Composite implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps

Let E be a real Banach space and let K be a nonempty closed convex subset of E  . Let {Ti}i=1N be N strictly pseudocontractive self-maps of K   such that F=⋂i=1NF(Ti)≠∅, where F(Ti)={x∈K:Tix=x}F(Ti)={x∈K:Tix=x} and let {αn}n=1∞,{βn}n=1∞⊂[0,1] be two real sequence satisfying the conditions:(i) ∑n=1∞(1−αn)=+∞;  (ii) ∑n=1∞(1−αn)2<+∞;(iii) ∑n=1∞(1−βn)<+∞;  (iv) (1−αn)(1−βn)L2<1,∀n⩾1, where L⩾1L⩾1 is common Lipschitz constant of {Ti}i=1N. For x0∈Kx0∈K, let {xn}n=1∞ be new implicit process defined byxn=αnxn−1+(1−αn)Tnyn,xn=αnxn−1+(1−αn)Tnyn,yn=βnxn−1+(1−βn)Tnxnyn=βnxn−1+(1−βn)Tnxn where Tn=Tn mod NTn=Tn mod N, then(i)limn→∞‖xn−p‖ exists, for all p∈F;(ii)lim infn→∞‖xn−Tnxn‖=0.The results of this paper generalize and improve the results of Osilike in 2004. In this paper, the proof methods of the main results are also different from that of Osilike.