81 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (2) 2019 Abstract Some cases of common fixed point theory for classes of generalized nonexpansive maps are studied. Also, we show that the Picard-Mann scheme can be employed to approximate the unique solution of a mixed-type Volterra-Fredholm functional nonlinear integral equation. Keywords: Banach space, common fixed point, strong convergence, condition ( ). 1. Introduction Let B be a non-empty subset of a Banach space M. A map T on B is called quasi- nonexpansive [1]. if ( ) ‖ ‖ ‖ ‖ ( ) where ( ) denoted the set of all fixed points of T. In 2008, Suzuki [2]. introduced a condition on T which is stronger than quasi-nonexpansive and weaker than nonexpansive, called condition ( ) and presented some results about fa fixed pointfor such maps. In 2009, Dhompongsa et al [3]. extended Suzaicr’s theorems to the general class of maps in Banach spaces. Garcial-Falset et al [4]. defined two generalization of condition ( ), called condition ( ) and condition ( ) And studied their asymptotic behavior as well as the existence of fixed points. On the other hand, Bruck [5]. introduced a map called firmly nonexpansive map in Banach space. Of course, every firmly nonexpansive is nonexpansive. To discuss about convergence theorem for two nonexpansive maps S and T on B to itself, Khan and Kim [6]. constricted the following iterative scheme to find a common fixed point of S and T: ( ) ( ) Where ( ) ( ) ( ) This scheme is independent of both Ishikawa scheme and Yao-Chen scheme [6]. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Convergence Comparison of two Schemes for Common Fixed Points with an Application Department of Mathematics, College of Education for Pure Sciences Ibn Al-Haitham, University of Baghdad, Baghdad,Iraq. Salwa Salman Abed Zahra Mahmood Mohamed Hasan salwaalbundi@yahoo.com zahramoh1990@gmail.com Article history: Received 13 Novrmber 2018, Accepted 11 December 2018, Publish May 2019 Doi: 10.30526/32.2.2146 mailto:salwaalbundi@yahoo.com mailto:salwaalbundi@yahoo.com mailto:/ mailto:/ 82 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 In this paper, we prove some convergence theorems for approximating common fixed points of firmly nonexpansive and maps satisfied condition ( ). 2. Preliminaries We will assume throughout this paper that ( ) is a uniformly convex Banach space and B is a non-empty closed convex subset of M. For maps the set of all fixed points of S and T will be denoted by ( ). A sequences ( ) in B is called: Picard-Mann hybrid [7]. ( ) ( ) Where ( ) ( ) Noor iterative scheme [8]. if ( ) ( ) ( ) ( ) Where ( ) ( ) ( ) , - Definition (1) [9]. A map said to be Lipschitz continuous or liLipschitzf such that ‖ ‖ ‖ ‖ If , then T is nonexpansive. Definition (2) [10]. A map is said to satisfying: 1-Condition ( ) if ‖ ‖ ‖ ‖ → ‖ ‖ ‖ ‖ . 2-Condition ( ) if ‖ ‖ ‖ ‖ → ‖ ‖ ‖ ‖ ( ) Defintion (3)[5]. A map is said to be firmly nonexpansive map if ‖ ‖ ‖( )( ) ( )‖ Definition (4)[11]. Two maps are called: 1-Condition (A) if there is a nondecreasing function , ) , ) ( ) ( ) ( ) such that : Either ‖ ‖ ( ( )) ‖ ‖ ( ( )) ( ) *‖ ‖ + ( ) ( ) 2-Condition ( ) if ‖ ‖ ‖ ‖ Definition (5)[12]. A map is called 1-Demiclosed at 0 if sequence ( ) such that ( ) converges weakly to ( ) and ( ) converges strongly to 0, then 2-Affine if B is convex and ( ( ) ) ( ) ( ) , - 83 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Definition (6)[7]. Let ( ) ( ) be two sequences of real numbers that converging to ‖ ‖ ‖ ‖ Then ( ) converges faster than ( ) Lemma (7)[13]. Let ( ) ∞ n=0 ( ) ∞ n=0 be nonnegative real sequences satisfying the inequality: ( ) Where ( ) ∑ Lemma (8)[10]. Let M be a uniformly convex Banach space and Suppose that ( ) ( ) are two sequences of M such that ‖ ‖ ‖ ‖ ‖ ( ) ‖ hold for some Then ‖ ‖ 3. Two Lemmas Lemma (9): Let B be a non-empty closed convex subset of a normed space M, be a firmly noninexpensivep and satisfying Lipschitz be satisfying condition ( ).Let 1-( ) be as in (1) where ( ) ( ) 2-( ) be as in (2) where ( ) ( ) ( ) , - If ( ) then ‖ ‖ ‖ ‖ ( ) Proof: Let ( ). By using condition( ), we have ‖ ‖ ‖ ‖ → ‖ ‖ ‖ ‖ Then 1-‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ,( ) -‖ ‖ ( )‖ ‖ ‖ ‖ ‖ ‖ Then ‖ ‖ exists ( ). 2-‖ ‖ ‖( ) ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ,( ) -‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ,( ) -‖ ‖ ‖ ‖ Now ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ 84 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 ‖ ‖ Then ‖ ‖ exists ( ). Lemma (10): Let M be a uniformly convex Banach space and B be a nonempty closed convex subset of M. Let: 1- be firmly nonexpansive map and satisfying Lipschitz, be affine and satisfying condition( ) ( ) be as in (1) . 2-T:B→B be firmly nonexpansive map and satisfying Lipschitz, S:B→B be satisfying condition( ) ( ) be as in (2). Suppose that condition ( ) holds. If ( ) , then ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ Proof: Let * ( ). 1- As proved by lemma (9), ‖ ‖ exists. Suppose that ‖ ‖ If there is nothing to prove. Now, suppose , Since, ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ‖ ‖ Then ‖ ‖ Next consider ‖ ‖ ( )‖ ‖ ‖ ‖ By applying lemma (9),we obtain n n Now ‖ ‖ ‖ ‖ ‖ ‖ ‖ ,( ) ‖ ( )‖ ‖ ‖ ‖ By applying Lemma (8) , we have ‖ ‖ Next, by using condition (I), we obtain ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ Thus ‖ ‖ 2- As proved by lemma (9), ‖ ‖ exists. Suppose that ‖ ‖ If there is nothing to prove. Now, suppose , Since ‖ ‖ ‖ ‖ and as proved by lemma (3.1) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖. Then, ‖ ‖ ‖ ‖ ‖ ‖ Moreover ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ 85 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 By applying lemma (9), we get ‖ ‖ Now ‖ ‖ ( )‖ ‖ ‖ ‖ Then, ‖ ‖ Since, ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ which implies to ‖ ‖ That gives ‖ ‖ , so ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ By lemma (9), we obtain: ‖ ‖ Next ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ Letting n→∞, we have: ‖ ‖ ‖ ‖ That means ‖ ‖ 4. Convergence and Equivalence Results Theorem (11): Let M be a uniformly convex Banach space. Let ( n) ( n) be as in lemma (10) and T, S satisfying condition (A). If ( ) , then ( n) ( n) converge strongly to a common fixed point of T and S. Proof: Now, we will show that ( ) is strong convergence. By lemma (10), ‖ ‖ exists. Suppose that ‖ ‖ From lemma (9), we have ‖ ‖ ‖ ‖ That gives ‖ ‖ ‖ ‖ Which means, ( ) ( ) → ( ) By using condition (A), we have ( ( ) ‖ ‖ Or ( ( ) ‖ ‖ In both situations, we obtain ( ( ) Since g is a non-decreasing function and ( ) It follows that ( ) Now to show that ( ) Is a Cauchy sequence in B. Let ( ) a positive integer , such that: ( ) 86 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 In particular. *‖ ‖ + Thus,it must exist ( ) such that ‖ ‖ . Now, , we obtain: ‖ ‖ ‖ ‖ ‖ ‖ Hence, ( ) Is Cauchy sequence in the B of M. Then ( ) converges to a point ( ) → ( ) Since F is closed, hence ( ) By utilizing the same procedure, we can prove (zn) convergence strongly. Theorem (12): Let be a firmly nonexpansive and satisfying lipschitz, satisfying condition ( ), with ( ) and, 1-( ) be as in (1) and ( ) ( ) satisfying ∑ . 2-( ) be as in (2) and ( ) ( ) ( ) , - satisfying ∑ Then ( ) & ( ) converge to a unique common fixed point ( ) Proof: 1-‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ,( ) -‖ ‖ Suppose ( ) ( ( ) )‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( ( ) )‖ ‖ By induction ‖ ‖ ∏ ( ( ) )‖ ‖ ‖ ‖ ( ) ∑ Since ∑ ( ) ∑ Thus ‖ ‖ 2-‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ,( ) -‖ ‖ Setting ( ) ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ ( )‖ ‖ ( )‖ ‖ Now ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( )‖ ‖ 87 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 ( )‖ ‖ , -‖ ‖ , -‖ ‖ By induction ‖ ‖ ∏ , -‖ ‖ ‖ ‖ ∑ Since ∑ ∑ Thus, ‖ ‖ Theorem (13): Let be a firmly nonexpansive mapping and satisfying lipachitz, satisfying condition( ) and be a common fixed point of S and T. Let ( ) and ( ) be the Picard-Mann and Noor iterations defined in (1) and (2). Suppose ( ) ( ) ( ) satisfied the following conditions: 1-( ) ( ) ( ) 2-∑ 3-∑ If ( ) ( ) are bounded, then the Picard-Mann iteration sequence ( ) converges strongly to ( ) and the Noor iteration sequence ( ) converges strongly to ( ). Proof: Since the range of T and S is bounded, let: *‖ ‖+ ‖ ‖ and *‖ ‖+ ‖ ‖ Then ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ Therefore ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( ) ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ( ‖ ‖) ‖ ‖ ( )‖ ‖ ‖ ‖ Since T is Lipschitzain and firmly nonexpansive, setting ( )‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ Then ‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ 88 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 ( )‖ ‖ ( ‖ ‖) ( ‖ ‖) ( )( ‖ ‖) ( )‖ ‖ ( )( ‖ ‖) ( ‖ ‖) ( )( ‖ ‖) ( )‖ ‖ ( )( ‖ ‖) Let ‖ ‖ ( )( ‖ ‖) and By applying lemma (7), we get: ‖ ‖ If ( ) then ‖ ‖ ‖ ‖ ‖ ‖ If ( ) then ‖ ‖ ‖ ‖ ‖ ‖ Theorem (14): Let be a firmly nonexpansive mapping and satisfying Lipschitz with and satisfying condition ( ). Suppose that the Picard-Mann and Noor iteration converge to the same common fixed point a * . Then picard-Mann iteration converges faster than Noor iteration. Proof: Let ( ) Then, for Picard-Mann iteration. ‖ ‖ ( )‖ ‖ ‖ ‖ Setting ( ) , then we have ( ( ) )‖ ‖ Next ‖ ‖ ‖ ‖ ‖ ‖ ( ( ) )‖ ‖ ( ( ) ) ‖ ‖ Let ( ( ) ) ‖ ‖ Now, Noor iteration. ‖ ‖ ( )‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ( )‖ ‖ ‖ ‖ ( ( ) )‖ ‖ Then ‖ ‖ ( )‖ ‖ ‖ ‖ ( ( ( ) )‖ ‖ Assume that ( ( ( ) ) ‖ ‖ ‖ ‖ 89 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 Let ‖ ‖ Now, ( ( ) ) ‖ ‖ ‖ ‖ ( ( )) ‖ ‖ ‖ ‖ Then, ( ) converges faster than ( ) . Example (15): Let , ) be an mappings defined by and Choose n n n with initial value 1 The Picard-Mann iteration converges faster than Noor iteration, as shown in Table 1. and Figure 1. Table 1. Numerical results corresponding to for 30 steps. n Picard-Mann Noor n Picard-Mann Noor 0 20 20 16 1.0000 1.0353 1 4.8000 13.8250 17 1.0000 1.0238 2 1.7600 9.6569 18 1.0000 1.0161 3 1.1520 6.8434 19 1.0000 1.0109 4 1.0304 4.9443 20 1.0000 1.0073 5 1.0061 3.6624 21 1.0000 1.0049 6 1.0012 2.7971 22 1.0000 1.0033 7 1.0002 2.2131 23 1.0000 1.0023 8 1.0000 1.8188 24 1.0000 1.0015 9 1.0000 1.5527 25 1.0000 1.0010 10 1.0000 1.3731 26 1.0000 1.0007 11 1.0000 1.2518 27 1.0000 1.0005 12 1.0000 1.1700 28 1.0000 1.0003 13 1.0000 1.1147 29 1.0000 1.0002 14 1.0000 1.0774 30 1.0000 1.0001 15 1.0000 1.523 Figure 1: Convergence behavior corresponding to for 30 steps. 90 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 5. Application The following mixed type of Volterra-Fredholm functional nonlinear integral equation that is appeared in [14]. We use theorem (14) to solve it: ( ) ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) ) ( ) Where: , - , - be an interval in , - , - , - , - continuous functions and G: , - , - . Assume that the following conditions are accomplished: i- (, - , - , - , - ) ii- (, - , - ) iii- such that | ( ) ( )| | | | | | | , - , - iv- such that | ( ) ( )| | | | ( ) ( )| | | , - , - v- ( )( ) ( ) (, - , -) Theorem (16)[14]. Suppose that conditions (i-v) are satisfied. Then, the equation (3) has a unique solution (, - , -) Theorem (17): We deem Banach space (, - , - ‖ ‖) ∑ Let ( ) be as shown in step (1) and a map is defined by ( ) ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) )) Suppose that the conditions (i-v) are accomplished. Then, the equation (3) has a unique solution (, - , -) and the Picard-Mann iteration converges to Proof: To prove Let ‖ ‖ ‖ ‖ | ( ) ( )| ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) ) | ( ) ( )| |∫ ∫ ( ( )) ∫ ∫ ( ( )) | |∫ ∫ ( ( )) ∫ ∫ ( ( )) ) | , ( )( ) ( )-‖ ‖ Since, ‖ ‖ ( )‖ ‖ ‖ ( ) ( )‖ 91 Lbn Al-Haitham Jour.for Pure & Appl .Sci. 32 (2) 2019 ( )‖ ‖ | ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) ( ( ) ∫ ∫ ( ( )) ∫ ∫ ( ( )) ) | ( )‖ ‖ [ ( )( ) ( )] ‖ ‖ * ( [ ( )( ) ( )]+‖ ‖ ‖ ‖ ∏* ( [ ( )( ) ( )]+ By condition (v), [ ( )( ) ( )] Now, under using theorem (12), we obtain that equation (3) has a unique solution (, - , -) and Picard-Mann iteration converges to . In the same scope you can see the results in [15]. and [16]. where Hasan and Abed established weak convergence theorems by using appropriate conditions for approximating common fixed points and equivalence between the convergence of the Picard-Mann iteration scheme and Liu el.at iteration scheme in Banach spaces. 6.Conclusion In the setting of 2-normed spaces [16]. we define firmly nonexpansive and generalized nonexpansive maps. Then, we study the convergence of Picard-Mann iteration and Noor iteration. References 1. Diaz, J.B.; Metcalf, F.T. On the structure of the set of subsequential limit points of successive approximation. Bull. Am. Math. Sco. 1967, 73, 516-519. 2. Suzuki, T. Fixed point theorems and convergence theorems for some generalized- nonexpansive mappings. J. Math. Anal. Appl. 2008, 340, 1088–1095. 3. Dhompongsa, S.; Inthakon, W.; Kaewkhao, A.; Edelstein, S. Method and Fixed point theorems for some generalized nonexpansive mappings. J.Math. Anal. Appl.2009, 350,12-17. 4. 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