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On the existence of common fixed points for semigroups of nonlinear mappings in modular function spaces. (English) Zbl 1311.47078

Let \(T_t: C\to C\), \(t\geq 0\), be a nonlinear mappings such that \(T_0(x)= x\), \(T_{t+s}(x)= T_t(T_s(x))\). In the present paper, the author proves, among others, the following:
Theorem 3.6. Let \(\rho\) be a nonzero regular convex function modular and \(L_\rho\) have the \(\rho\)-a.e. strong Opial property. Let \(C\) be a nonempty, \(\rho\)-a.e. compact convex subset of \(L_\rho\) such that \(\sup\{\rho(\beta(x- y)): x, y\in C\}< \infty\) for some \(\beta> 1\). Let \({\mathcal F}= \{T_t: t\geq 0\}\) be a \(\rho\)-contractive semigroup on \(C\). Then \({\mathcal F}\) has a unique common fixed point \(z\in C\) and for each \(u\in C\), \(\rho(T_t(u)- z)\to 0\).
Theorem 4.8. Let \(\rho\) be a nonzero regular convex function modular and satisfying the \((UUC1)\) condition. Let \(C\) be a nonempty, \(\rho\)-closed, \(\rho\)-bounded convex set. Let \({\mathcal F}= \{T_t: t\geq 0\}\) be a nonexpansive semigroup on \(C\). Then the set \(\text{Fix}({\mathcal F})\) of common fixed pints is nonempty \(\rho\)-closed and convex.

MSC:

47H20 Semigroups of nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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