On Olaleru’s open problem on Gregus fixed point theorem

Sirous Moradi () and Ali Farajzadeh ()

Journal of Global Optimization, 2013, vol. 56, issue 4, 1689-1697

Abstract: Let (X, d) be a complete metric space and $${TX \longrightarrow X }$$ be a mapping with the property d(Tx, Ty) ≤ ad(x, y) + bd(x, Tx) + cd(y, Ty) + ed(y, Tx) + fd(x, Ty) for all $${x, y \in X}$$ , where 0 > a > 1, b, c, e, f ≥ 0, a + b + c + e + f=1 and b + c > 0. We show that if e + f > 0 then T has a unique fixed point and also if e + f ≥ 0 and X is a closed convex subset of a complete metrizable topological vector space (Y, d), then T has a unique fixed point. These results extend the corresponding results which recently obtained in this field. Finally by using our main results we give an answer to the Olaleru’s open problem. Copyright Springer Science+Business Media, LLC. 2013

Keywords: Fixed point; Metrizable; Topological vector space (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10898-012-9960-x

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