Strong convergence to a common fixed point. of nonexpansive mappings semigroups

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1 Theoretical Mathematics & Applications, vol.3, no., 23, ISSN: (print), (online) Scienpress Ltd, 23 Strong convergence to a common fixed point of nonexpansive mappings semigroups Nguyen Buong and Nguyen Duc Lang 2 Abstract In this paper, we introduce some new iteration methods based on the hybrid method in mathematical programming and the descent-like method for finding a fixed point of a nonexpansive mapping and a common fixed point of a nonexpansive semigroup in Hilbert spaces. The main results in this paper modify and improve some well-known results in the literature. Mathematics Subject Classification : 4A65, 47H7, 47H2 Keywords: Metric projection; Fixed point; Nonexpansive Mappings and Semigroups Introduction Let H be a real Hilbert space with the scalar product and the norm denoted by the symbols.,. and., respectively, and let C be a nonempty closed Vietnamese Academy of Science and Technology Institute of Information Technology 8, Hoang Quoc Viet, Cau Giay, Ha Noi, Viet Nam 2 Viet Nam, Thainguyen University, University of Sciences, nguyenduclang22@yahoo.com Article Info: Received : May 6, 22. Revised : August 9, 22 Published online : April 5, 23

2 36 Strong convergence to a common fixed point... and convex subset of H. Denote by P C x the metric projection of an element x H onto C. It is well-known that P C is a nonexpansive mapping on H for any closed convex subset C in H. Recall that a mapping T is said to be nonexpansive on C, if T : C C and T x T y x y for all x, y C. We use F (T ) to denote the set of fixed points of T, i.e., F (T ) = {x C : x = T x}. We know that F (T ) is nonempty, if C is bounded, for more details see []. Let {T (t) : t > } be a nonexpansive semigroup on C, that is, () for each t >, T (t) is a nonexpansive mapping on C; (2) T ()x = x for all x C; (3) T (s + t) = T (s) T (t) for all s, t > ; and (4) for each x C, the mapping T (.)x from (, ) into C is continuous. Assume that F = t> F (T (t)). We know that F is a closed convex subset [2] and that F =, if C is bounded [3]. For finding a fixed point of a nonexpansive mapping T on C, Alber [4] proposed the following descent-like method: x n+ = P C (x n µ n (I T )x n ), n, x C, () where I denotes the identity mapping in H, and proved that if the sequence of positive real numbers {µ n } is chosen such that µ n as n and {x n } is bounded, then: (i) there exists a weak accumulation point x C of {x n }; (ii) all weak accumulation points of {x n } belong to F (T ); (iii) if F (T ) is a singleton, i.e., F (T ) = { x}, then {x n } converges weakly to x. Motivated by Solodov and Svaiter s algorithm [5], Nakajo and Takahashi [2] introduced the following strongly convergence iteration procedures: x C any element, y n = α n x n + ( α n )T x n, C n = {z C : y n z x n z }, Q n = {z C : x n x, z x n }, x n+ = P Cn Q n (x ), n, (2) where {α n } [, a] for some a [, ), for finding a fixed point of a nonex-

3 Nguyen Buong and Nguyen Duc Lang 37 pansive mapping T on C, and x C any element, y n = α n x n + ( α n )T n x n, C n = {z C : y n z x n z }, Q n = {z C : x n x, z x n }, x n+ = P Cn Qn (x ), n, (3) where where T n is defined by T n y = T (s)yds, for each y C, α n [,a] for some a [,) and { } is a positive real number divergent sequence, for finding a common fixed point of a nonexpansive semigroup {T (t) : t > }. Further, in 28, Takahashi, Takeuchi and Kubota [6] proposed a simple variant of (3) that has the following form: x H, C = C, x = P C x, y n = α n x n + ( α n )T n x n, C n+ = {z C n : y n z x n z }, x n+ = P Cn+ x, n. (4) They showed (Theorem 4.4 in [6]) that if α n a <, < < for all n and, then {x n } converges strongly to u = P F x. At the time, Saejung [7] considered the following analogue without Bochner integral: x H, C = C, x = P C x, y n = α n x n + ( α n )T (t n )x n, C n+ = {z C n : y n z x n z }, x n+ = P Cn+ x, n, (5) where α n a <, lim inf n t n =, lim sup n t n >, and lim n (t n+ t n ) =. Then {x n } converges strongly to u = P F x. If C H, then C n and Q n in (2)-(5) are two halfspaces. So, the projection x n+ onto C n Q n or C n+ in these methods can be found by an explicit formula

4 38 Strong convergence to a common fixed point... [5]. Clearly, if C is a proper subset of H, then C n and Q n in these algorithms are not two halfspaces. Then, the following problem is posed: how to construct the closed convex subsets C n and Q n and if we can express x n+ of the above algoritms in a similar form as in [5]? This problem is solved very recently in [8], [9] and []. In the works, C n and Q n in (2)-(3) are replaced by two halfspaces and y n is the right hand side of () with a modification. In this paper, using the idea, we present a new variant for (4)-(5) where C n+ becomes a halfspace H n+ defined below. More precisely, we consider the following algorithms: x H = H, y n = x n µ n (I T P C )x n, H n+ = {z H n : y n z x n z }, (6) for finding an element in F (T ); x n+ = P Hn+ x, n, x H = H, y n = x n µ n (I T n P C )x n ), H n+ = {z H n : y n z x n z }, (7) and x n+ = P Hn+ x, n ; x H = H, y n = x n µ n (I T (t n )P C )x n, H n+ = {z H n : y n z x n z }, x n+ = P Hn+ x, n, (8) for finding an element in F. We shall prove that iteration processes (6) and (7), (8) converge strongly to a fixed point of T and a common fixed point of {T (t) : t > } in sections 2 and 3, respectively. The symbols and denote weak and strong convergences, respectively. 2 Strong convergence to a fixed point of nonexpansive mappings We formulate the following facts needed in the proof of our results.

5 Nguyen Buong and Nguyen Duc Lang 39 Lemma 2. []. Let C be a nonempty closed convex subset of a real Hilbert space H. For any x H, there exists a unique z C such that z x y x for all y C, and z = P C x if and only if z x, y z for all y C. Theorem 2.2. Let C be a nonempty closed convex subset in a real Hilbert space H and let T be a nonexpansive mapping on C such that F (T ). Assume that {µ n } is a sequence in (a, ) for some a (, ]. Then, the sequences {x n } and {y n }, defined by (6), converge strongly to the same point u = P F (T ) x. Proof. First, note that y n z x n z is equivalent to y n x n, x n z 2 y n x n 2. Thus, H n is a halfspace. Next, we show that F (T ) H n for all n. It is clear that F (T ) = F (T P C ) := {p H : T P C p = p} for any mapping T from C into C. So, we have for each p F (T ) that y n p = ( µ n )x n + µ n T P C x n p = ( µ n )(x n p) + µ n (T P C x n T P C p) ( µ n ) x n p + µ n x n p = x n p. Therefore, p H n for all n. Further, since F (T ) is a nonempty closed convex subset of H, by Lemma 2., there exists a unique element u F (T ) such that u = P F (T ) x. From x n+ = P Hn+ x, we obtain that x n+ x z x for every z H n+. As u F (T ) H n+, we get x n+ x u x n. (9) Now, we show that lim x n+m x n =, ()

6 4 Strong convergence to a common fixed point... for each fixed integer m >. Indeed, from the definition of H n+, it implies that H n+ H n and hence we have that x n x x n+ x n. Therefore, there exists lim n x n x = c. Next, by Lemma 2., x n+m H n and x n = P Hn x, we get that Thus, x n x, x n+m x n. x n+m x n 2 = x n+m x 2 x n x 2 2 x n x, x n+m x n x n+m x 2 x n x 2 from that and lim n x n x = c, () is implied. So, {x n } is a Cauchy sequence. We assume that x n p H. On the other hand, from () and the following inequalities x n T P C x n = µ n y n x n we get a ( y n x n+m + x n+m x n ) 2 a x n+m x n, lim x n T P C x n =. So, p = T P C p. It means that p F (T ). Now, from (9) and Lemma 2., it implies that p = u. The strong convergence of the sequence {y n } to u is followed from lim y n x n = lim µ n x n T P C x n = and x n u. This completes the proof.

7 Nguyen Buong and Nguyen Duc Lang 4 3 Strong convergence to a common fixed point of nonexpansive semigroups Lemma 3. [2]. Let C be a nonempty bounded closed convex subset in a real Hilbert space H and let {T (t) : t > } be a nonexpansive semigroup on C. Then, for any h > ( lim sup sup T (h) t t y C t ) T (s)yds t t T (s)yds =. Theorem 3.2. Let C be a nonempty closed convex subset in a real Hilbert space H and let {T (t) : t > } be a nonexpansive semigroup on C such that F = t> F (T (t)). Assume that {µ n } is a sequence in (a, ] for some a (, ] and { } is a positive real number divergent sequence. Then, the sequences {x n } and {y n } defined by (7), converge strongly to the same point u = P F x. Proof. For each p F C, we have from (7) and p = P C p that ( y n p = ( µ n)(x n p) + µ n p) ( µ n ) x n p + µ n (T (s)p C x n T (s)p C p)ds ( µ n ) x n p + µ n = x n p. x n p ds Therefore, p H n. It means that F H n for all n. As in the proof of Theorem 2.2, we get that {x n } is well defined, it converges strongly to an element p H, and x n+ x u x, where u = P F x. Since lim x n =, () C

8 42 Strong convergence to a common fixed point... and P C is a nonexpansive mapping, we have that P Cx n = P C x n P C x n. So, we obtain from () that lim P Cx n =. (2) This together with () and x n p implies that the sequence {P C x n } also converges strongly to p. Since C is closed, we get p C. On the other hand, we have for each h > that ( T (h)p C x n P C x n T (h)p Cx n T (h) ) ( + T (h) ) + P C x n 2 P C x n ( + T (h) ). (3) Let C = {z C : z u 2 x u }. Since u = P F x C, we have from () and P C x n u = P C x n P C u x n u x n x + x u 2 x u. So, C is a nonempty bounded closed convex subset. It is easy to verify that {T (t) : t > } also is a nonexpansive semigroup on C. By Lemma 3., (3) and P C x n p, we get p = T (h)p for each h >. So, p F. Again, from () and p F, it implies that p = u and y n u as n. This completes the proof.

9 Nguyen Buong and Nguyen Duc Lang 43 Theorem 3.3. Let C be a nonempty closed convex subset in a real Hilbert space H and let {T (t) : t > } be a nonexpansive semigroup on C such that F = t> F (T (t)). Assume that {µ n } is a sequence in (a, ] for some a (, ] and {t n } is a sequence of positive real numbers satisfying the condition lim inf n t n =, lim sup n t n >, and lim n (t n+ t n ) =. Then, the sequences {x n } and {y n } defined by (8), converge strongly to the same point u = P F x. Proof. As in the proof of Theorems 2.2 and 3.2, we get x n+ x u x, lim x n T (t n )P C x n =, (4) lim P Cx n T (t n )P C x n =, (5) and the sequence {x n } and {P C x n } also converge strongly to p C. Without loss of generality, as in [7], let lim j t n j = lim j P C x nj T (t nj )P C x nj t nj =. (6) Now, we prove that p = T (t)p for a fixed t >. It is easy to see that Therefore, P C x nj T (t)p P C x nj T (t)p [t t nj ] l= T (lt nj )P C x nj T ((l + )t nj )P C x kj ([ ]) ([ ]) + t t T P C z nj T p t nj t nj ([ ]) + t T p T (t)p t kj t P C x nj T (t nj )P C x nj + P C x nj p t nj [ ) + t (t T ]t nj p p. t nj t t nj P C x nj T (t nj )P C x nj + P C x nj p + sup{ T (s)p p : s t nj }. This fact, together with (6) and property (4) for the semigroup, implies that lim P Cx nj T (t)p =. j

10 44 Strong convergence to a common fixed point... Therefore, p F. So, from (4), we have that the sequence {x n } converges strongly to u as n. The strong convergence of the sequence {y n } to u is followed from (8), (4), µ n (a, ] and x n u as n. The theorem is proved. References [] E.F. Browder, Fixed-point theorems for noncompact mappings in Hilbert spaces, Proceed. Nat. Acad. Sci. USA, 53, (965), [2] K. Nakajo and W. Takahashi, Strong convergence theorem for nonexpansive mappings and nonexpansive semigroup, J. Math. Anal. Appl., 279, (23), [3] R. DeMarr, Common fixed points for commuting contraction mappings, Pacific J. Math., 3, (963), [4] Ya.I. Alber, On the stability of iterative approximations to fixed points of nonexpansive mappings, J. Math. Anal. Appl., 328, (27), [5] M.V. Solodov and V.F. Svaiter, Forcing strong convergence of proximal point iterations in Hilbert space, Math. Program., 87, (2), [6] W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorem by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 34, (28), [7] S. Saejung, Strong convergence theorems for nonexpansive semigroups without Bochner integrals, Fixed Point Theory and Applications(FPTA), 28, (28), -7, Article ID 745, doi:.55/28/745. [8] Ng. Buong, Strong convergence theorem for nonexpansive semigroup in Hilbert space, Nonl. Anal., 72(2), (2), [9] Ng. Buong, Strong convergence theorem of an iterative method for variational inequalities and fixed point problems in Hilbert spaces, Applied Math. and Comp., , doi:.6/j.amc

11 Nguyen Buong and Nguyen Duc Lang 45 [] Ng. Buong, Strong convergence of a method for variational inequality problems and fixed point problems of a nonexpansive semigroup in Hilbert spaces, JAMI, (2), accepted. [] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35, (972), [2] T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl., 2, (997), 7-83.