3.1 Strong convergence of an iterative algorithm for asymptotically k-strictly pseudocontractive maps
We now introduce the following iterative algorithm analogous to one studied in [13].
Modified averaging Mann algorithm Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a given mapping. For arbitrary , our iteration sequence is given by
(3.1)
where and are suitable real sequences in satisfying some appropriate conditions that will be made precise in our strong convergence theorem.
We now prove the following convergence theorem.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically k-strictly pseudocontractive mapping with a sequence such that . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
;
-
(c4)
, and for some ϵ.
Then the modified averaging iteration sequence generated from by (3.1) converges strongly to a fixed point of T.
Proof Observe that (1.3) is equivalent to each of the following inequalities:
(3.2)
(3.3)
Let be arbitrary. Then, using (1.3), (2.3), (3.1) and (3.3), we obtain
(3.4)
Hence
Since , it follows from (3.4) that is bounded. Hence is also bounded. Furthermore, it follows from (2.2) that
(3.5)
From (3.1) and (3.5) we obtain
(3.6)
Since is bounded, then
and hence using condition (c4) and (3.6) in (3.4), we obtain
(3.7)
Since is bounded, we have that there exists such that
(3.8)
From (3.7) and (3.8) we obtain
(3.9)
To complete the proof, we now consider the following two cases.
Case 1. Suppose that is a monotone sequence, then we may assume that is monotone decreasing. Then exists and it follows from (3.9), conditions (c1) and that
(3.10)
Furthermore,
Hence
and
Observe also that since T is uniformly L-Lipschitzian, we obtain
(3.11)
Furthermore,
(3.12)
Since , then the demiclosedness property of , (2.4) and the usual standard argument yield that and converge weakly to some .
Since , and since , , and for some , then using (3.4) we obtain
(3.13)
Thus
where , and , with . Since converges weakly to , then , and this together with condition (c1) (i.e., ) implies that as . It now follows from Lemma 2.1 that converges strongly to . Consequently, converges strongly to .
Case 2. Suppose that is not a monotone decreasing sequence, then set , and let be a mapping defined for all for some sufficiently large by
Then τ is a non-decreasing sequence such that as and for . Using (c1) and (c2) in (3.9), we obtain
(3.14)
Following the same argument as in Case 1, we obtain
As in Case 1, we also obtain that and converge weakly to some in . Furthermore, for all , we obtain from (3.13) that
(3.15)
It follows from (3.15) that
Thus
Furthermore, for , we have if (i.e., ), because for . It then follows that for all we have
This implies , and hence converges strongly to . □
Corollary 3.1 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H with . Let be an asymptotically k-strictly pseudocontractive mapping with a sequence such that . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
;
-
(c4)
, and for some ϵ.
Then the modified averaging iteration sequence , generated from by
converges strongly to a fixed point of T.
Remark 3.1 Prototypes for our real sequences and are:
Corollary 3.2 Let C be a nonempty closed convex subset of a real Hilbert space, and let be an asymptotically nonexpansive mapping with a sequence . Let , and let and be sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, and for some ϵ.
Then the modified averaging iteration sequence , generated from by (3.1), converges strongly to a fixed point of T.
3.2 Demiclosedness principle and strong convergence results for uniformly Lipschitzian asymptotically pseudocontractive maps
Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. For uniformly L-Lipschitzian asymptotically pseudocontractive maps , we first prove that is demiclosed at 0 and then introduce a modified averaging Ishikawa iteration algorithm and prove that it converges strongly to a fixed point of without any compactness assumption on T or C and without further requirement on . Our demiclosedness principle does not require the boundedness of C imposed in the result of [18].
Theorem 3.2 Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping. Then is demiclosed at 0.
Proof Let be a sequence in C which converges weakly to p and converges strongly to 0. We prove that . Since converges weakly, it is bounded. For each , define by
(3.16)
Observe that for arbitrary but fixed integer , we have
Set
where , and . Then
and
Hence
Also
From (2.4) we obtain
Thus
and hence
(3.17)
Observe that
(3.18)
Equations (3.17) and (3.18) imply that
from which it follows that
Thus
Hence
It now follows that as . Since T is continuous, we have that as , and hence . □
We now introduce the following iterative algorithm for uniformly L-Lipschitzian asymptotically pseudocontractive maps.
Modified averaging Ishikawa algorithm For arbitrary , the sequence is given by
(3.19)
Theorem 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence , and . Let , and be real sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, where ;
-
(c4)
.
Then the sequence generated from an arbitrary by (3.19) converges strongly to a fixed point of T.
Proof Since T is asymptotically pseudocontractive, it follows that
(3.20)
and
(3.21)
Set
Then, for arbitrary , we obtain
Thus
(3.22)
From (3.22) we obtain
(3.23)
and
(3.24)
Observe that
(3.25)
Hence
and it follows, as in the proof of Theorem 3.1, that is bounded. Observe that
Hence
(3.26)
Furthermore,
Thus
(3.27)
Observe also that
(3.28)
Using (3.26) and (3.28), we obtain
It now follows from (3.27) that
(3.29)
Using (3.29) in (3.25), we obtain
(3.30)
Since is bounded, we have that there exists such that
(3.31)
From (3.30) and (3.31) we obtain
(3.32)
To complete the proof, we now consider the following two cases.
Case 1. Suppose that is a monotone sequence, then we may assume that is monotone decreasing. Then exists and it follows from (3.32), conditions (c1) and that
(3.33)
Furthermore,
Hence
Furthermore,
Thus
and
Observe also that since T is uniformly L-Lipschitzian, we obtain
(3.34)
Furthermore,
(3.35)
Since , then the demiclosedness property of , (2.4) and the usual standard argument yield that and converge weakly to some . Since , , and for some , then using (3.25) we obtain
(3.36)
Thus
where as , and with . It now follows from Lemma 2.1 that converges strongly to . Consequently, converges strongly to .
Case 2. Suppose that is not a monotone decreasing sequence, then set , and let be a mapping defined for all for some sufficiently large by
Then τ is a non-decreasing sequence such that as and for . Using (c1) and (c2) in (3.32), we obtain
(3.37)
Following the same argument as in Case 1, we obtain
As in Case 1 we also obtain that and converge weakly to some in .
Furthermore, for all , we obtain from (3.36) that
(3.38)
It follows from (3.38) that
Thus
Furthermore, for , we have if (i.e., ), because for . It then follows that for all we have
This implies , and hence converges strongly to . □
Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H with , and let be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence , and . Let , and be real sequences in satisfying the conditions:
-
(c1)
;
-
(c2)
;
-
(c3)
, where ;
-
(c4)
.
Then the sequence
generated from an arbitrary
by
converges strongly to a fixed point of T.