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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.

47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54F05Linearly, generalized, and partial ordered topological spaces
54H25Fixed-point and coincidence theorems in topological spaces
15A24Matrix equations and identities
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