# A fixed point result for mappings on the $\ell_{\infty}$-sum of a closed and convex set based on the degree of nondensifiability

### Gonzalo García

Universidad Nacional de Educación a Distancia (UNED), Elche, Spain

## Abstract

Let $C$ a non-empty, bounded, closed and convex subset of a Banach space $X$, and denote by $\ell_{\infty}(C)$ the $\ell_{\infty}$-sum of $C$. In the present paper, by using the degree of nondensifiability (DND), we introduce the class of $r$-$\Delta$-DND-contraction maps $f: \ell_{\infty}(C)\rightarrow X$ and prove that if $f(\ell_{\infty}(C))\subset C$ then there is some $x^{*}\in C$ with $f(x^{*},x^{*},\ldots,x^{*},\ldots)=x^{*}$. Our result, in the specified framework, generalizes other fixed point results for the so called generalized $r$-contraction and even other existing fixed point result based on the DND. Also, we derive a new Krasnosel’skiĭ-type fixed point result.

## Cite this article

Gonzalo García, A fixed point result for mappings on the $\ell_{\infty}$-sum of a closed and convex set based on the degree of nondensifiability. Port. Math. 79 (2022), no. 1/2, pp. 199–210

DOI 10.4171/PM/2081