JournalspmVol. 79, No. 1/2pp. 199–210

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

Let CC a non-empty, bounded, closed and convex subset of a Banach space XX, and denote by (C)\ell_{\infty}(C) the \ell_{\infty}-sum of CC. In the present paper, by using the degree of nondensifiability (DND), we introduce the class of rr-Δ\Delta-DND-contraction maps f:(C)Xf: \ell_{\infty}(C)\rightarrow X and prove that if f((C))Cf(\ell_{\infty}(C))\subset C then there is some xCx^{*}\in C with f(x,x,,x,)=xf(x^{*},x^{*},\ldots,x^{*},\ldots)=x^{*}. Our result, in the specified framework, generalizes other fixed point results for the so called generalized rr-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