In this section, we present our Korpelevich-like algorithm and consequently we will show its strong convergence.
3.1 Conditions assumptions
(A1) E is a uniformly convex and 2-uniformly smooth Banach space with a weakly sequentially continuous duality mapping;
(A2) C is a nonempty closed convex subset of E;
(A3) is an α-strongly accretive and L-Lipschitz continuous mapping with ;
(A4) is a sunny nonexpansive retraction from E onto C.
3.2 Parameters restrictions
(P1) λ, μ and γ are three positive constants satisfying:
-
(i)
, for some a, b with ;
-
(ii)
where K is the smooth constant of E.
(P2) is a sequence in such that and .
Algorithm 3.1 For given , define a sequence iteratively by
(3.1)
Theorem 3.2 The sequence generated by (3.1) converges strongly to , where is a sunny nonexpansive retraction of E onto .
Proof Let . First, from Lemma 2.2, we have for all . In particular, for all .
Since is α-strongly accretive and L-Lipschitzian, it must be -inverse-strongly accretive mapping. Thus, by Lemma 2.4, we have
Since and , we get for enough large n. Without loss of generality, we may assume that for all , , i.e., . Hence, is nonexpansive.
From (3.1), we have
(3.2)
By (3.1) and (3.2), we have
(3.3)
Hence, is bounded.
Set . From (3.1), we have for all . Then we have
and thus
It follows that
This together with Lemma 2.6 implies that
From (3.2), we have
(3.4)
From (3.1), (3.3) and (3.4), we obtain
Therefore, we have
Since and , we obtain
It follows that
Since A is α-strongly accretive, we deduce
which implies that
that is,
It follows that
(3.5)
Next, we show that
(3.6)
To show (3.6), since is bounded, we can choose a sequence of converging weakly to z such that
(3.7)
We first prove . It follows that
(3.8)
By Lemma 2.5 and (3.8), we have , it follows from Lemma 2.3 that .
Now, from (3.7) and Lemma 2.2, we have
Noticing that , we deduce that
Since and for all , we can deduce from Lemma 2.2 that
and
Therefore, we have
which implies that
(3.9)
Finally, we will prove that the sequence . As a matter of fact, from (3.1) and (3.9), we have
Applying Lemma 2.6 to the last inequality, we conclude that converges strongly to . This completes the proof. □