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Fixed Point Theorems for Approximating a Common Fixed Point for a Family of Nonself, Strictly Pseudo Contractive and Inward Mappings in Real Hilbert Spaces

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Fixed Point Theorems for Approximating a Common Fixed Point for a Family of Nonself, Strictly Pseudo Contractive and Inward Mappings in Real Hilbert Spaces Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 3657-3677 © Research India Publications http://www.ripublication.com Fixed Point Theorems for Approximating a Common Fixed Point for a Family of Nonself, Strictly Pseudo Contractive and Inward Mappings in Real Hilbert Spaces Mollalgn Haile Takele Department of Mathematics, University College of science Osmania University, Hyderabad, India & Bahir Dar University, Bahir Dar, Ethiopia B. Krishna Reddy Department of Mathematics, University College of Science Osmania University, Hyderabad, India Abstract In this paper, we introduce iterative methods of Mann type for approximating fixed point and common fixed point of a finite family of nonself, k-strictly pseudo contractive and inward mappings in real Hilbert spaces and we prove the weak and strong convergence theorems of the iterative methods. Keywords and phrases: Common fixed point; k-strictly pseudo contractive mapping; nonself mapping; Mann’s iterative method, uniformly convex; uniformly smooth Banach space; weak and strong convergence. 2000 subject Classification: 47H05, 47H09, 47J05; 47H10. 1. INTRODUCTION In this paper, our aim is to study a common fixed for the finite family of strictly pseudo contractive mappings in Banach spaces, in particular, in real Hilbert spaces. The class of pseudo contractive mappings will play a great role in many of the existence for 3658 Mollalgn Haile Takele and B. Krishna Reddy solutions for nonlinear problems and hence the class of strictly pseudo contractive mappings as a subclass has a significant role. Let E be a Banach space with its dual E* and K be non empty, closed and convex subset of E .Then a mappingTKE: is said to be strongly pseudo contractive if there exists a positive constant k and j()() x y J x y such that 2 Tx Ty,()j x y k x y (1.1) and a mappingTKE: is called pseudo contractive if there exists 2 such that Tx Ty,()j x y x y for all x, y K . (1.2) * Where JE: 2E is normalized duality mapping given by 2 2 Jx f E* :, f x x f where, .,. denotes the generalized duality pairing which is analogous to an inner product in Hilbert space. It can be seen that J is single valued if E is smooth. T is called α-strictly pseudo contractive if there exists )1,0( and such that, 2 2 Tx Ty,()jxy xy ()() ITxITy for all x, y K (1.3) T is called Lipschitzian if and only if Tx Ty L x y for x, y K some L 0 (1.4) and if L 1, T is contraction mapping, whereas, if L 1, the mapping is called nonexpansive. T is called contractive if Tx Ty x y for all x y and x, y K (1.5) T is called quasi-nonexpansive if Tx p x p ,x K and p is a fixed point of T (1.6) (1.1) and (1.2) are equivalent to x y 1( t)( x y)() kt Tx Ty for all x, y E and t 0 (1.7) and x y 1( t)( x y)() t Tx Ty for all x, y K and (1.8) respectively. Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3659 We see that every contraction mapping is nonexpansive, every nonexpansive mapping is pseudo contractive and strongly pseudo contractive as well but the converse is not true. In fact, if T is non expansive then 2 Tx Ty,()j x y Tx Ty jx jy x y x yj ()() x x y y J x y (1.9) Thus, T is pseudo contractive mapping. In Hilbert spaces, H (1.1) and (1.2) are reduced to the followings;x , y K 2 2 2 Tx Ty x y k()() I T x I T y for all x, y K , for some k )1,0( (1.10) 2 2 2 Tx Ty x y ()() I T x I T y x, y K (1.11) respectively. The class of pseudo contractive mappings is more general than the class of nonexpansive mappings and is strong linked with that of accretive mappings; That is, a mapping AKE: is said to be accretive if for all and t 0 x y (() x y t Ax Ay holds. (1.12) (1.12) is equivalent to there exists such that Ax Ay,j ( x y ) 0 for all . (1.13) We observe that A is accretive if and only if TIA is pseudo contractive mapping and the zero of A is the fixed point of T and vice versa. As a result, finding fixed point or common fixed point of pseudo contractive mapping (mappings) hence its subclass k-strictly pseudo contractive mappings is essential. We denote the set of fixed points of the mapping TKE: by F():() T x K T x xwhere K is a non empty closed and convex subset of a real Banach space E . We shall have the following definitions: Definition 1.1 A Banach space E is said to be Uniformly convex if for every 0 2 , there is a 0such that for all x, y S x E: x 1 , if x y () x y , then x y 1 . 2 3660 Mollalgn Haile Takele and B. Krishna Reddy If dimE 2, the modulus of convexity of E is a function defined by x y E (t ) inf 1 ,x y 1& x y t, x , y E ,0 t 2 , (1.14) 2 and E is said to be uniformly convex if E (t ) 0 for all 0t 2 . Definition 1.2 A uniformly smooth space, E is a normed space in which for every 0 there is 0 such that for x, y E , x 1, y , x y x y 2 y . If , then the modulus Smoothness of E is defined by x y x y E t)( Sup 1,x 1, y t , (1.15) 2 t)( and E is uniformly smooth if and only if lim E 0. t0 t Let p, q 1 be real numbers. Then E is said to be p-uniformly convex (respectively q- p uniformly smooth) if there is a constant c 0 such that E t)( ct (respectively q E t)( ct ). Hilbert spaces, the Lebesgue Lp and the sequences l p are examples of uniformly convex and uniformly smooth Banach spaces for p(1, ) (see, for example, in [14]). Definition 1.3. Let E be a Banach space, let S be defined by S x E: x 1 . Then E is said to have Gateaux differentiable norm if x ty x lim (1.16) t0 t exists for all x, y S . E is said to be smooth if its norm is Gateaux differentiable. If for each y S the limit in (1.16) exists uniformly for each x S , then the norm of E is said to be uniformly Gateaux differentiable. The norm of E is Frechet differentiable if for each the limit in (1.16) exists uniformly for each y S . The norm of E is uniformly Frechet differentiable if the limit (1.16) exists uniformly for each x, y S . Thus, it is clear that every Frechet differentiable norm is Gateaux differentiable, however, the converse is not in general true. This can be seen (See, for example, in [2]). Mann’s iterative process was first introduced by Mann in [10], since then ,considerably research works have been made to approximate fixed point of nonexpansive and k- Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3661 strictly pseudo contractive mappings , via Mann’s iterative method [10] (see, for example [15]). Mann’s iterative method generates a sequence xn according to the recursive formula, xn1 n x n (1 n )Tx n , n (0,1),n 0 , x0 K (1.17) If T is a nonexpansive self mapping on a nonempty, closed convex subset K of a real uniformly convex Banach space E with a Frechet differentiable norm with fixed point and if the sequence n )1,0( satisfying n(1 n ) , then converges to a fixed point of T weakly (See, for example [15]). Since the convergence is weak, it is expected the result to be slow, however, Nakajo and Takahashi in [13] proposed a modified Mann type algorithm for nonexpansive and self mapping on a non empty , closed and convex subset of a real Hilbert space H which is known as CQ algorithm and can be restated as; x0 K, yn n x n (1 n )Tx n , 2 2 2 Kn z K: yn z xn z 1( n )(k n) x n Tx n , (1.18) : , , 0 , Qn z K xn z x0 z x P x . n1 KQn n 0 Here PK the metric projection of the Hilbert space H on to a nonempty, closed convex subset K of H. Nakajo and Takahashi in [13] proved that the sequence generated by the algorithm (1.17) converges strongly to a fixed point of T provided that satisfies lim sup n 1. n Marino and Xu in [20] extended the CQ algorithm (1.17) for k-strictly pseudo contractive mapping in Hilbert spaces. They proved the following theorem; Theorem 1.1 MXu [20] Let K be a closed and convex subset of a Hilbert space H. Let TKK: be a κ-strict pseudo contractive mapping for some 0k 1and assume that F(T) is nonempty . Let be the sequence generated by Mann’s algorithm (1.17). 3662 Mollalgn Haile Takele and B. Krishna Reddy Assume that the sequence n is chosen such that k n 1 for all n and xn (n k )(1 n ) .
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