Abstract

In the setting of Hilbert spaces, we study Mann’s type method to approximate strong solutions of variational inequalities. We show that these solutions are fixed points of a nonexpansive mapping and/or a strongly quasinonexpansive mapping, depending on the coefficients involved in the algorithm.

1. Introduction

Let be a real Hilbert space with inner product and induce a norm . Let be a nonlinear mapping. denotes the fixed point set of .

The problem of approximating fixed points of nonlinear mapping has been widely investigated by many authors.

The starting point of the present paper is the following scheme introduced by Iemoto and Takahashi in [1]: where is a nonexpansive mapping and is a nonspreading mapping.

Depending on the coefficients and , in [1] was obtained weak convergence of the method either to a fixed point of or to a fixed point of or to a common fixed point of and .

Motivated by this result, with the aim of obtaining strong convergence, in [2] we introduced a viscosity in a previous scheme and we have been able to prove the following result, in which the most important role was played by the average type mappings and , .

Theorem 1 (see [2]). Let be a nonexpansive mapping and let be a nonspreading mapping. Let with , , and . Then(i)if , , and , then strongly converges to which is the unique point in that satisfies the variational inequality(ii)if and , then converges strongly to which is the unique point in that satisfies the variational inequality(iii)if and , then strongly converges to which is the unique point in that satisfies the variational inequality

At this point, three questions can be posed:(1)Is it possible to replace the coefficient with the term where is a linear, bounded, strongly positive operator? In doing so, we would be able to solve some minimization problem.(2)Is it possible to replace the type average mappings and with the original mappings and ?(3)Is it possible to replace the nonspreading mapping with a more general type of mappings?

We will see here that the answers are all positive.

A natural class of mappings containing the nonspreading mappings is given by the class of hybrid mappings. Following Aoyama et al. [3] we say that is an - mapping, with , if or equivalently

We denote by the family of - mappings defined on the space .

Note that particular choices of give important families of nonlinear mappings:(i) is the family of nonexpansive mappings (i.e., for all ).(ii) is the family of nonspreading mappings (i.e., , ).

Relevant properties of - mappings are the following:(-1)If and , then is quasinonexpansive (i.e., for all , for all ) (the proof is immediate by the definition).(-2)If is demiclosed in (for the proof see [1]).(-3)If is a closed and convex set (for the proof, see [4]).(-4)If , where is the type average mapping  (for the proof, see [5]).

A first improvement of Theorem 1 [2] that one can obtain is to replace with an - mapping. (This will be a corollary of our main result, thanks to (-1), (-3), and next Proposition 2.)

But we can do better. Inspired by the paper of Wongchan and Saejung [6], we can consider a mapping that has only the properties of being strongly quasinonexpansive and demiclosed.

We recall that the concept of strongly nonexpansive mapping was introduced by Bruck and Reich in 1977 [7] as follows: a mapping is said to be strongly nonexpansive if is nonexpansive and whenever is bounded and , it follows that .

To our knowledge, Saejung [8] in 2010 introduced the concept of strong quasi-nonexpansivity: a mapping is said strongly quasi-nonexpansive if , is quasinonexpansive and whenever is a bounded sequence such that for some .

In [7] was proved that an average mapping defined on a uniformly convex Banach space is strongly nonexpansive.

Following the same line on the proof, one can show that a type average mapping of a quasinonexpansive mapping is strongly quasinonexpansive. For the sake of completeness, we propose here the following proof, lightly different, valid in Hilbert spaces.

Proposition 2. Let be a quasinonexpansive mapping, where is a closed convex subset of . Then the type average mapping is a strongly quasinonexpansive mapping.

Proof. Suppose is a bounded sequence such that for some . Now SoNow, let be a subsequence of that admits limit, , as . Then, from the assumption that , it follows that as ; that is, . But thenBy the boundness of the sequence , this is sufficient, by the standard argument of compactness, to ensure as .

So, by the quasinonexpansivity of - mappings, it follows that in Theorem 1 [2] the assumption that is a strongly quasinonexpansive mapping instead of type average nonspreading (or also -) mapping is a better assumption.

Of course, one can ask if some important - mapping, as a nonspreading mapping, or a nonexpansive mapping, is already strongly quasinonexpansive. This is not always true, as shown in the following proposition.

Proposition 3. There exist nonexpansive mappings that are not strongly quasinonexpansive. Moreover, there exist nonspreading mappings that are not strongly quasinonexpansive.

Proof. Let be such that . Then is nonexpansive but not strongly quasinonexpansive.
Moreover let , where Define byOne can see that is a nonspreading mapping, distinguishing three cases: (, ), (, ), and (, ).
To see that is not strongly quasinonexpansive, take with .
Then . Moreover, define . Then and but .

Conversely, we have the following.

Proposition 4. There exist strongly quasinonexpansive mappings that are not - mappings for any (and so, thanks to (-4), that are not ever type average with -).

Proof. Let . Define Then define in linear way on each interval , .
One can see easily that and is strongly quasinonexpansive.
Moreover, from the fact that, for large , is defined almost as , one can prove that can not be - for any .

Finally, we show that the hypothesis on demiclosedness of is necessary in Theorem  2.3 (the main Theorem) of [6].

Proposition 5. There exist strongly quasinonexpansive mappings such that are not demiclosed in .

Proof. Let . Define byThen , is strongly quasinonexpansive, but is not demiclosed on ( is such that but ).

Reassuming, the main result of [6] has allowed us to improve Theorem 1 [2] with an emphasis on the concept of strong quasinonexpansivity instead of nonspreading.

Our reasoning is inspired by the ideas contained in [2, 6, 9].

The solid bases on which our proof rests are given by the following lemmas.

Lemma 6 (Xu’s Lemma [10]). Assume is a sequence of nonnegative numbers such that where is a sequence in and is a sequence in such that (1);(2);(3), .Then

Lemma 7 (Maingé’s Lemma [11]). Let be real sequence that has a subsequence which satisfies for all . Then there exists an increasing sequence of integers satisfying(1);(2), for all ;(3), for all .

2. Main Results

In all this section we denote by any bounded real sequence (so, e.g., ).

Theorem 8. Let be a Hilbert space. Suppose (i) is a nonexpansive mapping and is a strongly quasinonexpansive mapping such that is demiclosed in , for which ;(ii) is a linear bounded self-adjoint strongly positive operator, ;(iii) is a contraction, with (of course this is not a real restriction, but this is useful for convenience of notations);(iv) are two real sequences in .Consider the iteration schemeThen(1)if , , , and , then strongly converges to which is the unique point in that solves the variational inequality which is the optimality condition for the minimization problem  where is a potential function for (i.e., , for all );(2)if , , , and , then converges strongly to which is the unique solution in of the variational inequality(3)if , , and , then strongly converges to which is the unique solution in of the variational inequality

Proof. First of all, we see that is a bounded sequence.
Indeed, let and . Then (by linearity of ) (since , [9]) (from convexity) (from induction) .
Then is bounded.
Moreoversince .
Proof of (1). The the key is to prove that To see this, we calculate where ; definitively, .
Thanks to hypotheses on , we see that , , and . This is sufficient, from Xu’s Lemma, to conclude .
From this and (22), it follows immediately thatIndeed, .
Moreover and so from (25) and hypotheses , we have also From this we deduce also thatand this gives that any weak limit of is in , since is nonexpansive, and thus the Principle of Demiclosedness is satisfied.
Now we can show that , where is the unique solution in of variational inequality (18). We show first thatIndeed, let be a subsequence of such thatand . Then and so, from (30), and this is nonpositive by definition of .
So, Finally where Thanks to the hypotheses on and (29), from Xu’s Lemma again, we obtain .
Proof of (2). Let be the unique solution of variational inequality (20).
We want to show that . Now,At this point we distinguish two cases: or the sequence is definitively not increasing or not.
Alternative 1. is definitively not increasing, so , .
Then (35) furnishPutting , , and , we can rewrite (36) asso the thesis will follow again by Xu’s Lemma if we are able to show that(Note that until now we have not used the hypothesis of strong quasinonexpansivity of .) Now, since is definitively not increasing, there exists the . Then ThusFrom the strong quasinonexpansivity of , we deduceAt this point, by using the demiclosedness of in , we can proceed as in the Proof of (1) to show (38).
The statement is proved when Alternative 1 holds.
Alternative 2. () is not definitively not increasing; that is, there exists a subsequence () such thatFrom Maingé’s Lemma, it follows that there exists an increasing sequence of integers satisfying Then Retracing the same inequalities used to have (41) with instead of , we obtain Again the strong quasinonexpansivity yieldsAnd, from the demiclosedness of in , we deduce as above thatIncidentally, we observe that and so, from (47), it follows also that .
Now rewrite (35) with instead of : dividing by Passing to limsup and recalling the hypothesis and (48), we obtain . Equation (44) ensures that also .
Proof of (3). Let be the unique point in that satisfies variational inequality (21). Then Also now we distinguish two cases.
Alternative 1. is definitively not increasing, , .
Then there exists , so (52) furnish and so, by hypothesis , we deduceMoreover, But thenSince both of the addends are nonpositive and the limit of the sum is zero, it follows thatBut then the hypothesis implies thatFrom strong quasinonexpansivity of , it follows that Again , so, by (54) and (57),We show now thatIndeed, select a subsequence such that But by the demiclosedness of both and and by (59) and (60), one deduces that (61) is obtained.
Moreover, from (59) and (60) at once, it follows thatFinally we are able to show . Indeed, Putand the thesis follows again by Xu’s Lemma, taking account of (61) and (64).
Alternative 2. is not definitively not increasing; that is, there exists a subsequence such that , .
From Maingé’s Lemma, it follows that there exists an increasing sequence of integers satisfying (43) and (44).
Then HenceNow, retracing the same inequalities used to obtain (59) with instead of , we haveMoreover, we can rewrite (52) as and so, from (68) and the hypothesis ,Again, so, by (69) and (71),so alsoThe same reasoning used to have (61) can be now repeated with instead of obtainingAnalogously, we can repeat all the reasoning used to deduce (68) with instead of and obtain where So, from Xu’s Lemma, taking in account (74) and (75), we have Once again by (44) we deduce .