Halpern-type proximal point algorithm in complete CAT(0) metric spaces

Transcription

1 DOI: /auom A. Şt. Uiv. Ovidius Costaţa Vol. 24(3),2016, Halper-type proximal poit algorithm i complete CAT(0) metric spaces Mohammad Taghi Heydari ad Sajad Rajbar Abstract First, Halper-type proximal poit algorithm is itroduced i complete CAT(0) metric spaces. The, Browder covergece theorem is cosidered for this algorithm ad also we prove that Halper-type proximal poit algorithm coverges strogly to a zero of the operator. 1 Itroductio Oe of the most importat parts i oliear ad covex aalysis is mootoe operator theory. It has a essetial role i covex aalysis, optimizatio, variatioal iequalities, semigroup theory ad evolutio equatios. A zero of a maximal mootoe operator is a solutio of variatioal iequality associated to the mootoe operator also a equilibrium poit of a evolutio equatio govered by the mootoe operator as well as a solutio of a miimizatio problem for a covex fuctio whe the mootoe operator is a subdifferetial of the covex fuctio. Therefore existece ad approximatio of a zero of a maximal mootoe operator is the ceter of cosideratio of may recet researchers. The most popular method for approximatio of a zero of a maximal mootoe operator is the proximal poit algorithm which was itroduced by Martiet [30] ad Rockafellar [32]. Rockafellar [32] showed the weak covergece of the sequece geerated by the proximal poit algorithm to a zero Key Words: Hadamard space, Maximal mootoe operator, Halper-type proximal poit algorithm, w-covergece, Strog covergece Mathematics Subject Classificatio: Primary 47H05, 47J05; Secodary 47J20, 65K05. Received: Accepted:

2 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 142 of the maximal mootoe operator i Hilbert spaces. Güler s couterexample [21] showed that the sequece geerated by the proximal poit algorithm does ot ecessarily coverge strogly eve if the maximal mootoe operator is the subdifferetial of a covex, proper, ad lower semicotiuous fuctio. Xu [35] ad Kamimura ad Takahashi [23] itroduced a Halper-type proximal poit algorithm, which guaratees the strog covergece i Hilbert space. For some geeralizatio i Hilbert spaces the reader ca cosult [6, 12, 17, 24, 25, 33]. Proximal poit algorithm itroduced by Bačák [3] for the case of covex fuctios i Hadamard spaces. I the geeral cases, this algorithm is orgaized i Hadamard spaces by Khatibzadeh ad Rajbar [26] for the mootoe operators (also, see [31]). I this paper by usig of the duality theory itroduced i [2], we cosider maximal mootoe operators ad Halper-type proximal poit algorithm o Hadamard spaces ad prove the strog covergece of Halpertype proximal poit algorithm i this oliear versio of Hilbert spaces (i.e. complete CAT(0) spaces). 2 Prelimiaries Let (X, d) be a metric space ad x, y X. A geodesic path joiig x to y is a isometry c : [0, d(x, y)] X such that c(0) = x, c(d(x, y)) = y. The image of a geodesic path joiig x to y is called a geodesic segmet betwee x ad y. The metric space (X, d) is said to be a geodesic space if every two poits of X are joied by a geodesic, ad X is said to be a uiquely geodesic space if there is exactly oe geodesic joiig x ad y for each x, y X. A geodesic space (X, d) is a CAT(0) space if satisfies the followig iequality: CN iequality: If x, y 0, y 1, y 2 X such that d(y 0, y 1 ) = d(y 0, y 2 ) = 1 2 d(y 1, y 2 ), the d 2 (x, y 0 ) 1 2 d2 (x, y 1 ) d2 (x, y 2 ) 1 4 d2 (y 1, y 2 ). A complete CAT(0) space is called a Hadamard space. It is kow that CAT(0) spaces are uiquely geodesic spaces. For other equivalet defiitios ad basic properties, we refer the reader to the stadard texts such as [10, 14, 20, 22]. Some examples of CAT(0) spaces are pre-hilbert spaces (see [10]), R-trees (see [27]), Euclidea buildigs (see [13]), the complex Hilbert ball with a hyperbolic metric (see [19]), Hadamard maifolds ad may others. For all x ad y belog to a CAT(0) space X, we write (1 t)x ty for the uique poit z i the geodesic segmet joiig from x to y such that d(z, x) = td(x, y) ad d(z, y) = (1 t)d(x, y). Set [x, y] = {(1 t)x ty : t [0, 1]}, a subset C of X is called covex if [x, y] C for all x, y C. The followig techical lemma is well-kow i CAT(0) spaces.

3 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 143 Lemma 2.1. [16] Let (X, d) be a CAT(0) space. The, for all x, y, z X ad all t [0, 1] : (1) d 2 (tx (1 t)y, z) td 2 (x, z) + (1 t)d 2 (y, z) t(1 t)d 2 (x, y), (2) d(tx (1 t)y, z) td(x, z) + (1 t)d(y, z), I additio, by usig (1), we have d(tx (1 t)y, tx (1 t)z) (1 t)d(y, z). A cocept of covergece i complete CAT(0) spaces was itroduced by Lim [29] that is called -covergece as follows: Let (x ) be a bouded sequece i complete CAT(0) space (X, d) ad x X. Set r(x, (x )) := lim sup d(x, x ). The asymptotic radius of (x ) is give by r((x )) := if{r(x, (x )) : x X} ad the asymptotic ceter of (x ) is the set A((x )) := {x X : r(x, (x )) = r((x ))}. It is kow that i the complete CAT(0) spaces, A((x )) cosists of exactly oe poit (see [28]). A sequece (x ) i the complete CAT(0) space (X, d) is said -coverget to x X if A((x k )) = {x} for every subsequece (x k ) of (x ). The cocept of -covergece has bee studied by may authors (e.g. [16, 18]). Berg ad Nikolaev [4] have itroduced the cocept of quasiliearizatio for CAT(0) space X. They deote a pair (a, b) X X by ab ad called it a vector. The the quasiliearizatio map. : (X X) (X X) R is defied by ab, cd = 1 2 (d2 (a, d) + d 2 (b, c) d 2 (a, c) d 2 (b, d)), (a, b, c, d X). It ca be easily verified that ab, ab = d 2 (a, b), ba, cd = ab, cd ad ab, cd = ae, cd + eb, cd are satisfied for all a, b, c, d, e X. Also, we ca formally add compatible vectors, more precisely ac + cb = ab, for all a, b, c X. We say that X satisfies the Cauchy-Schwarz iequality if ab, cd d(a, b)d(c, d), (a, b, c, d X). It is kow ([4], Corollary 3) that a geodesically coected metric space is a CAT(0) space if ad oly if it satisfies the Cauchy-Schwarz iequality. Ahmadi Kakavadi ad Amii [2] have itroduced the cocept of dual space of a complete CAT(0) space X, based o a work of Berg ad Nikolaev [4], as follows. Cosider the map Θ : R X X C(X, R) defied by Θ(t, a, b)(x) = t ab, ax, (t R, a, b, x X), where C(X, R) is the space of all cotiuous real-valued fuctios o X. The the Cauchy-Schwarz iequality implies that Θ(t, a, b) is a Lipschitz fuctio

4 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 144 with Lipschitz semi-orm L(Θ(t, a, b)) = t d(a, b) (t R, a, b X), where L(ϕ) = sup{ ϕ(x) ϕ(y) d(x,y) : x, y X, x y} is the Lipschitz semi-orm for ay fuctio ϕ : X R. A pseudometric D o R X X is defied by D((t, a, b), (s, c, d)) = L(Θ(t, a, b) Θ(s, c, d)), (t, s R, a, b, c, d X). For a Hadamard space (X, d), the pseudometric space (R X X, D) ca be cosidered as a subspace of the pseudometric space of all real-valued Lipschitz fuctios (Lip(X, R), L). It is obtaied that D((t, a, b), (s, c, d)) = 0 if ad oly if t ab, xy = s cd, xy, for all x, y X [2, Lemma 2.1]. The, D ca impose a equivalet relatio o R X X, where the equivalece class of (t, a, b) is [t ab] = {s cd : D((t, a, b), (s, c, d)) = 0}. The set X = {[t ab] : (t, a, b) R X X} is a metric space with metric D([t ab], [s cd]) := D((t, a, b), (s, c, d)), which is called the dual space of (X, d). It is clear that [ aa] = [ bb] for all a, b X. Fix o X, we write 0 = [ oo] as the zero of the dual space. I [2], it is show that the dual of a closed ad covex subset of Hilbert space H with oempty iterior is H ad t(b a) [t ab] for all t R, a, b H. Note that X acts o X X by x, xy = t ab, xy, (x = [t ab] X, x, y X). Also, we use the followig otatio: αx + βy, xy := α x, xy + β y, xy, (α, β R, x, y X, x, y X ). Itroducig of a dual for a CAT(0) space implies a cocept of weak covergece with respect to the dual space which is amed w covergece i [2]. I [2], authors also showed that w-covergece is stroger tha -covergece. Ahmadi Kakavadi i [1] preseted a equivalet defiitio of w-covergece i complete CAT(0) spaces without usig of dual space, as follows: Defiitio 2.2. [1] A sequece (x ) i the complete CAT(0) space (X, d) w-coverges to x X if lim sup xx, xy = 0, y X. w-covergece is equivalet to the weak covergece i Hilbert space H, because if (.,.) is the ier product i Hilbert space H, the 2 xz, xy = d 2 (x, y) + d 2 (z, x) d 2 (z, y) = 2(x z, x y). We must otice that ay bouded sequece does ot have a subsequece that is w-coverget. It is obvious that covergece i the metric implies w-covergece, ad i [2] it has bee show that w-covergece implies - covergece but the coverse is ot valid (see [1]). However Ahmadi Kakavadi [1] proved the followig characterizatio of -covergece.

5 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 145 Lemma 2.3. [1] A bouded sequece (x ) i Hadamard space (X, d), - coverges to x X if ad oly if lim sup xx, xy 0, y X. I the sequel, we deote w-covergece by ad strog covergece by. 3 Maximal Mootoe Operators Let X be a complete CAT(0) space with dual X ad A : X 2 X be a multi-valued operator with domai D(A) := {x X : Ax }, rage R(A) := x X Ax, A 1 (x ) := {x X : x Ax} ad graph gra(a) := {(x, x ) X X : x D(A), x Ax}. Defiitio 3.1. Let X be a Hadamard space with dual space X. The multivalued operator A : X 2 X is mootoe if ad oly if for all x, y D(A), x Ax, y Ay. x y, yx 0, Defiitio 3.2. Let X be a Hadamard space with dual X. The multi-valued mootoe operator A : X 2 X is maximal if there exists o mootoe operator B : X 2 X such that gra(b) properly cotais gra(a), i.e. for ay (y, y ) X X, the iequality x y, yx 0, for all (x, x ) gra(a) implies y Ay. I this sectio, we show that the graph of a maximal mootoe operator is sequetialy weakly-strogly closed i X X, i.e. if (x, x ) gra(a) N, x x ad (x ) X coverges to x X i metric D the x Ax. Lemma 3.3. Let X be a Hadamard space with dual X the x y, yx D(x, y )d(x, y), for all x, y X, x, y X. Proof. let x, y X, x, y X. Cosider t, s R ad a, b, c, d X such

6 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 146 that x = [t ab] ad y = [s cd]. If x = y that is clear, suppose x y, the x y, yx = [t ab] [s cd], yx = t ab, yx s cd, yx = t ab, ya + t ab, ax s cd, yc s cd, cx = t ab, ax t ab, ay s cd, cx + s cd, cy (Θ(t, a, b)(x) Θ(s, c, d)(x)) (Θ(t, a, b)(y) Θ(s, c, d)(y)) = d(x, y) d(x, y) (Θ(t, a, b) Θ(s, c, d))(u) (Θ(t, a, b) Θ(s, c, d))(v) d(x, y) sup{ : u, v X, u v} d(u, v) = d(x, y)l(θ(t, a, b) Θ(s, c, d)) = d(x, y)d([t ab], [s cd]) = d(x, y)d(x, y ). Theorem 3.4. Let X be a Hadamard space with dual X ad A : X 2 X be a multi-valued maximal mootoe operator. Suppose (x, x ) gra(a) for all N such that (x ) is a bouded sequece i X that is w-coverget to x X ad (x ) X coverges to x X i metric D the x Ax. Proof. By Lemma 3.3, for all N ad all (y, y ) gra(a), we have x y, yx x y, yx = x x, yx + x y, yx + x y, xy = x x, yx + x y, xx x x, yx + x y, xx D(x, x )d(x, y) + x y, xx. Let, we get x y, yx x y, yx. (3.1) O the other had, by mootoicity of A, for all (y, y ) gra(a), we have 0 x y, yx, N, which, by (3.1), implies 0 x y, yx, (y, y ) gra(a).

7 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 147 Hece, the maximality of A implies x Ax. We say that a subset C of Hadamard space X is w-sequetially closed if for ay sequece (x ) C that x x, we have x C. It is clear that every w-sequetially closed subset of X is closed. By Theorem 3.4, it is easy to verify that if A : X 2 X be a multi-valued maximal mootoe operator the A 1 (x ) is a w-sequetially closed subset of Hadamard space X, for ay x X. 4 Halper-type Proximal Poit Algorithm Oe of the most importat problems i mootoe operator theory is fidig a zero of a maximal mootoe operator. This problem ca be formulated i Hadamard space as follows: F id x X, such that 0 A(x), (4.1) where A : X 2 X is a mootoe operator o the Hadamard space X ad 0 is the zero of dual space X. Let X be a Hadamard space with dual X ad A : X 2 X be a multivalued operator. We say that A satisfies the rage coditio if for every y X ad every α > 0, there exists a poit x X such that [α xy] Ax. It is kow that if A is a maximal mootoe operator o the Hilbert space H the R(I + λa) = H, λ > 0, where I is idetity operator. Thus, every maximal mootoe operator A o a Hilbert space satisfies the rage coditio. Lemma 4.1. If A is a mootoe operator o a Hadamard space X that satisfies the rage coditio the for every y X ad every α > 0, there exists a uique poit x X such that [α xy] Ax. Proof. If there exists x, z X such that [α xy] Ax ad [α zy] Az, the by mootoicity of A, we have 0 2 [α xy] [α zy], zx = 2α xy, zx 2α zy, zx = α(d 2 (y, z) d 2 (x, z) d 2 (y, x)) α(d 2 (x, z) + d 2 (y, z) d 2 (y, x)) = 2d 2 (x, z), which implies x = z. We do ot kow if every maximal mootoe operator A : X 2 X satisfies the rage coditio whe X is a Hadamard space.

8 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 148 Let A : X 2 X be a multi-valued maximal mootoe operator o the Hadamard space X with dual X that satisfies the rage coditio, (λ ) is a sequece of positive real umbers, (α ) is a sequece i ]0, 1[ ad u X. Halper-type proximal poit algorithm for maximal mootoe operator A i Hadamard space X is the sequece geerated by { [ 1 λ x +1 (α u (1 α )x )] Ax +1, (4.2) x 0 X. Note that, sice the operator A satisfies the rag coditio, the Halper-type proximal poit algorithm (4.2) is well-defied ad also (4.2) is accordace with the Halper-type proximal poit algorithm { x +1 (I + λ A) 1 (α u + (1 α )x ), x 0 X. where I is the idetity operator i a Hilbert space that is cosidered by [5, 6, 7, 8, 9, 25, 33, 35]. The aim of this sectio is to prove strog covergece of the sequece geerated by the Halper-type proximal poit algorithm (4.2) to a elemet of A 1 (0), where 0 is the zero of dual space. To this purpose, we eed to the followig lemmas. Lemma 4.2. Let X be a CAT(0) space, x, y X ad t ]0, 1[. The, yz, (tx (1 t)y)y t yz, xy, Proof. Let z X. By Lemma 2.1 we have, (z X). 2( yz, (tx (1 t)y)y t yz, xy ) = (d 2 ((tx (1 t)y), z) d 2 ((tx (1 t)y), y) d 2 (y, z)) t(d 2 (x, z) d 2 (x, y) d 2 (y, z)) td 2 (x, z) + (1 t)d 2 (y, z) t(1 t)d 2 (x, y) = 0, which implies the desired iequality. t 2 d 2 (x, y) d 2 (y, z) td 2 (x, z) + td 2 (x, y) + td 2 (y, z) Lemma 4.3. Let X be a CAT(0) space ad (x ) be a bouded sequece i X that -coverges to X. The, d 2 (x, y) lim if d 2 (x, y) (y X)

9 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 149 Proof. By Lemma 2.3, for all y X, we get d 2 (x, y) lim if that is the desired iequality. d 2 (x, y) = lim sup(d 2 (x, y) d 2 (x, y)) lim sup(d 2 (x, x) + d 2 (x, y) d 2 (x, y)) = lim sup xx, xy 0, Let X be a Hadamard space with dual X, A : X 2 X be a multivalued mootoe operator such that satisfies the rage coditio ad C > 0 ad u X are fixed. The by Lemma 4.1, for each t ]0, 1[ ad each x X, there exists a uique poit z t,x such that [ 1 C z t,x(tu (1 t)x)] A(z t,x ). Thus, for every t ]0, 1[, we ca defie the mappig S t : X X with S t (x) = z t,x x X. I the followig, we show that, for every t ]0, 1[, S t has a uique fixed poit z t X. Propositio 4.4. For each t ]0, 1[, S t has a uique fixed poit z t X. Proof. Let x, y X, the [ 1 S t (x)(tu (1 t)x)] A(S t (x)) ad [ 1 S t (y)(tu (1 t)y)] A(S t (y)). C C By mootoicity of A, we have 2 S t (x)(tu (1 t)x), S t (x)s t (y) 2 S t (y)(tu (1 t)y), S t (x)s t (y) which implies, Hece, = 2 S t (y)(tu (1 t)x), S t (x)s t (y) + 2 (tu (1 t)x)(tu (1 t)y), S t (x)s t (y), 2d 2 (S t (x), S t (y)) 2 (tu (1 t)x)(tu (1 t)y), S t (x)s t (y) 2d(tu (1 t)x, tu (1 t)y)d(s t (x)s t (y)). d(s t (x), S t (y)) d(tu (1 t)x, tu (1 t)y) (1 t)d(x, y) Thus, for each t ]0, 1[, S t is a cotractio. Cosequetly, by Baach s Cotractio Priciple, for each t ]0, 1[, S t has a uique fixed poit that is amed z t.

10 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 150 I the followig, we show that (z t ) coverges strogly to P A 1 (0)u, where P A 1 (0) is the metric projectio o A 1 (0). Theorem 4.5. [15] Let C be a oempty covex subset of a CAT(0) space X, x X ad u C. The u = P C x if ad oly if xu, uy 0, y C. We say that a Hadamard space X satisfies the coditio Q if every bouded sequece i X has a subsequece that is w-coverget. It is said that a Hadamard space (X, d) satisfies the (S) property if for ay (x, y) X X there exists a poit y x X such that [ xy] = [ yx x]. Hilbert spaces ad symmetric Hadamard maifold satisfy the (S) property (see [1], Defiitio 2.7). Lemma 2.8 of [1] implies that if a Hadamard space (X, d) satisfies the (S) property the it satisfies the coditio Q because every bouded sequece i a Hadamard space (X, d) has a -coverget subsequece. Also, the proper Hadamard spaces satisfy the coditio Q (see [1], Propositios 4.3 ad 4.4). The followig theorem is a versio of Browder covergece theorem (see [11]) for Halper-type proximal poit algorithm. Theorem 4.6. Let X be a Hadamard space with dual X that satisfies the coditio Q ad A : X 2 X be a multi-valued maximal mootoe operator that satisfies the rage coditio ad A 1 (0) is covex. The (z t ) coverges strogly to P A 1 (0)u as t 0. Proof. For each t ]0, 1[, [ 1 C z t (tu (1 t)z t )] A(z t ). By mootoicity of A, for all q A 1 (0), we have 0 2 z t (tu (1 t)z t ), qz t hece, which implies, = d 2 (tu (1 t)z t, q) d 2 (z t, q) d 2 (tu (1 t)z t, z t ), d 2 (z t, q) d 2 (tu (1 t)z t, q) td 2 (u, q) + (1 t)d 2 (z t, q), d 2 (z t, q) d 2 (u, q), q A 1 (0). (4.3) I particular, {z t } is bouded. Thus, by the coditio Q, there exists a subsequece of (z t ) that is w-coverget. Moreover, lim D([ 1 1 z t (tu (1 t)z t )], 0) = lim t 0 C t 0 C d(tu (1 t)z t t, z t ) = lim t 0 C d(u, z t) = 0. (4.4)

11 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 151 Now, if (t ) is a sequece i ]0, 1[ such that t 0 ad (z t ) is w-coverget to z. The, by (4.4) ad Theorem 3.4, we get z A 1 (0). Thus, by (4.4) ad Lemma 4.3, we obtai d 2 (z, q) lim if d 2 (z t, q) d 2 (u, q), q A 1 (0), which, for all q A 1 (0), implies, zu, zq = zu, zu + zu, uq = d 2 (z, q) d 2 (u, q) 0. Hece, by Theorem 4.5, z = P A 1 (0)u. The arbitrariess of the subsequece (z t ) of (z t ) esures that (z t ) ideed w-coverges to P A 1 (0)u, as t 0. Now, we prove the strog covergece of (z t ). By mootoicity of A ad Lemma 4.2, for all q A 1 (0), we have 0 2 z t (tu (1 t)z t ), qz t t z t u, qz t = t z t q, qz t + t qu, qz t, which implies I particular, d 2 (z t, q) qu, qz t q A 1 (0). d 2 (z t, P A 1 (0)u) (P A 1 (0)u)u, (P A 1 (0)u)z t. Lettig t 0, we get d 2 (z t, P A 1 (0)u) 0, that is the desired result. Lemma 4.7. [15] Let (X, d) be a CAT(0) space ad a, b, c X. The for each λ [0, 1], d 2 (λa (1 λ)b, c) λ 2 d 2 (a, c) + (1 λ) 2 d 2 (b, c) + 2λ(1 λ) ac, bc. Theorem 4.8. Let X be a Hadamard space with dual X that satisfies the coditio Q ad A : X 2 X be a multi-valued maximal mootoe operator such that satisfies the rage coditio ad A 1 (0) is covex. If ((x, x )) is a sequece i graph of A such that (x ) is bouded ad lim D(x, 0) = 0, the lim sup up, x p 0, where p = P A 1 (0)u.

12 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 152 Proof. For each t ]0, 1[, there exists a uique poit z t X such that [ 1 C z t (tu (1 t)z t )] A(z t ). By Theorem 4.6, as t 0, (z t ) coverges strogly to p = P A 1 (0)u. By mootoicity of A, for each t (0, 1) ad all N, we have 0 2 C z t (tu (1 t)z t ) x, x z t. which, by Lemma 4.7, implies d 2 (z t, x ) + 2 x, x z t d 2 (tu (1 t)z t, x ) t 2 d 2 (u, z t ) t 2 d 2 (u, x ) + (1 t) 2 d 2 (z t, x ) + 2t(1 t) ux, z t x = t 2 d 2 (u, x ) + (1 t) 2 d 2 (z t, x ) + 2t(1 t) uz t, z t x Thus, for each t (0, 1) ad all N, we get + 2t(1 t)d 2 (z t, x ) = t 2 d 2 (u, x ) + (1 t 2 )d 2 (z t, x ) + 2t(1 t) uz t, z t x. 2t(1 t) uz t, x z t t 2 d 2 (u, x ) + 2 x, z t x Hece, for each t (0, 1), we obtai lim sup uz t, x z t t 2 d 2 (u, x ) + 2D(x, 0)d(z t, x ). t 2(1 t) O the other had, by the cotiuity of d, lim sup d 2 (u, x ). (4.5) uz t, x z t up, x p as t 0, uiformly respect to. Therefore, for ay umber ε > 0, there exists δ > 0 such that up, x p ε + uz t, x z t, for all 0 < t < δ ad all N. This implies that lim sup up, x p ε + lim sup uz t, x z t ε + Lettig t 0, we get lim sup up, x p ε. Hece, as ε 0, we deduce lim sup up, x p 0. t 2(1 t) lim sup d 2 (u, x ).

13 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 153 Lemma 4.9. [34] Let (s ) be a sequece of oegative real umbers, (α ) a sequece of real umbers i [0, 1] with =1 α =, (u ) a sequece of oegative real umbers with =1 u <, ad (t ) a sequece of real umbers with lim sup t 0. Suppose that for all N. The lim s = 0. s +1 (1 α )s + α t + u, The followig lemma is a direct applicatio of the well-kow Stolz-Cesàro theorem. Lemma Suppose that (a ) ad (b ) are positive sequeces such that + =1 b m a = + ad lim =1 + b = 0, the lim a m + m = 0. =1 b Theorem Let X be a Hadamard space with dual X that satisfies the coditio Q ad A : X 2 X be a multi-valued maximal mootoe operator that satisfies the rage coditio ad A 1 (0) is covex. If (x ) geerated by (4.2) such that D 2 ([ 1 x +1(α u (1 α )x )], 0) D 2 1 ([ x (α 1u (1 α 1)x 1)], 0) λ λ 1 + θ, where (θ ) is a positive sequece with =1 θ <, =1 λ2 = ad α λ 0, the 2 (1) D([ 1 λ x +1 (α u (1 α )x )], 0) 0, as. Also, if (x k ) is a subsequece of (x ), w-coverges to x, the x A 1 (0). (2) If α 0 ad + =1 α = +, the (x ) coverges strogly to p = P A 1 (0)u. Proof. Let q A 1 (0). By mootoicity of A ad (4.2), we have which implies 0 [ 1 λ x +1 (α u (1 α )x )], qx +1, d 2 (x +1, q) + d 2 (x +1, α u (1 α )x )) d 2 (α u (1 α )x ), q). (4.6) Thus d 2 (x +1, q) d 2 (α u (1 α )x, q) α d 2 (u, q) + (1 α )d 2 (x, q) max{d 2 (u, q), d 2 (x 1, q)}.

14 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 154 So, (x ) is bouded. Let us to prove (1). By (4.6), we have that is d 2 (x +1, α u (1 α )x )) d 2 (α u (1 α )x ), q) d 2 (x +1, q) α d 2 (u, q) + d 2 (x, q) d 2 (x +1, q), λ 2 D 2 ([ 1 λ x +1 (α u (1 α )x )], 0) α d 2 (u, q) + d 2 (x, q) d 2 (x +1, q). Moreover, by assumptios, for all k >, we get D 2 ([ 1 λ k x k+1 (α k u (1 α k )x k )], 0) D 2 ([ 1 λ x +1 (α u (1 α )x )], 0) which, by (4.7), implies + k i=+1 θ i, (4.7) λ 2 D 2 ([ 1 λ k x k+1 (α k u (1 α k )x k )], 0) α d 2 (u, q) + d 2 (x, q) d 2 (x +1, q) + λ 2 k i=+1 Summig up from = 1 to k, after that dividig by k =1 λ2, the D 2 ([ 1 k =1 x k+1 (α k u (1 α k )x k )], 0) α d 2 (u, q) λ k + d2 (x 1, q) k k =1 λ2 =1 λ2 θ i. + k =1 λ2 i=+1 θ i k. =1 λ2 Lettig k the, by Lemma 4.10, we get D([ 1 λ k x k+1 (α k u (1 α k )x k )], 0) 0. Now, if (x k ) is a subsequece of (x ) that w-coverges to x, the by Theorem 3.4, we get x A 1 (0). For prove (2), by Theorem 4.8 ad part (1), we get lim sup up, x p 0, where p = P A 1 (0)u. (4.8)

15 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 155 O the other had, by (4.6), ad Lemma 4.7 we have d 2 (x +1, p) d 2 (α u (1 α )x ), p) α 2 d 2 (u, p) + (1 α ) 2 d 2 (x, p) + 2α (1 α ) up, x p (1 α )d 2 (x, p) + α (α d 2 (u, p) + (1 α ) up, x p ), which, by the Lemma 4.9, the assumptios ad (4.8), implies d(x +1, p) 0, that is the desired result. Theorem Let X be a Hadamard space with dual X that satisfies the coditio Q ad A : X 2 X be a multi-valued maximal mootoe operator such that satisfies the rage coditio ad A 1 (0) is covex. If (x ) geerated by (4.2) such that (i) (λ ) is a odecreasig sequece such that 1 =1 λ λ 1 <, (ii) α 0 ad + =1 α = +, the (x ) coverges strogly to p = P A 1 (0)u. Proof. By a proof similar to Theorem 4.11, (x ) is bouded. By mootoicity of A ad (4.2), we have [ x (α 1u (1 α 1 1)x 1)] [ x +1(α u (1 α )x )], x +1x λ λ 1 λ that is 1 x +1(α u (1 α )x ), x 1 +1x x (α 1u (1 α 1)x 1), x +1x λ λ 1 λ 2 which implies 1 d 2 1 (x +1, α u (1 α )x ) d 2 (x +1, α 1 u (1 α 1 )x 1 ) λ λ 1 λ 2 + α2 λ 2 d 2 (u, x ) 1 λ λ 1 d 2 (x, α 1 u (1 α 1 )x 1 ) 1 + d 2 (x +1, x ) + α2 λ λ 1 λ 2 d 2 (u, x ) 2 + d(x +1, x )d(x, α 1 u (1 α 1 )x 1 ) λ λ 1 1 λ 2 d 2 (x, α 1 u (1 α 1 )x 1 ) M( + α2 ), λ λ 1 λ 2

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19 Halper-type proximal poit algorithm i complete CAT(0) metric spaces 159 Mohammad Taghi Heydari, Departmet of Mathematics, College of Scieces, Yasouj Uiversity, Yasouj 75914, Ira. heydari@yu.ac.ir, ad Departmet of Mathematics, College of Scieces, Higher Educatio Ceter of Eghlid, Eghlid, Ira. mt heydari@yahoo.com Sajad Rajbar, Departmet of Mathematics, College of Scieces, Higher Educatio Ceter of Eghlid, Eghlid, Ira. srajbar@eghlid.ac.ir, srajbar74@yahoo.com. Correspodig author.