In this paper, we suppose that ψ from the definition of -contraction is continuous, strictly increasing. Let for all , we have that and is strictly increasing and onto. Consequently, we have that and is a bijection on .
Theorem 3.1 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition , and f be a Meir-Keeler contraction of C into itself. Let X be a left invariant -stable subspace of containing 1, be a strongly left regular sequence of means on X such that and be the sequence defined in (2.4) with . Suppose the sequences , , and in satisfy , . The following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
For arbitrary , generate a sequence by
(3.1)
Then converges strongly to , which is the unique solution of the variational inequality
Equivalently, we have , where P is the unique sunny nonexpansive retraction of C onto .
Proof First, we prove that is bounded. That is, if we take a point , we will show that for all . Let . It is obvious that . By induction we suppose that , where . Since is nonexpansive, we have
(3.2)
Since f is a Meir-Keeler contraction, we have that
That is, is bounded. By induction we have that is bounded, and so are the sequences , and . As is bounded, we have is also bounded. Denote , then it follows that
Since , we obtain that
(3.3)
Next, we will show that and by Lemma 2.3, we observe that
Setting , we see that
Then we compute
It follows that
From (i), (ii), (iv), (3.3) and Lemma 2.3, we have
Applying Lemma 2.7, we obtain and also
That is,
(3.4)
Next, we will show that the set of all limit points of is a subset of . Note that
It follows that
From (i), (ii), (iv) and (3.4), we have
(3.5)
Let p be a limit point of and be a subsequence of converging strongly to p. From Lemma 2.3, we obtain that
From (3.5), (2.4) and Corollary 2.2, we get .
We know that there exists a unique sunny nonexpansive retraction P of C onto , and from the Banach contraction mapping principle, we known that Pf has a unique fixed point q which by (2.3) is the unique solution of
(3.6)
Let be a subsequence of converging to . From the smoothness of E and (3.6), we have that
(3.7)
Finally, we show that the sequence converges strongly to . Now, we have
(3.8)
From Lemma 2.6, (3.8) and (2.2), we have
It follows that
where and . Now, from (i), (iii), (iv), (3.7) and Lemma 2.8, as . This proof is completed. □
Corollary 3.2 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition , and f be an α-contraction of C into itself. Let X be a left invariant -stable subspace of containing 1, be a strongly left regular sequence of means on X such that and be the sequence defined by (2.4) with . Suppose the sequences , , and in satisfy , . The following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
For arbitrary given , generate a sequence by
(3.9)
Then converges strongly to , which is the unique solution of the variational inequality
Equivalently, we have , where P is the unique sunny nonexpansive retraction of C onto .
Corollary 3.3 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition , and f be a Meir-Keeler contraction of C into itself. Let X be a left invariant -stable subspace of containing 1, be a strongly left regular sequence of means on X such that and be the sequence defined by (2.4) with . Suppose the sequences , and in satisfy , . The following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
.
For arbitrary , generate a sequence by
(3.10)
Then converges strongly to , which is the unique solution of the variational inequality
Equivalently, we have , where P is the unique sunny nonexpansive retraction of C onto .
Corollary 3.4 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and be a representation of S as Lipschitzian mappings from C into itself, with the uniform Lipschitzian condition , and f be an α-contraction of C into itself. Let X be a left invariant -stable subspace of containing 1, be a strongly left regular sequence of means on X such that and be the sequence defined by (2.4) with . Suppose the sequences , and in satisfy , . The following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
.
For arbitrary , generate a sequence by
(3.11)
Then converges strongly to , which is the unique solution of the variational inequality
Equivalently, we have , where P is the unique sunny nonexpansive retraction of C onto .
Corollary 3.5 Let C be a nonempty, compact and convex subset of a smooth Banach space E. Let S be a left reversible semigroup and be a representation of S as nonexpansive mappings from C into itself and f be an α-contraction of C into itself. Let X be a left invariant -stable subspace of containing 1, be a strongly left regular sequence of means on X such that and be the sequence defined by (2.4) with . Suppose the sequences , , and in satisfy , . The following conditions are satisfied:
-
(i)
and ;
-
(ii)
;
-
(iii)
;
-
(iv)
.
For arbitrary , generate a sequences by
(3.12)
Then converges strongly to , which is the unique solution of the variational inequality
Equivalently, we have , where P is the unique sunny nonexpansive retraction of C onto .