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A Kirk type characterization of completeness for partial metric spaces. (English) Zbl 1193.54047

Let (X,d) a metric space. A mapping f:XX is called a Caristi’s mapping if there exists a lower semicontinuous function ϕ:X[0,+) satisfying

d(x,fx)ϕ(x)-ϕ(fx),

for all xX. W. A. Kirk [Colloq. Math. 36, 81–86 (1976; Zbl 0353.53041)] proved that a metric space (X,D) is complete if and only if every Caristi’s mapping has a fixed point. A map p:X×X[0,+) is said to be a partial metric if for all x,y,xX: (i) x=yp(x,x)=p(x,y)=p(y,y); (ii) p(x,x)p(x,y); (iii) p(x,y)=p(y,x), (iv)p(x,z)p(x,y)+p(y,z)-p(y,y)· In this paper the author studies the relationship between the existence of fixed points for Caristi’s type mappings and completeness in a partial metric space.

MSC:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces
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