Fixed Point for Asymptotically Non-Expansive Mappings in 2-Banach Space Salwa S. Abed Rafah S. Abed Ali Department of Mathematics/College of Education for Pure Science(Ibn AL- Haitham) /Baghdad University Received in : 23 June 2013 , Accepted in : 4 December 2013 Abstract In this paper, we introduced some fact in 2-Banach space. Also, we define asymptotically non-expansive mappings in the setting of 2-normed spaces analogous to asymptotically non-expansive mappings in usual normed spaces. And then prove the existence of fixed points for this type of mappings in 2-Banach spaces. keywords: 2-Banach space, Non-expansive mapping, Asymptotically non-expansive mapping, Fixed point. 343 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Introduction and Preliminaries The concept of linear 2-normed spaces (breviary, 2-normed space) was initiated by S- Gahler in 1965 [1]. Other papers dealing with 2-normed spaces are [2], [3] and [4]. later on, several researchers studied 2-normed spaces using contractive map-pings (see [5] and [6] ) . Also, Mukti, Sahu and Baisnab [7] proved some fixed point theorems in 2-Banach spaces where mappings involved are of caristis type. The study of 2-normed spaces using asymptotically non-expansive mappings was not initiated by researcher. The purpose of this paper is to continue studying 2-normed spaces using asymptotically non-expansive mappings. Now, we recall the following definitions: Definition [8] Let X be a real linear space and ‖. , . ‖ be a nonnegative real valued function defined on X × X satisfying the following conditions: ‖𝑥, 𝑦‖ = 0 if and only if 𝓍𝓍 and y are linearly dependent in X. 1) ‖𝑥, 𝑦‖ = ‖𝑦, 𝑥‖ , for all 𝓍𝓍 ,y ϵX . 2) 3) ‖𝑥, 𝛼𝑦‖ = |𝛼|‖𝑥, 𝑦‖ , α ϵ R ,𝓍𝓍 , y ϵ X. 4) ‖𝑥, 𝑦 + 𝑧‖ ≤ ‖𝑥, 𝑦‖ + ‖𝑥, 𝑧‖ , for all 𝓍𝓍 , 𝑦 , 𝑧 ϵ X. Then (X ,‖. , . ‖) is called a 2-normed space. Note that the 2-normed space is Hausdorff space and ‖. , . ‖ is continuous function, for examples of 2-normed spaces see [1] . The ball in 2-normed space X with center x, radius r > 0 and is defined by Br(x) = { y, u ϵ X, || x – y, u || ˂ r}, and the open subset M of X is defined as follows: for any x ϵ M there is r > 0 such that Br(x) ⊂ M. therefore M is called closed subset of X if its complement is open. Definition [9] A sequence (xn) in a 2-normed space (X,||.,.||) is called a convergent sequence if there is, x ϵ X, such that lim𝑛→∞‖𝑥𝑛 − 𝑥, 𝑢‖= 0 , for all u ϵ X. Definition [9] A sequence (xn) in a 2-normed space (X,||.,.||) is called a Cauchy sequence if lim𝑚,𝑛→∞‖𝑥𝑚 − 𝑥𝑛, 𝑦‖ =0 , for all 𝑦 ∈ X. Definition [9] A linear 2-normed space X is said to be complete if every Cauchy sequence is convergent to an element of X. Then X is called a 2-Banach space. Definition [9] Let X be a 2-Banach space and T:X→X be a Mapping T is said to be continuous at x if for every sequence (𝓍𝓍n ) in X,( 𝓍𝓍n ) →𝓍𝓍 as n→∞ implies that {T(𝓍𝓍n )}→T(𝓍𝓍) as n→∞. We need to give some concepts in the setting of 2-normed space X as the first dual and the second dual of X are defined by X*= {f l f :X →ℝ , bounded linear function} , X**= { g l g : X*→ ℝ, bounded linear function}. respectively, then the mapping J: X→ X* , where J(x) = Fx(f) = f(x), f ϵ X* is called a natural embedding, so we say that X is reflexive if the natural embedding is an onto mapping. 344 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Definition [10] Let X be a 2-normed space, (𝓍𝓍n) be a sequence in X , Then (𝓍𝓍n) is said to be converges weakly to 𝓍𝓍 denoted by 𝓍𝓍n →𝓍𝓍 as n → ∞ , if f(𝓍𝓍n) → f(𝓍𝓍) as n → ∞. Definition [11] A 2-normed space (X,||.,.||) is said to be uniformly convex if for every ε ϵ (0,2] and u ≠ 0 in X , there exists α > 0 such that || 𝓍𝓍 , u || ≤ 1 ; || y , u || ≤ 1 and || 𝓍𝓍 – y , u || ≥ ε implies that || 1 2 ( 𝓍𝓍 + y ) , u || ≤ 1 - α. Definition Let X be a 2-normed space, Then we say that X satisfies Opial condition if for every bounded sequence (𝓍𝓍n) ϵ X converges weakly to 𝓍𝓍 ϵ X, Then lim𝑛→∞ inf�|xn − x , u|�˂ limn→∞ 𝑖𝑛𝑓�|xn − y , u|� for every x ≠ y & 𝑦 , 𝑢 𝜖 𝑋. Definition Let X be a 2-normed space. With the mapping T : X → X i- T is said to be Lipschitzian if there exists constant α ≥ 0 such that ||T(x) - T(y), u|| ≤ α ||x – y, u|| for all x ,y ,u ϵ X …(1.1) ii- If α = 1 then T is said to be non-expansive mapping such that ||T𝓍𝓍 – Ty , u || ≤ || 𝓍𝓍 – y , u || ; 𝓍𝓍 , y, u ϵ X …(1.2) Definition Let X be a 2-normed space, Then the mapping T :X→X is said to be Asymp-totically non- expansive mapping if there exists a positive sequence (kn) ϵ [1,∞) with lim𝑛→∞(𝑘𝑛) = 1, such that ||Tn𝓍𝓍 - Tny , u || ≤ kn || 𝓍𝓍 – y , u || …(1.3) for all 𝓍𝓍 , y , u ϵ X and n ≥1. . Definition i. If S a nonempty subset of a 2-normed space X and (xn) a bounded sequence in X. consider the functional ra (·, (xn)) : X × X → R+ defined by ra(x, (xn)) = lim𝑛→∞ 𝑠𝑢𝑝||𝑥𝑛 − 𝑥, 𝑢|| ; x, u ϵ X . ii. The infimum of ra(·, (xn)) over S is said to be the asymptotic radius of (xn) with respect to S and is denoted by ra(S, (xn)). A point z ∈ S is said to be an asymptotic center of the sequence (xn) with respect to S if ra(z, (xn)) = inf {ra(x, (xn)) : x ∈ S} The set of all asymptotic centers of (xn) with respect to S is denoted by Za(S,(xn)).if (xn) converges strongly to x ∈ S, then Za(S, (xn)) = {x}. Results in 2-Banach spaces We begin with the following : Theorem Let S be a nonempty closed convex subset of a uniformly convex 2-Banach space X and (xn) a bounded sequence in S such that Za(S, (xn)) = {z}. If (ym) is a sequence in S such that lim𝑚→∞ 𝑟𝑎 (ym, (xn)) = ra(S, (xn)), then lim𝑚→∞ 𝑦𝑚= z . 345 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Proof Suppose that (ym) does not converge strongly to z. Then there exists a subsequence (ymi) of (ym) such that ||ymi - z,u|| ≥ d > 0 for all i ∈ N, u ϵ S. By the uniform convexity of X, there exists ε > 0 such that (ra(S, (xn)) + ε)[1 – δx(d / ra(S,(xn)) + ε)] < ra(S, (xn)). Since ra(z, (xn)) = ra(S, (xn)), there exists no ∈ N such that ||xn – z ,u|| ≤ ra(S, (xn)) + ε for all n ≥ no. Since ra(ym, (xn)) → ra(S, (xn)) as m → ∞ and hence ra(ymi , (xn)) → ra(S, (xn)) as i→∞, then there exists an integer mo∈ N such that ||xn – ymi ,u|| ≤ ra(S, (xn)) + ε for all n ≥ mo . Since X is uniformly convex, ||xn – ( z + ymi) / 2 ,u|| ≤ [1−δx( d / (ra(S,(xn)))] (ra(S,(xn)) + ɛ) ˂ ra(S,(xn)) for all n ≥ max {no,mo} This implies that ra(( z + ymi / 2 ), (xn)) < ra(S, (xn)), which contradicts the uniqueness of the asymptotic center z. ■ Theorem Let S be a nonempty closed convex subset of a uniformly convex 2- Banach space. Then every bounded sequence (xn) in X has a unique asymptotic center with respect to S, i.e., Za(S, (xn)) = {z} and lim 𝑛→∞ sup‖𝑥𝑛 − 𝑧, 𝑢‖˂ lim𝑛→∞ sup ‖𝑥𝑛 − 𝑥, 𝑢‖ 𝑓𝑜𝑟 𝑥 ≠ 𝑧 , 𝑢 𝜖 𝑆. Theorem Let X be a uniformly convex 2-Banach space satisfying the Opial condition and S a nonempty closed convex subset of X. If (xn) is a sequence in S such that xn → z, then z is the asymptotic center of (xn) in S. Proof From Theorem 2-2, Za(S, (xn)) is singleton. Let Za(S, (xn)) = {x}, x ≠ z Since xn → z, by the Opial condition, lim𝑛→∞ 𝑠𝑢𝑝‖𝑥𝑛 − 𝑧, 𝑢‖˂ lim𝑛→∞ 𝑠𝑢𝑝‖𝑥𝑛 − 𝑥, 𝑢‖, u ϵ S. By theorem 2-2, we obtain limn→∞ sup‖xn − x, u‖ < lim𝑛→∞ 𝑠𝑢𝑝‖𝑥𝑛 − 𝑧, 𝑢‖. Therefore, z = x. ■ Theorem Let X be a uniformly convex 2-Banach space, let S be a nonempty closed convex subset of X and T:S→S an asymptotically non-expansive mapping. If (𝓍𝓍n) a bounded sequence in S such that lim𝑛→∞‖𝑥𝑛 − 𝑇𝑥𝑛, 𝑢‖ = 0 ; 𝑢 𝜖 𝑆 𝑎𝑛𝑑 Za (S,(𝓍𝓍n)) = { v } ,then v is the fixed point in S. Proof Define a sequence (ym) in S by ym = Tmv, m ϵ N. for integers n, m ϵ N, we have || ym - xn , u|| ≤ ||Tmv - Tmxn , u|| + ||Tmxn –Tm-1xn , u|| + … + ||Txn – xn , u|| ≤ km ||v – xn , u|| + (||Txn – xn , u|| + ∑ 𝑘𝑖 || 𝑥𝑛 – 𝑇𝑥𝑛 , 𝑢|| … (2.1)𝑚−1𝑖=1 Then by condition (2.1) we have ra (ym , (xn)) = lim𝑛→∞ 𝑠𝑢𝑝||𝑥𝑛 – 𝑦𝑚, 𝑢|| km ra(v , (xn)) ≤ 346 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 km ra (S , (xn)). = Hence ra (ym , (xn)) – ra (S ,( xn))| ≤ km ra (S , (xn)) - ra (S , (xn)) ≤ (km – 1) ra(S, (xn)) → 0 as m→∞. it follows from 2-1 that Tmv → v By the continuity of T we have Ty = T(lim𝑚→∞ 𝑇𝑚 𝑦) = lim𝑚→∞ 𝑇𝑚+1 𝑦 = 𝑦 . Theorem (Fixed Point Theorem) Let X be a uniformly convex 2-Banach space, S a nonempty closed convex bounded subset of X and T : S → S an asymptotically non-expansive mapping, Then T has a fixed point in S. Proof For fixed y ϵ S and r > 0 , set Ry = {r : there exists k ϵ N with S ∩ (⋂ 𝐵𝑟[𝑇𝑖 𝑦]) ≠∞𝑖=𝑘 Ø} and d = diam (S). Then d ϵ Ry. Hence Ry ≠ Ø. Let ro = inf { r : r ϵ Ry }, fot each 𝜀𝜀 > 0, we define Sɛ = ⋃ (⋂ 𝐵𝑟+ɛ∞𝑖=𝑘 ∞ 𝑘=1 [T i y]). Thus , for each ɛ > 0 , the set Sɛ ∩ S is nonempty and convex . The reflexivity of X implies that ∩ɛ>0 (𝑆𝜀� ∩ S) ≠ Ø Let 𝓍𝓍 ϵ ∩ɛ>0 (𝑆𝜀� ∩ S) and ƞ > 0 ,there exists an integer no such that 𝓍𝓍– Tny ,u|| ≤ ro + ƞ for all n ≥ no ,u ϵ S. || Now let 𝓍𝓍 ϵ ∩ɛ>0 (𝑆𝜀� ∩ S) and suppose that the sequence (T n𝓍𝓍) does not converge strongly to 𝓍𝓍 . Then there exists ɛ > 0 and a subse- quence (Tni𝓍𝓍) of (Tnx). such that ||Tni 𝓍𝓍 – 𝓍𝓍 ,u|| ≥ ɛ , for all i=1,2,… . Suppose kn is the Lipschitz constant of Tn . Then for m > n, we have ||Tn𝓍𝓍 – Tm𝓍𝓍 ,u|| ≤ kn ||𝓍𝓍 – Tm-n𝓍𝓍 ,u|| . Suppose that ro > 0 and choose α > 0 such that ( 1 -θx(( ɛ 𝑟𝑜+𝛼 )) (𝑟𝑜 + 𝛼) ˂ 𝑟𝑜 Select n such that || 𝓍𝓍– Tn 𝓍𝓍,u|| ≥ ɛ and kn = (ro + 𝛼 2 ) ≤ ro + α If no ≥ n, then m > no implies ||𝓍𝓍 – Tm-ny ,u|| ≤ ro + 𝛼 2 . Since ||Tn𝓍𝓍– Tmy ,u|| ≤ kn ||𝓍𝓍 – Tm-ny ,u|| kn (ro + 𝛼 2 ≤ ) ro + α ≤ And ||𝓍𝓍 – Tmy ,u|| ≤ ro + α It follows from the uniform convexity of X that for m > no ||½ ( 𝓍𝓍+ Tn𝓍𝓍) – Tmy, u|| ≤ (1 – θx( 𝜀 𝑟𝑜+𝛼 )) (ro + α) ˂ ro . This contradicts the definition of ro . Hence ro= 0 or T𝓍𝓍 = 𝓍𝓍. But ro = 0 implies that (Tny) is a Cauchy sequence and hence lim𝑛→∞ 𝑇 ny =𝓍𝓍 = T𝓍𝓍 . Therefore, the set ∩ɛ>0(𝑆𝜀� ∩ S) is a singleton that is a fixed point of T. ■ 347 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 Theorem Let X be a uniformly convex 2-Banach space, S a nonempty closed convex subset of X and T : S → S an asymptotically non-expansive mapping, u ∈ S, Then the follo-wing statements are equivalent: 1) T has a fixed point . 2) There exists a point xo ϵ S such that the sequence (Tnxo) is bounded. 3) There exists a bounded sequence (yn) in S such that lim𝑛→∞‖𝑦𝑛 − 𝑇𝑦𝑛, 𝑢‖ = 0 . Proof (1) ⟹ (2) and (1) ⟹ (3) follows easily. (3) ⟹ (1) Let (yn) be a bounded sequence in S such that lim𝑛→∞�|𝑦𝑛 − 𝑇𝑦𝑛, 𝑢|� = 0 Let Za (S,(yn)) = {v} ,therefore ,by theorem 2-4, implies that v is a fixed point of T. ■ Corollary Let S be a nonempty closed convex subset of a strictly convex 2-Banach space X and T : S → X a non-expansive mapping. Then F(T) is closed and convex. We have seen in a Corollary 2-7, that F(T) is closed and convex in strictly convex 2- Banach space for non-expansive mappings. However, we think that Corollary 2-7, is not true for asymptotically non-expansive mappings. In fact, we have: Theorem Let X be a uniformly convex 2-Banach space, S a nonempty closed convex bounded subset of X and T : S → S an asymptotically non-expansive mapping. Then F(T) is closed and convex. Proof The closedness of F(T) is obvious. To show convexity, it is sufficient to prove that z = (𝓍𝓍 + y ) / 2 ϵ F(T) for x, y and u ϵ F(T),for each n ϵ N, we have ||𝓍𝓍 – Tnz ,u|| = ||Tn𝓍𝓍 – Tnz ,u|| ≤ kn ||𝓍𝓍 – z ,u|| = ½ kn || 𝓍𝓍– y,u|| ||y – Tnz ,u|| = ||Tny – Tnz ,u|| ≤ kn ||y – z ,u|| = ½ kn ||𝓍𝓍 – y ,u|| By the uniform convexity of X , we have ||z – Tnz,u|| ≤ ½ [1 - λx(2 / kn )] kn ||𝓍𝓍 – y,u|| ≤ ½ [1 – λx (2 / kn )] kn diam (S) Hence Tnz → z as n → ∞ z is a fixed point of T , by the continuity of T. ■ 348 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 References 1. 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S., (2011), '' Some Sequence Spaces in 2-Normed Spaces Defined by Musielak-Orlisz Function'', Acta Univ. Sapientiae, Mathematical, 3, 1, pp. 97 - 109. 9. Mukti G., Mantu S. and Baisnab A. P., (2012), ''Caristi-Type Fixed Point Theor-ems in 2-Banach Space'', Gen, Math. Notes, 8, No. 1, pp. 1 – 5. 10. George B. and Lawrence N., (2000), Functional Analysis, Canada, General Pub-lishing Company. 11. Dominic Y. and Marudai M., (2012), ''Best Approximation in Uniformly Convex 2- Normed Space'' Int. Journal of Math. Analysis, 6, No. 21, pp. 1015 – 1021. 349 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014 بناخ -2النقطة الصامدة للتطبیقات شبة الالمتمددة في فضاء سلمان عبد سلوى رفاه ساجد عبدعلي جامعة بغداد/كلیة التربیة للعلوم الصرفة أبن الھیثم /قسم الریاضیات 2013كانون االول 4: في، قبل البحث 2013حزیران23 : في أستلم البحث الخالصة الالمتمددة شبھ تطبیقات عرفناایضا،. بناخ-2فضاء في الحقائق بعض قدمنا البحث خالل ھذا asymptotically non-expansive mapping للتطبیقات غیر المتمددة المعیاریة بطریقة مشابھة-2لفضاءاتا في بناخ.-2من التطبیقات في فضاء لھذا النمط نقاط صامدة و من ثم البرھنة عن وجود ،في الفضاءات المعیاریة العادیة ، نقطة صامدة.تطبیق ال متمدد، تطبیق شبة ال متمددبناخ، -2: فضاء مفتاحیة الكلمات ال 350 | Mathematics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@ÖÜ»€a@I1@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (1) 2014