Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity

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1 BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout Uform Covety D.R. SAHU, J.S. JUNG AND 3 R.K. VERMA Departmet of Appled Mathematcs, Shr Shakaracharya College of Egeerg ad Techology, Juwa, Dstt. Durg (C.G.) , Ida Departmet of Mathematcs, Dog A Uversty, Busa , Korea 3 Departmet of Mathematcs, Govt. Scece College, Pata, Pata Dstt. Durg (C.G.), Ida e-mal: sahudr@redffmal.com ad jugjs@mal.doga.ac.kr Abstract. Let C be a oempty closed cove subset of a refleve Baach space whose orm s uformly Gâteau dfferetable, T : C C a asymptotcally oepasve mappg ad P the suy oepasve retracto from C oto F (T ). I the paper, we troduce property (S) for mappg T as mmal codto for strog covergece to P of the sequece { } defed by ( ) T, N 0 +, where { } ad { } (,, ) are real sequeces satsfyg approprate codtos ad N 0 s suffcetly large atural umber. 000 Mathematcs Subject Classfcato: 47H09, 47H0. Itroducto Let C be a oempty subset of a real Baach space E ad let T : C C be a olear mappg. The mappg T s sad to be asymptotcally oepasve f for each, there ests a postve costat k wth lm k such that T T y k y for all, y C. T s oepasve f k for,,.

2 6 D.R. Sahu et al. Let C be a closed cove subset of E, T : C C be a oepasve mappg such that the set F (T ) of fed pots of T s oempty ad be a elemet of C. Browder [] proved that { t } defed by t ( t) T, 0 < < t + t t coverges strogly to a elemet of F (T ) whch s earest to F (T ) as t 0 case whe E s a Hlbert space. Rech [7] eteded Browder s result the framework of a uformly smooth Baach space. Lm ad Xu [6] partally eteded celebrated covergece theorem of Rech [7] for asymptotcally oepasve mappgs same framework. Recetly, usg a dea of Browder [], Shmzu ad Takahash [8] studed the covergece of the followg appromated sequece for a asymptotcally oepasve mappg the framework of a Hlbert space: a + T,,,, ( a) where { a } s a real sequece satsfyg 0 < a < ad a 0. Shoj ad Takahash [9] ad Jug et al. [5] eteded ther results uformly cove Baach spaces. I ths paper, we study estece ad strog covergece of the sequece { } defed by ( ) + a T,,,, () where T s a asymptotcally oepasve mappg o a oempty closed cove subset of a refleve Baach space whose orm s uformly Gâteau dfferetable. Our results mprove prevous kow results [5, 6, 7, 8, 9].. Prelmares ad Lemmas Let E be a Baach space ad E * be the dual space of E. The value of y E at E wll be deoted by, y. We also deote by J, the dualty mappg from E to E *, that s, { y E :, y, y } J ( ) * for each E.

3 Strog Covergece of Weghted Averaged Appromats 7 Recall that a Baach space E s sad to be smooth provded the lmt lm t 0 + ty t ests for each ad y S { E : }. I ths case, the orm of E s sad to be Gâteau dfferetable. It s sad to be uformly Gâteau dfferetable f for each y S, ths lmt s attaed uformly for S. The orm s sad to be Fréchet dfferetable f for each S, ths lmt s attaed uformly for y S. Fally, the orm s sad to be uformly Fréchet dfferetable f the lmt s attaed uformly for (, y) S S. I ths case E s sad to be uformly smooth. Sce the dual E * of E s uformly cove f ad oly f the orm of E s uformly Fréchet dfferetable, every Baach space wth a uformly cove dual s refleve ad has a uformly Gâteau dfferetable orm. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a asymptotcally oepasve mappg. Wthout loss of geeralty, we may assume that k for all,,. Set A : a T, C,,,, where { a } (,, ) are sequeces of real umbers such that () a 0 for all,,, () a, () a k β for all,, ad lm β. Now, for a C ad a postve teger, cosder a mappg T o C defed by T u ( ) A u, u C, + () where { } s a sequece of real umbers such that β 0 <, lm 0, ad lm sup <.

4 8 D.R. Sahu et al. Lemma. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a asymptotcally oepasve mappg. Let T be a mappg defed by (). The T has eactly oe fed pot C such that ( ) A + for all, (3) N 0 where N 0 s a suffcetly large atural umber. β < Proof. Sce lm sup, there ests a atural umber N 0 such that N, there ests the uque pot T defed by () satsfes u, v C ( ) β < for all N0. So, for each 0 C satsfyg + ( ) A, sce the mappg T u T v ( ) β u v for all. Lemma. Let C, T ad X be as Lemma. If F (T ) s oempty, the { } s bouded. Proof. Sce ( ) β < for all N0, for each ε > 0, there ests a atural umber 0 such that < ε for all ( ) 0. The for v F(T ), we have β mples for all 0. v + ( ) A v v ( ) β + v ( ) a T v ε v T v Let μ be a cotuous lear fuctoal o ad let a, a,, ). ( 0 a We wrte μ ( a ) stead of μ ( a0, a, a, ). We call μ a Baach lmt [] whe μ satsfes: μ μ ( ) ad μ a ) μ ( a ) ( + for all ( a 0, a, a, ). For a Baach lmt, μ we kow that lm f a μ ( a ) lm sup a for all ( a 0, a, a, ).

5 Strog Covergece of Weghted Averaged Appromats 9 Let { } be a bouded sequece E. The we ca defe a real-valued cotuous cove fucto o E by f ( z) μ z for all z E. The followg lemma s gve [3, 4, 0]. Lemma 3 [3, 4, 0]. Let C be a oempty closed cove subset of a Baach space whose orm s uformly Gâteau dfferetable. Let { } be a bouded sequece C, u C ad μ be a Baach lmt. The f ad oly f f ( u) m f ( z) z C μ z u, J( u) 0 for all z C. Let C be a cove subset of E, D a oempty subset of C ad P a retracto from C oto D, that s, P for all D. A retracto P s sad to be suy f P ( P + t( P) ) P for all C ad t 0. D s sad to be a suy oepasve retract of C f there ests a suy oepasve retracto of C oto D. For more detals, see [3]. The followg lemma s well kow (cf. [3]). Lemma 4 [3]. Let C be a cove subset of a smooth Baach space, D be a oempty subset of C ad P be a retracto form C oto D. The P s suy ad oepasve f P, J( z P) 0 for all C, z D. Lemma 5. Let C, T ad X be as Lemma. The β z for all z F(T ), where lm sup <. μ, J( z) μ Proof. Sce from (3) ( ) ( ) ( A ), we get for z F(T ) ad for each N0,

6 30 D.R. Sahu et al., J ( z) A, J ( z) A A z + z, J ( z) ( β ) z whch gves β z μ, J( z) μ z. Fally, we troduce the followg mmal property for covergece of sequece { } defed by (3): Defto. Let C be a oempty closed cove subset of a Baach space E ad T : C C be a mappg. The T s sad to satsfy the property (S) f the followg holds: for each bouded sequece } C, { lm T 0 mples M F(T ) φ. (S) where M { u C : f ( u) f f ( z) }. z C The eamples of such mappgs that satsfy the property (S) are oepasve mappgs uformly smooth Baach space (see [7]). 3. Ma results Theorem. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad P be the suy oepasve retracto form C oto F (T ). Let { a } (,, } be real sequeces satsfyg: a 0, a ad ak β,,,.

7 Let { } be a real sequece such that Strog Covergece of Weghted Averaged Appromats 3 β 0, lm 0 ad lm sup <. The (a) for ay C, there s eactly oe C such that + ( ) a T for all, N 0 where N 0 s a suffcetly large atural umber. (b) If { } s a appromate fed pot sequece for T,.e., lm T 0 follows that { } coverges strogly to P. Proof. (a) The result follows from Lemma. (b) From Lemma, t follows that { } s bouded. Defe a real-valued fucto o E by, t f ( z) μ z for all z E. The, sce f s cotuous ad cove, f (z) as z ad E s refleve, f attas ts fmum over C. Let u C such that f ( u) f z C f ( z). The M { u C : f ( u) f z C f ( z)} s oempty because u M. Sce { } s bouded ad lm T 0, by property (S), T has a fed pot M. Deote such a pot y. It follows from Lemma 3 that μ y, J( y) 0. Ths equalty ad Lemma 5 yelds μ y μ y, that s, μ y 0.

8 3 D.R. Sahu et al. Therefore, there s a subsequece { } of } whch coverges strogly to y F(T ). The, by Lemma 5, we have j { y, J( y P) y P. Ths equalty ad Lemma 4 yeld y P y P. From <, we have y P. To complete the proof, let } be aother { k subsequece of { } whch coverges strogly to z. We shall show that y z. Sce y P, t follows from Lemma 4 ad 5 that P, J ( z P) 0 ad z, J ( z P) z P, ad hece we have y z z y, J( z P) z, J ( z P) + P, J ( z P) y z. Thus, { } coverges strogly to P. Recall that a oempty subset D of C s sad to satsfy the Property (P) (cf. [6]) f the followg holds: where ω ω () s the weak ω -lmt set of T, that s, the set D mples ω ( ) D, (P) { y C : y weak lm T for some }. The followg lemma s crucal to prove our et ma result. Lemma 6. Let E be a refleve Baach space whose orm s uformly Gâteau dfferetable, C be a oempty closed cove subset of E ad T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k. Let { } be a bouded sequece wth lm T 0. Assume that every oempty weakly ω

9 Strog Covergece of Weghted Averaged Appromats 33 compact cove subset of C satsfyg the property (P) has a fed pot for T. The T satsfes property (S). Proof. Note that M { u C : f ( u) f f ( z) } z C s oempty closed cove bouded subset of C. Although M s ot ecessarly varat uder T, t does have the j property (P). I fact, f u M ad y weak lm T u belogs to the weak j ω -lmt set ω ω (u) of T at u, the from weak lower semcotuty of f ad lm T 0, we have j f ( y) lm f f ( T u) lm sup f ( T u) j lm sup lm sup k μ ( μ T u ) lm sup ( T T m m μm m u ) m m u μ m m u f z C f ( z). Thus, y M ad hece M satsfes the property (P). It follows from assumpto that T has a fed pot M. Therefore, T has satsfes property (S). Theorem. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k ad P be the suy oepasve retracto from C oto F (T). Let a } (,, ) be real sequeces satsfyg: { 0, ad a a Let { } be a real sequece such that a k β,,,. β 0, 0 ad lm sup <, ad let { } be a sequece C defed by (3) such that lm T 0. Assume that every oempty weakly compact cove subset of C satsfyg the property (P) has a fed pot for T. The } coverges strogly to P. { Proof. The result follows from Lemma 6 ad Theorem.

10 34 D.R. Sahu et al. I the case whe a for,,, Theorem ad, we have the followg corollares. Corollary. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad P be the suy oepasve retracto from C oto F (T). Let { a } be a real sequece such that β 0, lm 0 ad lm sup <, where β k. The (a) For ay C, there s eactly oe C such that + ( ) T for all N 0, (4) where N 0 s a suffcetly large atural umber. (b) If T 0 as, t follows that } coverges strogly to P. Corollary. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k ad P be the suy oepasve retracto from C oto F (T ). Let } be a real sequece such that { { β 0, 0 ad lm sup <, where β k. ad let { } be a sequece C defed by (4) such that lm T 0. Assume that every oempty weakly compact cove subset of C satsfyg the property (P) has a fed pot for T. The } coverges strogly to P. { I the case whe a for all ad a 0 otherwse, we have the followg: Corollary 3. Let C be a oempty closed cove subset of a refleve Baach space E whose orm s uformly Gâteau dfferetable, T : C C be a asymptotcally oepasve mappg wth Lpschtz costat k whch satsfes the property (S), ad

11 Strog Covergece of Weghted Averaged Appromats 35 P be the suy oepasve retracto from C oto F (T ). Let { } be a real sequece ( 0, ) such that ad let { } be a sequece C defed by k lm 0 ad lm sup < + ( ) T, N. 0 Suppose addto that lm T 0. The } coverges strogly to P. Remark. () Corollary ad eted Theorem of [5] ad Theorem of [9] from uformly cove Baach space to refleve Baach space, respectvely. () Corollary 3 mproves Theorem of Lm ad Xu [6], where the space s assumed to be uformly smooth. Ackowledgemet. The frst author wshes to ackowledge the facal support of Departmet of Scece ad Techology, Ida, made the program year , Project No. SR/FTP/MS 5/000. { Refereces. S. Baach, Theore des Operatos Leares, Moograph Mat., PWN, Warsazawa, 93.. F.E. Browder, Covergece of appromats to fed pots of o-epasve olear mappgs Baach spaces, Arch. Rato. Mech. Aal. 4 (967), R.E. Bruck ad S. Rech, Accretve operators, Baach lmts, ad dual ergodc theorems, Bull. Acad. Polo. Sc. 9 (98), K.S. Ha ad J.S. Jug, Strog covergece theorems for accretve operators Baach spaces, J. Math. Aal. Appl. 47 (990), J.S. Jug, D.R. Sahu ad B.S. Thakur, Strog covergece theorems for asymptotcally oepasve mappgs Baach spaces, Comm. Appl. Nolear Aal. 5 (998), T.C. Lm ad H.K. Xu, Fed pot theorems for asymptotcally oepasve mappgs, Nolear Aal. (994), S. Rech, Strog covergece theorems for resolvets of accretve operators Baach spaces, J. Math. Aal. Appl. 75 (980), T. Shmzu ad W. Takahash, Strog covergece theorems for asymptotcally oepasve mappgs, Nolear Aal. 6 (996), N. Shoj ad W. Takahash, Strog covergece of averaged appromats for oepasve mappgs Baach spaces, J. Appro. Theory 97 (999), W. Takahash ad Y. Ueda, O Rech s strog covergece theorems for resolvets of accretve operators, J. Math. Aal. Appl. 04 (984), Keywords ad phrases: Fed pot, asymptotcally oepasve mappg, uformly Gâteau dfferetable orm.