Weak and strong convergence to common fixed points of non-self nonexpansive mappings

SupposeK is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach spaceE withP as a nonexpansive retraction. LetT ₁,T ₂ andT ₃:K → E be nonexpansive mappings with nonempty common fixed points set. Letα _n ,β _n ,γ _n ,α _n ^′ ,β _n ^′ ,γ _n ^′ ,α _n ^′′ ,β _n ^′′ andγ _n ^′′ be real sequences in 0, 1] such thatα _n +β _n +γ _n =α _n ^′ +β _n ^′ +γ _n ^′ =α _n ^′′ +β _n ^′′ +γ _n ^′′ = 1, starting from arbitraryx ₁ ∈ K, define the sequencex _n by $$\left\{ \begin{gathered} z_n = P(\alpha ''_n T_1 x_n + \beta ''_n x_n + \gamma ''_n w_n ) \hfill \\ y_n = P(\alpha _n^\prime T_2 z_n + \beta _n^\prime x_n + \gamma _n^\prime v_n ) \hfill \\ x_{n + 1} = P(\alpha _n T_3 y_n + \beta _n x_n + \gamma _n u_n ) \hfill \\ \end{gathered} \right.$$ with the restrictions $\sum\limits_{n = 1}^\infty {\gamma _n }< \infty , \sum\limits_{n = 1}^\infty \gamma _n^\prime< \infty ,\sum\limits_{n = 1}^\infty {\gamma ''_n }< \infty $ . (i) If the dual E* ofE has the Kadec-Klee property, then weak convergence of ax _n to somex* ∈ F(T ₁) ∩F(T ₂) ∩ (T ₃) is proved; (ii) IfT ₁,T₂ andT ₃ satisfy condition (A′), then strong convergence ofx _n to some x* ∈F(T ₁) ∩F(T ₂) ∩ (T ₃) is obtained.