On Olaleru’s open problem on Gregus fixed point theorem

Let (X, d) be a complete metric space and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$${TX \longrightarrow X }$$</EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$${x, y \in X}$$</EquationSource> </InlineEquation>, 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