Ergodic approximations via matrix regularization approach

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R E S E A R C H

Open Access

Ergodic approximations via matrix

regularization approach

Giuseppe Marino

1,2*

_{, Filomena Cianciaruso}

1

_{and Nawab Hussain}

2

*_{Correspondence:}

[email protected]

1_{Dipartimento di Matematica,}

Universitá della Calabria, Arcavacata di Rende, CS 87036, Italy

2_{Department of Mathematics, King}

Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Abstract

In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence.

MSC: 47J20; 47J25; 49J40; 65J15

Keywords: monotone variational inequality; nonspreading mappings; iterative method; Cesaro means; Toeplitz matrix

1 Introduction and results

The results presented here will be proved in Section .

LetHbe a Hilbert space and letf :H→Hbe aρ-contraction. It is well known that if

T:H→His a quasi-nonexpansive mapping, the set of the ﬁxed pointsFix(T) is closed and convex and the variational inequality problem (VIP)

(I–f)x∗,y–x∗≥, ∀y∈Fix(T), (.)

admits a unique solution.

The interactions between fixed point theory and the calculus of variations are certainly not recent. In this direction Moudafi in  [], Xu in  [] and Marino and Xu [] in  have shown deep connections between the approximation of fixed points for nonlinear mappings and the approximation of solutions of variational inequalities.

In this paper our aim is to approximate onFix(T) the unique solution of a (VIP) when

Tis a nonspreading mapping,i.e.

Tx–Ty≤ x–y+ x–Tx,y–Ty, ∀x,y∈H. (.)

The class of nonspreading mappings has been introduced recently by Kohsaka and Taka-hashi in []. They observed that it contains the class ofﬁrmly nonexpansive-type mappings, introduced by the same authors in []

Note that a nonspreading mappingT isquasi-nonexpansive[].

The choice to consider quasi-nonexpansive mappings is made since these operators are widely used in the sub-gradient projection operator techniques to solve convex

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strained problems (see Maingé [] and the references contained therein). Nevertheless, the lack of continuity of a quasi-nonexpansive mapping makes it diﬃcult to prove asymp-totical stability of iterates ofT (see, for example, []). This property is a key for the con-vergence analysis of sequences generated by well-known iterative methods.

To ﬁll this gap, we employ regularizations that involve matrices (bn,k)n,k∈N, so-called

triangular-Toeplitz matrices, satisfying the following properties:

(h) bn,k≥for alln,k∈N;

(h) for anyn∈N,bn,k= fork>nand for alln∈N,(bn,k)k∈Nis a nonincreasing

sequence;

(h) for anyn∈N,n_k_=bn,k= , and

(h) for anyk∈N,bn,k→, asn→ ∞.

We will denote byTthe set of the triangular-Toeplitz matrices;Tis a subclass of the reg-ular matrices in the sense of Toeplitz(see []). The weak limits of a regularized sequence that involves a nonspreading mapping and a triangular-Toeplitz matrix are ﬁxed points ofT; more precisely, we will prove the following.

Lemma . Let T:H→H be a nonspreading mapping and let(bn,k)n,k∈N∈T.

Let(xn)n∈Nbe a bounded sequence in H;we deﬁne

yn:= n–

k=

bn,k+Tkxn. (.)

Then the weak limits of(yn)n∈Nare ﬁxed points of T,i.e.ωw(yn)⊂Fix(T).

In  Mann [], taking in account the works of Cesaro, Fejer and Toeplitz, considered the problem to construct a sequence in a convex and compact setCof a Banach spaceX

that converges to a ﬁxed point of a continuous transformationT:C→C. Mann introduced an inﬁnite triangular matrix (an,k)n,k∈N,

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

   . . . 

a, a,  . . . 

. . . .

an, an, . . . an,n . . . . . . .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

,

such that

(M) an,k≥, and an,k= , fork>n; (M) n

k=

an,k= , ∀n.

We will denote byMthe set of matrices satisfying (M) and (M). Let us observe thatT⊂M.

By using (an,k)n,k∈NMann introduced the following iteration process:

⎧ ⎨ ⎩

vn=

n k=an,kxk,

xn+=T(vn).

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It is very easy to check that the Cesaro matrix

C:=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

   . . . 

 



  . . . 

. . . .



n



n . . .



n . . . . . . .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

belongs toT. Moreover, by usingC, (.) becomes

vn+=

 – 

n+ 

vn+



n+ Tvn. (.)

However, to the best of our knowledge, there are not also convergence results for the original Mann iterative method. Therefore, in light of the above facts, we study an itera-tive method that implements two matrices: (an,k)n,k∈N∈Mto compute our iterations; and (bn,k)n,k∈N∈Tto regularize the mapping involved.

We will investigate on the followingMann-Toeplitz iteration process: letx∈H ⎧

⎨ ⎩

vn=

n k=an,kxk,

xn+=fμn

n–

k=bn,k+Tkvn,

(.)

where (an,k)n,k∈Nsatisﬁes (M) and (M),i.e.(an,k)n,k∈N∈M,Tis a nonspreading mapping,

f is aρ-contraction, andfμnis the averaged mappingfμn=μnf+ ( –μn)I(see [] for the regularization properties of nonspreading and averaged mappings).

The matrix (bn,k)n,k∈Nin (.) plays a regularization role similar to the Cesaro matrix. It is well known that the Cesaro mean preserves (or it increases) the regularity of a sequence. Therefore many recent papers, concerning the approximation of ﬁxed points of nonlin-ear mappings and sometimes solutions of a (VIP), combine well-known iterative methods (Mann-Dotson, viscosity, Halpern, and so on) with the Cesaro means.

The ﬁrst nonlinear ergodic theorem in Hilbert space was established by Baillon in [], who proved that, ifCis a nonempty closed convex subset of a Hilbert space andT:C→C

is a nonexpansive mapping withFix(T)=∅, the Cesaro means

Sn(x) = 

n

n–

k= Tk(x)

weakly converges to a ﬁxed point ofT.

Shimizu and Takahashi in [] studied by Cesaro means the convergence of an iteration process for a family of nonexpansive mappings in a Hilbert space. This result has been ex-tended by Shioji and Takahashi in [], by Song and Chen in [] in the setting of uniformly convex Banach spaces.

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Many references can be cited dealing with the approximation of solutions of (VIPs) and equilibrium problems. For instance in [, ], and [] the authors introduced iterative processes in Hilbert spaces by using the Cesaro means for ﬁnding a common element of the set of ﬁxed points of a nonexpansive mapping and the set of solutions of an equilibrium problem.

In  and in  Song (see [] and []) studied an implicit viscosity method for a continuous pseudocontractive mapping and a Mann iterations of its Cesaro mean for an asymptotically nonexpansive mapping in Banach spaces.

Nevertheless the Cesaro matrix is not the unique matrix with regularization properties. Toeplitz summability theorems proved that there exists a class of matrices including the Cesaro means with the same regularization property. Therefore we consider it relevant, in its generality, to put our attention on the class of triangular-Toeplitz matrices.

We denote byM∗an interesting subclass ofMof matrices such thatan,n→ asn→ +∞. An example of such matrices is

˜ A:=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

   . . . 

 



  . . . 

. . . .



n₍_n_–) _n₍_n_–) . . .  –_n . . .

. . . .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

. (.)

In Section  for this subclass of matrices we will prove the following.

Theorem . Let f :H→H be aρ-contraction and let fμnbe the averaged mapping fμn=

μnf+ ( –μn)I.Let T be a nonspreading mapping.

Let us choose(an,k)k,n∈N∈M∗and(bn,k)n,k∈N∈T.

Let us chooseμ<(–_(+_ρρ₎) and(μn)n∈N⊂(,μ)such that

(h) limn→∞μn= ,

n∈Nμn=∞.

(h) limn→∞–μann,n= .

Setτ=  –ρ.

Then the sequence generated by the Mann-Toeplitz iterative method(.)strongly con-verges to x∗∈Fix(T),that is,a unique solution of the variational inequality

(I–f)x∗,y–x∗≥, ∀y∈Fix(T).

Remark . The assumption (h) is easily satisﬁed. For example, choosingμn=_n andA˜ as in (.), we see that it holds.

If (γn)n∈Nis a sequence in [, ], we can deﬁne, recursively, a matrix (a˜n,k)n,k∈Nby

(a˜n,k)n,k∈N=

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

, n=k= ,

, n≥ andk>n,

γn–an–,k, n≥,k<n, ( –γn–), n≥,k=n.

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Then the Mann-Toeplitz iterative method (.) becomes

vn+=γnvn+ ( –γn)fμn n–

k=

bn,k+Tkvn. (.)

The use of this matrix of regularization will be that it permits us to improve the main the-orem of Dotson [], obtaining strong convergence instead of weak convergence. Indeed, we will prove the following.

Theorem . Let f :H→H be aρ-contraction and let fμnbe the averaged mapping fμn=

μnf + ( –μn)I. Let T be a nonspreading mapping.Let(bn,k)n,k∈N∈Tbe a

triangular-Toeplitz matrix.

Let us chooseμ<_(+(–_ρρ₎) and(μn)n∈N⊂(,μ)such that

(h) limn→∞μn= ,

n∈Nμn=∞.

Let us take(a˜n,k)n,k∈N∈Mdeﬁned by a sequence(γn)n∈N⊂[,α]⊂[, ) (see(.))such

that

(h) limn→∞γn= .

Then the sequence generated by v∈H and by the iteration(.)strongly converges to x∗∈

Fix(T),that is,the unique solution of the variational inequality

(I–f)x∗,y–x∗≥, ∀y∈Fix(T).

2 Proofs

First we prove Lemma ., which is a fundamental tool to prove the convergence of our methods.

Proof of Lemma. Note that since (bn,k)n,k∈N is nonincreasing, ≤bn,n≤bn,k, for any

k≤n, so (h) implies thatbn,n→, asn→ ∞. Moreover, following [], one easily obtains

Tk+x–Ty≤Tkx–Ty+Ty–y+ Tkx–Ty,Ty–y

+ Tkx–Tk+x,y–Ty, ∀k∈N.

Therefore the following inequality holds:

Tk+xn–Ty



–Tkxn–Ty



≤ Ty–y+ Tkxn–Ty,Ty–y

+ Tkxn–Tk+xn,y–Ty

. (.)

Firstly we work on the right side of the previous equation. Multiplying bybn,k+and

sum-ming up overkfrom  to (n– ), by (h) and (.) we obtain

n–

k=

bn,k+Ty–y+ 

n–

k= bn,k+

Tkxn–Ty,Ty–y

+  n–

k= bn,k+

Tkxn–Tk+xn,y–Ty

=Ty–y+  yn–Ty,Ty–y+ 

yn–

n–

k=

bn,k+Tk+xn,y–Ty

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We observe that

yn– n–

k=

bn,k+Tk+xn= n–

k=

bn,k+Tkxn– n–

k=

bn,k+Tk+xn

=bn,xn+ n–

k=

(bn,k+–bn,k)Tkxn+bn,nTnxn;

then, by the boundedness of (xn)n∈N and by the quasi-nonexpansivity ofT, there exists

M=M(y) such that, by (h), (.) becomes

Ty–y+ yn–Ty,Ty–y+ 

yn–

n–

k=

=Ty–y+ yn–Ty,Ty–y

+ 

bn, xn,y–Ty+ n–

k=

(bn,k+–bn,k)

Tkxn,y–Ty

+bn,n

Tnxn,y–Ty

≤ Ty–y+  yn–Ty,Ty–y+ M

bn,+

n–

k=

(bn,k+–bn,k) +bn,n

=Ty–y_{+ }_y

n–Ty,Ty–y+ Mbn,n.

In a similar manner, for the left side of (.), multiplying bybn,k+, summing overkfrom 

to (n– ) and using (h), we have

n–

k=

bn,k+Tk+xn–Ty



– n–

k=

bn,k+Tkxn–Ty



=bn,nTnxn–Ty–bn,xn–Ty+ n–

k=

(bn,k–bn,k+)Tkxn–Ty

≥bn,nTnxn–Ty



–bn,xn–Ty.

Therefore we have the following chain:

bn,nTnxn–Ty



–bn,xn–Ty

≤

n–

k=

bn,k+Tk+xn–Ty– n–

k=

bn,k+Tkxn–Ty

≤ Ty–y+  yn–Ty,Ty–y

+ 

yn–

n–

k=

+bn,xn–Ty

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and so, by the positivity ofbn,k, it follows that

bn,nTnxn–Ty≤ Ty–y+  yn–Ty,Ty–y+ Mbn,n+ Mbn,.

Now, ifw∈ωw(yn), bybn,n→ asn→+∞, we have the result that, for everyy∈H,

≤ Ty–y+  w–Ty,Ty–y

so, choosingy=w

Tw–w≤,

i.e.we have the thesis.

To obtain our results we also need of the following (well-known) lemmas; the ﬁrst is proved in [], the second is the discrete Jensen inequality.

Lemma .[] Assume(an)n∈Nis a sequence of nonnegative numbers for which

an+≤( –sn)an+δn, n≥,

where(sn)n∈Nis a sequence in(, )and(δn)n∈Nis a sequence inRsuch that

() limn→∞sn= and

∞

n=sn=∞;

() lim sup_n_→∞δn sn ≤or

∞

n=|δn|<∞.

Thenlimn→∞an= .

Lemma . Letψ :H→Rbe a convex function.Let {x, . . . ,xn} ⊂H and let positive

weights{a, . . . ,an} ⊂R,we have the result that

ψ n

i=aixi

n i=ai

≤

n

i=aiψ(xi) n i=ai

.

Proof of Theorem. It is well known that (see [])

fμnx–fμny ≤( –μnτ)x–y,

whereτ:=  –ρ.

Moreover, we will denoteyn=

n–

k=bn,k+Tkvn. SinceTis quasi-nonexpansive and (h) holds, for everyw∈Fix(T), as a result one ﬁnds that

yn–w ≤ n–

k=

bn,k+Tkvn–Tkw≤ n–

k=

bn,k+vn–w=vn–w.

Set

r:=max

x–w,

(I–f)w

τ

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we prove, by induction onn, that (xn)n∈Nand (vn)n∈N⊂B(w,r). Let us observe the follow-ing:

xn+–w ≤ fμnyn–w

≤ fμnyn–fμnw+fμnw–w

≤( –μnτ)yn–w+fμnw–w

≤( –μnτ)vn–w+μn(I–f)w

= ( –μnτ)

n

k=

an,k(xk–w)

+τ μn

(I–f)w

τ .

Sincex=v∈B(w,r), we immediately havex∈B(w,r) and then, by (M),v∈B(w,r).

Let us suppose thatxi∈B(w,r), for everyi= , . . . ,n. Then, by (M),vn∈B(w,r). There-fore by the previous results we have

xn+–w ≤( –μnτ)r+τ μnr=r

andvn+–w ≤r.

The next step is to prove that the weak cluster points of (xn)n∈Nare ﬁxed points ofT. First we note that

xn+–yn=fμnyn–yn

=μn(I–f)yn

≤μn(I–f)yn,

so, since (μn)n∈N is a null sequence (see (h)),ωw(xn) =ωw(yn). By the boundedness of (xn)n∈N, Lemma . ensures that the weak limits of the sequence (yn)n∈Nare contained in the set ﬁxed points ofT. The previous inequality shows thatωw(xn)⊂Fix(T).

Finally, ifx∗∈Fix(T) is the unique solution of a (VIP), as a result one ﬁnds that

xn+–x∗ =fμnyn–x

∗_±_{ –}_fμ

nx

∗

=fμnyn–fμnx∗

–μn(I–f)x∗

≤fμnyn–fμnx∗



– μn

(I–f)x∗,xn+–x∗

≤( –μnτ)vn–x∗



+ μn

–(I–f)x∗,xn+–x∗

= ( –μnτ)

n

k= an,k

xk–x∗



+ μn

–(I–f)x∗,xn+–x∗

≤( –μnτ) n

k=

an,kxk–x∗

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+ μn

–(I–f)x∗,xn+–x∗

= ( –μnτ)an,nxn–x∗ + ( –μnτ)

n–

k=

an,kxk–x∗+ μn

–(I–f)x∗,xn+–x∗

≤( –μnτ)xn–x∗



+r

n–

k=

an,k+ μn

–(I–f)x∗,xn+–x∗

= +μ_nτ– μnτxn–x∗



+ ( –an,n)r+ μn

–(I–f)x∗,xn+–x∗

= –μnτ( –μnτ)xn–x∗



+ ( –an,n)r+ μn

–(I–f)x∗,xn+–x∗

.

Since (xn)n∈Nis bounded, there exists a subsequencexnk zsuch that

lim sup

n

–(I–f)x∗,xn+–x∗

=lim

k

–(I–f)x∗,xnk+–x∗

=–(I–f)x∗,z–x∗

.

Since every weak cluster point of (xn)n∈Nis inFix(T),

lim sup

n

–(I–f)x∗,xn+–x∗

=–(I–f)x∗,z–x∗

≤.

By (h), (h), (h), and by using Lemma . we obtainxn→x∗.

Proof of Theorem . Also here we will denoteyn=

n–

k=bn,k+Tkvn. Since T is quasi-nonexpansive and (h) holds, for allw∈Fix(T), as a result one ﬁnds that

yn–w ≤ n–

k=

bn,k+Tkvn–Tkw≤ n–

k=

bn,k+vn–w=vn–w.

If

r:=max

v–w,

(I–f)w

τ

,

we prove, by induction onn, that (vn)n∈N⊂B(w,r). Setτ=  –ρ, let us observe the follow-ing:

vn+–w ≤γnvn–w+ ( –γn)fμnyn–w

≤γnvn–w+ ( –γn)fμnyn–fμnw+ ( –γn)fμnw–w

≤γnvn–w+ ( –γn)( –μnτ)yn–w+ ( –γn)fμnw–w

≤γnvn–w+ ( –γn)( –μnτ)vn–w+ ( –γn)μn(I–f)w

=γnvn–w+ ( –γn)( –μnτ)(vn–w)+τ μn( –γn)

(I–f)w

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Now, v ∈B(w,r). For induction, let us suppose that vi ∈B(w,r), for every i= , . . . ,n. Therefore by the previous results we have

vn+–w ≤γnr+ ( –γn)( –μnτ)r+τ μn( –γn)r=r

andvn+–w ≤r.

Now we want to prove that the weak cluster points of (vn)n∈Nare ﬁxed points ofT. First we note that

vn+–yn=γnvn+ ( –γn)fμnyn–yn

=γn(vn–yn) + ( –γn)μn(I–f)yn

≤γnvn–yn+μn(I–f)yn,

thus, since (μn)n∈N, (γn)n∈N are null sequences (see (h) and (h)), ωw(vn) =ωw(yn). By the boundedness of (vn)n∈N, Lemma . guarantees that the weak limits of the sequence (yn)n∈Nare contained in the set of ﬁxed points ofT. The previous inequality shows that

ωw(vn)⊂Fix(T).

Lastly, ifx∗∈Fix(T) is the unique solution of a (VIP), as a result one ﬁnds that

vn+–x∗ =γnvn+ ( –γn)fμnyn–x∗±( –γn)fμnx∗



=γn

vn–x∗

+ ( –γn)

fμnyn–fμnx

∗_{– ( –}_γ

n)μn(I–f)x∗



≤γn

vn–x∗

+ ( –γn)

fμnyn–fμnx∗



– ( –γn)μn

(I–f)x∗,xn+–x∗

≤γnvn–x∗



+ ( –γn)( –μnτ)vn–x∗



+ ( –γn)μn

–(I–f)x∗,vn+–x∗

= + ( –γn)

μ_nτ– μnτvn–x∗



+ ( –γn)μn

–(I–f)x∗,vn+–x∗

= – ( –γn)μnτ( –μnτ)vn–x∗ + ( –γn)μn

–(I–f)x∗,xn+–x∗

.

Since (vn)n∈Nis bounded, there exists a subsequencevnk zsuch that

lim sup

n

–(I–f)x∗,vn+–x∗

=lim

k

–(I–f)x∗,vnk+–x∗

=–(I–f)x∗,z–x∗

.

Since every weak cluster point of (vn)n∈Nis inFix(T),

lim sup

n

–(I–f)x∗,vn+–x∗

=–(I–f)x∗,z–x∗

≤.

(11)

Remark . If (bn,k)n,k∈N is generated recursively by iteration (.) and by a sequence (ηn)n∈Nsuch that

η= , ηn+≤ ηn  +ηn

(e.g.ηn=₍_n_+) η withη≥) then (bn,k)n,k∈Nbelongs toT.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript.

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU, for financial support.

Received: 7 January 2015 Accepted: 5 May 2015 References

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