Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence

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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 9 (2017), pp Research India Publications Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence Mamta Patel 1 and Sanjay Sharma 2 1,2 Bhilai Institute of Technology, Durg, Chhattisgarh, India. Abstract In this paper we first give a weak convergence theorem for pseudo nonspreading mappings and then we establish strong convergence for these mappings which is the generalization of the work recently done by Kurokawa and Takahashi.(2010). The results are the improvement of the work done by previous authors. Keywords: Nonspreading mapping, fixed point, demiclosed principle, strong convergence, weak convergence. Mathematics Subject Classifications: 47H1054H25 INTRODUCTION AND PRELIMINARIES: In this paper, Let H be a real Hilbert space and C be a complex convex subset of H.Throughout the paper,we denote "x n x and x n x" the strong and weak convergence of {x n }, respectively. Denote by F(t) the set of fixed points of a mapping T. Definition 1. Let T: C C be a mapping (1) T is said to be non expansive, if Tx Ty x y, x, y C. (2) T is said to be quasinonexpansive, if F(T) is nonempty and

2 5184 Mamta Patel and Sanjay Sharma Tx p x p, x C, p F(T). (3) T is said to be non spreading, if..(1) 2 Tx Ty 2 Tx y 2 + Ty p 2 x, y C..(2) It is easy to prove that T: C C is nonspreading if and only if Tx Ty 2 x y x Tx, y Ty x, y C..(3) (4) T: C H is said to be k strictly pseudononspreading in the terminology of Browder-Petryshyn, if there exists k [0,1) such that (5) Tx Ty 2 x y 2 + k x Tx (y Ty) x Tx, y Ty x, y C Definition 2. (1) If T: C C is a nonspreading mapping with F(t), then T is quasinonexpansive and F(T) is closed and convex. (2)Clearly every nonspreading mapping is k strictly pseudononspreading with k=0,but the inverse is not true. This can be seen from the following example. Example: Let R denote the set of all real numbers. Let T: R R be a mapping defined by x, x (, 0) Tx = {...(5) 2x, x [0, ). It is easy to see that k- strictly pseudononspreading mapping withk [0,1), but it is not nonspreading. Definition 3. (1)C be a mapping I-T is said to be demiclosed at ), if for any sequence {x n } H with x n x and (I T)x n 0, we have x = Tx. (2) A Banach space E is said to have Opial s property, if for any sequence {x n } E with x n x, we have lim inf x n x < lim inf x n y, y x (6) It is well known that each Hilbert space possesses Opial property. (3) A mapping Let S: C C is said to be semicompact, if for any bounded sequence {x n } C with

3 Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence 5185 lim x n Sx n =0, then there exists a subsequence {x ni } {x n } such that {x ni } converges strongly to some point x C. Lemma 1. Let E be a uniformly convex Banach space and let B r (0) = {x E: x r} be a closed ball with center 0 and radius r > 0.For any given sequence {x 1, x 2,.,x n.. } B r (0) and any given number sequence {λ 1, λ 2, λ n,.. } with λ i 0, i=1 λ i = 1, there exists a continuous strictly increasing and convex function g: [0,2r) [0, )with g(0) = 0 such that for any i, j N, i < j the following holds: n=1 λ n x n 2 n=1 λ n x n 2 λ i λ j g( x i x j ).. (7) Lemma 2. Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and T: C C be a strictly pseudononspreading mapping, (i) (ii) If F(T), then it is closed and convex. (I T) is demiclosed at origin. Lemma 3. Let T: C C be a strictly pseudononspreading mapping with k [0,1). Denote T β = βi + (1 β)t, where β [k, 1), then (i) F(T) = F(T β ); (ii) (iii) (iv) The following inequality holds T β x T β y 2 x y x T 1 β βx, y T β y, x, y C, (8) T β is a quasinonexpansive mapping, that is T β x p 2 x p 2, x C, p F(T)..(9) Lemma 4. Let C be a nonempty set and closed convex subset of a Hilbrt space H and let : C C R Be a bifunction satisfying conditions (A1), (A2), (A3), and (A4). Then for any r> 0 and x C, there exists z C such that (z, y) + 1 y z, z x 0, y C..(10) r Furthermore if for given r> 0 T r : C C C}...(11) by T r (x) = {z C: (z, y) + 1 y z, z x 0, y r

4 5186 Mamta Patel and Sanjay Sharma Then the following hold; (1) T r is single valued; (2) T r is firmly non expansive, that is T r x T r y 2 T r x T r y, x y ; (3)F(T r ) = Ω, where Ω is the set of solutions of the equilibrium problem; (4) Ω is a closed and convex subset of C. Concerning the weak and strong convergence problem for some kinds of iterative algorithms for nonspreading mappings, k-strictly pseudononspreading mappings and other kind of non-linear mappings have been considered in Osilike and Isiogugu[4],Igarashie at al, Iemoto and Takahashi, Kurokawa and Takahashi. The purpose of this paper is to propose anew iterative algorithm as a generalization for pseudononspreading mappings using weak and strong convergence. Under suitable conditions, the weak and strong convergence are proved. The results are extension and generalization of previous results. MAIN RESULTS We assume the following conditions satisfied, throughout this section. (1)H is a Hilbert space, C be nonempty and closed convex subset of H. (2) For each S i : C C, i = 1,2, & R j : C C, j = 1,2,. are k i & k j strictly pseudo nonspreading mapping with = sup i, 1 k i (0,1) and similarly k = sup j 1 k j (0,1). For given β, γ [k, 1), denoted by S i β = βi + (1 β)s i and R j γ = γi + (1 γ) R j For each i = 1,2 & j = 1,2.., it follows from (8)that S i,β x S i,β y 2 + R j,γ a R j,γ b 2 x y 2 + a b β x S i,βx, y S i,β y γ a R j,γa, b R j,γ b x, y, a, b C. (3) : C C R & φ: C C R are bifunction satisfying the conditions (A1) - (A4).The it follows from lemma 4 that the mapping defined in (11) is single valued, z = T r x, v = Q r a, F(T r ) = Ω, G(Q r ) = ξ where Ω and ξ are the solution set of the equilibrium problem and Ω and ξ are closed and convex subset of C. Theorem: Let H, C, { S i }{ R j }, k, β, γ, {S i β}, { R j γ},, φ, T r, Q r, Ω, ξ be the same as above.let {x n }, {a n }, {u n }and {c n } be the sequences defined by

5 Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence 5187 x 1, a 1 C, (x, y) + φ(a, b) + chosen arbitrarily 1 r y u n, u n x n + 1 r b c n, c n a n 0, y, b C x n+1 + a n+1 = α 0,n u n + δ 0,n c n + α i,n S i,β u n + δ j,n R jγ c n { i=1 j=1 (A) Where {α i,n } (0,1)& {δ j,n } (0,1) and {r n } & {d n } satisfies the following conditions (a) i=0 α i,n = 1 & j=0 δ j,n =1 for each n 1 (b) for each i, j 1, lim inf α 0,n α i,n > 0 & lim inf δ 0,n δ j,n > 0 (c) {r n } (0, ), and {d n } (0, ) and lim inf r n > 0, lim inf d n > 0 (I) If F = ( i=1 F( S i )) Ω 0 and F = ( j=1 F( {x n }, {a n }, {u n } and {c n } R j )) δ 0 then Converge weakly to some point x and a respectively where x, a F. (II) In addition, if there exists some positive integer m such that S m & R m such that S m & R m are semicompact, then the sequences {x n }, {u n } converge strongly to x F and the sequences {a n }and {c n } converge strongly to a F. Proof: First we prove the conclusion (I). The proof here is divided into three steps. Step1.We prove that the sequence {x n }, {a n }, {u n }and {c n }, {S i,β u n },{ R jγ c n }, i, j 1 are all bounded,and for each p, q F the limits lim x n p, lim u n p, lim a n q, lim c n q exists and lim x n p + lim a n q = lim u n p + lim c n q..(12) In fact it follows from lemma 4 thatu n = T rn x n, p = T rn p, c n = Q rn a n, q = Q rn q and u n p + c n q = T rn x n T rn p + Q rn a n Q rn q x n p + c n q n 1.(13) Since p, q F by lemma 3 p i=1 F( S i,β ) & q i=1 F( R j,γ ) Hence it follows that

6 5188 Mamta Patel and Sanjay Sharma x n+1 p + a n+1 q = α 0,n u n + α i,n S i,β u n p + δ 0,n c n + δ j,n i=1 j=1 R jγ c n q α 0,n u n p + i=1 α i,n S i,β u n p + δ 0,n c n q + j=1 δ j,n R jγ c n q (14) α 0,n u n p + α i,n u n p + δ 0,n c n q + δ j,n c n q i=1 j=1 = u n p + c n q x n p + c n q n 1 This implies that for each p, q F, the limits lim x n p, lim a n q, lim c n q exists and so {x n }, {a n }, {u n } and {c n }, are all bounded and (12) holds. Furthermore by (9) it is easy to see that for each i, j 1, {S i,β u n }, {S i,β x n }, {R jγ c n }, {R jγ a n } are also bounded. Step2. Next we prove that for each i, j 1the following holds lim x n S i x n + lim a n R j a n = lim u n S i u n + lim c n R j c n = 0 (15) Infact by lemma5. for any positive integer i, j 1 andp, q F, we have x n+1 p 2 + a n+1 q 2 = α 0,n (u n p) + α i,n (S i,β u n p) i=1 + δ 0,n (c n q) + δ j,n j=1 (R jγ c n q) 2 2 α 0,n u n p 2 + i=1 α i,n (S i,β u n p) 2 + δ 0,n c n q 2 + j=1 δ j,n (R jγ c n q) 2 α 0,n α i,n g( u n S i,β u n ) δ 0,n δ j,n h( c n R jγ c n ) (15)

7 Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence 5189 α 0,n u n p 2 + i=1 α i,n (u n p) 2 + δ 0,n c n q 2 + j=1 δ j,n (c n q) 2 α 0,n α i,n g( u n S i,β u n ) δ 0,n δ j,n h( c n R j,γ c n ) δ 0,n δ j,n h( c n R jγ c n ) δ 0,n δ j,n h( c n R jγ c n ) u n p 2 + c n q 2 α 0,n α i,n g( u n S i,β u n ) x n p 2 + a n q 2 α 0,n α i,n g( u n S i,β u n ) This shows that α 0,n α i,n g( u n S i,β u n ) + δ 0,n δ j,n h( c n R j,γ c n ) x n p 2 + a n q 2 x n+1 p 2 a n+1 q 2 0 as(n 0).(16) Since g, h are continuous and strictly increasing function with g(0) = 0. By condition (b), it yields that lim u n S i,β u n + lim c n R j,γ c n = 0 Therefore we have (17) lim u n S i u n + lim c n R j c n = lim u 1 β n S i,β u n + lim c 1 γ n R j,γ c n =0 1 1 (18) On the other hand, it follows from lemma4 that u n = T rn x n, c n = Q rn a n and for each p, q F u n p 2 + c n q 2 = T r n x n T r n p 2 + Q r n a n Q r n q 2 a n c n 2 } T r n x n T r n p, x n p + Q rn a n Q rn q, a n q = u n p, x n p + c n q, a n q = 1 2 { u n p 2 + x n p 2 + c n q 2 + a n q 2 x n u n 2 This shows that u n p 2 + c n q 2 x n p 2 + a n q 2 x n u n 2 a n c n 2 (19)

8 5190 Mamta Patel and Sanjay Sharma In view of (15) & (19) x n+1 p 2 + a n+1 q 2 u n p 2 + c n q 2 x n p 2 + a n q 2 x n u n 2 a n c n 2 That is, (20) x n u n 2 + a n c n 2 x n p 2 x n+1 p 2 + a n q 2 a n+1 q 2 0 as (n ) (21) In view of (21), (17) and (A) and noting that {x n S i,β u n } & {a n R j,γ c n } are bounded,we have x n S i,β u n } + a n R j,γ c n x n u n + a n c n + u n S i,β u n + c n R j,γ c n + S i,β u n S i,β x n + R j,γ c n R j,γ a n x n u n + a n c n + u n S i,β u n + c n R j,γ c n + { x n u n u 1 β n S i,β u n, x n S i,β u n } { a n c n c 1 γ n R j,γ c n, a n R j,γ c n } as (n ).(22) Therefore we have lim x n S i x n + lim a n R j a n 1 = lim 1 β x 1 n S i,β x n + lim 1 γ a n R j,γ a n = 0 (23) The conclusion is proved. Step3.Next we prove that the weak-accumulation point set W w (x n ) & V v (a n ) of the sequence{x n }, {a n } are singleton set and W w (x n ) F, V v (a n ) F

9 Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence 5191 Infact, for any w W w (x n ), v V v (a n ), there exists a subsequence {x ni } {x n }, {a nj } {a n } such that x ni w, a nj v. It follows from (19) that u ni w, c nj v.since u n = T rn x n, c n = Q rn a n we have from condition A2 that Since 1 1 y u ni, (u r ni x ni ) + b c nj, (c ni d nj a nj ) nj (y, u ni ) + φ (b c nj ) y, b C (24) ( 1 r ni ) ( u ni x ni ) + ( 1 d ) (c n nj j a nj ) 0 as(n ) and that u ni w, c nj v, it follows from condition A4, that (y, w) + φ(b, v) 0, y, b C (25) For any t (0,1), y, b C, letting y t = t y + (1 t)w, b t = b t + (1 t)v then y t, b t C 0 = (y t, y t ) + φ(b t, b t ) t{ (y t, y) + φ(b t, b)} + (1 t){ (y t, w) + φ(b t, v)} t{ (y t, y) + φ(b t, b)} This implies that (w, y) + φ(v, b) 0 y, b C.(26) This shows that w, v C are the solution to the equillibrium problem and that is w Ω, v ξ. On the other hand,by lemma2. for each i 1, I S i is demiclosed at 0. In view of (15),we know that w, v F.Due to the arbitrariness of w W w (x n ), v V v (a n ) with x y & a b.therefore there exists subsequences {x nk }, {x nl } in {x n }, {a nk }, {a nl }in {a n } such that x nk x and x nj y, and a nk a & a nl b. Since x, y, a, b C. by (12) the limits lim x n x & lim x n y and lim a n a & lim a n b exists. By using Opial property of H, we have liminf x n n k k x + liminf a n n k k a < lim x n n k k y + lim a n n k k b = lim x n y + lim a n b

10 5192 Mamta Patel and Sanjay Sharma < lim n j x n j x + lim n j a n j a = lim x n x + lim a n a = liminf n k x n k x + liminf n k a n k a (27) This is a contradiction. Therefore, W w (x n ), V v (a n ) are singleton. Without loss of generality, we assume that W w (x n ) = {x }, V v (a n ) = { a }and x n x & a n a.by using (A) and(15),we have u n x & a n a. This completes the proof of the conclusion (I). Next we prove the conclusion (II). Without loss of generality, we can assume S i & R j are semicompact. From (15) we have that x n S i x n + a n R J a n 0 as(n )..(28) Therefore, there exists a subsequence of {x ni } {x n }, & {a nj } {a n } such that x ni u C, & a nj c C. Since x ni x & a nj a we have x = u & a = b and so x ni x F, & a nj a F.By virtue of (12), we have here = lim u n x + lim c n a = 0 and = lim x n x + lim a n a = 0 (29) This completes the proof of the theorem. REFERENCES [1] A. Kangtunyakarn. (2012). Convergence theorem of common fixed points for a family of nonspreadingmappings in Hilbert space,. Optimization Letters,, vol. 6 (no. 5), pp [2] F. Kohsaka and W. Takahashi. (2008). Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Archiv dermathematik, vol. 91 (no. 2), pp [3] M. Eslamian and A. Abkar. (2011). One-step iterative process for a finite family ofmultivaluedmappings. Mathematical and Computer Modelling, vol. 54 (no. 1-2), pp

11 Fixed Point Theorem in Hilbert Space using Weak and Strong Convergence 5193 [4] M. O. Osilike andf. O. Isiogugu. (2011.). "Weak andstrong convergence theorems for nonspreading-type mappings in Hilbert spaces",. Nonlinear Analysis. Theory,Methods & Applications,, vol. 74, (no.5,), pp [5] S.-S. Chang, J. K. Kim, and X. R. Wang. (2010). Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. Journal of Inequalities and Applications, vol.2010, 14 pages. [6] S.Wang, G.Marino, and Y.C. Liou. (2012). "Strong convergence theoremsfor variational inequality, equilibrium and fixed point problems with applications". Journal of Global Optimization, vol.54 (no. 1), pp [7] U. Kamraksa and R. Wangkeeree. (2012). Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings. Journal of Global Optimization, vol. 54 (no. 1), pp [8] W.Takahashi,and F. Kohsaka. (2008). Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces". Archiv dermathematik, vol. 91 (no. 2), pp [9] X. Qin, S. Y. Cho, and S.M. Kang. (2010). Iterative algorithms for variational inequality and equilibrium problems with applications". Journal of Global Optimization, vol. 48 (no. 3), pp [10] Y. H. Zhao and S.-s. Chang. (2013). "Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces". Abstract and Applied Analysis, 24 August [11] Y. Shehu. (2012.). A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces. Journal of Global Optimization, vol. 54, (no. 3), pp ,.

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