1Department of Mathematics Yangtze Normal University, Chongqing, China
Vibroengineering PROCEDIA, Vol. 28, 2019, p. 201-205.
Received 19 September 2019; accepted 1 October 2019; published 19 October 2019
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) , Kirk (1965)  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 be a nonempty set, the function is called triangular if for all if or then
Definition 1.2 Let be a metric space and be a given mapping, if there exists a function such that , , then we say that is a -nonexpansive mapping.
Clearly, any nonexpansive mapping is a -nonexpansive mapping with for all .
Definition 1.3 Let be a mapping and be a function. We say that is a -admissible if ,
Definition 1.4  Let be a Hilbert space, is called demicompact if whenever is bounded and strongly convergent, then there exists a subsequence of which is strongly convergent.
Next our main results are presented.
2. Main results
Theorem 2.1 Let be a bounded closed convex subset of a Hilbert space , is triangular function, is a -nonexpansive mapping and it is -admissible. If the following conditions are satisfied:
(w1) there exists such that ;
(w2) there exists a sequence with such that for all , if , then , ;
(w3) if is satisfied moreover or then
Then has a fixed point.
Proof. Let such that . Take for all , , there . Now we fix , for each , from (w2), we may obtain
Also, for is -admissible, then is obtained. According to is a triangular function and (w1), then
Once again use (w2), then is also obtained. Continuously, we easily obtain:
Based on that is triangular, we may get:
So from Eq. (2) and for is -nonexpansive, we have:
Let , for is bounded we may get , hence is Cauchy sequence, it means there exists such that convergent to , that is:
Also, from Eq. (1) and (w3), we have:
Once again by Eq. (1), for is triangular, so we have:
Since is bounded, closed and convex in Hilbert , then it is weakly compact. Hence there exists a such that:
From Eqs. (6, 7), applying (w3) we have:
Next, we show that .
Indeed, according to , is -nonexpiansive and Eq. (5), we have:
Let in Eq. (9), utilize Eq. (4) we obtain , it implies that .
Finally, we show that is a fixed point of . If is any arbitrary point in , we have:
Since , then ,
So, based on the above inequality and Eq. (10), we get:
Setting in Eq. (11), we have:
Moreover, since , then:
So, in Eq. (13) as , for we have:
On the other hand, from Eq. (8) and since is -nonexpansive mapping, we have:
Hence by Eq. (14), we have:
And, due to is bounded, we have also:
So, by Eq. (12), we get , that is, is fixed point of .
Now, we provide a method for computation of that fixed point .
Theorem 2.2 Suppose all conditions of the Theorem 2.1 are satisfied. Then the Krasnoselskij iteration given by:
converges to a fixed point of .
Proof. Take the same as Theorem 2.1, and such that . From (w2) we get:
For is triangular, so:
Since is a -admissible, from Eq. (20) we have:
Once again for is triangular, by Eqs. (21) and (22) we have
Also, from (w2) we have
Continuously, we can obtain:
Also form Theorem 2.1, we know that is fixed point of , and Based on all conditions of Theorem 2.1 are satisfied in Theorem 2.2, similarly we have:
From Eqs. (25) and (26), for is triangular, then:
Also, by Eqs. (24) and (27), use is triangular, we get:
Based on Eq. (28), since is -nonexpansive mapping, then we have:
Continuously, we have , which implies that is monotone decrease bounded sequence. So exists.
Next, we prove that :
Also, on the other hand for any constant :
Adding Eq. (31) to Eq. (32) and let , we may obtain:
Since exists, in the above inequality let it results
For is demicompact, it results that there exists a strongly convergent subsequence such that , that is, . Also is convergent, it implies that . Hence that is convergent to .
This work was supported by the Educational Science Foundation of Chongqing.
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