Fixed points for \(\varphi\)-contractions in \(E\)-Banach spaces (Q2873979)
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scientific article; zbMATH DE number 6251076
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Fixed points for \(\varphi\)-contractions in \(E\)-Banach spaces |
scientific article; zbMATH DE number 6251076 |
Statements
28 January 2014
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contraction principle
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\(E\)-Banach space
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\(E\)-metric space
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fixed point
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\(\varphi\)-contraction mapping
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Fredholm-Volterra equation
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Krasnosel'skij's theorem
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Picard operator
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Riesz space
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Schauder's theorem
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sum of two operators
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vector Banach space
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vector metric space
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Fixed points for \(\varphi\)-contractions in \(E\)-Banach spaces (English)
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Let \(E\) be an ordered linear space, that is, a real linear space with an order relation \(\leq \) which satisfies the following properties: NEWLINE\[NEWLINEx\leq y\implies x+z\leq y+z,NEWLINE\]NEWLINE NEWLINE\[NEWLINEx\leq y\implies tx\leq ty, \quad t>0.NEWLINE\]NEWLINE An ordered linear space for which the partially ordered set \((E,\leq )\) forms a lattice is said to be a Riesz space. An \(E\)-metric is a metric on a nonempty set \(X\) taking its values in a Riesz space. An \(E\)-norm is defined in a similar fashion.NEWLINENEWLINEIn the paper under review, the author extends some of the existing results regarding the metric version of Banach's fixed point theorem for classes of \(\varphi\)-contractions to ones for \(E\)-metric spaces. Finally, the \(E\)-Banach space version of Krasnoselskii's fixed point theorem is established.
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