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

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 $f:{\ell }_{\infty }(C)\to X$ and prove that if $f({\ell }_{\infty }(C))\subset C$ then there is some $x^{\ast }\in C$ with $f(x^{\ast },x^{\ast },\dots ,x^{\ast },\dots )=x^{\ast }$. 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.