It. Joural of Math. Aalysis, Vol. 4, 00, o. 5, 37-45 Strog Covergece Theorems Accordig to a New Iterative Scheme with Errors for Mappig Noself I-Asymptotically Quasi-Noexpasive Types Narogrit Puturog Mathematics Program, Faculty of Sciece Udo Thai Rajabhat Uiversity, Udo Thai 4000, Thailad aputurog@hotmail.com Abstract I this paper, a ew iterative scheme with errors for mappig oself I- asymptotically quasi-oexpasive types ad oself I-asymptotically quasioexpasive types i Baach space is defied. The results obtaied i this paper exted ad improve upo those recetly aouced by S.S.Yao ad L.Wag [ Yao & Wag, (008) : Strog Covergece Theorems for Noself I-Asymptotically Quasi-Noexpasive Mappigs, published i Applied Mathematical Scieces 9(008 : 99-98) ] amog others. Mathematics Subject Classificatio: 46C05, 47H05, 47H09, 47H0 Keywords: Noself I-asymptotically quasi-oexpasive type mappig; Completely cotiuous; Uiformly covex Baach apace; Uiformly L- Lipschitzia. Itroductio Let X be a real Baach space ad let C be a oempty subset of X, P : X C be a oexpasive retractio of X oto C. A oself mappig T : C X is called asymptotically oexpasive [] if there exists a sequece {k } [, ) with k as such that for each, T(PT) x T(PT) y k x - y, for all x, y C. T is said to be uiformly L- Lipschitzia if there exists a costat L > 0 such that T(PT) x T(PT) y L x - y, for all x, y C.
38 N. Puturog T : C X is completely cotiuous [] if for all bouded sequeces {x } C there exists a coverget subsequece of {Tx }. Recallig that a Baach space X is called uiformly covex [5] if, for x + y every 0 < ε, there exists δ δ( ε) > 0 such that δ for every x, y S X ad x y ε, S X {x X : x }. Let T, I : C C, the T is called I-oexpasive o C [4] if Tx Ty Ix Iy for all x, y C. T is called I-asymptotically oexpasive o C if there exists a sequece { λ k } [0, ) with λ k 0 as k such that T k x T k y ( λ k +) I k x I k y, for all x, y C ad k,, 3, T is called I-asymptotically quasi-oexpasive o C [7] if there exists a sequece { λ k } [0, ) with λ k 0 as k such that T k x f ( λ k +) I k x f, for all x C ad k 0 ad for every f F(T) F(I), where φ F(T) F(I) be the set of all commo fixed poits of T ad I. Let T, I : C X, the T is called oself I-asymptotically quasi-oexpasive [6] if there exists a sequece {k } [, ) with k as such that for each, T(PT) x f k I(PI) x f, for all x C ad f F(T) F(I), where P is a retractio from X oto C. I 004, N.Shahzad [4] itroduced the cocept of I-oexpasive mappig i Baach space. I 007, S.Temir ad O.Gul [7] defied I-asymptotically quasioexpasive mappig ad studied the weak covergece theorems for I-asymptotically quasi-oexpasive mappig i Hilbert space. More recetly, S.S.Yao ad L.Wag [6] defied oself I-asymptotically quasi-oexpasive mappig ad proved some strog covergece theorems for such mappig i uiformly covex Baach spaces. The purpose of this paper is to itroduce the cocept of oself I-asymptotically quasi-oexpasive types ad strog covergece theorems, ad defie a ew iterative scheme with errors which modified the iterative scheme of S.S.Yao ad L.Wag [6].. Prelimiaries Let C be a oempty subset of a Baach space X. A subset C is called retract of X if there exists cotiuous mappig P : X C such that Px x for all x C. It is well kow that every closed covex subset of a uiformly covex Baach space is a retract. A mappig P : X C is called a retractio if P P. Note that if a mappig P is a retractio, the Pz z for all z i the rage of P.
Strog covergece theorems 39 Let X be a real Baach space ad let C be a oempty closed covex subset of X. Let P : X C be a retractio of X oto C ad let I : C X be a oself mappig ad T : C X be a oself I-asymptotically quasi-oexpasive type as defied by defiitio 3.. Algorithm. For a give x C, we compute the sequece {x } by the iterative scheme x + P(a I(PI) y + ( a b )x + b u ) y P(c T(PT) x + ( c d )x + d v ),, where {u }, {v } are bouded sequeces i C ad {a }, {b }, {c }, {d }, {a + b } ad {c + d } are appropriate sequeces i [0, ]. (.) If b 0 ad d 0, the (.) is reduced to the iterative scheme defied by S.S.Yao ad L.Wag [6], as follows: Algorithm. For a give x C, we compute the sequece {x } by the iterative scheme x + P(a I(PI) y + ( a )x ) y P(c T(PT) x + ( c )x ),, where {a }, {c } are appropriate sequeces i [0, ]. (.) Recallig a well-kow cocept, ad the followig essetial lemmas, i order to prove our mai results: Lemma. [3]. Let {a }, {b } ad { δ } be sequeces of oegative real umbers satisfyig the iequality a + ( + δ )a + b,,, It δ < ad b <, the lim a exists. Lemma. []. Let X be a real uiformly covex Baach space ad 0 p t q < for all positive itegers. Also suppose that {x } ad {y } are two sequeces of X such that limsup x r, limsup y r ad limsup t x + ( t )y r hold for some r 0, the lim x y 0. 3. Mai Results I this sectio, we provide proof of a covergece theorem for a ew iterative scheme with errors for a mappig of oself I-asymptotically quasioexpasive types. I providig such proof of our mai results, the followig defiitio ad lemmas are required:
40 N. Puturog Defiitio 3.. Let C be a oempty closed covex subset of real Baach space X. T : C X be a oself I-asymptotically quasi-oexpasive mappig, I : C X be a oself mappig, φ F(T) F(I) be the set of all commo fixed poits of T ad I. T is called a oself I-asymptotically quasi-oexpasive type mappig if T is uiformly cotiuous, ad limsup{sup( T(PT ) x q I( PI) x q )} 0, for all q F(T) F(I), where P is a retractio from X oto C. Lemma 3.. Let C be a oempty subset of a real Baach space X. Let T : C X be a oself I-asymptotically quasi-oexpasive type, I : C X be a oself asymptotically oexpasive mappig with sequece {k } [, ), (k ) <, P be a retractio from X oto C. Put G max {0, sup( T(PT ) x q I( PI ) x q ) },, q F(T) F(I), so that G <. Suppose that sequece {x } is geerated by (.) with b < ad d <. If F(T) F(I) φ, the lim x q exists for ay q F(T) F(I). Proof. Settig k + r. Sice (k ) <, so r <. Let q F(T) F(I), ad M sup{ u - q : }, M sup{ v - q : }. Usig (.) with b < ad d <, we have x + q P(a I(PI) y + ( a b )x + b u ) q a I(PI) y q + ( a b ) x q + b u q a k y q + ( a b ) x q + b u q a k P(c T(PT) x + ( c d )x + d v ) q + ( a b ) x q + b u q a k c (T(PT) x q) + ( c d )(x q) + d (v q) + ( a b ) x q + b M a k c ( T(PT) x q + a k ( c d ) x q + a k d v q + ( a b ) x q + b M a k c ( T(PT) x q - I(PI) x q + a k c I(PI) x q + a k ( c d ) x q + a k d v q + ( a b ) x q + b M
Strog covergece theorems 4 a k c sup { T(PT) x q - I(PI) x q } + a k k c x q + a k ( c d ) x q + a k d M + ( a b ) x q + b M a k c G + [a c k + a k ( c d ) + ( a b )] x q + a k d M + b M [a c ( + r ) + a ( + r ) a c ( + r ) a d ( + r ) + a b ] x q + a c ( + r )G + a d ( + r )M + b M [a c r + a c r + a r a d a d r + b ] x q + a c ( + r )G + a d ( + r )M + b M [ + r + r + r ] x q + ( + r )G + ( + r )d M + b M [ + (r + r )] x q + s, (3.) where s ( + r )G + ( + r )d M + b M. Sice r <, G <, b < ad d <, we see that ( r + r ) < ad s <. If follows from Lemma. that lim x q exists. This completes the proof. # Lemma 3.. Let X be a uiformly covex Baach space. Let C, T, I ad {x } be same as i Lemma 3.. Put G max {0, sup( T(PT ) x q I( PI) x q ) },, q F(T) F(I), so that G <. If T is uiformly L-Lipschitzia for some L > 0 ad F(T) F(I) φ, the lim Tx x lim Ix x 0. Proof. By Lemma 3., for ay q F(T) F(I), bouded. Assume lim x q t 0. Let M sup{ u q : } ad M sup{ v q : }. Usig (.) with b < ad d lim x <, we have q exists, the {x } is y q P(c T(PT) x + ( c d )x + d v ) q c T(PT) x q + ( c d ) x q + d v q c ( T(PT) x q - I(PI) x q ) + c I(PI) x q + ( c d ) x q + d M c sup { T(PT) x q - I(PI) x q ) + c k x q
4 N. Puturog + ( c d ) x q + d M c G + c ( + r ) x q + ( c d ) x q + d M c G + (c r + d ) x q + d M ( + r ) x q + G + d M ( + r ) x q + e, (3.) where e G + d M. Sice G < ad d <, so that e <. Takig lim sup o both sides i above iequality, we obtai lim sup y q t. (3.3) Sice I(PI) y q ( + r ) y q. Takig lim sup o both sides i above iequality ad usig (3.3), we have Sice lim sup I(PI) y q t. lim x + q t, the t lim P(a I(PI) y + ( a b )x + b u ) q lim a (I(PI) y q) + ( a b )(x q)+ b (u q) lim [ a (I(PI) y q)+( a )(x q) ] -lim b x q + Sice b <, so that lim b x q 0 ad lim b M 0, we have t lim [ a (I(PI) y q) + ( a )(x q) ]. Similarly, for proof (3.), we have Sice lim a (I(PI) y q) + ( a )(x q) lim x q + r <, lim ( + r )G 0 ad t lim(r + G < ad d r ) x q + lim ( + r )G + <, we see that lim ( + r )d M 0. We have lim a (I(PI) y q) + ( a )(x q) lim b M. lim ( + r )d M. lim(r+ r ) x q 0, lim x q t so that lim a (I(PI) y q) + ( a )(x q) t. It follows from Lemma. that, lim I(PI) y x 0. (3.4) Next, x - q x I(PI) y + I(PI) y q x I(PI) y + ( + r ) y q gives that t lim x - q lim if y q. By (3.3), we have
Strog covergece theorems 43 So that t lim y q t. lim P(c T(PT) x + ( c d )x + d v ) q lim ( c (T(PT) x q) + ( c )(x q) - + lim d v q lim d x q lim ( c (T(PT) x q) + ( c )(x q) Similarly, for proof (3.), we have t lim ( c (T(PT) x q) + ( c )(x q) lim x q t. So that lim ( c (T(PT) x q) + ( c )(x q) t. (3.5) Next, T(PT) x q T(PT) x q - I(PI) x q + I(PI) x q sup ( T(PT) x q - I(PI) x q ) + I(PI) x q Sice G <, we have lim G + ( + r ) x q G + x q + r x q. r < ad lim sup x q t, sup T(PT) x q t. (3.6) By (3.5), (3.6), lim sup x q t ad Lemma., we have lim T(PT) x x 0. (3.7) Also, I(PI) x x I(PI) x I(PI) y + I(PI) y x ( + r ) x y + I(PI) y x ( + r ) c (x T(PT) x ) + d ( x q + q v ) + I(PI) y x c ( + r ) x T(PT) x + d ( x q + M ) + I(PI) y x. Thus by (3.4), (3.7) ad d <, we have lim I(PI) x x 0. (3.8) Sice I(PI) x T(PT) x I(PI) x x + x - T(PT) x I(PI) x x + x T(PT) - x. It follows from (3.7) ad (3.8) that lim I(PI) x T(PT) x 0. (3.9) I additio, x + x a I(PI) y x + b u x a I(PI) y x + b ( u q + q x ) a I(PI) y x + b (M + q x ) Thus by (3.4) ad b <, we have
44 N. Puturog lim x + x 0. (3.0) Sice I(PI) y x + I(PI) y x + x x +, by (3.4) ad (3.0), we have lim I(PI) y x + 0. (3.) So, x + y x + I(PI) y + I(PI) y y x + I(PI) y + I(PI) y x + x y x + I(PI) y + I(PI) y x + y x x + I(PI) y + I(PI) y x + P(c T(PT) x + ( c d )x + d v ) x x + I(PI) y + I(PI) y x + c T(PT) - x x + d v x. Usig (3.4), (3.7), (3.) ad d <, we have lim x + y 0. (3.) Ix x Ix I(PI) y + I(PI) y I(PI) x + I(PI) x x Ix I(PI) y + I(PI) y I(PI) x + I(PI) x x ( + r ) x I(PI) y + (+ r ) y x + I(PI) x x. It follows from (3.8), (3.) ad (3.), we obtai lim Ix x 0. (3.3) Sice T is uiformly L-Lipschitzia for some L > 0, Tx x Tx T(PT) x + T(PT) x x Tx T(PT) x + T(PT) x x L x T(PT) x + T(PT) x x L[ x x - + x - T(PT) x + T(PT) x T(PT) x ] + T(PT) x x L x - T(PT) x + (L + L) x x - + T(PT) x x. Usig (3.7) ad (3.0), we have lim Tx x 0. (3.4) This completes the proof. # Theorem 3.3. Let X, C, T, I ad {x } be same as i Lemma 3.. Put G max {0, sup( T(PT ) x q I( PI) x q ) },, q F(T) F(I), so that G <. If I is completely cotiuous ad F(T) F(I) φ, the {x } is coverged strogly to a commo fixed poit of T ad I. Proof. From Lemma 3., we kow that lim x q exists for ay q F(T) F(I), the {x } is bouded. By Lemma 3., we have
Strog covergece theorems 45 lim Tx x 0 ad lim Ix x 0. (3.5) Suppose that I is completely cotiuous, ad otig that {x } is bouded: We coclude that subsequece { Ix } of {Ix j } exists, such that { Ix } coverges. j Therefore, from (3.5), { x } is coverged. Let x j r as j. From the j cotiuity of P, T, I ad (3.5), we have r Tr Ir. Thus, {x } is coverged strogly to a commo fixed poit r of T ad I. This completes the proof. # Refereces [] C.E. Chidume, E.U. Ofoedu ad H. Zegeye, Strog ad weak covergece theorems for asymptotically oexpasive mappigs, J. Math. Aal. Appl., 80(003), 364-374. [] J. Schu, Iterative costructio of fixed poit of asymptotically oexpasive mappigs, Joural of Mathematical Aalysis ad Applicatios, 58(99), 407-43. [3] K. K. Ta ad H. K. Xu, Approximatig fixed poits of oexpasive mappig by the Ishikawa iteratio process, Joural of Mathematical Aalysis ad Applicatios, 78(993), 30-308. [4] N. Shahzad, Geeralized I-oexpasive maps ad best approximatios i Baach spaces, Demostratio Math. XXXVII(3)(004), 597-600. [5] Robert E. Meggiso, A Itroductio to Baach Space Theory, Spriger- Verlag New York, 998. [6] S.S. Yao ad L. Wag, Strog covergece theorems for oself I- asymptotically quasi-oexpasive mappigs, Applied Mathematical Scieces, 9(008), 99-98. [7] S. Temir ad O. Gul, Covergece theorems for I- asymptotically quasioexpasive mappig i Hilbert space, J. Math. Aal. Appl. 39(007), 759-765. Received: Jauary, 00