1 Introduction
2 Preliminaries and lemmas
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Denote a pair \((a,b)\in X\times X\) by \(\overrightarrow{ab}\) and call it a vector. Quasi-linearization in \(\operatorname{CAT}(0)\) space X is defined as a mapping \(\langle\cdot,\cdot\rangle: (X\times X)\times(X\times X)\to\mathbb {R}\) such thatfor all \(a,b,c,d\in X\).$$ \langle\overrightarrow{ab},\overrightarrow{cd}\rangle=\frac {1}{2} \bigl(d^{2}(a,d)+d^{2}(b,c)-d^{2}(a,c)-d^{2}(b,d) \bigr) $$(2.2)
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We say that X satisfies the Cauchy–Schwarz inequality ifIt is well known [18, Corollary 3] that a geodesically connected metric space is a \(\operatorname{CAT}(0)\) space if and only if it satisfies the Cauchy–Schwarz inequality.$$ \langle\overrightarrow{ab}, \overrightarrow{cd} \rangle\le d(a, b) d(c, d),\quad \forall a, b, c, d \in X. $$(2.3)
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By using quasi-linearization, Ahmadi Kakavandi [19] proved that \(\{x_{n}\}\) Δ-converges to \(x \in X\) if and only if$$ \limsup_{n \to\infty}\langle\overrightarrow{xx_{n}}, \overrightarrow {xy}\rangle\le0,\quad \forall y \in X. $$(2.4)
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Let C be a nonempty closed convex subset of a complete \(\operatorname{CAT}(0)\) space X (i.e., a Hadamard space). The metric projection \(P_{C}: X\to C\) is defined by$$ u=P_{C}(x) \quad \Longleftrightarrow \quad d(u,x)=\inf\bigl\{ d(y,x):y \in C\bigr\} ,\quad x\in X. $$(2.5)