Abstract
First we present some interesting variants of Mann’s method. In the last section, we show that many existing results in the literature are concrete realizations of our general scheme under varying assumptions on the coefficients.
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1 Some results on very small variation of Mann’s method
Let H be a real Hilbert space, \((\alpha_{n})_{n\in\mathbb{N}}\subset (0,\alpha]\subset(0,1)\) and \((\beta_{n})_{n\in\mathbb{N}},(\mu_{n})_{n\in \mathbb{N}}\subset(0,1]\). In the sequel, we will use the following notation:
-
We say that \(\zeta_{n}=o(\eta_{n})\) if \(\frac{\zeta_{n}}{\eta_{n}}\to0\) as \(n\to\infty\).
-
We say that \(\zeta_{n}=O(\eta_{n})\) if there exist \(K,N>0\) such that \(N\leq \vert \frac{\zeta_{n}}{\eta_{n}}\vert \leq K\).
Iterative schemes to approximate fixed points of nonlinear mappings have a long history and they still are an active research area of the nonlinear operator theory.
Here we are interested in the Mann iterative method introduced by Mann [1] in 1953. The method generates a sequence \((x_{n})_{n\in\mathbb {N}}\) via the recursive formula
where the coefficients sequence \((\alpha_{n})_{n\in\mathbb{N}}\) is in the real interval \([0,1]\), T is a self-mapping of a closed and convex subset of C of a real Hilbert space H, and the value \(x_{0}\in C\) is chosen arbitrarily.
Mann’s method has been studied in the literature chiefly for nonexpansive mappings T (i.e., \(\|Tx-Ty\|\leq\|x-y\|\) for all \(x,y\in C\)). Due to Reich [2] it is known that if T is nonexpansive and \(\sum_{n}\alpha_{n}(1-\alpha_{n})=+\infty\), then the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by Mann’s algorithm (1.1) converges weakly to a fixed point of T. Thanks to the celebrated counterexample of Genel and Linderstrauss [3], we know that Mann’s algorithm fails, in general, to converge strongly also in the setting of a Hilbert space.
In order to ensure strong convergence, in the past years, the method has been modified in several directions: by Ishikawa [4] using a double convex-combination, by Halpern [5] using an anchor, by Moudafi [6] using a contraction mapping, by Nakajo and Takahashi [7] using projections. These are just a few (but extremely relevant) of such modifications.
In this section we propose a variation of Mann’s method (Theorem 1.1 and Theorem 1.3) which differs from all those present in the literature, and it is closest to the original method (1.1). Moreover, we give several corollaries that are concrete and meaningful applications.
In the next section we see that the proof of all these results can be obtained by a very general two-step iterative algorithm. In the last section we give the proof of our main theorem and compare the rate of convergence of our method with that of Halpern on a specific case.
To our knowledge, Theorem 1.1 below provides a method that is almost the Mann method but ensures strong convergence.
Theorem 1.1
Let \(\alpha_{n}, \mu_{n}\in(0,1]\) such that
-
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
\(|\mu_{n-1}-\mu_{n}|=o(\mu_{n})\), and \(|\alpha _{n-1}-\alpha_{n}|=o(\alpha_{n}\mu_{n})\).
Then the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by
strongly converges to a point \(x^{*}\in \operatorname{Fix}(T)\) with minimum norm \(\|x^{*}\|=\min_{x\in \operatorname{Fix}(T)}\|x\|\).
Taking \(\mu_{n}=1\) we obtain the following.
Corollary 1.2
Let \(\alpha_{n}\in(0,1]\) such that
Then the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by
strongly converges to a point \(x^{*}\in \operatorname{Fix}(T)\) with minimum norm \(\|x^{*}\|=\min_{x\in \operatorname{Fix}(T)}\|x\|\).
We can see \(x^{*}\) as the point in \(\operatorname{Fix}(T)\) nearest to \(0\in H\). If we search for the point in \(\operatorname{Fix}(T)\) nearest to an arbitrary \(u\in H\), then we have the following theorem.
Theorem 1.3
Under the same assumptions on the coefficients \(\alpha_{n},\mu_{n}\) of Theorem 1.1, the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by
strongly converges to a point \(x_{u}^{*}\in \operatorname{Fix}(T)\) nearest to u, \(\|x_{u}^{*}-u\|=\min_{x\in \operatorname{Fix}(T)}\|x-u\|\).
Taking again \(\mu_{n}=1\), we obtain the following.
Corollary 1.4
(Halpern’s method)
Under the same assumptions on the coefficients \(\alpha_{n}\) of Corollary 1.2, the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by
strongly converges to a point \(x_{u}^{*}\in \operatorname{Fix}(T)\) nearest to u, \(\|x_{u}^{*}-u\|=\min_{x\in \operatorname{Fix}(T)}\|x-u\|\) too.
If A is a δ-inverse strongly monotone operator with \(A^{-1}(0)\neq\emptyset\), then \((I-\delta A)\) is nonexpansive [8, p.419] with fixed points \(\operatorname{Fix}(I-\delta A)=A^{-1}(0)\). By Theorem 1.3 with \(T=(I-\delta A)\), we have the following.
Corollary 1.5
Under the same assumptions on the coefficients \(\alpha_{n}\), \(\mu_{n}\) of Theorem 1.3, the sequence \((x_{n})_{n\in\mathbb{N}}\) generated by
strongly converges to a point \(x_{A,u}^{*}\in A^{-1}(0)\) nearest to u, \(\|x_{A,u}^{*}-u\|=\min_{x\in A^{-1}(0)}\|x-u\|\).
The first interesting example of a monotone δ-inverse strongly monotone operator with \(A^{-1}(0)\neq\emptyset\) is the gradient of a convex function. Precisely, let \(\phi:H\to\mathbb{R}\) be a convex and Fréchet differentiable function. Let us suppose that \(\nabla\phi(x)\) is an L-Lipschitzian mapping. We are interested in approximate solutions of the variational inequality (in the sequel (VIP))
since it is the optimality condition for the minimum problem
In our hypotheses, \(\nabla\phi(x)\) is a \(\frac{1}{L}\)-inverse strongly monotone operator. Then the mapping \((I-\frac{1}{L} \nabla\phi)\) is nonexpansive, and it results that the following can be obtained by Corollary 1.5.
Corollary 1.6
Let \(\operatorname{Fix}((I-\frac{1}{L} \nabla\phi))\neq\emptyset\) and \(u\in H\). Let us suppose that
-
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
\(|\mu_{n-1}-\mu_{n}|=o(\mu_{n})\), and \(|\alpha _{n-1}-\alpha_{n}|=o(\alpha_{n}\mu_{n})\).
Then the sequence generated by the iteration
strongly converges to \(x^{*}\in \operatorname{Fix}((I-\frac{1}{L} \nabla\phi))\) that is the unique solution of the variational inequality
i.e., \(x^{*}\) is the solution of (1.2) nearest to u.
A further interesting result concerns a Tikhonov regularized-constrained least squares defined as follows:
that aims the (ill-posed) constrained least squares problem
where \(C=\bigcap_{n\in\mathbb{N}}\operatorname{Fix}(T_{n})\), A is a linear and bounded operator on H, \(b\in H\) and \((T_{n})_{n\in\mathbb{N}}\) are nonexpansive satisfying the following:
-
(h1)
\(T_{n}:H\to H\) are nonexpansive mappings, uniformly asymptotically regular on bounded subsets \(B\subset H\), i.e.,
$$\lim_{n\to\infty}\sup_{x\in B}\|T_{n+1}x-T_{n}x \|=0; $$ -
(h2)
it is possible to define a nonexpansive mapping \(T:H\to H\) with \(Tx:=\lim_{n\to\infty}T_{n}x\) such that if \(F:=\bigcap_{n\in\mathbb{N}} \operatorname{Fix}(T_{n})\neq\emptyset\), then \(\operatorname{Fix}(T)=F\).
Reich and Xu in [9] proved, among others, that the unique solution of (1.5) strongly converges, when \(\varepsilon\to0\), to the solution of (1.6) that has minimum norm. The optimality condition to solve (1.5) is to solve the following variational inequality:
where \(A^{*}\) is the adjoint of A.
In light of Reich and Xu’s results, it would be interesting to approximate a solution of (1.7) (for small ε).
Let \(B:=A^{*}A-A^{*}b\). Note that since B is firmly nonexpansive, i.e., 1-inverse strongly monotone so \(I-B\) is firmly nonexpansive [10], hence nonexpansive. We are able to prove the following results.
Theorem 1.7
Assume that
-
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
\(|\beta_{n}-\beta_{n-1}|=o(\alpha_{n}\beta_{n}\mu_{n})\), \(|\mu_{n}-\mu_{n-1}|=o(\alpha_{n}\beta_{n}\mu_{n})\) and \(|\alpha_{n}-\alpha _{n-1}|=o(\alpha_{n}\beta_{n}\mu_{n})\);
-
\(\vert \frac{1}{\beta_{n}}-\frac{1}{\beta_{n-1}}\vert =O(\alpha_{n}\mu_{n})\).
Let us suppose \(\lim_{n\to\infty}\frac{\beta_{n}}{\alpha _{n}\mu_{n}}=\tau\in(0,+\infty)\).
(These hypotheses are satisfied, for instance, by \(\alpha _{n}=\mu_{n}=\frac{1}{\sqrt[4]{n}}\), \(\beta_{n}=\frac{2}{\sqrt{n}}\), \(n\geq1\).)
Then \((x_{n})_{n\in\mathbb{N}}\) defined by (2.1), i.e.,
strongly converges to \(\tilde{x}\in \bigcap_{n\in\mathbb{N}} \operatorname{Fix}(T_{n})\) that is the unique solution of the variational inequality
i.e.,
Note that we do not assume any hypotheses on the commutativity of the mappings in spite of the main theorem in [9] (see also [11, 12]).
2 A general iterative method
In this section we study the convergence of the following general two-step iterative algorithm in a Hilbert space H:
where
-
\(D:H\to H\) is a σ-strongly monotone and L-Lipschitzian operator on H, i.e., D satisfies
$$\langle Dx-Dy,x-y\rangle\geq\sigma\|x-y\|^{2} \quad\mbox{and}\quad \|Dx-Dy\| \leq L\|x-y\|. $$ -
\(S:H\to H\) is a nonexpansive mapping.
-
\((W_{n})_{n\in\mathbb{N}}\) is defined on H and such that
-
(h1)
\(W_{n}:H\to H\) are nonexpansive mappings, uniformly asymptotically regular on bounded subsets \(B\subset H\), i.e.,
$$\lim_{n\to\infty}\sup_{x\in B}\|W_{n+1}x-W_{n}x \|=0, $$ -
(h2)
it is possible to define a nonexpansive mapping \(W:H\to H\), with \(Wx:=\lim_{n\to\infty}W_{n}x\) such that if \(F:=\bigcap_{n\in\mathbb{N}} \operatorname{Fix}(W_{n})\neq\emptyset\) then \(\operatorname{Fix}(W)=F\).
-
(h1)
-
The coefficients \((\alpha_{n})_{n\in\mathbb{N}}\subset(0,\alpha ]\subset(0,1)\), \((\beta_{n})_{n\in\mathbb{N}}\subset(0,1)\) and \((\mu _{n})_{n\in\mathbb{N}}\subset(0,\mu)\), where \(\mu<\frac {2\sigma}{L^{2}}\).
Remark 2.1
If \((T_{n})_{n\in\mathbb{N}}\) does not satisfy (h1) and (h2), then it is always possible to construct a family of nonexpansive mappings \((W_{n})_{n\in\mathbb{N}}\) satisfying (h1) and (h2) and such that \(\bigcap_{n\in\mathbb{N}} \operatorname{Fix}(T_{n})=\bigcap_{n\in\mathbb{N}} \operatorname{Fix}(W_{n})\) (see [13, 14]).
All the previous results easily follow from our main theorem below.
Theorem 2.2
Let H be a Hilbert space. Let D, S, \((W_{n})_{n\in\mathbb{N}}\) be defined as above. Then:
(1) Let \(\tau=\lim_{n\to\infty}\frac{\beta _{n}}{\alpha_{n}\mu_{n}}=0\). Assume that
-
(H1)
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
(H2)
\(\sup_{x\in B} \|W_{n}z-W_{n-1}z\|=o({\alpha _{n}\mu_{n}})\), where \(B\subset H\) is bounded;
-
(H3)
\(|\mu_{n-1}-\mu_{n}|=o(\mu_{n})\) and \(|\alpha_{n-1}-\alpha_{n}|=o(\alpha_{n}\mu_{n})\).
(These hypotheses are satisfied, for instance, by \(\alpha _{n}=\mu_{n}=\frac{1}{\sqrt{n}}\) and \(\beta_{n}=\frac{1}{n^{2}}\), \(n\geq1\).)
Then the sequence generated by iteration (2.1) strongly converges to \(x^{*}\in F\) that is the unique solution of the variational inequality
(2) Let us suppose \(\lim_{n\to\infty}\frac{\beta _{n}}{\alpha_{n}\mu_{n}}=\tau\in(0,+\infty)\). Assume that
-
(H1)
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
(H4)
\(\sup_{x\in B} \|W_{n}z-W_{n-1}z\|=o(\alpha_{n}\mu _{n}\beta_{n})\), where \(B\subset H\) is bounded;
-
(H5)
\(|\beta_{n}-\beta_{n-1}|=o(\alpha_{n}\beta_{n}\mu _{n})\), \(|\mu_{n}-\mu_{n-1}|=o(\alpha_{n}\beta_{n}\mu_{n})\) and \(|\alpha_{n}-\alpha _{n-1}|=o(\alpha_{n}\beta_{n}\mu_{n})\);
-
(H6)
\(\vert \frac{1}{\beta_{n}}-\frac{1}{\beta _{n-1}}\vert =O(\alpha_{n}\mu_{n})\).
(These hypotheses are satisfied, for instance, by \(\alpha _{n}=\mu_{n}=\frac{1}{\sqrt[4]{n}}\), \(\beta_{n}=\frac{2}{\sqrt{n}}\), \(n\geq1\) and \((W_{n})\) and \((\lambda_{n})\) are as in Stm1.)
Then \(x_{n}\to\tilde{x}\), as \(n\to\infty\), where \(\tilde{x}\in F\) is the unique solution of the variational inequality
(3) If \(\tau=\lim_{n\to\infty}\frac{\beta _{n}}{\alpha_{n}\mu_{n}}=\infty\) and \(\operatorname{Fix}(S)\bigcap F\neq\emptyset\). Let us suppose that
-
(H1)
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
(H2)
\(\sup_{x\in B} \|W_{n}z-W_{n-1}z\|=o({\alpha _{n}\mu_{n}})\), where \(B\subset H\) is bounded.
-
(H7)
\(|\mu_{n-1}-\mu_{n}|=o(\mu_{n})\), and \(|\alpha _{n-1}-\alpha_{n}|=o(\alpha_{n}\mu_{n})\).
If \(\beta_{n}\to\beta\neq0\), as \(n\to\infty\), and \(|\beta _{n-1}-\beta_{n}|=o(\alpha_{n}\mu_{n})\), then the sequence generated by iteration (2.1) strongly converges to \(x^{*}\in F\cap \operatorname{Fix}(S)\) that is the unique solution of the variational inequality
Proof
We give the proof in the next (and last) section. □
Proof of Theorem 1.1
It follows by Theorem 2.2(1) choosing \(D=I\), \(\mu=1\), \(W_{n}=T\) and \(S=I\). □
Proof of Theorem 1.3
If we take \(Dx=x-u\), \(S=I\), \(\mu=1\) and \(W_{n}=T\), the proofs follow by Theorem 2.2(1). □
Proof of Corollary 1.6
Easily follows by Corollary 1.5 when \(A=\nabla\phi\) and \(\delta=\frac{1}{L}\). □
Remark 2.3
It is interesting to note that the convergence in Corollary 1.6 can be obtained from Theorem 2.2(3) when \(S=W_{n}=(I-\delta A)\).
The last application of our main theorem concerns the problem to minimize a quadratic function over a closed and convex subset C of H
where h is a potential function for a contraction mapping f, i.e., \(h'= f\) on H (for references, one can read [15, 16]).
Let A be a strongly positive bounded linear operator on H, i.e., there exists \(\bar{\gamma}>0\) such that \(\langle Ax,x\rangle\geq \bar{\gamma}\|x\|^{2}\) for all \(x \in H\).
Let us take as a subset C the set of common fixed points of a given semigroup of nonexpansive mappings. Let \(\mathfrak{T}\) be a one-parameter continuous semigroup of nonexpansive mappings defined on H with a common fixed points set \(F\neq\emptyset\). Let \((\lambda_{n})_{n\in \mathbb{N}}\) be a sequence in \((0,1)\) such that \(\lim_{n\to\infty}\lambda_{n}=\lambda\in (0,1)\).
We know the fact, due to Suzuki [17], that \(W_{n}x:=\lambda_{n} T(1)x+(1-\lambda_{n})T(\sqrt{2})x\) is a nonexpansive mapping such that \(\operatorname{Fix}(W_{n})= \operatorname{Fix}(T(1))\cap \operatorname{Fix}(T(\sqrt{2}))=F\). Moreover,
Further, if x lies in a bounded set \(B\subset X\), the uniform asymptotic regularity on B follows. If \(Sx:=\lambda T(1)x+(1-\lambda)T(\sqrt{2})x\), then \(\operatorname{Fix}(S)=F\) and, for all \(x\in H\),
In light of [16, 18, 19], we consider
We are able to prove the following new convergence result.
Theorem 2.4
Let us suppose that
-
\(\lim_{n\to\infty}\alpha_{n}=0\), \(\sum_{n\in\mathbb{N}}\alpha_{n}\mu_{n}=\infty\);
-
\(|\lambda_{n+1}-\lambda_{n}|=o({\alpha_{n}\mu_{n}})\);
-
\(|\mu_{n-1}-\mu_{n}|=o(\mu_{n})\), and \(|\alpha _{n-1}-\alpha_{n}|=o(\alpha_{n}\mu_{n})\).
If \(\beta_{n}\to\beta\neq0\), as \(n\to\infty\), and \(|\beta _{n-1}-\beta_{n}|=o(\alpha_{n}\mu_{n})\), then the sequence generated by iteration (2.1), i.e.,
strongly converges to \(x^{*}\in F\) that is the unique solution of the variational inequality
which is the optimality condition to solve
Proof of Theorem 2.4
Easily follows by Theorem 2.2 statement 3 when \(S=\lambda T(1)+(1-\lambda)T(\sqrt{2})\), \(W_{n}=\lambda_{n} T(1)+(1-\lambda_{n})T(\sqrt{2})\) and \(D=A-f\) (see [16]). □
3 Proof of Theorem 2.2
Lemma 3.1
Let \((x_{n})_{n\in\mathbb{N}}\) be defined by iteration (2.1) and \((\alpha_{n})_{n\in\mathbb{N}}, (\beta_{n})_{n\in\mathbb{N}}\subset [0,1]\) and \((\mu_{n})_{n\in\mathbb{N}}\subset(0,\mu)\). Assume that
-
(H0)
\(\beta_{n}=O(\alpha_{n}\mu_{n})\)
holds. Then \((x_{n})_{n\in\mathbb{N}}\) and \((y_{n})_{n\in\mathbb{N}}\) are bounded.
Proof
Putting \(B_{n}:=(I-\mu_{n}D)\), we then have
i.e., \((I-\mu_{n}D)\) is a \((1-\mu_{n}\rho)\)-contraction (see [20]). Let \(z \in F\). Then, for sufficiently large \(N_{0}\) and for \(\gamma>0\), we have
So, by an inductive process, one can see that
As a rule \((y_{n})_{n\in\mathbb{N}}\) is bounded too. □
We recall the following lemma.
Lemma 3.2
In the hypotheses of Theorem 2.2(1), we have that the sequence generated by \(z_{0}\in H\) and the iteration
strongly converge to \(x^{*}\in F\) that is the unique solution of the variational inequality
Proof
The proof is given in [13, Theorem 2.6]. □
Proof of Theorem 2.2
Proof of 1. Let us note that since \(\tau=0\), then (H0) holds so \((x_{n})_{n\in\mathbb{N}}\) is bounded by Lemma 3.1. Let us consider the iteration generated by
By Lemma 3.2, \((z_{n})_{n\in\mathbb{N}}\) strongly converges to the unique solution of VIP (3.1). Then if we compute
Calling \(s_{n}:=\|x_{n}-z_{n}\|\), \(a_{n}:=\alpha_{n}\mu_{n}\rho\), we have that
Since \(\sum_{n}\alpha_{n}\mu_{n}=\infty\) and \(\tau=0\), we can apply Xu’s Lemma 2.5 in [19] to obtain the required result.
Proof of 2. It is not difficult to observe that, by Byrne [10], \((I-S)\) is a \(\frac{1}{2}\)-inverse strongly monotone operator, so \((\frac {1}{\tau}D+(I-S) )\) is a \(\frac{\sigma}{\tau}\)-strongly monotone operator. Then (VIP) (2.3) has a unique solution by the celebrated results of Browder and Petryshyn [21] and Deimling Theorem 13.1 in [22].
We next prove that \((x_{n})_{n\in\mathbb{N}}\) is asymptotically regular with respect to \((\beta_{n})_{n\in\mathbb{N}}\), i.e.,
In order to prove the previous limit, we first compute
By definition of \(y_{n}\) one obtains that
So, substituting (3.4) in (3.3), we obtain
Let us observe that by (H5)
and (H4) guarantees that
Putting \(s_{n}:=\|x_{n}-x_{n-1}\|\), \(a_{n}:=\mu_{n}\alpha_{n}\rho\) and \(b_{n}=\| W_{n}y_{n-1}-W_{n-1}y_{n-1}\|+(\alpha_{n}|\mu_{n}-\mu_{n-1}|+|\alpha_{n}-\alpha _{n-1}|+|\beta_{n}-\beta_{n-1}|)O(1)\), we can write (3.5) as
Thus (H1), (H4) and (H5) are enough to apply Xu’s Lemma 2.5 in [19] to assure that \((x_{n})_{n\in\mathbb{N}}\) is asymptotically regular. Moreover, dividing by \(\beta_{n}\) in (3.5), one observes that
Since (H1), (H4) and (H5) hold, by using again Xu’s Lemma 2.5 in [19], we have
Moreover, by the asymptotic regularity of \((x_{n})_{n\in\mathbb{N}}\), we show that the weak limits \(\omega_{w}(x_{n})\subset F\). Let \(p_{0}\in\omega _{w}(x_{n})\) and \((x_{n_{k}})_{k\in\mathbb{N}}\) be a subsequence of \((x_{n})_{n\in\mathbb{N}}\) weakly converging to \(p_{0}\). If \(p_{0}\notin F\), then by the Opial property of a Hilbert space
Since \(\beta_{n}\to0\) as \(n\to\infty\), then
which is absurd. Then \(p_{0}\in F\).
On the other hand,
so that we define
As a rule \(v_{n}:=\frac{x_{n}-x_{n+1}}{(1-\alpha_{n})\beta_{n}}\) is also a null sequence as \(n\to\infty\).
Now we prove that \(\omega_{w}(x_{n})=\omega_{s}(x_{n})\), i.e., every weak limit is a strong limit too. We only need to prove that \(\omega _{w}(x_{n})\subset\omega_{s}(x_{n})\).
Let us fix \(z\in \omega_{w}(x_{n})\), then \(z\in F\), and by (3.6) it results
Since the operator \((I-W_{n})\) is monotone for all \(n\in\mathbb{N}\), we obtain that
and so we can write
Let us note that
So, by the hypotheses, \(\|y_{n}-W_{n}y_{n}\|\to0\) as \(n\to\infty\). So if \((x_{n_{k}})_{k}\) is such that \(x_{n_{k}}\to z\) as \(k\to\infty\), it follows that
Since \(v_{n}\to0\) and \((I-W_{n})y_{n}\to0\) as \(n\to\infty\), then every weak cluster point of \((x_{n})_{n\in\mathbb{N}}\) (that lies in F) is also a strong cluster point.
We prove that \(\omega_{w}=\omega_{s}(x_{n})\) is a singleton. By the boundedness of \((x_{n})_{n\in\mathbb{N}}\), let \((x_{n_{k}})_{k\in\mathbb{N}}\) be a subsequence of \((x_{n})_{n\in \mathbb{N}}\) converging (weakly and strongly) to \(x'\). For all \(z\in F\), again by (3.6)
Passing to limit as \(k\to\infty\), we obtain
that is, (2.3) holds. Thus, since (2.3) cannot have more than one solution, it follows that \(\omega_{w}(x_{n})=\omega_{s}(x_{n})=\{\tilde{x}\}\) and this, of course, ensures that \(x_{n}\to\tilde{x}\) as \(n\to\infty\).
Now we investigate the case
Proof of 3. Let \(z \in F\cap \operatorname{Fix}(S)\). Then
So, by an inductive process, one can see that
By (3.5) in Proof of 2, we have
Let us observe that by (H7) we have
and (H2) guarantees that
Then, calling \(s_{n}=\|x_{n}-x_{n-1}\|\), \(a_{n}=\mu_{n}\alpha_{n}\rho\) and \(b_{n}=\| W_{n}y_{n-1}-W_{n-1}y_{n-1}\|+(\alpha_{n}|\mu_{n}-\mu_{n-1}|+|\alpha_{n}-\alpha _{n-1}|+|\beta_{n}-\beta_{n-1}|)O(1)\), we can write (3.7) as
and (H1), (H2) and (H7) are enough to apply Xu’s Lemma 2.5 in [19] and to assure that \((x_{n})_{n\in\mathbb{N}}\) is asymptotically regular.
For every \(v\in \operatorname{Fix}(S)\cap F\), it results that
So, by the boundedness we get
Then \(\|Sx_{n}-x_{n}\|\to0\) as \(n\to\infty\) and, by the demiclosedness principle, the weak cluster points of \((x_{n})_{n\in\mathbb{N}}\) are fixed points of S, i.e., \(\omega_{w}(x_{n})\subset \operatorname{Fix}(S)\). Let us show that there are more \(\omega_{w}(x_{n})\subset F\). If not, let \(p_{0}\in\omega_{w}(x_{n})\) and \(p_{0}\notin F\). By the Opial property of a Hilbert space, we have
which is absurd, so \(p_{0}\in F\). To conclude, if z is the unique solution of VIP (2.8), then
Since every weak cluster point of \((x_{n})_{n\in \mathbb{N}}\) lies in \(F\cap \operatorname{Fix}(S)\), then for an opportune subsequence \((x_{n_{k}})\rightharpoonup p_{0}\)
Thus, calling \(s_{n}:=\|x_{n}-z\|^{2}\), \(a_{n}= \alpha_{n}\mu_{n}\rho\) and \(b_{n}=2\alpha_{n}\mu_{n}\langle-Dz,x_{n+1}-z\rangle\), we can write
and by Xu’s Lemma 2.5 in [19], \(x_{n}\to z\) as \(n\to\infty\). □
To us, applications of Theorem 2.2 are a well-known problem, and also we know that there exist several iterative approaches to approximate the solutions. Nevertheless, our iterative scheme summarizes a lot of them assuming very simple hypotheses on the numerical sequences, and it can be applied to a wide class of mappings thanks to hypotheses (h1) and (h2). However the reader could ask for a comparison of scheme (2.1) and the well-known iterative approach cited here. We do not known the rate of convergence of our method, but it is enough to see the numerical examples in [13, 23] to conclude that it is not possible to compare two iterative schemes. However, for the sake of completeness, we include a very simple case which shows that our scheme is faster than Halpern’s scheme.
Example 3.3
Let \(H=\mathbb{R}\), \(u=1\), \(W_{n}x:=-x\) (hence \(F=\{0\}\)), \(Dx:=x-1\), \(Sx:=x-1\). Let \(\alpha_{n}=\frac{1}{2n}\), \(\mu_{n}=1\) and \(z_{1}=2\). Then Halpern’s iterative method
becomes
If \(\beta_{n}=\frac{1}{n^{2}}\), from our scheme (2.1) we obtain
Thus our iterative scheme is slightly faster as shown in Table 1 (see also [24]).
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Acknowledgements
The authors are grateful to the anonymous referees for their useful comments and suggestions. This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University under grant No. (29-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
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Hussain, N., Marino, G., Muglia, L. et al. On some Mann’s type iterative algorithms. Fixed Point Theory Appl 2015, 17 (2015). https://doi.org/10.1186/s13663-015-0267-8
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DOI: https://doi.org/10.1186/s13663-015-0267-8