#### Abstract

In the setting of Hilbert spaces, inspired by Iemoto and Takahashi (2009), we study a Mann’s method with viscosity to approximate strongly (common) fixed points of a nonexpansive mapping and a nonspreading mapping. A crucial tool in our results is the nonspreading-average type mapping.

#### Introduction

Let be a real Hilbert space with inner product that induces the norm . Let be a mapping. We denote by the set of fixed points of , .

A mapping is said to be(i)nonexpansive [1] (1967) if for all ;(ii)firmly nonexpansive [1] (1967) if for all ;(iii)firmly type nonexpansive [2] (2009) if for all for all ;(iv)strongly nonexpansive [3] (1977) if is nonexpansive and , bounded and then for all ;(v)nonspreading [4] (2008) if for all ;(vi)-strict nonspreading [5] (2011) if (vii)quasi-nonexpansive if for all and for all .

Of course,

firmly nonexpansive firmly type nonexpansive strongly nonexpansive nonexpansive quasi-nonexpansive nonspreading -strict pseudononspreading.

If is a nonempty, closed, and convex subset of , we denote by the metric projection on ; that is, for any , is the unique element in such that It is well known (see [1]) that is a firmly nonexpansive mapping and that is characterized by the variational inequality The firm nonexpansivity has many equivalent formulations.

Theorem 1. *Let a mapping. There are equivalents.*(1)* is firmly nonexpansive; that is, for all .*(2)*For each , the convex function defined by
is nonincreasing on .*(3)*The mapping is nonexpansive.*(4)* with is nonexpansive.*(5)* for all .*(6)*The mapping is firmly nonexpansive.*(7)* holds.*(8)*One has .*

*Proof. *The equivalences (1) to (5) are proved in [6]. The equivalences (1) and (6) are proved in [7]. Let us prove that (1) is equivalent to (8):
Finally, we prove that (1) is equivalent to (7). is firmly nonexpansive .

Two important classes of mappings containing the firmly nonexpansive mappings are the average mappings and the nonspreading mappings.

After [7], we say that is an nonexpansive-average mappings if for some , and is a nonexpansive mapping.

*Definition 2. *Let be a class of mappings. One says that is a -average mapping if
for some where is a mapping belonging to the class .

Of course .

The nonexpansive-average mapping* regularizes* a nonexpansive mapping according to the celebrated Schaefer’s result [8].

Theorem 3. *Any orbit of a nonexpansive-average mapping converges weakly to a fixed point of whenever such points exist.*

Here we are interested in nonspreading and nonspreading-average mappings.

Theorem 4. *Let be a mapping. The following are equivalent.*(1)* is nonspreading; that is, ;*(2)*One has ;*(3)*.**Moreover, let be a nonspreading mapping. Then*(a)* is closed and convex;*(b)* is demiclosed;*(c)*One has .**If is a nonspreading-average mapping, then one has the following.*(i)*
In particular is quasi firmly nonexpansive; that is,
*(ii)*, for all .*

*Proof. *The equivalence of (1) and (2) is proved in Lemma 3.2 of [9].

The equivalence of (1) and (3) follows by the fact that
The item (a) is proved in [4], while (b) and (c) are proved in [9].

The item (i) is proved in Theorem 3.1 of [5].

Now we prove (ii). Since
thus we need to show that
This follows by quasi-nonexpansivity of . Indeed

Recently, Song and Chai [2] in the general setting of Banach spaces obtained strong convergence of Halpern’s iteration for firmly type nonexpansive mapping . (Saejung in [10] noted that their proof seems to be questionable, but the result is true as a consequence of a more general result proved in [10]). Indeed, in [10] is proved the strong convergence of Halpern iteration for strongly nonexpansive mappings (and it is easy to see that the class of strongly nonexpansive mappings contains the class of firmly type nonexpansive mappings).

Osilike and Isiogugu [5] studied the Halpern iteration for -strict pseudo-non-spreading mappings. They showed that if one considers the -strict pseudo-non-spreading-average mapping, then Halpern’s iteration converges strongly to a fixed point of such a mapping.

On the other hand, Iemoto and Takahashi [9] approximated weakly fixed points of nonexpansive mappings and/or a nonspreading mapping in a Hilbert space using Moudafi’s iteration scheme [11]. Specifically, they proved the following result.

Theorem 5. *Let be a Hilbert space and let be a nonempty closed convex subset of . Let be a nonspreading mapping on into itself and let be a nonexpansive mappings on into itself such that . Define a sequence as follows:
**
for all , where , , are in . Then the following holds.*(I)*if and , then weakly converges to .*(II)*if and , then weakly converges to .*(III)*if and , then weakly converges to .*

In [12], the authors obtained strong convergence for the Halpern method by using type average mappings, with assumptions on the coefficients very similar to Theorem 5.

So one can ask if this result holds for Moudafi’s viscosity method [13]. We cannot take advantage of using the above positive results on Halpern’s iteration and invoke Suzuki’s result [14] that affirms that Halpern’s approximation convergence implies Moudafi’s viscosity approximation convergence. Indeed, as proved by Suzuki, this is true for nonexpansive mappings not for nonspreading mappings.

In spite of this we obtain the affirmative answer in our main result.

Our proofs took inspiration by [5, 12, 15, 16]. Related papers in which there are not nonspreading but other types of mappings or semigroups of nonexpansive mappings are [17–23].

#### 2. Main Results

In this section, we always will assume the following.(i) is a Hilbert space.(ii) is a closed and convex subset of .(iii) is a nonexpansive mapping.(iv) is an average mapping of , .(v) is a nonspreading mapping.(vi) is a nonspreading-average mapping of , .(vii) is a convex combination of and , .(viii).(ix) is a contraction; that is, , .(x) is a real sequence satisfying and .(xi) denote any bounded real sequence (so ).

The following lemmas are the keys to obtain our main result.

Lemma 6 (see [24]). *Assume that is a sequence of nonnegative numbers such that
**
where is a sequence in and is a sequence in and such that, *(1)*;
*(2)* and .**Then, .*

Lemma 7. *Let be the sequence defined by
**
Then, (i) is quasi nonexpansive; (ii) , and are bounded sequences.*

*Proof. *(i) Any convex combination of quasi nonexpansive mappings is, in turn, quasi nonexpansive. So is , since and are quasi nonexpansive (see Theorem 4, (i)).

(ii) We see that the boundedness of follows by the quasi nonexpansivity of . For this let . Then
The boundedness of is proved. The boundedness of the other sequences in (ii) follows by this last (since ).

Lemma 8. *Let be a bounded sequence in . Then one has the following. *(i)*If , then
where is the unique point in that satisfies the variational inequality
*(ii)*If , then
where is the unique point in that satisfies the variational inequality
*(iii)*If both and , then
where is the unique point in that satisfies the variational inequality
*

*Proof. *(i) Let satisfy (21). Let be a subsequence of for which
Select a subsequence of such that (this, of course, is possible by boundedness of ). From the assumption and demiclosedness of (see [1]) we have , and
so the claim follows by (21).

(ii) It follows as in (i) since is demicloded too (see Theorem 4, (b)).

(iii) Select a subsequence of such that
where satisfies (25). Now select a subsequence of such that . Then, by demiclosedness of both and , and by the hypotheses and , we obtain that ; that is, . So the claim follows by (25) and

Lemma 9 (see [6]). *Let be a nonempty closed convex subset of and let be a nonexpansive mapping. Then is -inverse strongly monotone; that is,
*

Lemma 10 (Maingé [25]). *Let be real sequence that has a subsequence which satisfies for all . Then the sequence of integers defined by has the following properties:*(1)*;*(2)* as ;*(3)*;*(4)*.*

Theorem 11. *Let
**
with and . Then one has the following.*(i)* and , then strongly converges to that is the unique point in that satisfies the variational inequality
*(ii)*, then converges strongly to that is the unique point in that satisfies the variational inequality
*(iii)*, then strongly converges to which is the unique point in that satisfies the variational inequality
*

*Proof. *By Lemma 7, we obtain that is bounded.*Proof of (i). *Let be as in (i) of Lemma 8; that is,*Step **1*. One has .*Proof of Step **1. *This immediately follows by the asymptotic regularity of . So we prove that is asymptotically regular; that is, :
So
where is such that , thanks to the assumptions and .

So if we put we have
From the assumption we deduce immediately . This is sufficient for Xu’s Lemma 6, to conclude that is asymptotically regular.*Step **2*. One has .*Proof of Step **2. *We define an auxiliary sequence by
Observe that
and so
hence we get
Now
and hence
Passing to , the last member goes to zero thanks to Step 1, to boundedness of and (41). So we obtain
From this immediately we have also .

From Step 2 and Lemma 8(i) we obtain
Moreover, from Step 2 and Lemma 8 we also have*Step **3*. One has .*Proof of Step **3*. By using the auxiliary sequence , we can write as
where is a bounded sequence and so
So putting , , and
one has easily that , , , and
Thus, we can rewrite (50) as
This is sufficient, for Xu’s Lemma 6, to conclude that . Lastly, by (49) immediately follows .*Proof of (ii).* Rewrite as
where is bounded (i.e., ).

Now,
and hence
Now we distinguish two alternatives.*Alternative **1. * is definitively nonincreasing.

Then there exists and so, passing to the in (56), we obtain
By Lemma 8(ii) it follows that
So
So, as in Step 1, thanks to (57), (58), and Xu’s Lemma, we obtain .*Alternative **2. * is not definitively nonincreasing.

This means that there exists a subsequence such that
Then, thanks to Maingé’s Lemma, we know that there exists a sequence of integers that satisfies the following.

(i) is nondecreasing, (ii) , (iii) , and (iv) , for all .

Consequently,
So
If we rewrite (56) as
then (62) implies that
and this, in turn, by using Lemma 8(ii), means that
At this point it is clear that we can continue as in Alternative 1 and we obtain .

Then (62) furnishes
and finally by property (iv) of Maingé’s Lemma, that is, , for all , we point out that .*Proof of (iii).* Let be as in (iii) of Lemma 8; that is,
Now,
so
From this we derive the following inequalities:
Now, also here consider two alternatives.* Alternative **1. * is definitively nonincreasing.

Then there exist and .

So, passing to the in (71), the assumption yields:
Moreover, so,
and so
so that from it follows at once that
From Lemma 8(iii) we obtain
Further, from , we get
Now we are able to show that .

Indeed,