In the sequel, we will use following results.
Lemma 2.1 [29]
Suppose that , are two sequences of nonnegative numbers such that, for some real number ,
for all . Then we have the following:
-
(1)
If , then exists.
-
(2)
If and has a subsequence converging to zero, then .
Lemma 2.2 [31]
For all and , the following well-known identity holds:
Now, we prove our main results.
Lemma 2.3 Let H be a Hilbert space. Then, for all , ,
(2.1)
where , , and .
Proof For any , , it can be easily seen that
(2.2)
Consider the following:
(2.3)
For all , we have
(2.4)
and
(2.5)
Substituting (2.4) and (2.5) in (2.3), we get
This completes the proof. □
Remark 2.1 Lemma 2.2 is now the special case of our result.
Theorem 2.1 Let K be a compact convex subset of a real Hilbert space H and , , be a family of continuous hemicontractive mappings. Let be such that and satisfying for some , .
Then, for arbitrary , the sequence defined by (1.9) converges strongly to a common fixed point in .
Proof Let . Using the fact that , are hemicontractive, we obtain
(2.6)
With the help of (1.9), Lemma 2.3 and (2.6), we obtain the following estimates:
(2.7)
Substituting (2.6) in (2.7), we get
Also, we have
(2.9)
Substituting (2.9) in (2.8), we get
which implies that
Thus, from the condition for some , , we obtain
(2.10)
for all fixed points . Moreover, we have
and thus, for all ,
Hence, for all , we obtain
(2.11)
for each , which implies that
for each . From (2.9), it further implies that
By the compactness of K, this immediately implies that there is a subsequence of which converges to a common fixed point of , say . Since (2.10) holds for all fixed points of , we have
and, in view of (2.11) and Lemma 2.1, we conclude that as , that is, as . This completes the proof. □
Theorem 2.2 Let H, K, , , be as in Theorem 2.1 and be such that and satisfying for some , .
If is the projection operator of H onto K, then the sequence defined iteratively by
for each converges strongly to a common fixed point in .
Proof The mapping is nonexpansive (see [2]) and K is a Chebyshev subset of H and so is a single-valued mapping. Hence, we have the following estimate:
which implies that
The set is compact and so the sequence is bounded. The rest of the argument follows exactly as in the proof of Theorem 2.1. This completes the proof. □
Theorem 2.3 Let K be a compact convex subset of a real Hilbert space H and , , be a family of Lipschitz hemicontractive mappings. Let be such that and satisfying for some , .
Then, for arbitrary , the sequence defined by (1.9) converges strongly to a common fixed point in .
Theorem 2.4 Let H, K, , , be as in Theorem 2.3 and be such that and satisfying for some , .
If is the projection operator of H onto K, then the sequence defined iteratively by
for each converges strongly to a common fixed point in .
Example For , we can choose the following control parameters: , and .