Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space (X, ‖ · ‖, ), T : C → C a monotone 1-Lipschitz mapping and x T(x), then the sequence of averages n ∑n−1 i=0 T i(x) converges weakly to a xed point of T . As a consequence, it is shown that the sequence of Picard’s iteration {Tn(x)} also converges weakly to a xed point of T. The results are new even in a Hilbert space. The Krasnosel’skĭıMann and the Halpern iteration schemes are studied as well.


Introduction
Let C be a bounded closed convex subset of a Banach space X. A mapping T : C → X is said to be nonexpansive if for each pair of elements x, y ∈ C, we have A mappping T is accretive if for each x, y ∈ C and λ ≥ , The study of xed points of nonexpansive mappings and null points of accretive mappings extends the classical theory of successive approximations for strict contractions based on the Banach contraction principle. It is well-known that if T : C → C is a strict contraction, the Picard sequence x n+ = T(xn) converges strongly to the unique xed point of T. In the case of nonexpansive mappings, the Picard sequence need not converge nor need the xed point be unique if it exists. However, if the space X is su ciently smooth (for example, a Hilbert space), for any α ∈ ( , ) and x ∈ C the iterative scheme de ned by x n+ = ( − α)xn + α T(xn) converges weakly to a xed point of a nonexpansive mapping T : C → C. This, and a more general scheme, x n+ = ( − αn)xn + αn T(xn) is now called the Krasnosel'skiȋ-Mann (or Mann) formula for nding xed points of nonexpansive mappings.
It turns out that if the space X is su ciently smooth and parameters {αn} are appropriately chosen, the above sequence converges strongly to a xed point of T which is the projection of the initial u to the set of xed points of T. The third kind of the approximation procedure that, in a sense, generalizes the Krasnosel'skiȋ-Mann method comes from the ergodic theory: the sequence of averages converges weakly to a xed point of T. This is the nonlinear ergodic theorem, proved in a Hilbert space by J.B. Baillon [2].
A wide variety of well-known algorithms applied in practical and theoretical problems are based on the above iterative schemes and their perturbative modi cations. These include the proximal point algorithm, the generalized Yosida approximation of the maximal monotone operator, the algorithms for solving split feasibility problems used frequently in signal processing and image reconstruction, and many others as described, for example, in [3][4][5][6][7].
The objective of this paper is to develop similar methods for mappings T : C → C that are monotone nonexpansive, that is, Here is a partial order on C that corresponds somehow to the norm of X. Very often, such ordering is given by a cone in X with certain properties. In this paper, following the recent works of M.A. Khamsi and his collaborators, we take a more general view and consider partial orders on X such that the order intervals are closed and, moreover, the partial order is compatible with the linear structure of X (for details, see Section 2). It turns out that many results in the theory of nonexpansive mappings are valid for monotone nonexpansive mappings too, despite the fact that those need not even be continuous. The interplay between the order and metrical structure of the space is very fruitful as observed, for example, by A. Ran and M. Reurings who applied a generalization of the Banach contraction principle to matrix equations (see [8]). The existential part of the theory of monotone nonexpansive mappings was recently studied in [9] (see also [10][11][12]). It is shown there that if X is a topological space with a partial order for which order intervals are compact, then every directed subset of X has a supremum. It allows one to apply methods based on the Knaster-Tarski (Abian-Brown) theorem.
In this paper we concentrate on the iterative procedures for nding xed points. One of the long-standing problems in the theory of nonexpansive iterations is whether the Krasnosel'skiȋ-Mann and the Halpern methods, as well as the nonlinear ergodic theorem, hold true in all uniformly convex Banach spaces. The usual requirements here are some kind of smoothness like the Fréchet or Gateaux di erentiable norm, or possessing a weakly continuous duality map (see, eg., [13][14][15][16][17]).
We prove that in the case of monotone nonexpansive mappings, uniform convexity is su cient to obtain weak convergence of the above algorithms to a xed point of T. The paper is organized as follows. In Section 2 we show the demiclosedness principle for monotone nonexpansive mappings which is one of the main tools in our analysis. Sections 3 and 4 are devoted, respectively, to Krasnosel'skiȋ-Mann's and Halpern's iterations. Our main result-the nonlinear ergodic theorem-is proved in Section 5: if C is a bounded closed convex subset of an ordered uniformly convex space (X, · , ), T : C → C a monotone nonexpansive mapping and x T(x), then the sequence of averages n n− i= T i (x) converges weakly to a xed point of T. As a consequence, we show that the Picard iteration {T n (x)} also converges weakly to a xed point of T. Our results, except the Krasnosel'skiȋ-Mann iteration (see [11,Theorem 3.3]), appear to be new even in a Hilbert space.

Demiclosedness principle in uniformly convex spaces
Let X be a Banach space with a partial order compatible with the linear structure of X, that is, for every x, y, z ∈ X and λ ≥ . It follows that all order intervals [x, →] = {z ∈ X : x z} and [←, y] = {z ∈ X : z y} are convex. Moreover, we will assume that each [x, →] and [←, y] is closed. We will say that (X, · , ) is an ordered Banach space. A sequence {xn} is said to be an approximate xed point sequence (a.f.p.s. for short) for a mapping T if T(xn) − xn → as n → ∞, see [18] The following lemma can be proved in much the same way as is the case of nonexpansive mappings (see, e.g., [19,Prop. 10.1]). Choose s > such that s < ε r . Hence, for su ciently large n, By the triangle inequality and monotone nonexpansivity of T, we get By the uniform convexity of X, we have By the triangle inequality, (2.1) and (2.2), we obtain un − vn ≤ un − (wn + T(wn)) + vn − (wn + T(wn)) Letting n → ∞, we get r ≤ r( − δ(s)), a contradiction and this completes the proof.
In what follows we will use the following observation (see [20,Lemma 3.1]). Suppose that {xn} is a monotone sequence that has a cluster point, i.e., there exists a subsequence {xn j } that converges to g (with respect to the strong or weak topology). Since the order intervals are (weakly) closed, we have g ∈ [xn , →) for each n, i.e., g is an upper bound for {xn}. If g is another upper bound for {xn}, then xn ∈ (←, g ] for each n, and hence g g . It follows that {xn} converges to g = sup{xn}. The following result is a basic tool in our consideration. Let s be a real number such that s < ε r , then s < ε vn j for su ciently large j. By the de nition of uniform convexity of X, for su ciently large j, we have wn j ≤ r + n j ( − δ(s)).
It is not di cult to see that {wn j } is monotone and converges weakly to x. By Lemma 1, {wn j } is an approximate xed point sequence that contradicts the de nition of r.

Lemma 2. [20]. Let C be a bounded closed convex subset of an ordered uniformly convex space (X, · , ) and T : C → C a monotone nonexpansive mapping. Suppose that {xn} is a sequence de ned by (3.1) and x T(x ).
Then lim n→∞ xn − T(xn) = . The above theorem should be compared with Theorem 2 in [23], which asserts the same conclusion in a uniformly convex space with a Fréchet di erentiable norm without the assumption about monotonicity of the mapping T (it is required that ∞ n= αn( − αn) = ∞ there). In the case of monotone nonexpansive mappings we can drop the assumption about the Fréchet di erentiability of the norm, thus uniform convexity is su cient to obtain the weak convergence of Krasnosel'skiȋ-Mann iteration (3) in this case.

Theorem 2. Let C be a nonempty bounded closed convex subset of an ordered uniformly convex space
Theorem 2 complements [11,Theorem 3.3], where a similar result was proved in ordered Banach spaces with the Opial property (see also [24][25][26] for some generalizations).

Halpern iteration
Let C be a convex subset of a Banach space X and T : C → C a nonexpansive mapping. The following iteration process is known as the Halpern iteration process [1]: where {αn} is a sequence in ( , ). The rst result extending Halpern's insights outside Hilbert space is Corollary 2 of the early paper by S. Reich [27]. Using the tools of Section 2 we can prove a counterpart of Xu's result concerning weak convergence of Halpern's iterations for monotone nonexpansive mappings in ordered uniformly convex spaces.  Proof. We shall use induction to prove our conclusion. By assumption, x u T(x ). Since a partial order is compatible with the linear structure of X, we have Since T is monotone, we have u T(x ) T(x ). Thus (4.3) is true for n = . Now suppose it is true for any k ∈ N, that is, u T(x k+ ) and x k x k+ . It follows that T(x k ) T(x k+ ). By assumption, {αn} is decreasing and hence Since T is monotone, u T(x k+ ) T(x k+ ). Therefore, (4.3) is true for all n ∈ N. Notice that our result covers a natural iteration scheme x ∈ C, x T(x ), x n+ = n x + ( − n )T(xn).

Nonlinear ergodic theory
A standard way to regularize a nonconvergent sequence {yn} is to consider its averages xn = n n i= y i . Baillon showed in [2] that if T is a nonexpansive mapping acting on a nonempty bounded closed convex subset of a Hilbert space then the sequence of averages converges weakly to a xed point of T, see also [28][29][30]. This result, known as the nonlinear ergodic theorem, was generalized to L p spaces by Baillon [31] and to uniformly convex spaces with a Fréchet di erentiable norm by R.E. Bruck [32] and Reich [23,33]. A long-standing problem is to drop the assumption about a Fréchet di erentiability of the norm. In this section we are able to do it in the case of monotone nonexpansive mappings.
We will need the following variant of Lemma 1 (see also [34,Lemma 3.17]). Since T is a monotone nonexpansive mapping, xn , yn ≤ for all n ∈ N. By the de nition of the modulus of convexity, we have for every x, y ∈ C with x , y ≤ . Therefore, We are now in a position to prove the nonlinear ergodic theorem for monotone nonexpansive mappings in any ordered uniformly convex Banach space.

Open Problem
Finally, we raise the following question: is there a direct proof of Theorem 5?