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18 Fixed Point Theory and Applications For t with 0
Fixed Point Theory and Applications 19 Therefore, we have w1 − tni ,g ≥ 〈w1 − tni ,Aw1〉 ≥ 〈w1 − tni ,Aw1〉 tni − uni − w1 − tni , Ayni λni w1 − tni ,Aw1 tni − uni − Ayni − λni w1 − tni ,Aw1 tni − uni − Atni Atni − Ayni − λni 〈w1 − tni ,Aw1− Atni〉 w1 − tni ,Atni− Ayni − ≥ w1 − tni ,Atni− Ayni − w1 − tni , tni − uni λni . v − tni , tni − uni λni 3.55 Hence we obtain 〈w1 − w, g〉 ≥0asi → ∞. Since T is maximal monotone, we have w ∈ T −10 and hence w ∈ VIC, A. We next show that w ∈ N i1 FixTi. To see this, we observe that we may assume by passing to a further subsequence if necessary λni,k → λk for k 1, 2,...,N. Let W be the W-mapping generated by T1,T2,...,TN and λ1,λ2,...,λN.ByLemma 2.5,weknowthatWis nonexpansive and N i1 FixTi FixW. it follows from Lemma 2.6 that Wnix −→ Wx, ∀x ∈ C. 3.56 Assume w /∈ FixW. Since xni ⇀wand w / Ww, it follows from the Opial condition, 3.42, and 3.56 that lim inf i →∞ xni − w < lim infxni − Ww i →∞ ≤ lim inf i →∞ {xni − Wnixni Wnixni − Wxni Wxni − Ww} ≤ lim infxni − w, i →∞ 3.57 which is a contradiction. Hence, we have w ∈ FixW N i1 FixTi. This implies w ∈ Ω. Therefore, we have lim sup fu0 − u0,xn − u0 lim fu0 − u0,xnj − u0 n →∞ j →∞ fu0 − u0,w− u0 ≤ 0. 3.58 Finally, we show that xn → u0, where u0 PΩfu0.
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