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Fixed points of nonlinear and asymptotic contractions in the modular space. (English) Zbl 1152.47043

The main result of the paper is the following fixed point theorem for nonlinear contractions in modular spaces. Theorem 2.1.Let \(X_{\rho}\) be a \(\rho\)-complete modular space, where \(\rho\) satisfies the following condition:
\[ \rho(2x_n)\to 0 \text{ as } n\to +\infty \text{ whenever } \rho(x_n)\to 0 \text{ as } n\to +\infty. \]
Assume that \(\psi:\mathbb{R}_+\to \mathbb{R}_+\) is an increasing, upper semicontinuous function such that \(\psi(t)<t\) for each \(t>0\). Let \(B\) be a \(\rho\)-closed subset of \(X_{\rho}\) and \(T:B\to B\) be an operator such that there exist \(c,l\in \mathbb{R}_+\) with \(c>l\) so that
\[ \rho(c(Tx-Ty))\leq \psi(\rho(l(x-y))) \text{ for all } x,y\in B. \]
Then \(T\) has a fixed point.
An asymptotic version of this result is also given.

MSC:

47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
54H25 Fixed-point and coincidence theorems (topological aspects)
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References:

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