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.
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Moradi, S., Farajzadeh, A. On Olaleru’s open problem on Gregus fixed point theorem. J Glob Optim 56, 1689–1697 (2013). https://doi.org/10.1007/s10898-012-9960-x
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DOI: https://doi.org/10.1007/s10898-012-9960-x