Iterative Algorithm for a Split Equilibrium Problem and Fixed Problem for Finite Asymptotically Nonexpansive Mappings in Hilbert Space

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1 Flomat 31:5 (017), DOI 10.98/FIL170543W Publshed by Faculty of Sceces ad Mathematcs, Uvesty of Nš, Seba Avalable at: Iteatve Algothm fo a Splt Equlbum Poblem ad Fxed Poblem fo Fte Asymptotcally Noexpasve Mappgs Hlbet Space Sheg Hua Wag a, M Jag Che b a Depatmet of Mathematcs ad Physcs, Noth Cha Electc Powe Uvesty, Baodg , Cha b Depatmet of Mathematcs ad Sceces, Shjazhuag Uvesty of Ecoomcs, Shjazhuag , Cha Abstact. I ths pape, we popose a teatve algothm fo fdg the commo elemet of soluto set of a splt equlbum poblem ad commo fxed pot set of a fte famly of asymptotcally oexpasve mappgs Hlbet space. The stog covegece of ths algothm s poved. 1. Itoducto Thoughout ths pape, let R deote the set of all eal umbes, N deote the set of all postve tege umbes, H be a eal Hlbet space ad C be a oempty closed covex subset of H. Let T : C C be a mappg. If thee exsts a sequece {k } [1, ) wth lm k = 1 such that T x T y k x y, x, y C, we call T a asymptotcally oexpasve mappg. If k 1, the T s sad to be a oexpasve mappg. The set of fxed pots of T s deoted by Fx(T). Let F : C C R be a bfucto. The equlbum poblem fo F s to fd z C such that The set of all solutos of (1.1) s deoted by EP(F),.e., F(z, y) 0, y C. (1.1) EP(F) = {z C : F(z, y) 0, y C}. May poblems physcs, optmzato, ad ecoomcs ca be educed to fd the soluto of (1.1); see [1 4]. I 1997, Combettes ad Hstoaga [5] toduced a teatve scheme of fdg the soluto of (1.1) ude the assumpto that EP(F) s o-empty. Late o, may teatve algothms ae cosdeed to fd the elemet of Fx(S) EP(F); see [6 8]. Recetly, some ew poblems called splt vaatoal equalty poblems ae cosdeed by some authos. Ceso et al. [9] tally studed ths class of splt vaato equalty poblems. Let H 1 ad H be two eal Hlbet spaces. Gve the opeatos f : H 1 H 1 ad : H H, bouded lea opeato 010 Mathematcs Subject Classfcato. Pmay 54E70; Secoday 47H5 Keywods. splt equlbum poblems; oexpasve mappgs; splt feasble soluto poblems; Hlbet spaces. Receved: 17 Febuay 015; Accepted: 10 Septembe 015 Commucated by Ljubom Ćć Emal addess: sheg-huawag@hotmal.com (Sheg Hua Wag)

2 S.H. Wag, M.J. Che / Flomat 31:5 (017), A : H 1 H, ad oempty closed covex subsets C H 1 ad Q H, the splt vaatoal equalty poblem s fomulated as follows: ad such that fd a pot x C such that f (x ), x x 0, x C y = Ax Q solves (y ), y y 0, y Q. Afte vestgatg the algothm of Ceso et al., Moudaf [10] toduced a ew teatve scheme to solve the followg splt mootoe vaatoal cluso: ad such that fd x H 1 such that 0 f (x ) + B 1 (x ) y = Ax H sovles 0 (y ) + B (y ), whee B 1 : H H s a set-valued mappgs fo = 1,. I 013, Kazm ad Rzv [11] cosdeed a ew class of splt poblem called splt equlbum poblem. Let F 1 : C C R ad F : Q Q R be two bfuctos ad A : H 1 H be a bouded lea opeato. The splt equlbum poblem s to fd x C such that ad such that F 1 (x, x) 0, x C, (1.) y = Ax Q sovles F (y, y) 0, y Q. (1.3) The set of all solutos of (1.) ad (1.3) s deoted by Ω,.e., Ω = {z C : z EP(F 1 ) such that Az EP(F )}. O splt equlbum poblem, the teested autho also may efe to [1, 13] whch the autho gave a teatve algothm to fd the commo elemet of sets of solutos of the splt equlbum poblem ad heachcal fxed pot poblem. To the kowledge of autho, the splt equlbum poblems ad fxed pot poblems fo asymptotcally oexpasve mappgs have ot bee vestgated lteatue by fa. I ths pape, sped by the esults [11 13], we popose a teatve algothm to fd the commo elemet of soluto sets of a splt equlbum poblem ad commo fxed pots set of a fte famly of asymptotcally oexpasve mappgs Hlbet spaces ad pove the stog covegece fo the algothm.. Pelmaes Let H be a Hlbet space ad C be a oempty closed ad covex subset of H. Fo each pot x H, thee exsts a uque eaest pot of C, deoted by P C x, such that x P C x x y fo all y C. Such a P C s called the metc pojecto fom H oto C. It s well kow that P C s a fmly oexpasve mappg fom H oto C,.e., P C x P C y P C x P C y, x y, x, y H. Futhe, fo ay x H ad z C, z = P C x f ad oly f x z, z y 0, y C. A mappg A : C H s called a α-vese stogly mootoe f thee exsts α > 0 such that x y, Ax Ay α Ax Ay, x, y H. Fo each λ (0, α], I λa s a oexpasve mappg of C to H; see [14]. Cosde the followg vaatoal equalty o vese stogly mootoe mappg A: fd u C such that v u, Au 0, v C.

3 S.H. Wag, M.J. Che / Flomat 31:5 (017), The set of solutos of the vaatoal equalty s deoted by VI(C, A). It s kow that u VI(C, A) u = P C (u λau) fo ay λ > 0. Let S : C C be a mappg. It s kow that S s oexpasve f ad oly f the complemet I S s 1 -vese stogly mootoe; see [15]. Let T : C C be a asymptotcally oexpasve mappg wth the sequece {k }. The fo ay (x, ˆx) C Fx(T), we have whch s obtaed dectly fom the followg T x x x T x, x ˆx + (k 1) x ˆx, N, (.1) k x ˆx T x T ˆx = T x ˆx = T x x + (x ˆx) = T x x + x ˆx + T x x, x ˆx. Let F be a bfucto of C C to R satsfyg the followg codtos: (A1) F(x, x) = 0 fo all x C; (A) F s mootoe,.e., F(x, y) + F(y, x) 0 fo all x, y C; (A3) fo each x, y, z C, lm t 0 F(tz + (1 t)x, y) F(x, y); (A4) fo each x C, y F(x, y) s covex ad lowe semcotuous. Lemma.1 [16] Let C be a oempty closed covex subset of a Hlbet space H ad let F : C C R be a bfucto whch satsfes the codtos (A1)-(A4). Fo x H ad > 0, defe a mappg T F : H C by The T s well defed ad the followg hold: (1) T F s sgle-valued; () T F s fmly oexpasve,.e., fo ay x, y H, (3) Fx(T F ) = EP(F); (4) EP(F) s closed ad covex. T F (x) = {z C : F(z, y) + 1 y z, z x 0, y C}. (.) T F x T F y T F x T F y, x y ; Lemma. [17] Let F : C C R be a bfucto satsfyg the codtos (A1)-A(4). Let T F ad Ts F be defed as Lemma.1 wth, s > 0. The, fo ay x, y H, oe has T F x Ts F y x y + 1 s T F x x. Lemma.3 [8] Let F : C C R be a fuctos satsfyg the codtos (A1)-(A4) ad let T F s ad T F t be defed as Lemma.1 wth s, t > 0. The the followg holds: fo all x H. Ts F x Tt F x s t T s x T t x, T s x x s Lemma.4 [18] Let {x } ad {y } be bouded sequeces a Baach space E ad let {β } be a sequece [0, 1] wth 0 < lm f β lm sup β < 1. Suppose x +1 = β y + (1 β )y fo all teges 0 ad lm sup ( y +1 y x +1 x ) 0. The, lm y x = 0. Lemma.5 [19] Let T be a asymptotcally oexpasve mappg o a closed ad covex subset C of a eal Hlbet space H. The I T s demclosed at ay pot y H. That s, f x x ad x Tx y H, the x Tx = y.

4 S.H. Wag, M.J. Che / Flomat 31:5 (017), Lemma.6 [0] Assume that {α } s a sequece of oegatve umbes such that α +1 (1 a )α + a t, 0, whee {a } s a sequece (0, 1) ad {t } s a sequece R such that (1) =1 a = ; () ethe lm sup t 0 o =0 a t <. The lm α = Ma Results Theoem 3.1 Let H 1 ad H be two eal Hlbet spaces ad C H 1 ad Q H be oempty closed ad covex subsets. Let F : C C R ad G : Q Q R be two bfuctos satsfyg (A1-A4) ad assume that G s uppe semcotuous the fst agumet. Let f : C C be ρ-cotacto ad {T } l : C C be l asymptotcally oexpasve mappgs wth the same sequece {k } satsfyg the codto that lm sup x K T +1 x T x = 0 (Γ) fo ay bouded subset K of C ad each = 1,, l. Let A : H 1 H be a bouded lea opeato. Suppose that Fx(T) Ω, whee Fx(T) = l Fx(T ) ad Ω = {v C : v EP(F) such that Av EP(G)}. Let {α } (0, 1) be a sequece. Defe the sequece {x } by the followg mae: x 0 C ad u = T F (I γa (I Ts G )A)x, x +1 = α f (x ) + (1 α l ) T l u, N, whee { } [, ) wth > 0, {s } [s, ) wth s > 0, γ (0, 1/L ], L s the spectal adus adus of the opeato A A ad A s the adjot of A. If the cotol sequeces {α }, { }, {s } ad {k } satsfy the followg codtos: () lm α = 0, =1 α = ; () =1 α α 1 <, =1 1 <, =1 s s 1 < ; k () lm 1 α = 0, the {x } geeated by (3.1) stogly coveges to z = P Fx(T) Ω f (z). Remak 3.1. Fo each N, A (I T G )A s a 1 -vese stogly mootoe mappg. I fact, sce T G L s s (fmly) oexpasve ad I Ts G s 1 -vese stogly mootoe, we have A (I T G s )Ax A (I T G s )Ay = A (I T G s )(Ax Ay), A (I T G s )(Ax Ay) = (I T G s )(Ax Ay), AA (I T G s )(Ax Ay) L (I T G s )(Ax Ay), (I T G s )(Ax Ay) = L (I T G s )(Ax Ay) L Ax Ay, (I T G s )(Ax Ay) = L x y, A (I T G s )(Ax Ay), fo all x, y H, whch mples that A (I T G s )A s a 1 L -vese stogly mootoe mappg. Note that γ (0, 1 L ]. Thus I γa (I T G s )A s oexpasve. (3.1)

5 S.H. Wag, M.J. Che / Flomat 31:5 (017), Poof. We fst show that {x } s bouded. Let p Fx(S) Ω. The p = T F p ad (I γa (I T G s )A)p = p. Thus we have u p = T F (I γa (I T G s )A)x T F (I γa (I T G s )A)p (I γa (I T G s )A)x (I γa (I T G s )A)p x p. Take ɛ (0, 1 ρ). Sce (k 1)/α 0 as, thee exsts N N such that fo all N, (k 1) < ɛα. Let T = 1 l l T fo each N. It s easy to see that T x T y k x y fo all x, y C ad N. Fom (3.1) ad (3.) t follows that, fo all > N, (3.) x +1 p = α ( f (x ) f (p)) + α ( f (p) p) + (1 α )(T u p) α f (x ) f (p) + α f (p) p + (1 α ) T u p α ρ x p + α f (p) p + (1 α )k u p = (1 α (1 ρ)) x p + α f (p) p + (1 α )(k 1) u p (1 α (1 ρ)) x p + α f (p) p + α ɛ x p = (1 α (1 ρ ɛ)) x p + α f (p) p max{ x p, 1 f (p) p }. 1 ρ ɛ (3.3) By ducto, we see that, fo all > N, x p max { x N p, 1 1 ρ ɛ f (p) p }. It follows that {x } s bouded ad so s {u }. Next we pove that lm x +1 x = 0. Sce (I γa (I T G s )A) s oexpasve, by Lemma. we have u +1 u = T F +1 (I γa (I T G s +1 )A)x +1 T F (I γa (I T G s )A)x (I γa (I T G s +1 )A)x +1 (I γa (I T G s )A)x + +1 T F +1 (I γa (I Ts G +1 )A)x +1 (I γa (I Ts G +1 )A)x x +1 x + (I γa (I Ts G +1 )A)x (I γa (I Ts G )A)x + +1 σ +1 = x +1 x + γ A (Ts G Ts G +1 )Ax + +1 σ +1, whee σ = sup{ N} T F (I γa (I T G s )A)x (I γa (I T G s )A)x. Futhe, by Lemma.3 we get u +1 u x +1 x + γ A Ts G Ax Ts G +1 Ax + +1 σ +1 x +1 x + γ A ( s s +1 Ts G s Ax Ts G +1 Ax, Ts G Ax Ax ) σ +1 x +1 x + γ A ( s s +1 ) 1 η s + +1 σ +1, (3.4) whee η = sup N T G s Ax T G s +1 Ax, T G s Ax Ax.

6 S.H. Wag, M.J. Che / Flomat 31:5 (017), Let y = x +1 α f (x ) 1 α. The fom (3.1) ad (3.4) t follows that y +1 y x +1 x = T +1 u +1 T u x +1 x T +1 u +1 T +1 u x +1 x + T +1 u T u [ ( s s +1 (k +1 1) x +1 x + k γ A s + T +1 u T u η ) 1 [ ( s s +1 (k +1 1)( x +1 + x ) + k γ A s + 1 l l l T +1 u T u. η ) ] σ ] σ +1 Sce the mappgs {T } l satsfy the codto (Γ), by the codto () we get lm sup( y +1 y x +1 x ) = 0. Hece, by Lemma.4 we coclude that whch mples that Futhe, by (3.4) we get Fom (3.1) ad (3.5) t follows that lm y x = 0, lm x +1 x = 0. (3.5) lm u +1 u = 0. (3.6) lm T u x = 0. (3.7) Now we pove that lm T x x 0 fo each {1,, l}. To show ths, we fst pove that lm u x = 0. Sce A (I T G s )A s 1 L -vese stogly mootoe, by (3.1) we have u p = T F (I γa (I T G s )A)x T F (I γa (I T G s )A)p (I γa (I T G s )A)x (I γa (I T G s )A)p = (x p) γ(a (I T G s )Ax A (I T G s )Ap) = x p γ x p, A (I T G s )Ax A (I T G s )Ap + γ A (I T G s )Ax A (I T G s )Ap x p γ L A (I T G s )Ax A (I T G s )Ap + γ A (I T G s )Ax A (I T G s )Ap = x p + γ(γ 1 L ) A (I T G s )Ax A (I T G s )Ap = x p + γ(γ 1 L ) A (I T G s )Ax

7 S.H. Wag, M.J. Che / Flomat 31:5 (017), Thus we have x +1 p α f (x ) p + (1 α )k u p α f (x ) p + (1 α )k [ x p + γ(γ 1 L ) A (I Ts G )Ax ] = α f (x ) p + (1 α )(1 + θ + θ ) x p + γ(1 α )k(γ 1 L ) A (I Ts G )Ax α f (x ) p + x p + (1 α )(θ + θ ) x p + γ(1 α )k(γ 1 L ) A (I Ts G )Ax, whee θ = k 1. Theefoe, γ(1 α )k ( 1 L γ) A (I T G s )Ax α f (x ) p + x x +1 ( x p + x +1 p ) + (1 α )(θ + θ ) x p. Sce α 0, k 1 ad both { f (x )} ad {x } ae bouded, by (3.5) we have whch mples that lm A (I Ts G )Ax = 0, (3.8) lm (I TG s )Ax = 0. (3.9) Sce T F s fmly oexpasve ad (I γa (T G s I)A) s oexpasve, by (3.1) we have u p = T F ( x + γa (T G s I)Ax ) T F (p) u p, x + γa (T G s I)Ax p = 1 { u p + x + γa (T G s I)Ax p u p [x + γa (Ts G I)Ax p] } = 1 { u p (I + γa (Ts G I)A)x (I γa (Ts G I)A)p u x γa (Ts G I)Ax } 1 { u p x + p u x γa (Ts G I)Ax } = 1 { u p x + p [ u x + γ A (Ts G I)Ax γ u x, A (T G s I)Ax ]}, whch mples that u p x p u x + γ u x A (T G s I)Ax. (3.10)

8 Now, fom (3.1) ad (3.10) we get Hece, S.H. Wag, M.J. Che / Flomat 31:5 (017), x +1 p α f (x ) p + (1 α ) T u p α f (x ) p + (1 α )k u p α f (x ) p + (1 α )k [ x p u x + γ u x A (T G s I)Ax ] = α f (x ) p + (1 α )(1 + θ + θ ) x p (1 α )k u x + (1 α )k γ u x A (T G s I)Ax ] α f (x ) p + (θ + θ ) x p (1 α )k u x + x p + k γ u x A (T G s I)Ax ] (1 α )k u x α f (x ) p + x x +1 ( x p + x +1 p ) + (θ + θ ) x p + k γ( u + x ) A T G s I)Ax. Sce α 0, k 1 ad {x } ad {u } ae bouded, by (3.5) ad (3.10) we have lm u x = 0. (3.11) Combg (3.5) ad (3.11), by u x +1 u x + x x +1 we see that lm u x +1 = 0. (3.1) Note that 1 l l (T u u ) = (T u u ) By (.1) ad (3.13), fo each = 1,, l, we have 1 l T u u 1 l l = l T u u = 1 1 α [ x+1 u + α (u f (x )) ]. l T u u, u p + (k 1) u p [ x+1 u, u p + α u f (x ), u p ] + (k 1) u p. (1 α ) (3.13) (3.14) Sce α 0 ad k 1, fom (3.1) ad (3.14) t follows that, fo each = 1,, l, lm T u u = 0. (3.15) Let k = sup N k <. Cosequetly, by (3.6) ad (3.15) we get that, fo each = 1,, l, T u u T u T +1 u + T +1 u T +1 u +1 + T +1 u +1 u +1 + u +1 u k u T u + T +1 u +1 u +1 + (1 + k ) u +1 u 0, as.

9 Futhe, we have, fo each = 1,, l, S.H. Wag, M.J. Che / Flomat 31:5 (017), T x x T x T u + T u u + u x (k 1 + 1) u x + T u u 0, as. (3.16) Sce P Fx(S) Ω f s a cotacto, thee exsts a uque z Fx(S) Ω such that z = P Fx(S) Ω f (z). Sce {x } s bouded, we ca choose a subsequece {x k } of {x } such that lm sup f (z) z, x z = lm f (z) z, x z. k As {x k } s bouded, thee s a subsequece {x k } of {x k } covegg weakly to some w C. Wthout loss of geealty, we ca assume that x k w. Now we show that w l Fx(T ). I fact, sce each x T x 0 ad x k w, by Lemma.3 we obta that w Fx(T ). So w Fx(T) = l Fx(T ). Next we show that w Ω. By (3.1), u = T F (I γa (I Ts G )A)x, that s F(u, y) + 1 y u, u x 1 y u, γa (T G s I)Ax 0, y C. Fom the mootocty of F t follows that 1 y u, γa (T G s I)Ax + 1 y u, u x F(y, u ), y C. Replacg wth k the above equalty, we have 1 k y u k, γa (T G s I)Ax k + 1 k y u k, u x k F(y, u k ), y C. Sce u k x k 0, A (T G k I)Ax k 0 ad x k w 0 as k, we have F(y, w) 0, y C. Fo ay 0 < t 1 ad y C, let y t = ty + (1 t)w. The we have y t C. Futhe, we have 0 = F(y t, y t ) tf(y t, y) + (1 t)f(y t, w) tf(y t, y). So F(y t, y) 0. Let t 0, oe has F(w, y) 0,.e., w EP(F). Next we show that Aw EP(G). Sce A s bouded lea opeato, Ax k follows that Ts G Ax k Aw. By the defto of T G Ax k, we have Aw. The fom (3.9) t G(T G s Ax k, y) + 1 s k y T G s Ax k, T G s Ax k Aw 0, y C. (3.1) Sce each G s uppe semcotuous the fst agumet, takg lm sup to (3.1) as k, we get whch mples that Aw EP(G). Theefoe, w Ω. By the popety o P Fx(T) Ω, we have G(Aw, y) 0, y C, lm sup f (z) z, x z = lm f (z) z, x k z k = f (z) z, w z 0. (3.13)

10 S.H. Wag, M.J. Che / Flomat 31:5 (017), Sce α 0, thee exsts N 1 N such that ( ρ)α < 1 fo all N 1. Now, by (3.1) we have, fo all > N 1, x +1 z = α f (x ) + (1 α )T u p So whee M = sup N x z. Put ad The (1 α ) T u p + α f (x ) z, x +1 z [(1 α )k ] u p + α f (x ) f (z), x +1 z + α f (z) z, x +1 z [(1 α )k ] x p + ρα x z x +1 z + α f (z) z, x +1 z [(1 α )k ] x p + ρα ( x z + x +1 z ) + α f (z) z, x +1 z = [(1 α )(k 1) + (1 α )] x z + ρα x z + ρα x +1 z + α f (z) z, x +1 z = [1 ( ρ)α + α + (1 α ) (k 1) + (1 α ) (k 1)] x z + ρα x +1 z + α f (z) z, x +1 z [1 ( ρ)α + α + (k 1) + (k 1)] x z + ρα x +1 z + α f (z) z, x +1 z. x +1 z 1 ( ρ)α 1 ρα x z + α + (k 1) + (k 1) 1 ρα M + α 1 ρα f (z) z, x +1 z = (1 (1 ρ)α 1 ρα ) x z + α + (k 1) + (k 1) 1 ρα M + α 1 ρα f (z) z, x +1 z, s = (1 ρ)α 1 ρα δ = α + (k 1) + (k 1) (1 ρ)α M ρ f (z) z, x +1 z. x +1 z (1 s ) x z + s δ. Note that s 0, =1 s = ad lm sup δ 0. By theoem.6 we coclude that lm x z = 0. Ths completes the poof. I Theoem 3.1, f T T, the the codto (Γ) s educed to asymptotcally egula ad we get the followg Coollay 3.1 Let H 1 ad H be two eal Hlbet spaces ad C H 1 ad Q H be oempty closed covex subsets. Let F : C C R ad G : Q Q R be two bfuctos satsfyg (A1-A4) ad assume that G s uppe semcotuous the fst agumet. Let f : C C be ρ-cotacto ad T : C C be a asymptotcally oexpasve mappg wth the sequece {k } satsfyg the codto that lm sup x K T +1 x T x = 0

11 S.H. Wag, M.J. Che / Flomat 31:5 (017), fo ay bouded subset K of C. Assume that T s asymptotcally egula ad suppose that Fx(T) Ω, whee Ω = {v C : v EP(F) ad Av EP(G)}. Let {α } (0, 1) be a sequece. Let A : H 1 H be a bouded lea opeato. Defe the sequece {x } by the followg mae: x 0 C ad { u = T F (I γa (I T G s )A)x, x +1 = α f (x ) + (1 α )T u, N, whee { } (, ) wth > 0, {s } [s, ) wth s > 0, γ (0, 1/L ], L s the spectal adus adus of the opeato A A ad A s the adjot of A. If the cotol sequeces {α } ad {k } satsfy the followg codtos: () lm α = 0, =1 α = ; () =1 α α 1 <, =1 1 <, =1 s s 1 < ; k () lm 1 α = 0, the {x } stogly coveges to z = P Fx(T) EP(F) f (z). I Coollay 3.1, f A 0, the we get the followg Coollay 3. Let H 1 ad H be two eal Hlbet spaces ad C H 1 ad Q H be oempty closed covex subsets. Let F : C C R be a bfucto satsfyg (A1-A4). Let f : C C be ρ-cotacto ad T : C C be a asymptotcally oexpasve mappg wth the sequece {k } satsfyg the codto that lm sup x K T +1 x T x = 0 fo ay bouded subset K of C. Assume that T s asymptotcally egula ad suppose that Fx(T) EP(F). Let {α } (0, 1) be thee sequece. Defe the sequece {x } by the followg mae: x 0 C ad { u = T F x, x +1 = α f (x ) + (1 α )T u, N, whee { } [, ) wth > 0. If the cotol sequeces {α }, { }, {s } ad {k } satsfy the followg codtos: () lm α = 0, =1 α = ; () =1 α α 1 <, =1 1 < ; k () lm 1 α = 0, the {x } stogly coveges to z = P Fx(T) EP(F) f (z). I Coollay 3., f F(x, y) 0 ad s 1, the u = P C x = x ad we get the followg Coollay 3.3 Let H 1 ad H be two eal Hlbet spaces ad C H 1 ad Q H be oempty closed covex subsets. Let f : C C be ρ-cotacto ad T : C C be a asymptotcally oexpasve mappg wth the sequece {k } satsfyg the codto that lm T +1 x T x = 0 sup x K fo ay bouded subset K of C. Assume that T s asymptotcally egula ad suppose that Fx(T). Let {α } (0, 1) be thee sequece. Defe the sequece {x } by the followg mae: x 0 C ad { x+1 = α f (x ) + (1 α )T x, N. If the cotol sequeces {α }, { } ad {k } satsfy the followg codtos: () lm α = 0, =1 α < ; () =1 α α 1 < ; () lm k 1 α = 0,

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