Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces

An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common fixed point of a nonexpansive semigroup {T (s)} s in Hilbert spaces. We prove that both approaches converge strongly to a common fixed point of {T (s)} s. Such common fixed point x is the unique solution of some variational inequality, which is the optimality condition for some minimization problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm common fixed point of {T (s)} s. 1 Introduction Let H be a real Hilbert space with inner product, and norm, respectively. Let C be a nonempty closed convex subset of H. Recall a mapping f : C H is called to a contraction if, for all x, y C, there exists ρ [, 1) such that f(x) f(y) ρ x y. A mapping T : C C is said to be nonexpansive if T x T y x y, x, y C. Denote the set of fixed points of T by F ix(t ). Let A be a strongly positive bounded linear operator on H, i.e., there exists a constant γ > such that Ax, x γ x 2 for all x H. Key Words: Common fixed point; Variational inequality; Nonexpansive Semigroup; Algorithms; Minimum norm. Mathematics Subject Classification: 47H5; 47H1; 47H17. Received: January, 21 Accepted: December, 21 331

332 Yonghong Yao, Yeong-Cheng Liou Iterative methods for nonexpansive mappings are widely used to solve convex minimization problems, see, for instance, [1],[2], [4]-[6], [9],[11]-[14],[16]- [3], [32], [33]. A typical problem is to minimize a function over the set of fixed points of a nonexpansive mapping T, min x F ix(t ) 1 Ax, x x, b. (1.1) 2 In [27], Xu proved that the sequence {x n } defined by x n+1 = α n b + (1 α n A)T x n, n strongly converges to the unique solution of (1.1) under certain conditions. Recently, Marino and Xu [17] introduced the viscosity approximation method x n+1 = α n γf(x n )+(1 α n A)T x n, n and proved that the sequence {x n } converges strongly to the unique solution of the variational inequality (A γf)z, x z, x F ix(t ) which is the optimality condition for the minimization problem min x F ix(t ) 1 Ax, x h(x) 2 where h is a potential function for γf (i.e., h = γf on H). In this paper, we focus on nonexpansive semigroup {T (s)} s. Recall that a family S := {T (s)} s of mappings of C into itself is called a nonexpansive semigroup if it satisfies the following conditions: (S1) T ()x = x for all x C; (S2) T (s + t) = T (s)t (t) for all s, t ; (S3) T (s)x T (s)y x y for all x, y C and s ; (S4) for all x H, s T (s)x is continuous. We denote by F ix(t (s)) the set of fixed points of T (s) and by F ix(s) the set of all common fixed points of S, i.e. F ix(s) = s F ix(t (s)). It is known that F ix(s) is closed and convex (Lemma 1 in [1]). Algorithms for nonexpansive semigroups have been considered by some authors, please consult [3], [7],[8],[1],[15], [31]. The following interesting problem arises: Can one construct some more general algorithms which unify the above algorithms? On the other hand, we also notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H 1 and a

Some unified algorithms... 333 bounded linear operator R : H 1 H 2, where H 2 is another Hilbert space. The C-constrained pseudoinverse of R, R C, is then defined as the minimum-norm solution of the constrained minimization problem R C (b) := arg min Rx b x C which is equivalent to the fixed point problem x = P C (x λr (Rx b)) where P C is the metric projection from H 1 onto C, R is the adjoint of R, λ > is a constant, and b H 2 is such that P R(C) (b) R(C). It is therefore another interesting problem to invent some algorithms that can generate schemes which converge strongly to the minimum-norm solution of a given problem. In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common fixed point of a nonexpansive semigroup {T (s)} s in Hilbert spaces. We prove that both approaches converge strongly to a common fixed point of {T (s)} s. Such common fixed point x is the unique solution of some variational inequality, which is the optimality condition for some minimization problem. As special cases of the above two algorithms, We obtain two schemes which both converge strongly to the minimum norm common fixed point of {T (s)} s. 2 Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. The metric (or nearest point) projection from H onto C is the mapping P C : H C which assigns to each point x C the unique point P C x C satisfying the property x P C x = inf x y =: d(x, C). y C It is well known that P C is a nonexpansive mapping and satisfies x y, P C x P C y P C x P C y 2, x, y H. Moreover, P C is characterized by the following properties: x P C x, y P C x, (2.1)

334 Yonghong Yao, Yeong-Cheng Liou and x y 2 x P C x 2 + y P C x 2, for all x H and y C. We need the following lemmas for proving our main results. Lemma 2.1. ([23]) Let C be a nonempty bounded closed convex subset of a Hilbert space H and let {T (s)} s be a nonexpansive semigroup on C. Then, for every h, lim t sup x C 1 t t T (s)xds T (h) 1 t t T (s)xds =. Lemma 2.2. ([12]) Let C be a closed convex subset of a real Hilbert space H and let S : C C be a nonexpansive mapping. Then, the mapping I S is demiclosed. That is, if {x n } is a sequence in C such that x n x weakly and (I S)x n y strongly, then (I S)x = y. Lemma 2.3. ([18]) Let {x n } and {y n } be bounded sequences in a Banach space X and let {γ n } be a sequence in [, 1] with < lim inf n β n lim sup n β n < 1. Suppose that x n+1 = (1 γ n )x n + γ n y n for all n and lim sup n ( y n y n 1 x n x n 1 ). Then, lim n y n x n =. Lemma 2.4. ([26]) Assume {a n } is a sequence of nonnegative real numbers such that a n+1 (1 γ n )a n + δ n γ n, where {γ n } is a sequence in (, 1) and {δ n } is a sequence such that (1) n=1 γ n = ; (2) lim sup n δ n or n=1 δ nγ n <. Then lim n a n =. 3 Main results In this section we will show our main results.

Some unified algorithms... 335 Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = {T (s)} s : C C be a nonexpansive semigroup with F ix(s). Let f : C H be a ρ-contraction (possibly non-self). Let A be a strongly positive linear bounded self-adjoint operator on H with coefficient γ >. Let { } <t<1 be a continuous net of positive real numbers such that lim t = +. Let γ and β be two real numbers such that < γ < γ/ρ and β [, 1). Let the net {x t } be defined by the following implicit scheme: x t = P C [tγf(x t ) + βx t + ((1 β)i ta) 1 λt T (s)x t ds], t (, 1). (3.1) Then, as t +, the net {x t } strongly converges to x F ix(s) which is the unique solution of the following variational inequality: x F ix(s), (γf A)x, x x, x F ix(s). (3.2) In particular, if we take f = and A = I, then the net {x t } defined by (3.1) reduces to x t = P C [βx t + (1 β t) 1 λt T (s)x t ds], t (, 1). (3.3) In this case, the net {x t } defined by (3.3) converges in norm to the minimum norm fixed point x of F ix(s), namely, the point x is the unique solution to the minimization problem: x = arg min x F ix(s) x. (3.4) Proof. First, we note that the net {x t } defined by (3.1) is well-defined. We define a mapping Gx := P C [tγf(x) + βx + ((1 β)i ta) 1 λt T (s)xds], t (, 1). It follows that Gx Gy tγ(f(x) f(y)) + β(x y) + ((1 β)i ta) 1 λt (T (s)x T (s)y)ds tγ f(x) f(y) + β x y + ((1 β)i ta) 1 λt (T (s)x T (s)y)ds tγρ x y + β x y + (1 β t γ) x y = (1 ( γ γρ)t) x y.

336 Yonghong Yao, Yeong-Cheng Liou This implies that the mapping G is a contraction and so it has a unique fixed point. Therefore, the net {x t } defined by (3.1) is well-defined. Take p F ix(s). By (3.1), we have x t p = P C [tγf(x t ) + βx t + ((1 β)i ta) 1 λt T (s)x t ds] p t(γf(x t ) Ap) + β(x t p) + ((1 β)i ta)( 1 λt T (s)x t ds p) t γf(x t ) Ap + β x t p + (1 β γt) 1 λt T (s)x t T (s)p ds tγ f(x t ) f(p) + t γf(p) Ap + β x t p + (1 β γt) x t p tγρ x t p + t γf(p) Ap + β x t p + (1 β γt) x t p. It follows that x t p 1 γf(p) Ap γ γρ which implies that the net {x t } is bounded. Set R := 1 γ γρ γf(p) Ap. It is clear that {x t} B(p, R). Notice that 1 λt T (s)x t ds p x t p R. Moreover, we observe that if x B(p, R) then T (s)x p T (s)x T (s)p x p R, i.e., B(p, R) is T (s)-invariant for all s. Set y t = tγf(x t ) + βx t + ((1 β)i ta) 1 λt T (s)x t ds. From (3.1), we

Some unified algorithms... 337 deduce T (τ)x t x t = P C [T (τ)x t ] P C [y t ] T (τ)x t y t T (τ)x t T (τ) 1 λt T (s)x t ds + T (τ) 1 λt T (s)x t ds 1 λt T (s)x t ds + 1 λt T (s)x t ds y t x t 1 λt T (s)x t ds + T (τ) 1 λt T (s)x t ds 1 λt T (s)x t ds + 1 λt T (s)x t ds y t 2t 1 β γf(x t) A By Lemma 2.1, we deduce for all τ < λt T (s)x t ds + T (τ) 1 λt T (s)x t ds 1 λt T (s)x t ds. lim T (τ)x t x t =. (3.5) t Note that x t = P C [y t ]. By using the property of the metric projection (2.1), we have Therefore, x t p 2 = x t y t, x t p + y t p, x t p y t p, x t p = t γf(x t ) Ap, x t p + β x t p 2 + ((1 β)i ta)( 1 λt T (s)x t ds p), x t p β x t p 2 + (1 β γt) x t p 2 x t p 2 +tγ f(x t ) f(p), x t p + t γf(p) Ap, x t p [1 ( γ γρ)t] x t p 2 + t γf(p) Ap, x t p. 1 γ γρ γf(p) Ap, x t p, p F ix(s).

338 Yonghong Yao, Yeong-Cheng Liou From this inequality, we have immediately that ω w (x t ) = ω s (x t ), where ω w (x t ) and ω s (x t ) denote the set of weak and strong cluster points of {x t }, respectively. Let {t n } (, 1) be a sequence such that t n as n. Put x n := x tn, y n := y tn and := n. Since {x n } is bounded, without loss of generality, we may assume that {x n } converges weakly to a point x C. Also y n x weakly. Noticing (3.5) we can use Lemma 2.2 to get x F ix(s). We can rewrite (3.1) as (A γf)x t = 1 t ((1 β)i ta)[x t 1 λt T (s)x t ds] + 1 t (x t y t ). Therefore, (A γf)x t, x t p = 1 β t + 1 A [ 1 λt (I T (s))x t (I T (s))p, x t p ds] λt [x t T (s)x t ]ds, x t p + 1 t x t y t, x t p. Noting that I T (s) is monotone and x t y t, x t p, so (A γf)x t, x t p 1 A λt [x t T (s)x t ]ds, x t p = Ax t A λt T (s)x t ds, x t p A x t 1 λt T (s)x t ds x t p t 1 β A γf(x t) A 1 Taking the limit through t := t ni, we have λt (A γf)x, x p = lim i (A γf)x ni, x ni p. T (s)x t ds x t p. Since the solution of the variational inequality (3.2) is unique. Hence ω w (x t ) = ω s (x t ) is singleton. Therefore, x t x. In particular, if we take f = and A = I, then it follows that x t x = P F ix(s) (), which implies that x is the minimum norm fixed point of S. As a matter of fact, by (3.2), we deduce x, x x, x F ix(s),

Some unified algorithms... 339 that is, x 2 x, x x x, x F ix(s). Therefore, the point x is the unique solution to the minimization problem This completes the proof. x = arg min x. x F ix(s) Next we introduce an explicit algorithm for finding a solution of minimization problem (3.4). This scheme is obtained by discretizing the implicit scheme (3.1). We will show the strong convergence of this algorithm. Theorem 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = {T (s)} s : C C be a nonexpansive semigroup with F ix(s). Let f : C H be a ρ-contraction (possibly non-self) with ρ [, 1). Let A be a strongly positive linear bounded self-adjoint operator on H with coefficient γ >. Let γ and β be two real numbers such that < γ < γ/ρ and β [, 1). Let the sequence {x n } be generated iteratively by the following explicit algorithm: x n+1 = (1 γ n )x n +γ n P C [α n γf(x n )+βx n +((1 β)i α n A) 1 T (s)x n ds], n, (3.6) where {α n } and {γ n } are real number sequence in [, 1] and { } is a positive real number. Suppose that the following conditions are satisfied: (i) lim n α n = and n=1 α n = ; (ii) lim n = and lim n 1 = ; (iii) < lim inf n γ n lim sup n γ n < 1. Then the sequence {x n } strongly converges to x F ix(s) which is the unique solution of the variational inequality (3.2). In particular, if we take f = and A = I, then the sequence {x n } generated by (3.6) reduces to x n+1 = (1 γ n )x n +γ n P C [βx n +(1 α n β) 1 T (s)x n ds], n. (3.7) In this case, the sequence {x n } converges in norm to the minimum norm fixed point x of F ix(s).

34 Yonghong Yao, Yeong-Cheng Liou Proof. Take p F ix(s). From (3.6), we have x n+1 p (1 γ n ) x n p + γ n (α n γf(x n ) Ap + β x n p +(1 β γα n ) 1 ) T (s)x n T (s)p ds (1 γ n ) x n p + γ n (α n γρ x n p + α n γf(p) Ap + β x n p ) +(1 β α n γ) x n p = [1 ( γ ργ)α n γ n ] x n p + α n γ n γf(p) Ap. It follows that by induction x n p max{ x p, γf(p) Ap }. γ γρ Set y n = P C [α n γf(x n ) + βx n + ((1 β)i α n A)z n ] for all n, where z n = 1 λn T (s)x n ds. Hence, we have y n y n 1 α n γf(x n ) + βx n + ((1 β)i α n A)z n α n 1 γf(x n 1 ) βx n 1 ((1 β)i α n 1 A)z n 1 and = γα n (f(x n ) f(x n 1 )) + γ(α n α n 1 )f(x n 1 ) + β(x n x n 1 ) z n z n 1 = 1 +((1 β)i α n A)(z n z n 1 ) + (α n 1 α n )Az n 1 γα n f(x n ) f(x n 1 ) + α n α n 1 ( γf(x n 1 ) + Az n 1 ) +β x n x n 1 + (1 β α n γ) z n z n 1 + 1 1 [T (s)x n T (s)x n 1 ]ds + ( 1 1 1 ) T (s)x n 1 ds 1 1 T (s)x n 1 ds + 1 1 1 T (s)x n T (s)x n 1 ds + 1 1 T (s)x n 1 T (s)p ds x n x n 1 + 2 1 x n 1 p. 1 [T (s)x n 1 T (s)p]ds

Some unified algorithms... 341 Therefore, y n y n 1 γα n ρ x n x n 1 + α n α n 1 ( γf(x n 1 ) + Az n 1 ) +β x n x n 1 + (1 β α n γ) x n x n 1 + 2 1 x n 1 p [1 ( γ γρ)α n ] x n x n 1 + M( α n α n 1 + 1 ), where M > is a constant such that Hence, we get sup{ γf(x n 1 ) + Az n 1, 2 x n 1 p } M. n lim sup( y n y n 1 x n x n 1 ). n This together with Lemma 2.3 imply that Therefore, Note that lim y n x n =. n lim x n+1 x n = lim γ n y n x n =. n n T (τ)x n x n T (τ)x n T (τ) 1 + T (τ) 1 + 1 T (τ) 1 +2 x n 1 T (s)x n ds T (s)x n ds 1 T (s)x n ds x n T (s)x n ds 1 T (s)x n ds T (s)x n ds T (s)x n ds. (3.8)

342 Yonghong Yao, Yeong-Cheng Liou From (3.6), we have x n 1 T (s)x n ds x n x n+1 + x n+1 1 T (s)x n ds x n x n+1 + (1 γ n ) x n 1 T (s)x n ds It follows that +γ n α n γ f(x n ) + γ n β x n 1 +γ n α n A 1 T (s)x n ds. T (s)x n ds x n 1 T (s)x n ds 1 { x n x n+1 + γ n α n γ f(x n ) (1 β)γ n +γ n α n A 1 } T (s)x n ds. (3.9) From (3.8), (3.9) and Lemma 2.1, we have lim n T (τ)x n x n = for every τ. (3.1) Notice that {x n } is a bounded sequence. Let x be a weak limit of {x n }. Putting x = P F ix(s) (I A + γf). Then there exists R such that B(x, R) contains {x n }. Moreover, B(x, R) is T (s)-invariant for every s ; therefore, without loss of generality, we can assume that {T (s)} s is a nonexpansive semigroup on B(x, R). By the demiclosedness principle (Lemma 2.2) and (3.1), we have x F ix(s). Therefore, lim sup γf(x ) Ax, x n+1 x = lim n n γf(x ) Ax, x x. Finally, we prove x n x. Set u n = tγf(x n )+βx n +((1 β)i ta) 1 λn T (s)x n ds. It follows that y n = P C [u n ] for all n. By using the property of the metric projection (2.1), we have y n u n, y n x.

Some unified algorithms... 343 So, y n x 2 = y n x, y n x that is, = y n u n, y n x + u n x, y n x u n x, y n x = α n γ f(x n ) f(x ), y n x + α n γf(x ) Ax, y n x +β x n x, y n x + ((1 β)i α n A)(z n x ), y n x α n γ f(x n ) f(x ) y n x + α n γf(x ) Ax, y n x +β x n x y n x + (1 β γα n ) z n x y n x [1 ( γ γρ)α n ] x n x y n x + α n γf(x ) Ax, y n x 1 ( γ γρ)α n x n x 2 + 1 2 2 y n x + α n γf(x ) Ax, y n x, y n x 2 [1 ( γ γρ)α n ] x n x 2 + 2α n γf(x ) Ax, y n x. By the convexity of the norm, we have x n+1 x 2 (1 γ n ) x n x 2 + γ n y n x 2 [1 ( γ γρ)α n γ n ] x n x 2 + 2α n γ n γf(x ) Ax, y n x. Hence, all conditions of Lemma 2.4 are satisfied. Therefore, we immediately deduce that x n x. In particular, if we take f = and A = I, then it is clear that x = P F ix(s) () is the unique solution to the minimization problem x = arg min x F ix(s) x. This completes the proof. Acknowledgment The first author was supported in part by Colleges and Universities Science and Technology Development Foundation (2913) of Tianjin and NSFC 1171279. The second author was supported in part by NSC 99-2221-E-23-6. References [1] F. E. Browder, Convergence of approximation to fixed points of nonexpansive nonlinear mappings in Hilbert spaces, Arch. Rational Mech. Anal. 24 (1967), 82-9.

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