEWLINE\]NEWLINE (\(\varphi: [0,\infty) \to [0,\infty)\) is a continuous and nondecreasing function with \(\varphi(t) = 0\) if and only if \(t = 0\)). In this case the sequence \((x_n)\) convergence strongly to the unique point \(q \in C\) such that \(q = P_{\text{Fix}\, T} f(q)\).NEWLINENEWLINEPS. The author describes in detail properties of general Riemannian manifolds, however, does not give the definition of Hadamard (Hadamard-Cartan) manifolds.