Fixed point problems of nonexpansive mappings for nononvex set in Hilbert spaces

Xianbing Wu1

1Department of Mathematics Yangtze Normal University, Chongqing, China

1Corresponding author

Vibroengineering PROCEDIA, Vol. 28, 2019, p. 201-205. https://doi.org/10.21595/vp.2019.21034
Received 19 September 2019; accepted 1 October 2019; published 19 October 2019

Copyright © 2019 Xianbing Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Creative Commons License
Abstract.

In this paper, we introduce a new concept of W-nonexpansive mappings and obtain fixed point theorems for nonexpansive mappings for non-convex set. Our results resolve fixed pointed problem that nonexpansive mappings be not on closed convex set, and it extends fixed point theorems for nonexpansive mappings.

Keywords: W-nonexpansive mapping, Hilbert space, fixed point theorem, non-convex set.

1. Introduction and preliminaries

Fixed point theory is widely applied in engineering. Browder (1965) [1], Kirk (1965) [2] obtained fixed points theorem for nonexpansive mapping. Non-expansion fixed point theory has made great progress, large number of results are obtained by authors (e.g. See [3-11]). let’s come up with some definitions.

Definition 1.1 Let X be a nonempty set, the function W:X×X[0,) is called triangular if for all x,yX, if W(x,y)1,W(y,z)1 or W(y,x)1,W(y,z)1, then W(x,z)1.

Definition 1.2 Let (X,d) be a metric space and T:XX be a given mapping, if there exists a function W:X×X[0,) such that Wx,ydTx,Tydx,y, x,yX, then we say that T is a W-nonexpansive mapping.

Clearly, any nonexpansive mapping is a W-nonexpansive mapping with W(x,y)=1 for all x,yX.

Definition 1.3 Let T:XX be a mapping and W:X×X[0,) be a function. We say that T is a W-admissible if W(x,y)1W(Tx,Ty)1, x,yX.

Definition 1.4 [4] Let H be a Hilbert space, T:HH is called demicompact if whenever {xn}H is bounded and {Txn-Txn} strongly convergent, then there exists a subsequence {xnk} of {xn} which is strongly convergent.

Next our main results are presented.

2. Main results

Theorem 2.1 Let E be a bounded closed convex subset of a Hilbert space H, W:E×E[0,) is triangular function, T:EE is a W-nonexpansive mapping and it is W-admissible. If the following conditions are satisfied:

(w1) there exists x0E such that W(x0,Tx0)1;

(w2) there exists a sequence {sj}[0,1) with limnsj=1 such that for all x,yE, if W(x,y)1, then W(x,(1-sj)x+sjy)1, sj{sj};

(w3) if {xn}E is satisfied W(x0,xn)1, moreover xnx* or xnx*E, then W(xn,x*)1.

Then T has a fixed point.

Proof. Let x0X such that α(x0,Tx0)1. Take xn+1j=(1-sj)x0+sjTxnj for all j, nN, there x0=x0j. Now we fix j, for each jN, from (w2), we may obtain W(x0j,x1j)1.

Also, for T is W-admissible, then W(Tx0j,Tx1j)1 is obtained. According to W is a triangular function and (w1), then W(x0j,Tx1j)1.

Once again use (w2), then W(x0j,x2j)1 is also obtained. Continuously, we easily obtain:

(1)
W x 0 j , x n j 1 ,         n N .

Based on that W is triangular, we may get:

(2)
W x n j , x m j 1 ,       n , m N , n < m .

So from Eq. (2) and for T is W-nonexpansive, we have:

(3)
x n j - x m j = s j T x n - 1 j - T x m - 1 j s j W ( x n - 1 j , x m - 1 j ) T x n - 1 j - T x m - 1 j
          s j x n - 1 j - x m - 1 j . . . s j n x 0 j - x m - n j .

Let n, for E is bounded we may get xnj-xmj0, hence {xnj} is Cauchy sequence, it means there exists xj*E such that {xnj} convergent to xj*, that is:

(4)
l i m n x n j - x j * = 0 .

Also, from Eq. (1) and (w3), we have:

(5)
W x n j , x j * 1 .

Once again by Eq. (1), for W is triangular, so we have:

(6)
W x 0 j , x j * 1 .

Since E is bounded, closed and convex in Hilbert H, then it is weakly compact. Hence there exists a x*E such that:

(7)
x j * x * ,         ( j ) .

From Eqs. (6, 7), applying (w3) we have:

(8)
W ( x j * , x * ) 1 .

Next, we show that xj*=(1-sj)x0j+sjTxj*.

Indeed, according to xnj=(1-sj)x0+sjTxn-1j, T is W-nonexpiansive and Eq. (5), we have:

(9)
x j * - 1 - s j x 0 j + s j T x j * = x j * - x n j + x n j - 1 - s j x 0 j + s j T x j *
        = x n j - x j * + s j T x n - 1 j - T x j * x n j - x j * + s j W x n j , x j * T x n - 1 j - T x j *
        x n j - x j * + s j x n - 1 j - x j * .

Let n in Eq. (9), utilize Eq. (4) we obtain xj*-1-sjx0j+sjTxj*0, it implies that xj*=(1-sj)x0j+sjTxj*.

Finally, we show that x* is a fixed point of T. If y is any arbitrary point in H, we have:

(10)
x j * - y 2 = x j * - x * + x * - y 2 = x j * - x * 2 + x * - y 2 + 2 x j * - x * , x * - y .

Since xj*x*, then 2xj*-x*, x*-y0, (j).

So, based on the above inequality and Eq. (10), we get:

(11)
l i m j x j * - y 2 - x j * - x * 2 = x * - y 2 .

Setting y=Tx* in Eq. (11), we have:

(12)
l i m j x j * - T x * 2 - x j * - x * 2 = x * - T x * 2 .

Moreover, since xj*=(1-sj)x0j+sjTxj*, then:

(13)
T x j * - x j * = T x j * - ( 1 - s j ) x 0 j - s j T x j * = ( 1 - s j ) T x j * - x 0 j .

So, in Eq. (13) as j, for limjsj=1 we have:

(14)
T x j * - x j * 0 .

On the other hand, from Eq. (8) and since T is W-nonexpansive mapping, we have:

T x j * - T x * W ( x j * , x * ) T x j * - T x * x j * - x * .

Thus:

(15)
x j * - T x * x j * - T x j * + T x j * - T x * x j * - T x j * + x j * - x * ,

in turn:

(16)
x j * - T x * - x j * - x * x j * - T x j * .

Hence by Eq. (14), we have:

(17)
l i m j x j * - T x * - x j * - x * l i m j x j * - T x j * = 0 .

And, due to E is bounded, we have also:

(18)
l i m j x j * - T x * 2 - x j * - x * 2
          = l i m j x j * - T x * - x j * - x * x j * - T x * + x j * - x * 0 .

So, by Eq. (12), we get x*-Tx*2=0, that is, x* is fixed point of T.

Now, we provide a method for computation of that fixed point x*.

Theorem 2.2 Suppose all conditions of the Theorem 2.1 are satisfied. Then the Krasnoselskij iteration {xn}0 given by:

(19)
x n + 1 = 1 - s x n + s T x n ,         s { s j } j N ,         n = 0,1 , 2 , ,

converges to a fixed point of T.

Proof. Take the same x0E as Theorem 2.1, and such that W(x0,Tx0)1. From (w2) we get:

(20)
W ( x 0 , ( 1 - s ) x 0 + s T x 0 ) = W ( x 0 , x 1 ) 1 .

For W is triangular, so:

(21)
W ( T x 0 , x 1 ) 1 .

Since T is a W-admissible, from Eq. (20) we have:

(22)
W ( T x 0 , T x 1 ) 1 .

Once again for W is triangular, by Eqs. (21) and (22) we have W(x1,Tx1)1.

Also, from (w2) we have W(x1,(1-s)x1+sTx1)=W(x1,x2)1.

Continuously, we can obtain:

(23)
W x n , x m 1 ,       n , m N ,         n < m .

Hence:

(24)
W x 0 , x n 1 .

Also form Theorem 2.1, we know that x* is fixed point of T, and Based on all conditions of Theorem 2.1 are satisfied in Theorem 2.2, similarly we have:

(25)
W x 0 , x j * 1 ,
(26)
W x j * , x * 1 .

From Eqs. (25) and (26), for W is triangular, then:

(27)
W x 0 , x * 1 .

Also, by Eqs. (24) and (27), use W is triangular, we get:

(28)
W x n , x * 1 .

Based on Eq. (28), since T is W-nonexpansive mapping, then we have:

(29)
T x n - T x * W x n , x * T x n - T x * x n - x * .

So:

(30)
x n + 1 - x * = ( 1 - s ) ( x n - x * ) + s ( T x n - T x * )
            1 - s x n - x * + s T x n - T x *
            ( 1 - s ) ( x n - x * ) + s x n - x * = ( x n - x * ) .

Continuously, we have xn+1-x*x0-x*, which implies that {xn+1-x*} is monotone decrease bounded sequence. So limnxn+1-x* exists.

Next, we prove that xn-Txn0:

(31)
x n + 1 - x * 2 = 1 - s x n - x * - s T x n - T x * 2
            = ( 1 - s ) 2 x n - x * 2 + s 2 T x n - T x * 2 + 2 ( 1 - s ) s x n - x * , T x n - T x *
            ( 1 - s ) 2 x n - x * 2 + s 2 x n - x * 2 + 2 ( 1 - s ) s x n - x * , T x n - T x *
            = ( ( 1 - s ) 2 + s 2 ) x n - x * 2 + 2 ( 1 - s ) s x n - x * , T x n - T x * .

Also, on the other hand for any constant λ:

(32)
λ 2 x n - T x n 2 = x n - x * - T x n - T x * 2
            = λ 2 x n - x * 2 + λ 2 T x n - T x * 2 - 2 λ 2 x n - x * , T x n - T x *
            λ 2 x n - x * 2 + λ 2 x n - x * 2 - 2 λ 2 x n - x * , T x n - T x *
            = 2 λ 2 x n - x * 2 - 2 λ 2 x n - x * , T x n - T x *

Adding Eq. (31) to Eq. (32) and let λ2(1-s)s , we may obtain:

(33)
x n + 1 - x * 2 + λ 2 x n - T x n 2 ( 1 - s ) 2 + s 2 + 2 λ 2 x n - x * 2
            + ( 2 ( 1 - s ) s - 2 λ 2 ) x n - x * , T x n - T x *
            ( 1 - s ) 2 + s 2 + 2 λ 2 + 2 1 - s s - 2 λ 2 x n - x * 2 = x n - x * 2 .

It implies λ2xn-Txn2xn-x*2-xn+1-x*2.

Since limnxn+1-x* exists, in the above inequality let n, it results λ2xn-Txn20.

It means xn-Txn0.

For T is demicompact, it results that there exists a strongly convergent subsequence {xni}{xn} such that xnix*F(T), that is, xnix*0. Also xnx* is convergent, it implies that xnx*0. Hence that {xn} is convergent to x*F(T).

Acknowledgements

This work was supported by the Educational Science Foundation of Chongqing.

References

  1. Browder F. E. Nonexpansive nonlinear operators in a Banach space. Proceedings of the National Academy of Sciences of the United States of America, Vol. 54, 1965, p. 1041-1044. [Publisher]
  2. Kirk W. A. A fixed point theorem for mappings which do not increase distance. American Mathematical Monthly, Vol. 72, 165, p. 1004-1006. [Publisher]
  3. Browder F. E. Construction of fixed points of nonlinear mappings in Hilbert space. Journal of Mathematical Analysis and Applications, Vol. 20, Issue 2, 1967, p. 197-228. [Publisher]
  4. Amini-Harandi A., Fakhar M. Approximate fixed points of α-nonexpamsive mappings. Journal of Mathematical Analysis and Applications, Vol. 462, Issue 2, 2018, p. 1168-1173. [Publisher]
  5. Shimizu T., Takahashi W. Strong convergence to common fixed points of families of nonexpansive mapping. Journal of Mathematical Analysis and Applications, Vol. 222, Issue 1, 1997, p. 71-83. [Publisher]
  6. Tang J. F., Chang Shih-Sen General split feasibility problems for two families of nonexpansive mappings in Hilbert spaces. Acta Mathematica Scientia, Vol. 36, Issue 2, 2016, p. 602-603. [Publisher]
  7. O’hara J. G., Pillay P., Xu H. K. Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis: Theory, Methods and Applications, Vol. 54, Issue 8, 2003, p. 1417-1426. [Publisher]
  8. Lee B. S., Chang Shih-Sen Fixed points for nonexpansive fuzzy mappings in locally convex space. Fuzzy Sets and Systems, Vol. 76, Issue 2, 1995, p. 247-251. [Publisher]
  9. Hierro A. F. R. L. D., Shahzad N. New fixed point theorem under R-contractions. Fixed Point Theory and Applications, Vol. 2015, 2015, p. 176. [CrossRef]
  10. Samet B., Vetro C., Vetro P. Fixed point theorems for α-ψ-contractive type mappings. Nonlinear Analysis, Vol. 75, 2012, p. 2154-2165. [Publisher]
  11. Karapinar E., Samet B. Generalized α-ψ contractive type mappings and related fixed point theorems with applications. Abstract and Applied Analysis, Vol. 2012, 2012, p. 793486. [Publisher]