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 mapping : EKT is said to be strongly pseudo contractive if there exists a positive constant k and yxJyxj )()( such that 2 TyTx )(, yxkyxj (1.1)
and a mapping : EKT is called pseudo contractive if there exists 2 such that TyTx )(, yxyxj for all , Kyx . (1.2)
* Where EJ 2: E is normalized duality mapping given by 2 2 Jx * ,: fxxfEf 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 TyTx )(, )()( yTIxTIyxyxj for all , Kyx (1.3)
T is called Lipschitzian if and only if TyTx yxL for , Kyx 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 TyTx yx for all yx and , Kyx
(1.5)
T is called quasi-nonexpansive if Tx pxp , Kx and p is a fixed point of T (1.6) (1.1) and (1.2) are equivalent to
1( )( yxtyx kt TyTx )() for all , Eyx and t 0 (1.7)
and
1( )( tyxtyx TyTx )() for all , Kyx 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 TyTx yxj )(, TyTx jyjx yxyxyx (1.9)
Thus, T is pseudo contractive mapping.
In Hilbert spaces, H (1.1) and (1.2) are reduced to the followings;
2 2 2 TyTx )()( yTIxTIkyx for all , Kyx , for some k )1,0( (1.10)
2 2 2 TyTx )()( yTIxTIyx , Kyx (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 : EKA is said to be accretive if for all and t 0
tyxyx AyAx )(( holds. (1.12)
(1.12) is equivalent to there exists yxJyxj )()( such that
AyAx yxj 0)(, for all , Kyx . (1.13)
We observe that A is accretive if and only if AIT 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 : EKT by )(:)( xxTKxTF where 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 20 , there is a 0such that for all , xExSyx 1: , if yxyx )( , then yx 1 . 2 3660 Mollalgn Haile Takele and B. Krishna Reddy
If E 2dim , the modulus of convexity of E is a function defined by yx E t 1inf)( &1, tEyxtyxyx 20,,, , (1.14) 2 and E is said to be uniformly convex if E t 0)( for all t 20 . Definition 1.2 A uniformly smooth space, E is a normed space in which for every 0 there is 0 such that for ,1,, yxEyx , 2 yyxyx .
If , then the modulus Smoothness of E is defined by yxyx E t)( Sup ,1,1 tyx , (1.15) 2
t)( and E is uniformly smooth if and only if lim E 0. t0 t Let qp 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 xExS 1: . Then E is said to have Gateaux differentiable norm if
x ty x lim (1.16) t0 t exists for all , Syx . E is said to be smooth if its norm is Gateaux differentiable.
If for each Sy the limit in (1.16) exists uniformly for each Sx , 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 Sy . The norm of E is uniformly Frechet differentiable if the limit (1.16) exists uniformly for each , Syx . 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, 1 xx nnn Tx nnn 1,0(,)1( ),n 0 , 0 Kx (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 nn )1( , 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;
0 Kx , xy nnn Txnn ,)1( 2 2 2 n : n n zxzyKzK 1( n )( ) xk Txnnn , (1.18) ,0,,: n n 0 zxzxKzQ xPx . n1 QK nn 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 n 1sup . 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 : KKT be a κ-strict pseudo contractive mapping for some k 10 and 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
n k n )1()( . Then the sequence xn converges weakly to a fixed point of
T. We notice that, if T is nonexpansive, then κ = 0 and Theorem 1.1 reduces to Reich’s theorem in Hilbert space setting (see, for example [15]). Since the convergence is weak, in order to get strong convergence Marino and Xu modified Mann’s algorithm to CQ algorithm for strict pseudo contractive mappings and they proved strong convergence in the following theorem;
Theorem 1.2 MXu[12] Let K be a closed and convex subset of a Hilbert space H. Let : KKT be a κ-strict pseudo contractive mapping for some k 10 and assume that F(T) is nonempty. Let be the sequence generated by the following CQ algorithm;
0 Kx , xy nnn Txnn ,)1( 2 2 2 n : n n zxzyKzK 1( n )( ) xk Txnnn , (1.19) ,0,,: n n 0 zxzxKzQ xPx . 1 QCn nn 0
Assume that the sequence is chosen such that n 1 for all . Then the sequence
converges strongly to TF xP 0)( .
However, the above results are for self mappings. On the other hand, in many cases nonself or family of nonself, k –strictly pseudo contractive mappings arises when the domain is a proper subset of the given space, thus finding a fixed point or a common fixed point (if it exists) is essential and is our purpose in this paper to modify Mann’s algorithm to approximate a fixed point and a common fixed point of the class of nonself and k-strictly pseudo contractive mappings in a real Hilbert space which is more general class than the class of nonexpansive mappings. Attempts retraction have been made to approximate fixed point of nonself mappings by using projection for sunny nonexpansive of the real Banach spaces E on to its closed and convex subset K of E.
Definition 1.4 Let E be a real Banach space. Then Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3663 a) a subset K of E is said to be a retract of E if there exists a continuous map : KEP such that )( xxP for all Kx . b) a map is said to be a retraction if 2 PP . It follows that if a map P is a retraction, then )( yyP for all y in the range of P. c) a map is said to be sunny if P Px xt )(( PxPx for all Ex and t 0 . d) a subset K of E is said to be a sunny nonexpansive retract of E if there exists a sunny nonexpansive retraction of E onto K and it is said to be a nonexpansive retract of E if there exists a nonexpansive retraction of E onto K.
For example, in Hilbert space H, the metric projection PK is a sunny nonexpansive retraction from H onto any closed convex subset K of H. As a result, a number of research efforts have been made to find iterative methods for approximating a fixed point or a common fixed point (when it exists) for nonexpansive, pseudo contractive mappings and k-strictly pseudo contractive mappings as well via projection for sunny nonexpansive retraction. In particular, attempts to modify the Mann iteration method [14] for nonexpansive mappings and strict pseudo contractions so that strong convergence is guaranteed have recently been made. (See, for example, in [12, 14, 18-19] and their references). To mention a few; Recently, Zhou in [20] modified the Mann’s iterative process [14] for nonself and k- strictly pseudo contractive mapping to have strong convergence in Hilbert spaces setting. He proved the following theorem. Theorem 1.3 Zhou [20] Let K be a closed and convex subset of a real Hilbert space H. Let : HKT be a k-strictly pseudo contractive and nonself mapping such that TF )( is non empty. For Ku and sequences n and n in 1,0 . Let the sequence xn be defined by
1 Kxx , xPy nnKn )1(( Txnn ), 1 nn ux Tynn n ,1,)1(
Where PK is the projection of H onto the closed and convex subset K, then if and satisfy the following conditions; 3664 Mollalgn Haile Takele and B. Krishna Reddy
(i) lim n 0 and k n 1 for all n 1; (ii) n n n1
n (iii) 1 nn , 1 nn or lim 1, then the sequence xn n n1 n1 n1
Converges strongly to a fixed point x of T, and TF )( uPx . Motivated by the result of Zhou in [20], Hao in [7] extended to uniformly convex Banach spaces. He proved strong convergence theorems for approximating a common fixed point of a finite family of k-strict pseudo-contractive mappings in Banach spaces via metric projection, however, the computation of metric projection is costly, even in Hilbert spaces it requires another approximation method. Thus, it is our purpose in this paper to introduce an iterative method for approximating fixed point or a common fixed point of nonself, k-strictly pseudo contractive and inward mapping or family of mappings without the computation of metric projection. On the other hand, in 2015, Colao and Marino in [3] introduced Krasnoselskii-Mann’s iterative method for approximating a fixed point of nonself, nonexpansive mapping with additional assumptions; the mapping to be inward and strictly convexity of the domain in lowering the requirement of metric projection.
Definition 1.5 A mapping : HKT is said to be inward (or to satisfy the inward condition) if for any Kx ,Tx IK &1:)()( Kucxucxx and T is said to ______satisfy weakly inward condition ifTx IK x)( ( closurethe of IK x)( .
Definition 1.6 A subset K of E is said to be strictly convex if for any tyxKyx 10,,, ,tx Kyt )int()1( , that is ,no line segment joining any two points of K totally lies on the boundary of K. They also proved weak and strong convergence for the following theorem.
Theorem 1.4 CM [3] ) Let K be a convex, closed and nonempty subset of a Hilbert space H and : HKT be a nonself mapping and let for any given Kx , xh )1(:0inf)( Kxx .Then the algorithm defined by Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3665
0 Kx 1 0 max xh 0 ,)(, 2 xx )1( Tx n1 nn nn n1 ,max xh nn 1 is well defined and assume that; K is strictly convex set and T is nonexpansive, nonself and inward mapping with F(T) is non empty, then the sequencexn converges weakly to )( and moreover if , then the convergence is strong . TFFp n )1( n1 More recently, Takele and Reddy in [5&6] extended the theorem of CM to approximate a common fixed point of family of nonself, nonexpansive and inward mappings in real Hilbert spaces and real uniformly convex Banach spaces respectively. They also proved weak and strong convergence theorems. To be precise we put their main theorem in [5] as follow.
Theorem 1.5 TR [5] Let , 21 ,... N : HKTTT be family of, nonself, nonexpansive and inward mappings on a non empty, closed and strictly convex subset K of a real Hilbert N space H with is non empty. , and if for each we define TFF k )( TT ModNkk )( 0 Kx k1
k : Kh by k xh )1(:0inf)( k KxTx , )1,0( be fixed .Then the sequence xn given by
1 Kx 1 xh 11 )(,max )1( n1 xx nn xT nnn n1 ,max xh nnn 11
is well-defined and if n )1,0(]1,[ for some )1,0( , then converges N weakly to some element p of . Moreover, if and (F,K) TFF k )( k1 satisfies S-condition, then the convergence is strong. Our concern is that is it possible to approximate fixed point of the family of nonself, k- strictly pseudo contractive mappings which is more general than the family of nonexapansive mappings? Our objective in this paper is to approximate a fixed point and a common fixed point (if it exists) for the family of nonself, k-strictly pseudo 3666 Mollalgn Haile Takele and B. Krishna Reddy contractive and inward mappings which is more general class than the class of nonexpansive mappings in real Hilbert spaces which is a positive answer to our concern.
2. PRELIMINARY CONCEPTS Let K be a non-empty, closed and convex subset of a real Banach space E. Let : EKT be nonself mapping; in the sequel, we shall need the following assumptions; Lemma 2.1. (See, for example, Proposition 2.1 in [9]) Suppose K is a closed and convex subset of a Hilbert space H. Let : KKT be a mapping on K. Then (a) if T is a κ-strict-pseudo contractive, then T satisfies the Lipschitz condition
k 1 TyTx yx , for all , Kyx ; 1 k (b) if T is a κ-strict pseudo contractive, then the mapping I − T is demi closed (at 0).
That is, if xn is a sequence in K such that n xx weakly and x Txnn 0 strongly, thenTx x; (c) if T is a κ-quasi-strict pseudo contraction, then the fixed point set TF )( of T is closed.
Remark: The proposition can be easily extended to for nonself mapping : HKT with the same conclusion.
Definition 2.1. A sequence xn in K is said to be Fejer monotone with respect to a subset F of K if , n1 n , nxxxxFx .
Lemma 2.2. (See, for example, (MARINO and Xu [12], lemma 1.1) Let H be a Hilbert 2 2 2 2 space. Then, for all 1,0 , )1( xyx y )1()1( yx , for all , Hyx .
Definition 2.2 Let F, K be two closed and convex nonempty sets in a Hilbert space H and KF . For any sequencen Kx , if xn converges strongly to an element
,\ n xxFKx , implies that is not Fejer-monotone with respect to the set , we called that, the pair (F, K) satisfies S-condition Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3667
Lemma 2.3 (see in [5,3])Suppose , 21 ,... N : EKTTT be nonself mappings. If for each
2,1 ,..., Nk we define k : Kh by k xh )1(:0inf)( k KxTx , then
a) k xhKx 1,0)(, and k xh 0)( if and only if k )( KxT ;
b) kk (, xhKx ),1 , then k kk )()1( KxTx ;
c) If Tk is inward mapping, then , k xhKx 1)(, ;
d) If k KxT , then k (1()( )) kk KxTxhxxh .
3. MAIN RESULT
Suppose , 21 ,... N : HKTTT be family of nonself and k-strictly pseudo contractive mappings on a non empty, closed and convex subset, K of a real Hilbert space H, our objective is to introduce an iterative method for common fixed point of the family and determine conditions for convergence of the iterative method. In lowering the requirement of metric projection calculation, which is expensive in many cases, we impose the condition that the mappings to be inward, and as similar to Lemma 2.1 in[9] we prove the following lemma which will be useful to prove the main results. We will prove our main theorem, which is given below;
Lemma 3.1 (Extension of lemma 2.1 for nonself mapping). Let K be a closed and convex subset of a Hilbert space H. Let : HKT be a mapping on K. Then if T is a κ-strict-pseudo contractive, then T satisfies the Lipschitz condition,
k 1 TyTx yx , for all , Kyx . 1 k Proof Let for every , Kyx , since T is a κ-strict-pseudo contractive mapping for some k 10 we have the following,
2 2 2 TyTx yxkyx TyTx )()( 2 2 )1( kyxk TyTx yxk ,2 TyTx 2 2 )1( kyxk TyTx 2 yxk TyTx 3668 Mollalgn Haile Takele and B. Krishna Reddy
2 2 Thus, k)1( TyTx 2 yxk TyTx yxk 0)1( .
Solving the quadratic inequality for TyTx by completing the square method as
2 2 2 2k k 2 k)1( 2 k 2 TyTx yx TyTx yx yx yx k)1( k)1( 2 k k)1()1( 2
1 2 yx . k)1( 2
2 2 k yx Which implies that . TyTx yx 2 k)1( k)1(
k 1 Therefore, TyTx yx for all Kyx ., 1 k Theorem 3.2 Let K be non-empty, closed and convex subset of a real Hilbert space H. Let : HKT be k-pseudo contractive for some k 10 , nonself and inward mapping with non-empty fixed pint set F(T). Let : Kh be defined for all Kx by xh x )1(:0inf)( Tx K. Let the sequence xn be generated by the algorithm
1 kKx ,, 1 xh 1 ,)(,max ,)1( n1 xx nn Txnn n1 xh nn 1 .,max
Then is well- defined and if , then converges weakly (n k)( n )1 n1 to some TFx )( .
Proof: By lemma 2.3 is well defined and in K. And also n is nondecreasing and k n 1, hence by theorem 1.1 in [12], converges weakly to some element .
Theorem 3.3 Let K be non-empty, closed and strictly convex subset of a real Hilbert space H. Let be k-pseudo contractive for some , nonself and inward mapping with non-empty fixed pint set F(T). Let be defined for all Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3669
Kx by xh x )1(:0inf)( Tx K. Let the sequencexn be generated by the algorithm
1 kKx ,, 1 xh 1 ,)(,max ,)1( n1 xx nn Txnn n1 xh nn 1 .,max
Then is well- defined and if (n k)( n )1 , then converges strongly n1 to some TFx )(
Proof Let TFp )( . Then from lemma 2.1, lemma 2.2 and T is k-strictly pseudo contractive, it can be shown thatn px is decreasing and bounded below, hence converges as follow;
2 2 n1 xpx nn )1( Txnn p 2 nn px 1()( )(Txnn P) 2 2 2 nn px )1( Txnn P n )1( x Txnnn (3.1) 2 2 2 2 nn px 1( )( nn xkpx Txnn n )1() x Txnnn 2 2 2 n px 1( )( nn ) xk Txnn n px
Thus is decreasing and bounded below, hence converges.
Thus lim n px exists, let lim n rpx for some r 0, then lim n1 rpx . n n n
Hence, cancelation of terms in the right hand-side and convergence of we
2 2 2 have 1( )( nn ) xk Txnn ( n n1 pxpx ) . (3.2) n1 n1 Again from the lemma 2.1 we have
nn n Mppxppxx 1 and
k 1 TpTxTx p Mppx (3.3) n n 1 k n 2 3670 Mollalgn Haile Takele and B. Krishna Reddy
hold for some MM 21 .0,
Thus, the sequencesxn andTxn are bounded, hence x Txnn is bounded. (3.4)
Since ( n k)( n )1 , we have lim 1( )( nn k 0) , sincen k and n n n1 is nondecreasing, we must have lim n 0)1( equivalentlylimn 1. (3.5) n n
Also, we have 1 xx nn 1( )( x Txnnn , thus lim 1 xx nn 0. (3.6) n
Thus xn is Cauchy sequence in H, which converges in H.
Since by lemma 2.3 xn is in K and K is closed, convex subset of the Hilbert space H.
Thus n Kxx .
Hence by lemma 2.3(a) xh 1)( , thus by definition of h we see that for any ([ xh ), )1 , x Tx K.)1( (3.7)
Since n limlim 1 xh nn 1)(,max , there must exist a subsequence xn of xn n n k such that xh n is increasing and lim xh n 1)( , hence for any 1, k k k
x )1( Tx K (3.8) nk nk
Now let 21 ((, xh ),1), 21 and 1x 1)1( Tx 1 Kz ,
2 x 2 )1( Tx 2 Kz
In particular, if 21 ],[ , hence (( xh ), )1 and xz )1( Tx K (3.9)
Since n Kxx and T is Lipschitzian, hence continuous and by lemma 2.3 (d)
x )1( Tx xz )1( Tx K (3.10) nk nk
Similarly , 21 Kzz , since is arbitrary, 21 ],[ Kzz , Since K is strictly convex
zz 21 , hence x Tx.
Therefore, n TFxx )( in norm, this complete the proof of the theorem 3.4. Remark Now we give example for k- strictly pseudo contractive, nonself and inward mapping that illustrate our result. Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3671
1 Let K 1, H . Let : KT byTx 2x . Thus 2
2 2 2 2 2 TyTx 422 3 yxyxyxyx
2 1 2 33 yxyx 3 (3.12) 2 1 2 yxyxyx )22()( 3 2 1 2 yxyx ()( TyTx 3 1 This shows that T is k strictly pseudo contractive for k )1,0( . 3 We see that T is nonself and inward mapping, indeed,
x x 1 1 Tx xx 22 xucxx )( , where c 12 andu , K 2 2 2 4 . Thus T is k-strictly pseudo contractive, nonself and inward mapping with fixed point 0. Theorem 3.4 Let K be non-empty, closed and convex subset of a real Hilbert space H. Let : HKA be non self mapping such that AI is k-pseudo contractive for some k 10 , and inward mapping with non-empty fixed pint set. Let : Kh be defined for all Kx by xh x )1(:0inf)( Ax K. Let the sequence xn be generated by the algorithm
1 kKx ,, 1 xh 1 ,)(,max 1( )( ,) n1 xx nn n xAI n n1 xh nn 1 .,max
Then is well- defined and if (n k)( n )1 , then the sequence n1 converges weakly to some AN )( . Moreover, if K is strictly convex and
(n k)( n )1 , then the convergence is strong. n1 3672 Mollalgn Haile Takele and B. Krishna Reddy
The proof of theorem 3.4 is the same as the proof of theorem 3.2 and 3.3 for AIT .
Lemma 3.5 Let : HKT be k-strictly pseudo contractive for some k )1,0( and
k )1,( , then : HKT defined by xxT )1( Tx is non expansive and
TFTF )()( .
Proof: let , Kyx and k )1,( , then by lemma 2.2 in [12] and T is k-strictly pseudo contractive we will have the following;
2 2 xyTxT Tx y )1(())1(( Ty 2 yx 1()( )( TyTx ) 2 2 2 yx )1( TyTx yx ()()1( TyTx (3.13) 2 2 2 yx )1( yxkyx TyTx )()( 2 yx TyTx )()()1( 2 2 2 yx 1( )( yxk TyTx )()() yx
2 Since k , we have 1( )( yxk TyTx 0)()() .
Thus yxyTxT , hence T is nonexpansive and TFx )( if and only if TFx )( . which completes the proof of the lemma.
Theorem 3.6 Let , 21 ,... N : HKTTT be family of nonself, k-strictly pseudo contractive and inward mappings on a non-empty, closed strictly convex subset K of a real Hilbert space H. Let )1,( TT and for 2,1 ,..., , k n n ModN)( Ni N xh )1(:0inf)( KxTx . Let TFF )( is nonempty. Then the i i i i1 sequence xn given by Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3673
1 , kKx 1 xh 11 ,)(,max is well-defined and if n )1,0(]1,[ for some 1 xx )1( xT n nn n nn n1 ,max xh nnn 1 N )1,0( xn converges weakly to some element p of TFF k )( . Moreover, if k1 and (F, K) satisfies S-condition, then the convergence is strong. n )1( n1
Proof suppose that for each i ,Ti is inward mapping.
Then for each Kx , i ( ), &1 KucxucxxT , thus
)1()( xTxxT i i xucxx )()1( (3.14) 1( )( xucx ) ( ), ,1)1( Kuccdxudx
Thus T is inward mapping, hence T N is family of nonself, nonexpansive and k i i1 inward mappings. Thus by the theorem of 1.5, lemma 3.1 and lemma 3.6 we complete the proof.
1 For example, let K 1, H , and let 21 :, KTT be defined by 2
1 , xx 0 1 1 2)( xxT , 2 xT )( 2 .Thus T1 and T are nonself, -strictly 2 3 xx 10, pseudo contractive and inward mappings with common fixed point 0.
Theorem 3.7 Suppose , 21 ,... N : HKAAA be family of, nonself on a non-empty, closed and convex subset K of a Hilbert space H , such that for each 2,1 ,..., Ni ,
AI i is k- strictly pseudo contractive and inward mapping. Let k )1,( for each Kx , )1( xAxA and NAA )(mod , i i i i N xh )1(:0inf)( KxAx . Let ANF )( is nonempty. Then the i i i i1 sequence xn given by 3674 Mollalgn Haile Takele and B. Krishna Reddy
1 , kKx 1 xh 11 ,)(,max is well-defined and if n )1,0(]1,[ for some 1 xx )1( xA nnn n n n1 ,max xh nnn 1 N )1,0( xn converges weakly to some element p of ANF i )( . Moreover, if i1 and (F,K) satisfies S-condition, then the convergence is strong. n )1( n1
Proof: Let AIT )( . Then from theorem 3.5, 3.7 and TFAN )()( we complete i i i i the proof of the theorem.
Corollary 3.8 Suppose : HKT is nonself, nonexpansive and inward mapping on a non-empty closed convex subset K of a Hilbert space H.
Let 1 Kx , xh x )1(:0inf)( Tx K. Then the sequence xn given by
1 Kx 1 xh 1 0,)(,max is well-defined and converges weakly to TFF )( , )1( n1 xx nn Txnn n1 ,max xh nn 1
Which agrees with Krasnoselskii-Mann type algorithm for 0−strictly Pseudo contractive mapping.
Corollary 3.9. Let , 21 ,... N : HKTTT be family of, nonself, nonexpansive and inward mappings on a non empty, closed and strictly convex subset K of a real Hilbert space N H with is non empty. , and if for each we define TFF i )( TT ModNnn )( 1 Kx i1
i : Kh by i xh )1(:0inf)( i CxTx , )1,0( be fixed .Then the sequence given by
1 Cx 1 xh 11 )(,max )1( n1 xx nn xT nnn n1 ,max xh nnn 11 Fixed Point Theorems for Approximating a Common Fixed Point for a Family… 3675
is well-defined and if n )1,0(]1,[ for some )1,0( , then xn converges N weakly to some element p of . Moreover, if and (F,K) TFF i )( n )1( i1 n1 satisfies S-condition, then the converges is strong
Proof. For k 0 , each i 2,1, ,..., NiforT is 0-pseudo contractive mapping, hence the corollary is the same as the theorem of TR [5].
Remark: If each , 21 ,... N : HCTTT are self-mappings, then, for all i ,0)(, ixhx .
Thus n , the above iterative methods reduces to Mann’s iterative method and converges weakly, since ,0 and 0, and )1( . Concluding remark: Our theorems generalize the theorem of CM to the class of k- strictly pseudo contractive mappings which is more general than the class of nonexpansive mappings. Our theorems also extend the theorem of MXu [12] to nonself mapping and for the family of nonself mappings as well. We also lower the requirement for metric projection computation ,which is costly and requires another approximation technique. Finally, we raise the following open question; Open question 1. Is it possible to extend our results to more general spaces such as; uniformly convex Banach spaces, uniformly smooth Banach spaces? If so under what conditions? Open question 2. Is it possible to extend our results to the class of pseudo contractive mappings which is more general than the class of strictly pseudo contractive mappings? If so under what conditions? Open question 3. Is it possible to extend our results to infinite family of k-strictly pseudo contractive mappings? If so under what conditions? Authors’ contributions Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Competing interests The authors declare that they have no competing interests.
3676 Mollalgn Haile Takele and B. Krishna Reddy
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3678 Mollalgn Haile Takele and B. Krishna Reddy