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Implicit iteration scheme for two phi-hemicontractive operators in arbitrary Banach spaces

Abstract

The purpose of this paper is to characterize conditions for the convergence of the implicit Ishikawa iterative scheme with errors in the sense of Agarwal et al. (J. Math. Anal. Appl. 272:435-447, 2002) to a common fixed point of two ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space.

MSC:47H09, 47J25.

1 Introduction and preliminaries

Let K be a nonempty subset of an arbitrary Banach space E and E be its dual space. The symbols T and F(T) stand for the self-map of K and the set of fixed points of T. We denote by J the normalized duality mapping from E to 2 E defined by

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

Definition 1.1 [14]

  1. (i)

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

    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 constant t>1 such that for each xK and qF(T), there exists j(xq)J(xq) satisfying

    T x T 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,yK, there exists j(xy)J(xy) satisfying

    T x T y , j ( x y ) x y 2 ϕ ( x y ) .
  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 xK and qF(T), there exists j(xq)J(xq) satisfying

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

Clearly, each strictly hemicontractive operator is ϕ-hemicontractive.

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in the case that 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 have generalized this result of Chidume in various directions [412].

In 2001, Xu and Ori [13] introduced the following implicit iteration process for a finite family of nonexpansive mappings { T i :iI} (here I={1,2,,N}), with { α n } a real sequence in (0,1), and an initial point x 0 K:

x 1 = ( 1 α 1 ) x 0 + α 1 T 1 x 1 , x 2 = ( 1 α 2 ) x 1 + α 2 T 2 x 2 , x N = ( 1 α N ) x N 1 + α N T N x N , x N + 1 = ( 1 α N + 1 ) x N + α N + 1 T N + 1 x N + 1 ,

which can be written in the following compact form:

x n =(1 α n ) x n 1 + α n T n x n for all n1,
(XO)

where T n = T n ( mod N ) (here the modN function takes values in I). Xu and Ori [13] proved that the process converges weakly to a common fixed point of a finite family in a Hilbert space. They remarked further that it is yet unclear what assumptions on the mappings and the parameters { α n } are sufficient to guarantee the strong convergence of the sequence { x n }.

In [14], Osilike proved the following results.

Theorem 1.2 [[14], Theorem 2]

Let E be a real Banach space and K be a nonempty closed convex subset of E. Let { T i :iI} be N strictly pseudocontractive self-mappings of K with F= i = 1 N F( T i ). Let { α n } n = 1 be a real sequence satisfying the conditions:

( i ) 0 < α n < 1 , ( ii ) n = 1 ( 1 α n ) = , ( iii ) n = 1 ( 1 α n ) 2 < .

For arbitrary x 0 K, define the sequence { x n } by the implicit iteration process (XO). Then { x n } converges strongly to a common fixed point of the mappings { T i :iI} if and only if lim n infd( x n ,F)=0.

It is well known that α n =1 1 n 1 2 , ( 1 α n ) 2 =. Hence the results of Osilike [14] need to be improved.

The purpose of this paper is to characterize conditions for the convergence of the implicit Ishikawa iterative scheme with errors in the sense of Agarwal et al. [15] to a common fixed point of two ϕ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our studying results improve and generalize most results in recent literature [4, 5, 79, 12].

2 Main results

The following results are now well known.

Lemma 2.1 [16]

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

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

Lemma 2.2 [17]

Let { θ n } be a sequence of nonnegative real numbers, and let { λ n } be a real sequence satisfying

0 λ n 1, n = 0 λ n =.

Suppose that there exists a strictly increasing function ϕ:[0,)[0,) with ϕ(0)=0. If there exists a positive integer n 0 such that

θ n + 1 2 θ n 2 λ n ϕ( θ n + 1 )+ σ n + γ n

for all n n 0 , with σ n 0, nN, σ n =0( λ n ) and n = 0 γ n <, then lim n θ n =0.

Now we prove our main results.

Theorem 2.3 Let K be a nonempty convex subset of an arbitrary Banach space E, and let T,S:KK be two uniformly continuous with F(T)F(S) and ϕ-hemicontractive mappings. Suppose that { u n } n = 1 and { v n } n = 1 are bounded sequences in K and { a n } n = 1 , { b n } n = 1 , { c n } n = 1 , { a n } n = 1 , { b n } n = 1 and { c n } n = 1 are sequences in [0,1] satisfying conditions

  1. (i)

    a n + b n + c n = a n + b n + c n =1,

  2. (ii)

    lim n b n = lim n c n = lim n b n =0,

  3. (iii)

    n = 1 c n <, and

  4. (iv)

    n = 1 b n =.

For any x 0 K, define the sequence { x n } n = 1 inductively as follows:

y n = a n x n 1 + b n S x n + c n u n , x n = a n x n 1 + b n T y n + c n v 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 T and S,

  2. (b)

    lim n T y n =q,

  3. (c)

    { T y n } n = 1 is bounded.

Proof Since T and S are ϕ-hemicontractive, then the common fixed point of F(T)F(S) is unique. Suppose that p and q are all common fixed points of T and S, then

p q 2 = p q , j ( p q ) = T p T q , j ( p q ) p q 2 ϕ ( p q ) < p q 2 ,

which is a contradiction. So, we denote the unique fixed point q.

Suppose that lim n x n =q. Then (ii) and the uniform continuity of T and S yield that

lim n y n = lim n [ a n x n 1 + b n S x n + c n u n ]=q,

which implies that lim n T y n =q. Therefore { T y n } n = 1 is bounded.

Put

M 1 = x 0 q+ sup n 1 T y n q+ sup n 1 u n q+ sup n 1 v n q.

Obviously, M 1 <. It is clear that x 0 p M 1 . Let x n 1 p M 1 . Next we prove that x n p M 1 .

Consider

x n q = a n x n 1 + b n T y n + c n v n q = a n ( x n 1 q ) + b n ( T y n q ) + c n ( v n q ) ( 1 b n ) x n 1 q + b n T y n q + c n v n q ( 1 b n ) M 1 + b n T y n q + c n v n q = ( 1 b n ) [ x 0 q + sup n 1 T y n q + sup n 1 u n q + sup n 1 v n q ] + b n T y n q + c n v n q x 0 q + ( ( 1 b n ) sup n 1 T y n q + b n T y n q ) + sup n 1 u n q + ( ( 1 b n ) sup n 1 v n q + b n v n q ) x 0 q + ( ( 1 b n ) sup n 1 T y n q + b n sup n 1 T y n q ) + sup n 1 u n q + ( ( 1 b n ) sup n 1 v n q + b n sup n 1 v n q ) = x 0 q + sup n 1 T y n q + sup n 1 u n q + sup n 1 v n q = M 1 .

So, from the above discussion, we can conclude that the sequence { x n q } n 0 is bounded. Since S is uniformly continuous, so { S x n q } n = 1 is also bounded. Thus there is a constant M 2 >0 satisfying

M 2 = sup n 1 x n q + sup n 1 S x n q + sup n 1 T y n q + sup n 1 u n q + sup n 1 v n q .
(2.2)

Denote M= M 1 + M 2 . Obviously, M<. Let w n =T y n T x n for each n1. The uniform continuity of T ensures that

lim n w n =0.
(2.3)

Because

y n x n = b n ( S x n x n 1 ) + b n ( x n 1 T y n ) + c n ( u n x n 1 ) c n ( v n x n 1 ) b n S x n x n 1 + b n x n 1 T y n + c n u n x n 1 + c n v n x n 1 2 M 2 ( b n + c n + b n + c n ) 0

as n.

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

x n q 2 = a n x n 1 + b n T y n + c n v n q 2 = a n ( x n 1 q ) + b n ( T y n q ) + c n ( v n q ) 2 ( 1 b n ) 2 x n 1 q 2 + 2 b n T y n q , j ( x n q ) + 2 c n v n q , j ( x n q ) ( 1 b n ) 2 x n 1 q 2 + 2 b n T y n T x n , j ( x n q ) + 2 b n T x n q , j ( x n q ) + 2 c n v n q x n q ( 1 b n ) 2 x n 1 q 2 + 2 b n T y n T x n x n q + 2 b n x n q 2 2 b n ϕ ( x n q ) + 2 M 2 c n ( 1 b n ) 2 x n 1 q 2 + 2 M b n w n + 2 b n x n q 2 2 b n ϕ ( x n q ) + 2 M 2 c n .
(2.4)

Consider

x n q 2 = a n x n 1 + b n T y n + c n v n q 2 = a n ( x n 1 q ) + b n ( T y n q ) + c n ( v n q ) 2 a n x n 1 q 2 + b n T y n q 2 + c n v n q 2 x n 1 q 2 + M 2 ( b n + c n ) ,
(2.5)

where the first inequality holds by the convexity of 2 .

Substituting (2.5) in (2.4), we get

x n q 2 [ ( 1 b n ) 2 + 2 b n ] x n 1 q 2 + 2 M b n ( w n + M ( b n + c n ) ) + 2 M 2 c n 2 b n ϕ ( x n q ) = ( 1 + b n 2 ) x n 1 q 2 + 2 M b n ( w n + M ( b n + c n ) ) + 2 M 2 c n 2 b n ϕ ( x n q ) x n 1 q 2 + M b n ( 3 M b n + 2 w n + 2 M b n ) + 2 M 2 c n 2 b n ϕ ( x n q ) = x n 1 q 2 + b n l n + 2 M 2 c n 2 b n ϕ ( x n q ) ,
(2.6)

where

l n =M ( 3 M b n + 2 w n + 2 M b n ) 0
(2.7)

as n.

Denote

θ n = x n q , λ n = 2 b n , σ n = b n l n , γ n = 2 M 2 c n .

Condition (i) assures the existence of a rank n 0 N such that λ n =2 b n 1 for all n n 0 . Now, with the help of conditions (ii), (iii), (2.3), (2.7) and Lemma 2.2, we obtain from (2.6) that

lim n x n q=0,

completing the proof. □

Using the method of proof in Theorem 2.3, we have the following result.

Corollary 2.4 Let E, K, T, S, { u n } n = 1 , { v n } n = 1 , { x n } n = 1 and { y n } n = 1 be as in Theorem  2.3. Suppose that { a n } n = 1 , { b n } n = 1 , { c n } n = 1 , { a n } n = 1 , { b n } n = 1 and { c n } n = 1 are sequences in [0,1] satisfying conditions (i), (ii), (iv) and c n =o( b n ). Then the conclusions of Theorem  2.3 hold.

Proof From the condition c n =o( b n ), set c n = b n t n , where lim n t n =0. Substituting (2.5) in (2.4), we get

x n q 2 [ ( 1 b n ) 2 + 2 b n ] x n 1 q 2 + 2 M b n ( w n + M ( b n + c n + t n ) ) 2 b n ϕ ( x n q ) x n q = ( 1 + b n 2 ) x n 1 q 2 + 2 M b n ( w n + M ( b n + c n + t n ) ) 2 b n ϕ ( x n q ) x n 1 q 2 + M b n ( 2 w n + M ( 3 b n + 2 c n + 2 t n ) ) 2 b n ϕ ( x n q ) = x n 1 q 2 2 b n ϕ ( x n q ) x n q + b n l n ,
(2.8)

where

l n =M ( 2 w n + M ( 3 b n + 2 c n + 2 t n ) ) 0
(2.9)

as n.

It follows from Lemma 2.2 that lim n x n q=0. □

Corollary 2.5 Let K be a nonempty convex subset of an arbitrary Banach space E, and let T:KK be a uniformly continuous and ϕ-hemicontractive mapping. Suppose that { u n } n = 1 and { v n } n = 1 are bounded sequences in K and { a n } n = 1 , { b n } n = 1 , { c n } n = 1 , { a n } n = 1 , { b n } n = 1 and { c n } n = 1 are sequences in [0,1] satisfying conditions

  1. (i)

    a n + b n + c n = a n + b n + c n =1,

  2. (ii)

    lim n b n = lim n c n = lim n b n =0,

  3. (iii)

    c n =o( b n ), and

  4. (iv)

    n = 1 b n =.

For any x 0 K, define the sequence { x n } n = 1 inductively as follows:

y n = a n x n 1 + b n T x n + c n u n , x n = a n x n 1 + b n T y n + c n v 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)

    lim n T y n =q,

  3. (c)

    { T y n } n = 1 is bounded.

Corollary 2.6 Let E, K, T, { u n } n = 1 , { v n } n = 1 , { x n } n = 1 and { y n } n = 1 be as in Corollary  2.5. Suppose that { a n } n = 1 , { b n } n = 1 , { c n } n = 1 , { a n } n = 1 , { b n } n = 1 and { c n } n = 1 are sequences in [0,1] satisfying conditions (i), (ii), (iv) and n = 1 c n <. Then the conclusions of Corollary  2.5 hold.

Corollary 2.7 Let K be a nonempty convex subset of an arbitrary Banach space E, and let T,S:KK be two uniformly continuous and ϕ-hemicontractive mappings. Suppose that { α n } n = 1 , { β n } n = 1 are any sequences in [0,1] satisfying

  1. (i)

    lim n β n =0= lim n α n ,

  2. (ii)

    n = 1 α n =.

For any x 0 K, define the sequence { x n } n = 1 inductively as follows:

y n = ( 1 β n ) x n 1 + β n S x n , x n = ( 1 α n ) x n 1 + α n T y n , n 1 .

Then the following conditions are equivalent:

  1. (a)

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

  2. (b)

    lim n T y n =q,

  3. (c)

    { T y n } n = 1 is bounded.

Corollary 2.8 Let K be a nonempty convex subset of an arbitrary Banach space E, and let T:KK be a uniformly continuous and ϕ-hemicontractive mapping. Suppose that { α n } n = 1 , { β n } n = 1 are any sequences in [0,1] satisfying

  1. (i)

    lim n β n =0= lim n α n ,

  2. (ii)

    n = 1 α n =.

For any x 0 K, define the sequence { x n } n = 1 inductively as follows:

y n = ( 1 β n ) x n 1 + β n T x n , x n = ( 1 α n ) x n 1 + α 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)

    lim n T y n =q,

  3. (c)

    { T y n } n = 1 is bounded.

Corollary 2.9 Let K be a nonempty convex subset of an arbitrary Banach space E, and let T:KK be a uniformly continuous and ϕ-hemicontractive mapping. Suppose that { α n } n = 1 is any sequence in [0,1] satisfying

  1. (i)

    lim n α n =0,

  2. (ii)

    n = 1 α n =.

For any x 0 K, define the sequence { x n } n = 1 inductively as follows:

x n = α n x n 1 +(1 α n )T x n ,n1.

Then the following conditions are equivalent:

  1. (a)

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

  2. (b)

    lim n T x n =q,

  3. (c)

    { T x n } n = 1 is bounded.

All of the above results are also valid for Lipschitz ϕ-hemicontractive mappings.

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

    MathSciNet  Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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 

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

    MathSciNet  MATH  Google Scholar 

  6. 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

    MathSciNet  Article  MATH  Google Scholar 

  7. 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 

  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.

    MathSciNet  Article  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. 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

    MathSciNet  Article  MATH  Google Scholar 

  11. Xu YG: 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

    MathSciNet  Article  MATH  Google Scholar 

  12. Xue ZQ: 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 

  13. Xu HK, Ori R: An implicit iterative process for nonexpansive mappings. Numer. Funct. Anal. Optim. 2001, 22: 767–773. 10.1081/NFA-100105317

    MathSciNet  Article  MATH  Google Scholar 

  14. Osilike MO: Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps. J. Math. Anal. Appl. 2004, 294(1):73–81. 10.1016/j.jmaa.2004.01.038

    MathSciNet  Article  MATH  Google Scholar 

  15. Agarwal RP, Cho YJ, Li J, Huang NJ: Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q -uniformly smooth Banach spaces. J. Math. Anal. Appl. 2002, 272: 435–447. 10.1016/S0022-247X(02)00150-6

    MathSciNet  Article  MATH  Google Scholar 

  16. Chang SS, Cho YJ, Kim JI: Some results for uniformly L -Lipschitzian mappings in Banach spaces. Appl. Math. Lett. 2009, 22(1):121–125. 10.1016/j.aml.2008.02.016

    MathSciNet  Article  MATH  Google Scholar 

  17. Yang LP: A note on a paper ‘Convergence theorem for the common solution for a finite family of ϕ -strongly accretive operator equations’. Appl. Math. Comput. 2012, 218: 10367–10369. 10.1016/j.amc.2012.04.037

    MathSciNet  Article  MATH  Google Scholar 

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Lv, G., Rafiq, A. & Xue, Z. Implicit iteration scheme for two phi-hemicontractive operators in arbitrary Banach spaces. J Inequal Appl 2013, 521 (2013). https://doi.org/10.1186/1029-242X-2013-521

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Keywords

  • implicit iterative scheme
  • ϕ-hemicontractive mappings
  • Banach spaces