# Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**Let H be a Hilbert space and $f:H\to \left[-\infty ,\infty \right]$ a proper lower semicontinuous convex function. For each $x\in X$, if $\left\{{J}_{{\mu}_{n}f}x\right\}$ is bounded by some sequence $\left\{{\mu}_{n}\right\}\subset \left[0,\infty \right]$ such that ${\mu}_{n}\to \infty $, then $argminf\ne \varnothing $ and

## 2. Preliminaries

**Theorem**

**2**

**Theorem**

**3**

**.**Let X be a complete admissible CAT(1) space, $x,y,z\in X$, and $t\in [0,1]$. Then,

## 3. Main Results

- (a)
- $\mathrm{M}-\underset{n\to \infty}{lim}argmin{f}_{n}=argminf$;
- (b)
- For all $b\in X$, there exists $\left\{{b}_{n}\right\}$ such that ${b}_{n}\to b$ and ${lim\; sup}_{n\to \infty}{f}_{n}\left({b}_{n}\right)\le f\left(b\right)$;
- (c)
- For any subsequence $\left\{{f}_{{n}_{i}}\right\}$ of $\left\{{f}_{n}\right\}$ and a $\Delta $-convergent sequence $\left\{{c}_{i}\right\}$ whose $\Delta $-limit is $c\in X$, it holds that $f\left(c\right)\le {lim\; inf}_{i\to \infty}{f}_{{n}_{i}}\left({c}_{i}\right)$.

- $\phi $ is increasing;
- $\phi $ is continuous;
- $\phi \left(d\right(\xb7,x\left)\right)$ is strictly convex for all $x\in X$;
- $\phi \left(t\right)-kt\to \infty $ as $t\to \infty $, for all constants $k\in \mathbb{R}$.

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

- $\phi $ is increasing;
- $\phi $ is continuous;
- $\phi \left(d\right(\xb7,x\left)\right)$ is strictly convex for all $x\in X$;
- $\phi \left(t\right)\to \infty $ as $t\to \pi /2$.

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Corollary**

**4.**

## 4. Applications to Hilbert Spaces

**Theorem**

**6.**

**Theorem**

**7**

**.**Let H be a Hilbert space, $\left\{{C}_{n}\right\}$ a sequence of nonempty closed convex subsets of X, and C a nonempty closed convex subset of X. If $\left\{{C}_{n}\right\}$ converges to C in the sense of Mosco, then for each $x\in X$,

## Author Contributions

## Funding

## Conflicts of Interest

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Kimura, Y.; Shindo, K. Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces. *Axioms* **2022**, *11*, 21.
https://doi.org/10.3390/axioms11010021

**AMA Style**

Kimura Y, Shindo K. Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces. *Axioms*. 2022; 11(1):21.
https://doi.org/10.3390/axioms11010021

**Chicago/Turabian Style**

Kimura, Yasunori, and Keisuke Shindo. 2022. "Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces" *Axioms* 11, no. 1: 21.
https://doi.org/10.3390/axioms11010021