- Research
- Open Access
Fixed points in countably Hilbert spaces
- Nashat Faried^{1}Email author and
- Hany A El-Sharkawy^{1}
https://doi.org/10.1186/s13660-016-0973-8
© Faried and El-Sharkawy 2016
Received: 31 August 2015
Accepted: 15 January 2016
Published: 27 January 2016
Abstract
Studying fixed points of nonlinear mappings in Hilbert spaces is of paramount importance (see, e.g., (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967)). We extend the notions of weakly contractive and asymptotically weakly contractive nonself-mappings defined on a closed convex proper subset of (into) a real Hilbert space to a real countably Hilbert space. Using the notion of metric projection on countably Hilbert spaces, we study iterative methods for approximating fixed points of nonself-maps. Moreover, we prove convergence theorems with estimates of convergence rates. Furthermore, we also establish the stability of the methods with respect to perturbations of the operators and with respect to the perturbations of the constraint sets.
Keywords
- Hilbert Space
- Banach Space
- Nonexpansive Mapping
- Hausdorff Distance
- Nonempty Closed Convex Subset
1 Introduction
Definition 1.1
(Uniformly convex space [2–6])
A normed linear space E is called uniformly convex if for any \(\epsilon\in(0,2]\), there exists \(\delta=\delta(\epsilon)> 0\) such that if \(x,y \in E\) with \(\|x\|=1\), \(\|y\|=1\), and \(\|x-y\| \geq\epsilon\), then \(\|\frac{1}{2} (x+y)\| \leq1-\delta\).
Definition 1.2
Definition 1.3
(Uniformly smooth space [2–6])
Definition 1.4
Let K be a nonempty convex subset of a real normed linear space E. For strict contraction self-mappings of K into itself, with a fixed point in K, a well-known iterative method ‘the celebrated Picard method’ has successfully been employed to approximate such fixed points. If, however, the domain of a mapping is a proper subset of E (and this is the case in several applications) and if it maps K into E, this iteration method may not be well defined. In this situation, for Hilbert spaces and uniformly convex uniformly smooth Banach spaces, this problem has been overcome by the introduction of the metric projection in the recursion formulas (see, e.g., [7–9]). The advantage of this is that if K is a nonempty closed convex subset of a Hilbert space H and \(P_{K} : H \to K\) is the metric projection of H onto K, then \(P_{K}\) is nonexpansive. This fact characterizes Hilbert spaces and unfortunately is not available in general Banach spaces.
Definition 1.5
(Metric projection [2, 7, 8, 10])
Our purpose in this paper is to study, in countably Hilbert spaces, the classes of weakly contractive and asymptotically weakly contractive nonself-maps, which are important extensions of the classes of maps studied by Alber and Guerre-Delabriere [7] and by Chidume et al. [10]. Then, assuming the existence of fixed points for maps in our classes of operators and using several results of Alber and Guerre-Delabriere [7], we prove convergence theorems with estimates of the convergence rates. Our theorems give analogue versions of some results of [7, 10] in countably Hilbert spaces.
2 Preliminaries
Let K be a nonempty proper subset of a real Banach space E. A map \(A:K\rightarrow K\) is called a strict contraction if there exists \(k \in[0,1)\) such that \(\|Ax-Ay\| \leq k\|x-y\|\) for all \(x,y \in K\), and A is called nonexpansive if, for arbitrary \(x, y \in K\), \(\|A x - Ay\| \leq\|x - y\|\). The map A is called asymptotically nonexpansive if, for all \(x, y \in K\), we have \(\|A^{n} x - A^{n} y\| \leq k_{n} \|x - y\|\) for all \(n \geq1\), where \(\{k_{n}\}\) is a sequence of real numbers such that \(\lim_{n\to\infty} k_{n} = 1\). It is obvious that for asymptotically nonexpansive mappings, we may assume that \(k_{n} \geq1\) and that \(k_{i+1} \leq k_{i}\), \(i = 1, 2, \ldots\) (see, e.g., [3]).
A mapping A is called weakly contractive of the class \(C_{\psi(t)}\) on a nonempty set K in a Banach space E if there exists a continuous and nondecreasing function \(\psi(t)\) defined on \(\mathbb{R}^{+}\) with \(\psi(t)>0\) for all \(t\in\mathbb{R}^{+} \setminus\{0\}\), \(\psi(0)=0\), and \(\lim_{t \rightarrow \infty} \psi(t)=+ \infty\) such that \(\|Ax-Ay\| \leq\|x-y\| -\psi(\| x-y\|)\) for all \(x,y \in K\).
Definition 2.1
(Asymptotically weakly contractive [10])
Definition 2.2
(Countably normed space [11–14])
Remark 2.3
([14])
If E is a countably normed space, the completion of E in the norm \(\|{\cdot}\|_{n}\) is denoted by \(E_{n}\). Then, by definition, \(E_{n}\) is a Banach space. Also in the light of Remark 2.3, we can assume that \(E\subset\cdots\subset E_{n+1} \subset E_{n} \subset\cdots\subset E_{1}\).
Remark 2.4
In the light of Remark 2.3, we can easily see that the topology of a countably normed space is generated by the neighborhood base consisting of all sets of the form \(U_{r, \epsilon} = \{ x : x \in E ; \|x\|_{r} < \epsilon\}\) for a positive integer r. Moreover, it is obvious that a nonempty subset K of a countably normed space E is bounded in E if and only if K is bounded in each \(\|{\cdot}\|_{i}\).
Proposition 2.5
([11])
Let E be a countably normed space. Then E is complete if and only if \(E = \bigcap_{n = 1}^{\infty} E_{n}\).
Each Banach space \(E_{n}\) has a dual, which is a Banach space and denoted by \(E_{n}^{*}\).
Proposition 2.6
([11])
The dual of a countably normed space E is given by \(E^{*} = \bigcup_{n = 1}^{\infty} E_{n}^{*}\), and we have the following inclusions: \(E_{1}^{*}\subset\cdots\subset E_{n}^{*} \subset E_{n+1}^{*} \subset\cdots \subset E^{*}\). Moreover, for \(f\in E_{n}^{*}\), we have \(\|f\|_{n} \geq\|f\|_{n+1}\).
Example 2.7
([13])
For \(1 < p < \infty\), the space \(\ell^{p+0} := \bigcap_{q > p} \ell^{q}\) is a countably normed space. In fact, we can easily see that \(\ell^{p+0} = \bigcap_{n} \ell^{p_{n}}\) for any choice of a monotonic decreasing sequence \(\{p_{n}\}\) converging to p. Using Proposition 2.5 and the fact that \(\ell^{p_{n}}\) is a Banach space for every n, it is clear now that the countably normed space \(\ell ^{p+0}\) is complete.
Definition 2.8
(Countably Hilbert space [11, 12])
A linear space H equipped with a countable system of compatible norms \(\|{\cdot}\|_{n}\) is said to be countably Hilbert space if each \(\|{\cdot}\|_{n}\) is an inner product norm and E is complete with respect to its topology.
Remark 2.9
Remark
If \(\beta= (\beta_{n})\) is a sequence of positive numbers, the dual of the Hilbert space \(\ell^{2} (\beta) := \{ x = \sum_{n=1}^{\infty} x_{n} \frac{e_{n}}{\beta_{n}} : \sum_{n=1}^{\infty} |x_{i}|^{2} < \infty\}\) is the Hilbert space \(\ell^{2} (\beta)\).
Example 2.10
The space \(E := \bigcap_{i = 1}^{\infty} \ell^{2} (\beta^{i})\) is a countably Hilbert space, where \(\beta^{i} = (\beta _{n}^{i})_{n\in\mathbb{N}}\) satisfies \(\beta_{n}^{i} \leq\beta_{n}^{i+1}\), so that the Hilbert spaces \(H_{i} = \ell^{2} (\beta^{i})\) follow the inclusion \(H_{i+1} \subseteq H_{i}\) for all i.
Example
The Köthe space \(\ell^{2} [\|e_{n}\|_{i} ] := \{ x = (x_{n}) : \|x\|_{i} = ( \sum_{n =1}^{\infty} |x_{n}|^{2} \| e_{n}\|_{i}^{2} )^{\frac{1}{2}} < \infty, i \in\mathbb{N} \} \) with unit basis identified by \((e_{n})\) is an example of a countably Hilbert space \((E, \|{\cdot}\|_{i})_{i = 1}^{\infty}\) that has an unconditional basis \((e_{n})\) (see [15]).
Let E be a countably normed space. In [13], E is called uniformly convex if \((E_{i}, \|{\cdot}\|_{i})\) is uniformly convex for all i, that is, if for each i, \(\forall \epsilon> 0\) \(\exists \delta _{i}(\epsilon) >0 \) such that if \(x, y \in E_{i}\) with \(\|x\|_{i} = 1 = \|y\|_{i}\) and \(\|x - y\|_{i} \ge\epsilon\), then \(1 - \|\frac{x + y}{2}\|_{i} > \delta_{i}\). The space E is called uniformly smooth if \((E_{i}, \|{\cdot}\|_{i})\) is uniformly smooth for all i, that is, if for each i whenever given \(\epsilon>0\) there exists \(\delta_{i} > 0\) such that if \(\|x\|_{i} = 1\) and \(\|y\|_{i} \leq\delta_{i}\), then \(\|x+y\|_{i} +\|x-y\|_{i} < 2 + \epsilon\|y\|_{i}\).
Following these two notions, we see that any countably Hilbert space E is uniformly convex and uniformly smooth because each of its \(H_{i}\) is a Hilbert space.
- (i)
E is uniformly convex if and only if for each i, \(\delta _{E_{i}} (\epsilon) > 0 \) for all \(\epsilon\in(0,2]\).
- (ii)
E is uniformly smooth if and only if \(\lim_{t\to0^{+}} \frac {\rho_{E_{i}}(t)}{t} = 0\) for all i.
- (iii)For each n, let \(E_{n}\) be the completion of E in the norm \(\| {\cdot}\|_{n}\), and \(E_{n}^{*}\) its dual. Then for each i, we have: for every \(\tau> 0\),$$\rho_{E_{i}}(\tau) = \sup \biggl\{ \frac{\tau\epsilon}{2} - \delta _{E_{i}^{*}}(\epsilon) : 0 < \epsilon\leq2 \biggr\} . $$
- (iv)
E is uniformly smooth if and only if \(E_{i}^{*}\) is uniformly convex for all i.
- (v)
E is uniformly convex if and only if \(E_{i}^{*}\) is uniformly smooth for all i.
Lemma 2.11
- 1.
\(\lambda_{n} \rightarrow0\) as \(n \rightarrow\infty\);
- 2.the estimate of convergence rateis satisfied, where Φ is defined by \(\Phi(t)=\int \frac{dt}{\psi(t)}\), and \(\Phi^{-1}\) is the inverse function to Φ.$$ \lambda_{n} \leq \Phi^{-1} \Biggl(\Phi( \lambda_{1})-\sum_{j=1}^{n-1} \alpha_{j} \Biggr) $$(2)
Lemma 2.12
- 1.
\(\lambda_{n} \rightarrow0\) as \(n \rightarrow\infty\);
- 2.there exists a subsequence \(\{\lambda_{n_{l}}\}\subset \{\lambda_{n}\}\), \(l=1,2,\ldots\) , such that$$\begin{aligned}& \lambda_{n_{l}}\leq \psi^{-1} \biggl(\frac{1}{\sum_{m=1}^{n_{l}}\alpha_{m}}+ \frac{\bar {\gamma}_{n_{l}}}{\alpha_{n_{l}}} \biggr), \quad \bar{\gamma}_{n_{l}} = \gamma _{n_{l}} + \beta_{n_{l}} M, M>0, \end{aligned}$$(4)$$\begin{aligned}& \lambda_{n_{l}+1}\leq \psi^{-1} \biggl(\frac{1}{\sum_{m=1}^{n_{l}}\alpha_{m}}+ \frac{\bar {\gamma}_{n_{l}}}{\alpha_{n_{l}}} \biggr)+\bar{\gamma}_{n_{l}}, \end{aligned}$$(5)$$\begin{aligned}& \lambda_{n} \leq\lambda_{n_{l}+1} - \sum _{m=n_{l}+1}^{n-1}\frac{\alpha_{m}}{{\mathcal{A}}_{m}}, \quad n_{l}+1 \leq n \leq n_{l+1}, {\mathcal{A}}_{m}=\sum _{i=1}^{m-1}\alpha_{i}, \end{aligned}$$(6)$$\begin{aligned}& \lambda_{n+1} \leq\lambda_{1} -\sum _{m=1}^{n} \frac{\alpha_{m}}{{\mathcal{A}}_{m}} \leq \lambda_{1} ,\quad 1 \leq n \leq n_{1}-1, \end{aligned}$$(7)$$\begin{aligned}& 1 \leq n_{1} \leq s_{\mathrm{max}}=\max \Biggl\{ s:\sum _{m=1}^{s}\frac{\alpha_{m}}{{\mathcal{A}}_{m}} \leq \lambda_{1} \Biggr\} . \end{aligned}$$(8)
Lemma 2.13
A version of the following theorem in countably Hilbert spaces is very important and will be used in the proofs of our main results.
Theorem 2.14
([13])
3 Main results ‘successive approximations in a countably Hilbert space’
In this section we give new versions for some definitions of [7, 10] in countably Hilbert spaces and prove our main theorems.
Definition 3.1
(Weakly contractive)
Let K be a nonempty subset of a real countably Hilbert space E. A mapping \(A : K \rightarrow K\) is called weakly contractive of the class \(C_{\psi(t)}\) if there exists a continuous and nondecreasing function \(\psi(t)\) defined on \(\mathbb{R}^{+}\) with \(\psi(t)>0\) \(\forall t\in\mathbb{R}^{+} \setminus \{0\}\), \(\psi(0)=0\), and \(\lim_{t \rightarrow \infty} \psi(t)=+ \infty\) such that for each i, we have \(\|Ax-Ay\|_{i} \leq\|x-y\|_{i} -\psi(\|x-y\|_{i})\) \(\forall x,y \in K\).
Definition 3.2
(Asymptotically weakly contractive)
Theorem 3.3
Proof
From (9) it follows that \(\|Ax_{n}\|_{i} - \|Ax^{*}\|_{i} \leq\|x_{n}\|_{i} + \| x^{*}\|_{i} \leq\|x_{1}\|_{i} + 2 \|x^{*}\|_{i} + \|x^{*}\|_{i}\). Since \(Ax^{*} = x^{*}\), we have \(\|Ax_{n}\|_{i} \leq\|x_{1}\|_{i} + 4 \|x^{*}\|_{i} = R_{i}'\), and therefore \(\{Ax_{n}\}\) is bounded.
Now, we present stability theorems for the perturbed approximations. First, we study the iterative method with perturbed operators \(A_{n}:K\rightarrow E\).
Theorem 3.4
Proof
Thus, the sequence of positive numbers defined by \(\lambda_{n}^{i}=\|y_{n}-x^{*}\|_{i}\), \(n\geq1\), satisfies the recursive inequality \(\lambda_{n+1}^{i}\leq\lambda_{n}^{i}-\psi(\lambda_{n}^{i})+\gamma_{n}^{i}\). Then the assertion \(\lambda_{n}^{i}=\|y_{n}-x^{*}\|_{i} \rightarrow0\) as \(n \rightarrow\infty\) and estimates (11)-(15) follow from Lemma 2.12 with \(\alpha_{i}=1\) and \(\beta_{i} = 0\) for all \(i \geq1\). □
Let us suppose that, instead of an exact set K, we have a sequence of perturbed sets \(K_{n} \subset E\), \(n\geq1\), such that the Hausdorff distance \({\mathcal{H}}(K_{n},K) \leq \sigma_{n}\), that is, ‘\({\mathcal{H}}(K_{n},K)\) tends to 0 as n tends to ∞’. Let \(D(A)\) be any domain for the operator A that contains both K and the perturbed sets \(K_{n}\)and such that the Hausdorff distance ‘\({\mathcal{H}}(K_{n},K)\) tends to 0 as n tends to ∞’.
Theorem 3.5
Proof
Now we work in a system of perturbed operators \(A_{n}\) and perturbed sets \(K_{n}\) to approximate a fixed point \(x^{*}\) of the operator A on K.
Theorem 3.6
Proof
Theorem 3.7
Remark 3.8
Observe that (27) for \(A_{n}\) is similar to the condition of weak contraction of A. At the same time, (28) is a standard condition of proximity between \(A_{n}\) and A in each point of K.
Proof of Theorem 3.7
Theorem 3.9
Proof
Theorem 3.10
Proof
- (i)
Assume that the given iteration starting at arbitrary \(y_{1} \in K\) is bounded, say by \(M > 0\); then \(\{A(P_{K}A)^{n-1} y_{n}\}\) is bounded, and hence, using (35), we get that \(\{A_{n}(P_{K}A_{n})^{n-1} y_{n}\}\) is bounded. Thus, by the nonexpansive property of \(P_{K}\) in each \(H_{i}\) and (35) we get \(\|(P_{K}A_{n})^{n} y_{n} - (P_{K}A)^{n} y_{n}\|_{i} \rightarrow 0\) as \(h_{n}\), \(\delta_{n} \to0\). Therefore, by (41) all the conditions of Lemma 2.12 are satisfied with \(\alpha_{i} = 1\) \(\forall i\geq1\).
- (ii)
Assume that the assumption \(\lim_{n \to\infty} \|(P_{K}A_{n})^{n} y_{n} - (P_{K}A)^{n} y_{n}\|_{i} = 0\) is satisfied. Then setting \(\lambda_{n}^{i} := \|y_{n} - x^{*}\|_{i}\), from (41) we get by Lemma 2.12 that the conclusions hold.
Declarations
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Authors’ Affiliations
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