Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or"almost have"fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations.

Recall that an operator T : X → X is firmly nonexpansive (see, e.g., [2], [10], and [11] for further information) if (∀x ∈ X)(∀y ∈ X) Tx − Ty 2 ≤ x − y, Tx − Ty and that a set-valued operator A : X ⇒ X is maximally monotone if it is monotone, i.e., for all (x, x * ) and (y, y * ) in the graph of A, we have x − y, x * − y * ≥ 0 and if the graph of A cannot be properly enlarged without destroying monotonicity.(We shall write dom A = x ∈ X Ax = ∅ for the domain of A, ran A = A(X) = x∈X Ax for the range of A, and gr A for the graph of A.) These notions are equivalent (see [13] and [9]) in the sense that if A is maximally monotone, then its resolvent J A := (Id +A) −1 is firmly nonexpansive, and if T is firmly nonexpansive, then T −1 − Id is maximally monotone.(Here and elsewhere, Id denotes the identity operator on X.)The Minty parametrization (see [13] and also [2,Remark 23.22(ii)]) states that if A is maximally monotone, then (3) gr A = (J A x, x − J A x) x ∈ X .

The aim of this note is to provide approximate fixed point results for compositions and convex combinations of finitely many firmly nonexpansive operators.
The first main result (Theorem 4.6) substantially extends a result by Bauschke [1] on the compositions of projectors to the composition of firmly nonexpansive mappings.The second main result (Theorem 5.5) extends a result by Bauschke, Moffat and Wang [3] on the convex combination of firmly nonexpansive operators from Euclidean to Hilbert space.The remainder of this section provides the standing assumptions used throughout the paper.
Even though the main results are formulated in the given Hilbert space X, it will turn out that the key space to work in is the product space (4) This product space contains an embedding of the original space X via the diagonal subspace ( 5) We also assume that we are given m firmly nonexpansive operators T 1 , . . ., T m ; equivalently, m resolvents of maximally monotone operators A 1 , . . ., A m : We now define various pertinent operators acting on X m .We start with the Cartesian product operators Denoting the identity on X m by Id, we observe that (9) Of central importance will be the cyclic right-shift operator and for convenience we set We also fix strictly positive convex coefficients (or weights) (λ i ) i∈I , i.e. ( Let us make X m into the Hilbert product space ( 13) The orthogonal complement of ∆ with respect to this standard inner product is known (see, e.g., [2, Proposition 25.4(i)]) to be ( 14) Finally, given a nonempty closed convex subset C of X, the projector (nearest point mapping) onto C is denoted by P C .It is well known to be firmly nonexpansive.

Properties of the Operator M
In this section, we collect several useful properties of the operator M, including its Moore-Penrose inverse (see [12] and e.g.[2, Section 3.2] for further information.).To that end, the following result-which is probably part of the folklore-will turn out to be useful.

Proposition 2.1
Let Y be a real Hilbert space and let B be a continuous linear operator from X to Y with adjoint B * and such that ran B is closed.Then the Moore-Penrose inverse of B satisfies Proof.Take y ∈ Y. Define the corresponding set of least squares solutions (see, e.g., [2, Proposition 3.25]) by Before we present various useful properties of M, let us recall the notion of a rectangular (which is also known as star or 3* monotone, see [6]) operator.A monotone operator B : Then the following hold.
(i) M is continuous, linear, and maximally monotone with dom M = X.
(ii) M is rectangular.
(ix): Note that P ∆ ⊥ = Id −P ∆ and that Thus, by (viii) and (16), Re-arranging this expression in terms of powers of R and simplifying leads to ∅, otherwise.
Remark 2.4 Denoting the symmetric part of M by M + = 1 2 M + 1 2 M * and defining the quadratic form associated with M by q M : x → 1 2 x, Mx , we note that [1, Proposition 2.3] implies that3 ran M + = dom q * M = ∆ ⊥ .Applying the Brezis-Haraux result to our given operators A and M, we obtain the following.

Corollary 2.6
The operator A + M is maximally monotone and ran(A + M) = ∆ ⊥ + ran A.

Proof.
Since each A i is maximally monotone and recalling Theorem 2.2(i), we see that A and M are maximally monotone.On the other hand, dom M = X.Thus, by the well known sum theorem for maximally monotone operators (see, e.g, [2, Corollary 24.4(i)]),A + M is maximally monotone.Furthermore, by Theorem 2.2(ii)&(iv), M is rectangular and ran M = ∆ ⊥ .The result therefore follows from Fact 2.5.

Composition
We now use Corollary 2.6 to study the composition.
(v): Let ε > 0. In view of (iv), there exists x ∈ X such that This, (35), and the triangle inequality imply that This completes the proof.

Asymptotic Regularity
The following notions (taken from Bruck and Reich's seminal paper [7]) will be very useful to obtain stronger results.Definition 4.1 ((strong) nonexpansiveness and asymptotic regularity) Let S : X → X. Then:

Convex Combination
In this section, we use our fixed weights (λ i ) i∈I (see (12)) to turn X m into a Hilbert product space different from X considered in the previous sections.Specifically, we set (39) Y := X m with x, y = ∑ i∈I λ i x i , y i so that x 2 = ∑ i∈I λ i x i 2 .We also set (40) (ii) Q coincides with the projector P ∆ .
(i): Clearly, dom M = X and (∀x ∈ X) Rx = x .Thus, R is nonexpansive and therefore M = Id −R is maximally monotone (see, e.g., [2, Example 20.27]).(ii): See [2, Example 24.14] and [1, Step 3 in the proof of Theorem 3.1] for two different proofs of the rectangularity of M.(iii): The definitions of M and R and the fact that R * is the cyclic left shift operator readily imply that ker M = ker M * = ∆.

Fact 2 . 5 (
Brezis-Haraux) (See[6] and also, e.g.,[2, Theorem 24.20].)Suppose A and B are monotone operators on X such that A + B is maximally monotone, dom A ⊆ dom B, and B is rectangular.Then int ran(A + B) = int(ran A + ran B) and ran(A + B) = ran A + ran B.

Fact 5 . 1 (Corollary 5 . 2 Proposition 5 . 3
See[2, Proposition 28.13].)In the Hilbert product space Y, we have P ∆ = Q.In the Hilbert product space Y, the operator Q is firmly nonexpansive and strongly nonexpansive.Furthermore, Fix Q = ∆ = ∅, 0 ∈ ran(Id −Q), and Q is asymptotically regular.Proof.By Fact 5.1, the operator Q is equal to the projector P ∆ and hence firmly nonexpansive.Now apply Fact 4.2(i) to deduce that Q is strongly nonexpansive.It is clear that Fix Q = ∆ and that 0 ∈ ran(Id −Q).Finally, recall Corollary 4.4 to see that Q is asymptotically regular.In the Hilbert product space Y, the operator T is firmly nonexpansive.

i 2 and x 2 =
∑ i∈I λ 2 i .Since Q is nonexpansive, we must have that Qx 2 ≤ x 2 , which is equivalent to strongly nonexpansive if S is nonexpansive and whenever (x n ) n∈N and (y n ) n∈N are sequences in X such that (x n − y n ) n∈N is bounded and x n − y n − Sx n − Sy n → 0, it follows that (x n − y n ) − (Sx n − Sy n ) → 0. (iii) S is asymptotically regular if (∀x ∈ X) S n x − S n+1 x → 0. Suppose that each T i is asymptotically regular.Then T m T m−1 • • • T 1 is asymptotically regular as well.Proof.Theorem 3.1(v) implies that 0 ∈ ran(Id −T m T m−1 • • • T 1 ).The conclusion thus follows from Corollary 4.5.Let C 1 , ..., C m be nonempty closed convex subsets of X.Then the composition of the corresponding projectors,P C m P C m−1 • • • P C 1 is asymptotically regular.Proof.For every i ∈ I, the projector P C i is firmly nonexpansive, hence strongly nonexpansive, and Fix P C i = C i = ∅.Suppose that (∀i ∈ I) T i = P C i , which is thus asymptotically regular by Corollary 4.4.Now apply Theorem 4.6.