Let be a nonempty closed convex subset of a Hilbert space and let be a sequence of -strict pseudocontractions mappings on into itself with . Assume that Let be a bifunction satisfying (A1), (A2), (A3), (A4), and . Let and be sequence generated by and
Assume that for all , where is a small enough constant, and is a sequence in with Let for any bounded subset of and let be a mapping of into itself defined by and suppose that Then the sequences and converge weakly to an element of .
Pick . Then from the definition of in Lemma 2.4, we have , and therefore . It follows from (3.1) that
Since for all , we get that is, the sequence is decreasing. Hence exists. In particular, is bounded. Since is firmly nonexpensive, is also bounded. Also (3.2) implies that
Taking the limit as yields that
Since is bounded, it follows that
We apply Lemma 2.2 to get
Next, we claim that . Indeed, let be an arbitrary element of . Then as above
Therefore, from (3.2), we have
So, from the existence of , we have
Next, we claim that . since is bounded and is reflexive, is nonempty. Let be an arbitrary element. Then a subsequence of converges weakly to . Hence, from (3.11) we know that As , we obtain that . Let us show . Since , we have
By (A2), we have
From (A4), we have
Then, for and , from (A1), and (A4), we also have
Taking and using (A3), we get
and hence . Since is a strict pseudocontraction mapping, by Lemma 2.1() we know that the mapping is demiclosed at zero. Note that and . Thus, . Consequently, we deduce that . Since is an arbitrary element, we conclude that .
To see that and are actually weakly convergent, we take Since exist for every , by (2.2), we have
Hence and proof is completed.