Fixed point theorems in ordered \(L\)-spaces. (English) Zbl 1086.47026

An ordered \(L\)-space \((X,\rightarrow ,\leq )\) consists of a nonempty set \(X\), a convergence structure \(\rightarrow\) on the set of the sequences in \(X\) and an ordering relation \(\leq\) on \(X\), the two being connected by some conditions of compatibility. In this paper, the authors establish several sufficient conditions in order that an operator \(f:X\rightarrow X\) defined on an ordered \(L\)-space to be a Picard operator (that is, to have a unique fixed point \(x^*\) and, for each \(x\in X\), the sequence \((f^n(x))\) of successive approximations to converge to \(x^*\)). Finally, some applications to matrix equations are considered.


54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54E70 Probabilistic metric spaces
15A24 Matrix equations and identities


Zbl 1060.47056
Full Text: DOI


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