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

Let \((X,d)\) a metric space. A mapping \(f:X\to X\) is called a Caristi’s mapping if there exists a lower semicontinuous function \( \phi :X\to [0,+\infty )\) satisfying
\[ d(x,f x)\leq \phi (x)-\phi (f x), \]
for all \(x\in X\). 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 \(p:X\times X\to [0,+\infty )\) is said to be a partial metric if for all \(x,y,x \in X\): (i) \(x=y \Leftrightarrow p(x,x)=p(x,y)=p(y,y)\); (ii) \(p(x,x)\leq p(x,y)\); (iii) \(p(x,y)=p(y,x)\), (iv)\( p(x,z)\leq p(x,y)+ p(y,z)-p(y,y).\) 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.


54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems


Zbl 0353.53041
Full Text: DOI EuDML


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