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Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces

Abstract

The purpose of this paper is to characterize the conditions for the convergence of the iterative scheme in the sense of Agarwal et al. (J. Nonlinear Convex. Anal. 8(1): 61-79, 2007), associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

Dedication

Dedicated to Professor Wataru Takahashi on the occasion of his seventieth birthday

1 Preliminaries

Let K be a nonempty subset of an arbitrary Banach space X, and let X be its dual space. Let T:XX be an operator. The symbols D(T) and R(T) stand for the domain and the range of T, respectively. We denote F(T) by the set of fixed points of a single-valued mapping T:KK. We denote by J the normalized duality mapping from X to 2 X defined by

J(x)= { f X : x , f = x 2 = f 2 } .

Let T:D(T)XX be an operator.

Definition 1 T is called L-Lipschitzian if there exists L0 such that

TxTyLxy

for all x,yD(T). If L=1, then T is called non-expansive, and if 0L<1, T is called contraction.

Definition 2 [13]

  1. (i)

    T is said to be strongly pseudocontractive if there exists a t>1 such that for each x,yD(T), there exists j(xy)J(xy) satisfying

    Re T x T y , j ( x y ) 1 t x y 2 .
  2. (ii)

    T is said to be strictly hemicontractive if F(T) and if there exists a t>1 such that for each xD(T) and qF(T), there exists j(xy)J(xy) satisfying

    Re T x q , j ( x q ) 1 t x q 2 .
  3. (iii)

    T is said to be ϕ-strongly pseudocontractive if there exists a strictly increasing function ϕ:[0,)[0,) with ϕ(0)=0 such that for each x,yD(T), there exists j(xy)J(xy) satisfying

    Re T x T y , j ( x y ) x y 2 ϕ ( x y ) xy.
  4. (iv)

    T is said to be ϕ-hemicontractive if F(T) and if there exists a strictly increasing function ϕ:[0,)[0,) with ϕ(0)=0 such that for each xD(T) and qF(T), there exists j(xy)J(xy) satisfying

    Re T x q , j ( x q ) x q 2 ϕ ( x q ) xq.

Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.

For a nonempty convex subset K of a normed space X,S:KK and T:KK,

  1. (a)

    the Mann iteration scheme [4] is defined by the following sequence { x n }:

    where { b n } is a sequence in [0,1];

  2. (b)

    the sequence { x n } defined by

    where { b n }, { b n } are sequences in [0,1] is known as the Ishikawa [2] iteration scheme;

  3. (c)

    the sequence { x n } defined by

    where { b n }, { b n } are sequences in [0,1], is known as the Agarwal-O’Regan-Sahu [5] iteration scheme;

  4. (d)

    the sequence { x n } defined by

    where { b n }, { b n } are sequences in [0,1], is known as the modified Agarwal-O’Regan-Sahu iteration scheme.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of L p (or l p ) into itself. Afterwards, several authors generalized this result of Chidume in various directions [3, 612].

The purpose of this paper is to characterize conditions for the convergence of the iterative scheme in the sense of Agarwal et al. [5] associated with nonexpansive and ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results improve and generalize most results in recent literature [1, 3, 5, 6, 8, 9, 11, 12].

2 Main result

The following result is now well known.

Lemma 3 [13]

For all x,yX and j(x+y)J(x+y),

x + y 2 x 2 +2Re y , j ( x + y ) .

Now, we prove our main result.

Theorem 4 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, let S:KK be nonexpansive, and let T:KK be a uniformly continuous ϕ-hemicontractive mapping such that S and T have the common fixed point. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [0,1] satisfying conditions

  1. (i)

    lim n (1 b n )= lim n b n =0,

  2. (ii)

    n = 1 (1 b n )=.

For any x 1 K, define the sequence { x n } n = 1 inductively as follows:

{ y n = b n x n + ( 1 b n ) T x n , x n + 1 = b n S x n + ( 1 b n ) T y n , n 1 .
(2.1)

Then the following conditions are equivalent:

  1. (a)

    { x n } n = 1 converges strongly to the common fixed point q of S and T.

  2. (b)

    { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

Proof First, we prove that (a) implies (b).

Since T is ϕ-hemicontractive, it follows that F(T) is a singleton. Let F(S)F(T)={q} for some qK.

Suppose that lim n x n =q. Then the continuity of S and T yields that

lim n S x n =q= lim n T x n

and

lim n y n = lim n [ b n x n + ( 1 b n ) T x n ] =q.

Thus, lim n T y n =q. Therefore, { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

Second, we need to show that (b) implies (a). Suppose that { S x n } n = 1 , { T x n } n = 1 and { T y n } n = 1 are bounded.

Put

M 1 = x 1 q+ sup n 1 S x n q+ sup n 1 T x n q+ sup n 1 T y n q.

It is clear that x 1 q M 1 . Let x n q M 1 . Next, we will prove that x n + 1 q M 1 .

Note that

x n + 1 q = b n S x n + ( 1 b n ) T y n q = b n ( S x n q ) + ( 1 b n ) ( T y n q ) b n S x n q + ( 1 b n ) T y n q ( b n + ( 1 b n ) ) M 1 = M 1 .

Thus, we can conclude that the sequence { x n q } n 1 is bounded, and hence, there is a constant M>0 satisfying

M= sup n 1 x n q+ sup n 1 S x n q+ sup n 1 T x n q+ sup n 1 T y n q.
(2.2)

Let w n =T y n T x n + 1 for each n1. The uniform continuity of T ensures that

lim n w n =0,
(2.3)

because

y n x n + 1 = b n ( x n T x n ) + ( 1 b n ) ( S x n T y n ) b n x n T x n + ( 1 b n ) S x n T y n 2 M ( b n + ( 1 b n ) ) 0 as  n .

By virtue of Lemma 3 and (2.1), we infer that

x n + 1 q 2 = b n S x n + ( 1 b n ) T y n q 2 = b n ( S x n q ) + ( 1 b n ) ( T y n q ) 2 b n 2 S x n q 2 + 2 ( 1 b n ) Re T y n q , j ( x n + 1 q ) b n 2 x n q 2 + 2 ( 1 b n ) Re T y n T x n + 1 , j ( x n + 1 q ) + 2 ( 1 b n ) Re T x n + 1 q , j ( x n + 1 q ) b n 2 x n q 2 + 2 ( 1 b n ) T y n T x n + 1 x n + 1 q + 2 ( 1 b n ) x n + 1 q 2 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q b n 2 x n q 2 + 2 M ( 1 b n ) w n + 2 ( 1 b n ) x n + 1 q 2 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q .
(2.4)

The real function f:[0,)[0,), f(t)= t 2 is increasing and convex. For all a[0,1] and t 1 , t 2 >0, we have

( ( 1 a ) t 1 + a t 2 ) 2 (1a) t 1 2 +a t 2 2 .

Hence,

x n + 1 q 2 = b n S x n + ( 1 b n ) T y n q 2 = b n ( S x n q ) + ( 1 b n ) ( T y n q ) 2 b n S x n q 2 + ( 1 b n ) T y n q 2 b n x n q 2 + ( 1 b n ) M 2 ,
(2.5)

where the second inequality holds by the convexity of 2 .

By substituting (2.5) in (2.4), we get

x n + 1 q 2 ( b n 2 + 2 b n ( 1 b n ) ) x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q = ( 1 ( 1 b n ) 2 ) x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q x n q 2 + 2 M ( 1 b n ) ( w n + M ( 1 b n ) ) 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q = x n q 2 + ( 1 b n ) l n 2 ( 1 b n ) ϕ ( x n + 1 q ) x n + 1 q ,
(2.6)

where

l n =2M ( w n + M ( 1 b n ) ) 0,
(2.7)

as n.

Let δ=inf{ x n + 1 q:n0}. We claim that δ=0. Otherwise, δ>0. Thus, (2.7) implies that there exists a positive integer N 1 such that l n <ϕ(δ)δ for each n N 1 . In view of (2.6), we conclude that

x n + 1 q 2 x n q 2 ϕ(δ)δ(1 b n ),n N 1 ,

which implies that

ϕ(δ)δ n = N 1 (1 b n ) x N 1 q 2 ,
(2.8)

which contradicts (ii). Therefore, δ=0. Thus, there exists a subsequence { x n i + 1 } n = 1 of { x n + 1 } n = 1 such that

lim i x n i + 1 =q.
(2.9)

Let ϵ>0 be a fixed number. By virtue of (2.7) and (2.9), we can select a positive integer i 0 > N 1 such that

x n i 0 + 1 q<ϵ, l n <ϕ(ϵ)ϵ,n n i 0 .
(2.10)

Let p= n i 0 . By induction, we show that

x p + m q<ϵ,m1.
(2.11)

Observe that (2.6) means that (2.11) is true for m=1. Suppose that (2.11) is true for some m1. If x p + m + 1 qϵ, by (2.6) and (2.10), we know that

ϵ 2 x p + m + 1 q 2 x p + m q 2 + ( 1 b p + m ) l p + m 2 ( 1 b p + m ) ϕ ( x p + m + 1 q ) x p + m + 1 q < ϵ 2 + ( 1 b p + m ) ϕ ( ϵ ) ϵ 2 ( 1 b p + m ) ϕ ( ϵ ) ϵ = ϵ 2 ( 1 b p + m ) ϕ ( ϵ ) ϵ < ϵ 2 ,

which is impossible. Hence, x p + m + 1 q<ϵ. That is, (2.11) holds for all m1. Thus, (2.11) ensures that lim n x n =q. This completes the proof. □

Taking S=I in Theorem 4, we get the following.

Corollary 5 Let K be a nonempty closed and convex subset of an arbitrary Banach space X, and let T:KK be a uniformly continuous ϕ-hemicontractive mapping. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [0,1] satisfying conditions (i)-(ii) of Theorem  4. For any x 1 K, define the sequence { x n } n = 1 inductively as follows:

{ y n = b n x n + ( 1 b n ) T x n , x n + 1 = b n x n + ( 1 b n ) T y n , n 1 .

Then the following conditions are equivalent:

  1. (a)

    { x n } n = 1 converges strongly to the unique fixed point q of T.

  2. (b)

    { T x n } n = 1 is bounded.

Remark 6

  1. 1.

    All the results can also be proved for the same iterative scheme with error terms.

  2. 2.

    The known results for strongly pseudocontractive mappings are weakened by the ϕ-hemicontractive mappings.

  3. 3.

    Our results hold in arbitrary Banach spaces, where as other known results are restricted for L p (or l p ) spaces and q-uniformly smooth Banach spaces.

  4. 4.

    Theorem 4 is more general in comparison to the results of Agarwal et al. [5] in the context of the class of ϕ-hemicontractive mappings. Theorem 4 extends convergence results coercing ϕ-hemicontractive mappings in the literature in the framework of Agarwal-O’Regan-Sahu iteration process (see also [1421]).

3 Applications

Theorem 7 Let X be an arbitrary real Banach space, S:XX be nonexpansive, and let T:XX be uniformly continuous ϕ-strongly accretive operators, respectively. Suppose that { b n } n = 1 and { b n } n = 1 are sequences in [0,1] satisfying conditions (i)-(ii) of Theorem  4. For any x 1 X, define the sequence { x n } n = 1 inductively as follows:

{ y n = b n x n + ( 1 b n ) ( f + ( I T ) x n ) , x n + 1 = b n ( f + ( I S ) x n ) + ( 1 b n ) ( f + ( I T ) y n ) , n 1 ,

where fX, and I is the identity operator. Then the following conditions are equivalent:

  1. (a)

    { x n } n = 1 converges strongly to the solution of the system Sx=f=Tx.

  2. (b)

    { ( I S ) x n } n = 1 , { ( I T ) x n } n = 1 and { ( I T ) y n } n = 1 are bounded.

Proof Suppose that x is the solution of the system Sx=f=Tx. Define G, G :XX by Gx=f+(IS)x and G x=f+(IT)x, respectively. Since S and T are nonexpansive and uniformly continuous ϕ-strongly accretive operators, respectively, so are G and G , then x is the common fixed point of G and G . Thus, Theorem 7 follows from Theorem 4. □

Example 8 Let X=R be the reals with the usual norm and K=[0,1]. Define S:KK by

Sx=sinxfor all xK

and T:KK by

Tx=xtanxfor all xK.

By the mean value theorem, we have

| T ( x ) T ( y ) | sup c ( 0 , 1 ) | T ( c ) | |xy|for all x,yK.

Noticing that T (c)=1 sec 2 (c) and 1< sup c ( 0 , 1 ) | T (c)|=2.4255. Hence,

| T ( x ) T ( y ) | L|xy|for all x,yK,

where L=2.4255. It is easy to verify that T is ϕ-hemicontractive mapping with ϕ:[0,)[0,) defined by ϕ(t)=tan(t) for all t[0,). Moreover, 0 is the common fixed point of S and T. Let { b n } n = 1 and { b n } n = 1 be sequences in [0,1] defined by

b n =1 1 n and b n = 1 n ,n1.

Then { x n } n = 1 defined by (2.1) in Theorem 4 converges to 0, which is the common fixed point of S and T.

References

  1. Chidume CE: Iterative approximation of fixed point of Lipschitz strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1987, 99: 283–288.

    MathSciNet  MATH  Google Scholar 

  2. Ishikawa S: Fixed point by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

    Article  MathSciNet  MATH  Google Scholar 

  3. Liu LW: Approximation of fixed points of a strictly pseudocontractive mapping. Proc. Am. Math. Soc. 1997, 125: 1363–1366. 10.1090/S0002-9939-97-03858-6

    Article  MathSciNet  MATH  Google Scholar 

  4. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 26: 506–510.

    Article  MathSciNet  MATH  Google Scholar 

  5. Agarwal RP, O’Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal. 2007, 8(1):61–79.

    MathSciNet  MATH  Google Scholar 

  6. Ciric LB, Ume JS: Ishikawa iterative process for strongly pseudocontractive operators in Banach spaces. Math. Commun. 2003, 8: 43–48.

    MathSciNet  MATH  Google Scholar 

  7. Liu LS: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194: 114–125. 10.1006/jmaa.1995.1289

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu Z, Kim JK, Kang SM: Necessary and sufficient conditions for convergence of Ishikawa iterative schemes with errors to ϕ -hemicontractive mappings. Commun. Korean Math. Soc. 2003, 18(2):251–261.

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu Z, Xu Y, Kang SM: Almost stable iteration schemes for local strongly pseudocontractive and local strongly accretive operators in real uniformly smooth Banach spaces. Acta Math. Univ. Comen. 2008, LXXVII(2):285–298.

    MathSciNet  MATH  Google Scholar 

  10. Schu J: On a theorem of C.E. Chidume concerning the iterative approximation of fixed points. Math. Nachr. 1991, 153: 313–319. 10.1002/mana.19911530127

    Article  MathSciNet  MATH  Google Scholar 

  11. Xue Z: Iterative approximation of fixed point for ϕ -hemicontractive mapping without Lipschitz assumption. Int. J. Math. Math. Sci. 2005, 17: 2711–2718.

    MathSciNet  MATH  Google Scholar 

  12. Zhou HY, Cho YJ: Ishikawa and Mann iterative processes with errors for nonlinear ϕ -strongly quasi-accretive mappings in normed linear spaces. J. Korean Math. Soc. 1999, 36: 1061–1073.

    MathSciNet  MATH  Google Scholar 

  13. Xu HK: Inequality in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K

    Article  MathSciNet  MATH  Google Scholar 

  14. Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508

    Article  MathSciNet  MATH  Google Scholar 

  15. Takahashi W: Nonlinear Functional Analysis - Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.

    MATH  Google Scholar 

  16. Takahashi W, Yao J-C: Weak and strong convergence theorems for positively homogeneous nonexpansive mappings in Banach spaces. Taiwan. J. Math. 2011, 15: 961–980.

    MathSciNet  MATH  Google Scholar 

  17. Tan KK, Xu HK: Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces. J. Math. Anal. Appl. 1993, 178: 9–21. 10.1006/jmaa.1993.1287

    Article  MathSciNet  MATH  Google Scholar 

  18. Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987

    Article  MathSciNet  MATH  Google Scholar 

  19. Hussain N, Rafiq A: On modified implicit Mann iteration method involving strictly hemicontractive mappings in smooth Banach spaces. J. Comput. Anal. Appl. 2013, 15(5):892–902.

    MathSciNet  MATH  Google Scholar 

  20. Hussain N, Rafiq A, Ciric LB: Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 160

    Google Scholar 

  21. Khan SH, Rafiq A, Hussain N: A three-step iterative scheme for solving nonlinear ϕ -strongly accretive operator equations in Banach spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 149

    Google Scholar 

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Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the first author acknowledges with thanks DSR, KAU for financial support.

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Hussain, N., Sahu, D. & Rafiq, A. Iteration scheme for common fixed points of hemicontractive and nonexpansive operators in Banach spaces. Fixed Point Theory Appl 2013, 247 (2013). https://doi.org/10.1186/1687-1812-2013-247

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