Microsoft Word - 500-509 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|500    Common Fixed Points in Modular Spaces Salwa Salman Abed  salwaalbundi@yahoo.com Dept. of Mathematics/College of Education forPure Science (Ibn Al-Haitham) University of Baghdad Sada Emad AbdulKarrar kararemad1982@gmail.com Dept. of Mathematics/ College of Education for Pure Science (Ibn Al- Haitham) University of Baghdad Abstract In this paper,there are   new considerations about the dual of a modular spaces and weak convergence. Two common fixed point theorems for a 𝑃-non-expansive mapping defined on a star-shaped weakly compact subset are proved, Here the conditions of affineness, demi-closedness and Opial's property play an active role in the proving our results. Keywords: Modular spaces, fixed points, best approximations. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|501    1. Introduction and Preliminaries Dotson 1 proved existence of fixed points for non-expansive self-mappings of star- shaped subsets of Banach spaces(under appropriate conditions). Subrahmanyam 2 and Habinak 3 used the concept of Banach operator to generalize Dotson's theorem and its application to invariant approximation. Recently, Abed [4] introduced the notion of best approximation in modular spaces and gave conditions to existences of proximinal and Chebysev sets in finite dimension modular spaces. Also, Abed and  Abdul Sada [5-7]  proved a theorem of Brosowski-Meinaraus type on invariant approximation, proved that two fixed point theorems for compact set-valued mappings in modular spaces with an application on invariant best approximation. The object of the present paper is to extend and unified the above results [2], [3], [4] and others to modular spaces. For other results in this field see [8]- [10] Definition (1.1)[5]: Let𝑀 be a linear space over𝐹 𝑅 𝑜𝑟 ₵ . A function𝛾:𝑀 → 0, ∞ is called modular if i.𝛾 𝑣 0if and only if𝑣 0; ii.𝛾 𝛼𝑣 = 𝛼 𝑣 for 𝛼 ∈𝐹 with |𝛼| 1, for all 𝛼 ∈ 𝐹; iii.𝛾 𝛼𝑣 𝛽𝑢 𝛾 𝑣 + 𝛾 𝑢 iff 𝛼, 𝛽 0, for all ,∈ 𝑀. If (iii) replaced by (iii´) 𝛾 𝛼𝑣 𝛽𝑢 𝛼𝛾 𝑣 +𝛽𝛾 𝑢 , for 𝛼, 𝛽 0, 𝛼 𝛽 1, for all 𝑣,u ∈𝑀 Then 𝑀 modular 𝛾 is called con𝑣ex modular. Definition 1.2 [6] A modular 𝛾 defines a corresponding modular space,𝑡ℎ𝑒𝑛,the space𝑀 given by 𝑀 𝑣 ∈ 𝑀: 𝛾 𝛼𝑣 → 0 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 𝛼 → 0 . Remark 1.1[6] by condition (iii) above, if u 0 then 𝛾 𝛼𝑣 =𝛾 𝛽𝑣 𝛾 𝛽𝑣 , for all 𝛼, 𝛽 𝑖𝑛 𝐹, 0 𝛼 𝛽.this shows that 𝛾 is increasing function. Definition 1.3[6] The 𝛾-ball, 𝐵 𝑢 centered at 𝑢 ∈ 𝑀 with radius 𝑟 0 as 𝐵 𝑢 𝒗 ∈ 𝑀 ; 𝛾 𝑢 𝑣 𝑟 . The class of all 𝛾-balls in a modular space𝑀 generates a topology which makes 𝑀 Hausdorff topological linear space. Every 𝛾-ball is convex set, therefore every modular space locally convex Hausdorff topological vector space [4]. Definition 1.5[6] Let 𝑀 be a modular spase. a) A sequence 𝑣 ⊂ 𝑀 is said to be 𝛾 -convergent to 𝑣 ∈ 𝑀 and write 𝑣 → 𝑣 if 𝛾 𝑣 𝑣 → 0 as n→ ∞. b) A sequence {𝑣 } is called 𝛾ــ Cauchy whenever 𝛾(𝑣 -𝑣 )→ 0 as , 𝑚 → ∞. c) 𝑀 is called 𝛾ــ complete if any 𝛾ــ Cauchy sequence in 𝑀 is 𝛾ــ convergent. d) A subset 𝐵⊂𝑀 is called 𝛾ــ closed if for any sequence 𝑣 ⊂𝐵𝛾ــ convergent to ∈ 𝑀 , we ha𝑣e 𝑣 ∈ 𝐵. e) A 𝛾ــ closed subset 𝐵⊂𝑀 is called 𝛾ــ compact if any sequence { 𝑣 } ⊂𝐵 has a 𝛾ــ convergent subsequence. f) A subset 𝐵⊂ 𝑀 is said to be 𝛾ــ bounded if 𝑑𝑎𝑖𝑚 𝐵 ∞ , where 𝑑𝑎𝑖𝑚 𝐵 sup 𝛾 𝑣 𝑢 ; 𝑣, 𝑢 ∈ 𝐵 is called the 𝛾ــ diameter of 𝐵. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|502    Definition (1.6) [7] Let 𝑀 be a modular space and 𝐴 ⊆ 𝑀 𝑆:𝐴 → 𝐴, 𝑆 is called contraction mapping if ∃ h ∈ 0 , 1 for all 𝑣 ,𝑢 in 𝑀 . Such that 𝛾 𝑆𝑣 𝑆𝑢 ℎ 𝑣 𝑢 and if h 1 then 𝑆 is called a non –expansive mapping. Definition (1.7): Let 𝑀 be a modular space and 𝑃, 𝑆: 𝑀 → 𝑀 be a mapping then 𝑆 is said to be 𝑃 ــ contraction if there exists h∈ 𝟶, 1 such that 𝛾 𝑆𝑣 𝑆𝑢 ℎ 𝛾 𝑃𝑣 𝑃𝑢 ∀ 𝑣, 𝑢 in 𝑀 . If h 1 in (1.7), then 𝑆 is called 𝑃ــ non– expansive mapping. Definition (1.8) a) A function 𝑆: 𝑀 → 𝑁 (where𝑀 , 𝑁 are modular spaces ) is said to be continuous at a point 𝑣 ∈ 𝑀 if 𝛾 𝑆𝑣 𝑆𝑣 → 0 as n→ ∞ whenever 𝛿 𝑣 𝑣 → 0 as n → ∞. b) A mapping 𝑆: 𝑀 → 𝑁 is said to be affine if ∀𝑣, 𝑢 in 𝑀 and ∀𝜆 , 0 𝜆 1, 𝑆 𝜆𝑣 1 𝜆 𝑢 𝜆𝑆 𝑣 1 𝜆 𝑆 𝑢 . Definition (1.9): A two mappings 𝑆 and 𝑃 on 𝑀 are said to be commute if 𝑆𝑃𝑣 𝑃𝑆𝑣 ∀ 𝑣∈ 𝑀 . The purpose of this article is to prove the completeness of dual space of a modular space and to give some related concepts and properties, also, to prove the existence of common fixed points for pair mapping 𝑆, 𝑃 where 𝑆 is 𝑃 non expansive. 2. Dual of a modular space let 𝑃 be a linear functional with domain in a modular space 𝑀 and range in the scalar field 𝐾 𝑃:𝐷 𝑃 → 𝐾, 𝑃 is bounded linear functional 𝑐 such that for all 𝑣 ∈ 𝐷 𝑃 , 𝛾 𝑃𝑣 𝑐𝛾 𝑣 . The set of all bounded linear functional on 𝑀 , 𝑴𝜸  is linear space with point-wise operations. In the following, we reform some concepts about dual space in the setting of modular spaces, we begin with following: Proposition (2.1): Let 𝑃 ∈ 𝑴𝜸  , define 𝜸 : 𝑴𝜸  → 𝑅 ∋ 𝜸 𝑃 sup 𝛾 𝑃𝑣 ∶ 𝛾 𝑣 1 then i. 𝜸 𝛼𝑃 𝜸 𝑃 , for 𝛼 ∈ K with |𝛼| 1 ii. 𝜸 𝛼𝑃 𝛽𝑄 𝜸 𝑃 𝜸 𝑄 , iii. 𝜸 𝑃 0 iff 𝑃 0. Proof: For 𝑖 𝜸 𝛼𝑃 sup 𝛾 𝛼𝑃𝑣 sup 𝛾 𝑃𝑣 𝜸 𝑃 . For (ii) 𝜸 𝛼𝑃 𝛽𝑄 sup 𝛾 𝛼𝑃𝑣 𝛽𝑄𝑣 sup 𝛾 𝑃𝑣 𝛾 𝑄𝑣 sup 𝛾 𝑃𝑣 sup 𝛾 𝑄𝑣 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|503    𝜸 𝑃 𝜸 𝑄 For (iii) 𝜸 𝑃 0 iff sup 𝛾 𝑃𝑣 ∶ 𝛾 𝑣 1 iff 𝛾 𝑃𝑣 0 for all𝑣 iff 𝑃 0. A modular 𝜸 defines a corresponding modular space,𝑖. 𝑒.,the space𝑴𝜸  given by 𝑴𝜸  𝑣 ∈ 𝑀: 𝜸 𝛼𝑃 → 0 𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟 𝛼 → 0 Theorem (2.2): 𝑴𝜸  is complete modular space. Proof: We consider an arbitrary Cauchy sequence 𝑆 in 𝑴𝜸  and show that 𝑆 converges to a 𝑆∈ 𝑴𝜸  Since 𝑆 is Cauchy, for every ϵ 𝟢 there is an L such that 𝜸 𝑆 𝑆 ∈ , 𝑛, 𝑚 𝐿 , For any 𝑣 ∈ 𝑀 and 𝑛, 𝑚 𝐿 , this implies that |𝑆 𝑣 𝑆 𝑣| | 𝑆 𝑆 𝑣| 𝛾 𝑆 𝑆 𝛾 𝑣 ∈ 𝛾 𝑣 . … (2.1) Now, for any fixed point 𝑣 and given ∈ we may choose ∈ ∈ so that ∈ 𝛾 𝑣 ∈ . Then from (2.1), we have |𝑆 𝑣 𝑆 𝑣| ∈ and 𝑆 𝑣 is Cauchy in 𝐾. By completeness of 𝐾, 𝑆 𝑣 converges, say, 𝑆 𝑣 → 𝑟. Clearly, the limit 𝑟 ∈ 𝐾 depends on the choice of 𝑣 ∈ 𝑀 . This defines a functional 𝑆: 𝑀 → 𝐾 where 𝑟 𝑆𝑣 .The functional 𝑆 is linear since lim → 𝑆 𝛼𝑣 𝛽𝑧 lim → 𝛼𝑆 𝑣 𝛽𝑆 𝑧 𝛼 lim → 𝑆 𝑣 𝛽 lim → 𝑆 𝑧. We prove that 𝑆 is bounded and 𝑆 →𝑆, that is 𝜸 𝑆 𝑆 → 𝟢. Since (2.1) holds for every 𝑚 𝐿 and 𝑆 𝑣 → 𝑆, we may let 𝑚→ ∞. Using the continuity of the modular, then for every 𝑛 𝐿 and all 𝑣 ∈ 𝑀 . |𝑆 𝑣 𝑆𝑣| 𝑆 𝑣 lim → 𝑆 𝑣 lim → |𝑆 𝑣 𝑆 𝑣| 𝜖𝛾 𝑣 … (2.2) This shows that 𝑆 𝑆 with 𝑛 𝐿 is a bounded linear functional. Since 𝑆 is bounded, 𝑆 𝑆 𝑆 𝑆 is bounded, that is, 𝑆∈ 𝑴𝜸  . Furthermore, if in (2.2) we take the supremum over all 𝑣 of modular 𝟣, we obtain 𝛾 𝑆 𝑆 𝜖, 𝑛 𝐿. Hence 𝜸 𝑆 𝑆 → 𝟢. This completes proof. Definition (2.3): A sequence 𝑣 in a modular space 𝑀 is said to be weakly convergent if there is an 𝑣  𝑀 such that for every 𝑃 𝑴𝜸  lim → 𝛾 𝑃𝑣 𝑃𝑣 0 This denoted by 𝑣 → 𝑣. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|504    Proposition (2.4): In a modular space 𝑀 , every convergent sequence is weakly convergent. Proof: By definition, 𝑣 → 𝑣 means 𝛾 𝑣 𝑣 → 0 and implies that for every P∈ 𝑴𝜸,  |𝑃 𝑣 𝑃 𝑣 | |𝑃 𝑣 𝑣 | 𝛾 𝑃 𝛾 𝑣 𝑣 → 0. This shows that 𝑣 → 𝑣. Note that, the converse of proposition (2.4) is not necessary true. To show this recall the usual case is in a normed space.In the following some other needed properties of weak convergence are given: Proposition (2.5): Let 𝑣 be weakly convergent sequence in a modular space 𝑀 , say 𝑣 → 𝑣 Then: i. The weak limit 𝑣 of 𝑣 is unique. ii. Every subsequence of 𝑣 converges weakly to 𝑣. Proof: For (i), suppose that 𝑣 𝒘 → 𝑣 as well as 𝑣 𝒘 → 𝑢. Then P 𝑣 → 𝑃 𝑣 as well as P 𝑣 → 𝑃 𝑢 . Since P 𝑣 is a sequence of numbers, its limit is unique. Hence 𝑃 𝑣 𝑃 𝑢 , that is, for every P∈ 𝑴𝜸.  We have 𝑃 𝑣 𝑃 𝑢 𝑃 𝑣 𝑢 𝟢.This implies 𝑣 𝑢 𝟢 and shows that the weak limit is unique. Part (ii) follows from the fact that P 𝑣 is convergent sequence of numbers. So that every subsequence of P 𝑣 converges and has same limit as the sequence. Definition (2.6): 𝐴 a subset of a modular space 𝑀 is said to be weakly compact if every sequence in 𝑀 has a weak convergent subsequence. Definition (2.7): Let 𝑀 , 𝑁 be two modular spaces and 𝑆 : 𝑀 𝑁 be mappings then: i. 𝑆 is continuous if𝑣  𝑣  𝑆(𝑣 )  𝑆 (𝑣). ii. 𝑆 is weakly continuous if 𝑣 → 𝑣  𝑆 𝑣 → 𝑆 (𝑣). Definition (2.8): Let 𝑀 be a modular space, 𝐴⊆𝑀 and 𝑆: 𝐴→ 𝑀 be a mapping, 𝑆 is called demi-closed of 𝑣 ∈ 𝐴, if for every sequence 𝑣 in 𝐴 such that 𝑣 → 𝑣 and 𝑣 → 𝑢 ∈ 𝑀 then 𝑢 𝑆𝑣 and 𝑆 is demi closed on 𝐴 if it is demi-closed of each 𝑣 in 𝐴. Definition (2.9): Let 𝑀 be a modular space, 𝑀 is said to be Opial if for every sequence 𝑣 in 𝑀 weakly convergent to 𝑣 ∈ 𝑀 the inequality lim → 𝑖𝑛𝑓 𝛾 𝑣 𝑣 lim → 𝑖𝑛𝑓𝛾 𝑣 𝑢 holds for all 𝑢 𝑣. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|505    3. Common fixed point for commuting mappings Mongkolkeha, Sintunavarat and Kumamstudy[11]and [12] proved the existence theorems of fixed points for contraction mappings in modular metric spaces with condition γ 𝑃 𝑣 ∞ to guarantee the existence and uniqueness of the fixed points. We start with following Proposition (3.1): Let 𝑃 be a continuous self-mapping of a complete modular space 𝑀 , 𝛾 if 𝑆: 𝑀 → 𝑀 is 𝑃- contraction mapping which commutes with 𝑃 and 𝑆 𝑀 ⊆ 𝑃 𝑀 and ∃ 𝑣 ∈ 𝑀 such that γ 𝑃 𝑣 ∞ then 𝐹 𝑃 ∩ 𝐹 𝑆 singleton. Proof: Suppose p 𝑎 𝑎 for some 𝑎 ∈ 𝑀 , define 𝑆: 𝑀 → 𝑀 by 𝑆 𝑣 𝑎 ∀ 𝑣 ∈ 𝑀 then 𝑆 𝑃 𝑣 𝑎 and 𝑃 𝑆 𝑣 𝑃 𝑎 for all 𝑣 ∈ 𝑀 so 𝑆 𝑃 𝑣 𝑃 𝑆 𝑣 , ∀ 𝑣 ∈ 𝑀 and 𝑆 commutes with 𝑃 moreover 𝑆 𝑣 𝑎 𝑃 𝑎 ∀ 𝑣 ∈ 𝑀 so that 𝑆 𝑀 ⊆ 𝑃 𝑀 . Finally, ∀𝑎 ∈ 0,1 , ∀ 𝑣,𝑢 in 𝑀 we have 𝛾 𝑆 𝑣 , 𝑆 𝑢 𝛾 𝑎, 𝑎 𝟶 𝑎 𝛾 𝑃 𝑣 , 𝑃 𝑢 . This completes the proof. Now, it is easy to show that the following needed lemma. Lemma (3.2): Let 𝑀 be a modular space, 𝑆: 𝑀 → 𝑀 be mapping, and 𝑢 ∈ 𝑀 . If 𝑆 ℎ𝑢 1 ℎ 𝑣 ℎ𝑆𝑢 1 ℎ 𝑣, ∀𝑣 ∈ 𝑀 and ℎ ∈ 0,1 , then 𝑢 is a fixed point. Theorem (3.3): Let ∅ 𝐴 weakly compact subset of a complete modular space 𝑀 . Let 𝑝 be a continuous and affine mapping on 𝑀 with p 𝐴 𝐴, 𝑆: 𝐴 →𝐴 be an 𝑃- non – expansive mapping commutes with 𝑃. If 𝐴 is star-shaped with respect to 𝑆,and there is some 𝑣 ∈ 𝐴 𝛾 𝑆 𝑣 ∞ and 𝑃 𝑆 is demi-closed on 𝑀 , then 𝐹 𝑆 ∩ 𝐹 𝑃 ∅. Proof: Since 𝐴 is star-shaped with respect to 𝑢∈ 𝐴, then 𝑆: 𝐴 → 𝐴, we define 𝑆 on 𝐴 for any 𝑣 in 𝐴 by, 𝑆 𝑣 ℎ 𝑆𝑣 1 ℎ 𝑢 and there is 𝑢 ∈ 𝐴, and the sequence ℎ → 1 as 𝑛 → ∞, 𝟶 ℎ 1 such that 1 ℎ 𝑢 ℎ 𝑆𝑣 ∈ 𝐴 ∀ 𝑣,𝑢 ∈ 𝐴. It is clear that 𝑆 ∶ 𝐴 → 𝐴. Note that 𝑆 𝐴 ⊆ 𝐴 and 𝑆 𝐴 ⊆ 𝑝 𝐴 . Since 𝑆 commutes with 𝑃 and 𝑃 is affine mapping, for each 𝑣 ∈ 𝐴. 𝑆 𝑃𝑣 ℎ 𝑆𝑝𝑣 1 ℎ 𝑃𝑢 ℎ 𝑃𝑆𝑣 1 ℎ 𝑃𝑢 𝑃 ℎ 𝑆𝑣 1 ℎ 𝑃𝑆 𝑣 ∋ 𝑆 commutes with 𝑃. Further, we observe that for each 𝑛 1, 𝑆 is 𝑃- non-expansive mapping, IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|506    𝛾 𝑆 𝑣 𝑆 𝑢 𝛾 ℎ 𝑆𝑣 1 ℎ 𝑢 ℎ 𝑆𝑢 1 ℎ 𝑢 ℎ 𝛾 𝑆𝑣 𝑆𝑢 ℎ 𝛾 𝑃𝑣 𝑃𝑢 ∀ 𝑣, 𝑢 ∈ 𝐴 hence 𝑆 is 𝑃- contraction. Thus by proposition (3.1), there is a unique 𝑣 ∈ 𝐴 such that 𝑣 𝑆 𝑃𝑣 for all 𝑛 1. Since 𝐴 is weakly compact, there is a subsequence 𝑣 of sequence 𝑣 which converges weakly to some 𝑣𝟶 ∈ 𝐴. Since 𝑃 is a continuous affine mapping then 𝑃 is weakly continuous and so, since S𝑣 and 𝑃𝑣 𝑣 . Now, 𝑃 𝑆 𝑣 𝑃𝑣 𝑆𝑣 𝑣 𝑢 𝑣 1 𝑢 𝑣 Therefore 𝑃 𝑆 𝑣 1 𝑢 𝑣 Thus 𝑃 𝑆 𝑣 1 𝛾 𝑢 𝑣 1 𝛾 𝑣 𝛾 𝑢 . Since 𝐴 is bounded, 𝑣 ∈ 𝐴 implies 𝛾 𝑣 is bounded and so by the fact that ℎ → 1, We have 𝛾 𝑃 𝑆 𝑣 → 𝟶 Now, since 𝑃 𝑆 is demi-closed then 𝑃 𝑆 𝑣𝟶 𝟶 and thus 𝑃𝑣𝟶 𝑣𝟶 𝑆𝑣𝟶. Hence, 𝐹 𝑆 ∩ 𝐹 𝑃 ∅. Another common fixed point theorem will be given for Opial's space. Theorem (3.4): Let∅ 𝐴 weakly compact subset of Opia's complete modular space 𝑀 . Let 𝑃 be a continuous and affine mapping on 𝑀 with 𝑃 𝐴 𝐴, 𝑆: 𝐴 → 𝐴 be 𝑃- non- IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|507    expansive mapping commutes with 𝑃. If 𝐴 has star-shaped with respect to 𝑆, then 𝐹 𝑆 ∩ 𝐹 𝑃 ∅. Proof: Since 𝐴 has star-shaped then 𝑆:𝐴→ 𝐴 and there is 𝑢 ∈ 𝐴 and the sequence ℎ → 1, as 𝑛→ ∞, 𝟶 ℎ 1 ∋ 1 ℎ 𝑢 ℎ 𝑆𝑣 ∈ 𝐴 for all 𝑣∈ 𝐴. Now, define 𝑆 on 𝐴 for any 𝑣 in 𝐴 by, 𝑆 𝑣 ℎ 𝑆𝑣 1 ℎ 𝑢 and there is 𝑢∈ 𝐴, it is clear that 𝑆 : 𝐴 → 𝐴. Note that 𝑆 𝐴 ⊆ 𝐴 and 𝑆 𝐴 ⊆ 𝑝 𝐴 . Since 𝑆 commutes with 𝑝 and 𝑝 is affine mapping, for each 𝑣 ∈ 𝐴. 𝑆 𝑃𝑣 ℎ 𝑆𝑃𝑣 1 ℎ 𝑃𝑢 ℎ 𝑃𝑆𝑣 1 ℎ 𝑃𝑢 𝑃 ℎ 𝑆𝑣 1 ℎ 𝑢 𝑃𝑆 𝑣 Thus each ℎ commutes with 𝑃. Further observe that for each 𝑛 1, 𝑆 is 𝑃 – non-expansive mapping. 𝛾 𝑆 𝑣 𝑆 𝑢 𝛾 ℎ 𝑆𝑣 1 ℎ 𝑢 ℎ 𝑆𝑢 1 ℎ 𝑢 ℎ 𝛾 𝑆𝑣 𝑆𝑢 ℎ 𝛾 𝑃𝑣 𝑃𝑢 ∀ 𝑢∈ 𝐴, hence 𝑆 is 𝑃- contraction. Thus by proposition (3.1), there is a unique 𝑣 ∈ 𝐴 such that 𝑣 𝑆 𝑣 𝑃𝑣 for all 𝑛 1. Since 𝐴 is weakly compact, there is a subsequence 𝑣 of sequence 𝑣 which converges weakly to some 𝑣𝟶 ∈ 𝐴. Since 𝑃 is a continuous affine mapping then 𝑃 is weakly continuous and so we have: 𝑃𝑣𝟶 lim → 𝑃𝑣 lim → 𝑣 𝑣𝟶 Since 𝑆𝑣 and 𝑃𝑣 𝑣 , we have: 𝑃 𝑆 𝑣 𝑃𝑣 𝑆𝑣 𝑣 𝑢 𝑣 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|508    𝑃 𝑆 𝑣 1 𝑢 𝑣 Therefore 𝑃 𝑆 𝑣 1 𝑢 𝑣 . Thus 𝛾 𝑃 𝑆 𝑣 1 𝛾 𝑢 𝑣 1 𝛾 𝑣 𝛾 𝑢 . Since 𝐴 is bounded by 𝐴 is weakly compact, 𝑣 ∈ 𝐴 implies 𝛾 𝑣 is bounded and so by the fact that ℎ → 1, we have 𝛾 𝑃 𝑆 𝑣 → 0 Now, since 𝑀 is Opial space and suppose that, 𝑆𝑣𝟶 𝑣𝟶 we have: lim → 𝑖𝑛𝑓𝛾 𝑣 𝑣𝟶 lim → inf 𝛾 𝑣 𝑆𝑣𝟶 lim → inf 𝛾 𝑆𝑣 𝑃 𝑆 𝑣 𝑆𝑣𝟶 lim → inf 𝛾 𝑆𝑣 𝑆𝑣𝟶 lim → 𝑖𝑛𝑓𝛾 𝑃 𝑆 𝑣 , since 𝑣 𝑃 𝑆 𝑣 𝑆𝑣 . And thus lim → 𝑖𝑛𝑓𝛾 𝑣 𝑣𝟶 lim → inf 𝛾 𝑆𝑣 𝑆𝑣𝟶 But on the other hand, we have lim → inf 𝛾 𝑆𝑣 𝑆𝑣𝟶 lim → inf 𝛾 𝑃𝑣 𝑃𝑣𝟶 lim → inf 𝛾 𝑣 𝑣𝟶 This is a contradiction. Hence 𝑣𝟶 ∈ 𝐹 𝑆 ∩ 𝐹 𝑃 ⇒ 𝐹 𝑆 ∩ 𝐹 𝑃 ∅. Acknowledgements: We would like to acknowledge the generous help of editors and we are grateful to the referees for their constructive input. References [1] JR.Dotson, , W. G., "Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces", J. London Math. Soc., 4(2),. 408-410. 1972 full/3.408.4-http://onlinelibrary.wiley.com/doi/10.1112/jlms/s2 [2] P. V.Subrahmanyam, , "Remarks on some fixed point theorems related to Banach contraction principle", J. Math. Phys. Sci.8 (1974), 445457; Erratum, J. Math.Phys. Sci. 9, 195,1975 [3] L. Habiniak, "Fixed point theorems and invariant approximation", J. Approx. Theory, 56, pp. 241-244,1989. [4] S.S. Abed, , "On invariant best approximation in modular spaces", Global Journal of Pure and Applied Mathematics, 13,. 9,. 5227-5233, 2017. http://www.ripublication.com/gjpam17/gjpamv13n9_102.pdf [5] S.S.Abed, , K.A. Abdul Sada, ," An Extension of Brosowski- Meinaraus Theorem in Modular Spaces", Inter. J. of Math. Anal., Hikari Ltd., 11, 18, 877 – 882, 2017 https://doi.org/10.12988/ijma.2017.77101 [6] Abed, S.S. Abdul Sada, K,.A. "Approximatively Compactness and Best Approximation in Modular Spaces" accepted in Conf. of Scie. Coll., Nahrain University. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1822 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/  www.ihsciconf.org   Mathematics|509    [7] Abed, S.S. Abdul Sada, K.A., " Fixed-Point and Best Approximation Theorems in Modular Spaces", to appear in Inter. J. of Pur. And Appli. Math. [8] Farajzadeh A.P. Mohammadi, M.B., Noor, MA, "Fixed point theorems in modular spaces". Math. Commun. 16, 13-20, 2011 https://pdfs.semanticscholar.org/2997/b07df8ee456156b991895e2652f1ca897880.pdf [9] K. Kuaket, and P. Kumam, ,"Fixed points of asymptotic point wise contractions in modular spaces. Appl. Math. Let. 24,1795-1798(2011). https://doi.org/10.1016/j.aml.2011.04.035 [10] Marwan A. Kutbi, Latif, A., "Fixed points of multivalued maps in modular function spaces". Fixed Point Theory Appl. 2009, doi:10.1155/2009/786357 [11] Mongkolkeha, C., Sintunavarat W., Kumam. P., " Fixed point theorems for contraction mappings in modular metric spaces", Fixed Point Theory and Applications, Springer 2011, 2011:93 . /http://www.fixedpointtheoryandapplications.com [12] C. Mongkolkeha; W. Sintunavarat and P. Kumam., " Fixed point theorems for contractionmappings in modular metric spaces Fixed Point Theory Appl. 2011", Fixed Point Theory and Applications, Springer, 2012. /http://www.fixedpointtheoryandapplications.com/content/2012