Convergence of a Sequence of Sets in a Hadamard Space and the Shrinking Projection Method for a Real Hilbert Ball

and Applied Analysis 3 Theorem 2.1 Kirk 1 . LetU be a bounded open subset of a Hadamard space X and S : clU → X a nonexpansive mapping. Suppose that there exists p ∈ U such that every x in the boundary ofU does not belong to p, Sx \ {Sx}. Then, S has a fixed point in clU. Let C be a nonempty closed convex subset of a Hadamard space X. Then, for each x ∈ X, there exists a unique point yx ∈ C such that d x, yx infy∈Cd x, y . The mapping x → yx is called a metric projection onto C and is denoted by PC. We know that PC is nonexpansive; see 13, pages 176-177 . Let {xn} be a bounded sequence in a metric space X. For x ∈ X, let r x, {xn} lim sup n→∞ d x, xn , r {xn} inf x∈X r x, {xn} . 2.3 The asymptotic center of {xn} is a set of points x ∈ X satisfying that r x, {xn} r {xn} . It is known that the asymptotic center of {xn} consists of one point for every bounded sequence {xn} in a Hadamard space; see 3 . The following property of asymptotic centers is important for our results. Theorem 2.2 Dhompongsa et al. 3 . Let C be a closed convex subset of a Hadamard spaceX and {xn} a bounded sequence in C. Then, the asymptotic center of {xn} is included in C. The notion of Δ-convergence was firstly introduced by Lim 6 in a general metric space setting. Following 5 , we apply it to Hadamard spaces. Let {xn} be a sequence in X. We say that {xn} is Δ-convergent to x ∈ X if x is the unique asymptotic center of any subsequence of {xn}. We know that every bounded sequence {xn} in a Hadamard space X has a Δ-convergent subsequence; see 5, 14 . 3. Convergence of a Sequence of Sets Let {Cn} be a sequence of closed convex subsets of a Hadamard space X. As an analogy of Mosco convergence in Banach spaces 15 , we introduce a new concept of set convergence. First let us define subsets d-LinCn and Δ-LsnCn of X as follows: x ∈ d-LinCn if and only if there exists {xn} ⊂ X such that {d xn, x } converges to 0 and that xn ∈ Cn for all n ∈ N. On the other hand, y ∈ Δ-LsnCn if and only if there exist a sequence {yi} ⊂ X and a subsequence {ni} of N such that {yi} has an asymptotic center {y} and that yi ∈ Cni for all i ∈ N. If a subset C0 of X satisfies that C0 d-LinCn Δ-LsnCn, it is said that {Cn} converges to C0 in the sense of Δ-Mosco, and we write C0 ΔM-limn→∞Cn. Since the inclusion d-LinCn ⊂ Δ-LsnCn is always true, to obtain C0 is a limit of {Cn} in the sense of Δ-Mosco, it suffices to show that Δ-LsnCn ⊂ C0 ⊂ d-LinCn. It is easy to show that, if every Cn is convex, then so is d-LinCn. Moreover, we know that d-LinCn is always closed. Therefore,ΔM-limn→∞Cn is closed and convex whenever {Cn} is a sequence of closed convex subsets of X. The following lemma is essentially obtained in 5 as the Kadec-Klee property in CAT 0 spaces. Wemodify it to a suitable form for our purpose. For the sake of completeness, we give the proof. 4 Abstract and Applied Analysis Lemma 3.1. Let X be a Hadamard space and {xn} a sequence in X. Suppose that {xn} is Δconvergent to x ∈ X and {d xn, p } converges to d x, p for some p ∈ X. Then, {xn} converges to x. Proof. Let {Δ x, p, xn } be comparison triangles in E2 for n ∈ N with an identical geodesic segment p, x . Then, we have that |x − p| E2 d x, p , |xn − p|E2 d xn, p , and |xn − x|E2 d xn, x for all n ∈ N. We know that {xn} is bounded in E2. Let {xni} be an arbitrary subsequence of {xn} converging to y ∈ E2. Then, by assumption, we have that ∣ ∣y − p∣ E2 lim i→∞ ∣ ∣xni − p ∣ ∣ E2 lim i→∞ d ( xni , p ) d ( x, p ) ∣ ∣x − p∣ E2 . 3.1 Let P P p,x be ametric projection of E2 onto a closed convex set p, x . Since P is continuous, we have that {Pxni} converges to Py ∈ E2. Let z ∈ p, x ⊂ X be a point corresponding to z Py ∈ p, x ⊂ E2. Using the CAT 0 inequality, we have that r {xni} lim sup i→∞ d x, xni lim sup i→∞ |x − xni |E2 ≥ lim sup i→∞ |Pxni − xni |E2 lim sup i→∞ |z − xni |E2 ≥ lim sup i→∞ d z, xni , 3.2 and hence r z, {xni} ≤ r {xni} . By the uniqueness of the asymptotic center of {xni}, we obtain that z x, and thus z x. Since ∣x − y∣ E2 ∣z − y∣ E2 ∣Py − y∣ E2 ≤ ∣ 1 − t x tp − y∣ E2 3.3 for every t ∈ 0, 1 ⊂ R, it follows that ∣x − y∣2 E2 ≤ ∣∣ 1 − t x tp − y∣2 E2 1 − t ∣x − y∣2 E2 t ∣p − y∣2 E2 − t 1 − t ∣x − p∣2 E2 1 − t ∣x − y∣2 E2 t2 ∣p − x∣2 E2 , 3.4 and thus |x − y| E2 ≤ t|p − x| E2 . Tending t ↓ 0, we obtain that x y. Since any convergent subsequence {xni} of a bounded sequence {xn} in E2 has a limit x, we have that {xn} converges to x. Thus we have that d xn, x |xn − x|E2 → 0 as n → ∞, and hence {xn} converges to x ∈ X. Now we state the main theorem of this section. Using a sequence of metric projections corresponding to a sequence of closed convex subsets, we give a characterization ofΔ-Mosco convergence in a Hadamard space. Abstract and Applied Analysis 5and Applied Analysis 5 Theorem 3.2. Let X be a Hadamard space and C0 a nonempty closed convex subset of X. Then, for a sequence {Cn} of nonempty closed convex subsets in X, the following are equivalent: i {Cn} converges to C0 in the sense of Δ-Mosco; ii {PCnx} converges to PC0x ∈ X for every x ∈ X. Proof. We first show that i implies ii . Fix x ∈ X, and let pn PCnx for n ∈ N. Since PC0x ∈ C0 d-LinCn, there exists {yn} ⊂ X such that yn ∈ Cn for all n ∈ N and that {yn} converges to PC0x. By the definition of metric projection, we have that d x, pn ≤ d x, yn for n ∈ N. Thus, tending n → ∞, we have that lim sup n→∞ d ( x, pn ) ≤ lim n→∞ d ( x, yn ) d x, PC0x . 3.5 It also follows that {pn} is bounded. Let {pni} be an arbitrary subsequence of {pn} and p0 an asymptotic center of {pni}. Then, for fixed > 0, it holds that d ( x, pni ) ≤ d x, PC0x 3.6 for sufficiently large i ∈ N. Since the closed ball with the center x and the radius d x, PC0x is convex, by Theorem 2.2, we have that d x, p0 ≤ d x, PC0x , and hence d ( x, p0 ) ≤ d x, PC0x . 3.7 On the other hand, since p0 ∈ Δ-LsnCn C0, we have that d x, PC0x ≤ d x, p0 , and therefore we have that d x, PC0x d x, p0 , which implies that p0 PC0x. Since all subsequences of {pn} have the same asymptotic center PC0x, {pn} is Δ-convergent to PC0x. Let us show that lim infn→∞d x, pn ≥ d x, PC0x . If it were not true, then there exists a subsequence {pni} of {pn} satisfying that lim infn→∞d x, pn limi→∞d x, pni < d x, PC0x . Let p ∈ X be an asymptotic center of {pni}. For > 0, we have that d x, pni ≤ δ for sufficiently large i ∈ N, where δ limi→∞d x, pni . Since the closed ball with the center x and the radius δ is convex, we have d x, p ≤ δ , and hence d x, p ≤ δ limi→∞d x, pni . Since p ∈ Δ-LsnCn C0, we get that d x, PC0x > lim i→∞ d ( x, pni ) ≥ dx, p ≥ d x, PC0x , 3.8 a contradiction. Therefore, we obtain that d x, PC0x ≤ lim inf n→∞ d ( x, pn ) ≤ lim sup n→∞ d ( x, pn ) ≤ d x, PC0x , 3.9 and thus {d x, pn } converges to d x, PC0x . Using Lemma 3.1, we have that {pn} converges to PC0x. Hence ii holds. Next we suppose ii and show that i holds. By assumption, for y ∈ C0, a sequence {PCny} converges to PC0y y. Since PCny ∈ Cn for all n ∈ N, we have that y ∈ d-LinCn, and hence C0 ⊂ d-LinCn. Let z ∈ Δ-LsnCn. Then, there exist {zi} ⊂ X and {ni} ⊂ N such that 6 Abstract and Applied Analysis zi ∈ Cni for all i ∈ N and z is an asymptotic center of {zi}. Since each Cni is convex, from the definition of metric projection, it follows that d ( z, PCni z ) ≤ d ( z, 1 − t PCni z ⊕ tzi ) 3.10 for t ∈ 0, 1 and i ∈ N. Then, we have that d ( z, PCni z )2 ≤ d ( z, 1 − t PCni z ⊕ tzi )2 ≤ 1 − t d ( z, PCni z )2 td z, zi 2 − t 1 − t d ( PCni z, zi )2 , 3.11