×

Sequences of iterates in locally convex spaces. (English) Zbl 0624.47060

Nonlinear functional analysis and its applications, Proc. NATO Adv. Study Inst., Maratea/Italy 1985, NATO ASI Ser., Ser. C 173, 195-206 (1986).
[For the entire collection see Zbl 0583.00019.]
Let \(\Delta\) be a nonempty set and \(R^{\Delta}=\prod_{\alpha \in \Delta}R_{\alpha}\) be the product of \(\Delta\) copies of the real line with product topology and usual algebraic operations. A partial ordering is defined by the cone \(R_+^{\Delta}=\{f:f(\alpha)\geq 0,\alpha \in \Delta \}\). Let E be a real Hausdorff locally convex space whose topology is generated by a family \(\{\rho_{\alpha}:\alpha \in \Delta \}\) of continuous seminorms. Define \(\rho:E\to R_+^{\Delta}\) by \([\rho (x)](\alpha)=\rho_{\alpha}(x)\), \(x\in E\), \(\alpha\in \Delta\). Let C be a subset of E. A mapping T:C\(\to C\) is generalized nonexpansive if there exists a number k, \(0<k<1\) such that for all x,y\(\in C\), \[ \rho (Tx- Ty)\leq \max \{\rho (x-y),1/2[\rho (x-Tx)+\rho (y-Ty)],\quad 1/2[\rho (x- Ty)+\rho (y-Tx)],k\rho (x-Ty),k\rho (y-Tx)\}. \] Define S:C\(\to C\) as \(S=\alpha_ 0I+\alpha_ 1T+\alpha_ 2T^ 2+...+\alpha_ kT^ k\) where \(\alpha_ i\geq 0,\alpha_ 0>0,\alpha_ 1>0\) and \(\sum^{k}_{i=0}\alpha_ i=1\). The authors have proved the following results regarding fixed points of generalized nonexpansive mappings T thus generalizing existing results on fixed points.
(1) If E is a Hausdorff locally convex space and C is a convex subset then \(Sx=x\) iff \(Tx=x.\)
(2) If E is a quasicomplete strictly convex space, C a closed convex subset and TC is contained in a compact subset of E then for each \(x\in C\), the sequence \(\{S^ n(x)\}\) converges to a fixed point of T.
(3) If E is a semireflexive generalized Hilbert space, C a closed convex weakly sequentially compact subset, T continuous, F(T)\(\neq \emptyset\) and I-S is demiclosed then for any \(x_ 0\in C,S^ n(x_ 0)\rightharpoonup y\in F(T)=F(S).\)
To illustrate the significance of these results, an example of a nonnormable, locally convex space and a nonexpansive mapping with a fixed point is also given.
Reviewer: I.Beg

MSC:

47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
46A03 General theory of locally convex spaces

Citations:

Zbl 0583.00019