To introduce our iterative algorithm for solving the split feasibility problem in real Hilbert spaces, firstly, we shall assume that problem (1) is consistent, namely, its solution set, denoted by S, is nonempty. Secondly, we need to define a special W-mapping Wnn≥1 as follows:(17)Un,n+1=IUn,n=1−ηnI+ηnPCUn,n+1Un,n−1=1−ηn−1I+ηn−1PCUn,n⋮Un,k=1−ηkI+ηkPCUn,k+1Un,k−1=1−ηk−1I+ηk−1PCUn,k⋮Un,2=1−η2I+η2PCUn,3Un,1=1−η1I+η1PCUn,2Wn=Un,1,where ηi=1/n−i+11≤i≤n. From Lemma 4, we know that Wn is nonexpansive.
Now, we will introduce our iterative algorithms for the split feasibility problem.
Proof.Since ψ:H1⟶H1 is a contraction mapping with δ∈0,1 and the fact that PS is nonexpansive, it is clear that PSψ:H1⟶S is also a contraction mapping. By Banach fixed point theorem, there exists x∗∈S, such that x∗=PSψx∗.
Since x∗∈S, that is, x∗∈C and Ax∗∈Q. By the definition of Wn, we have x∗=Wnx∗. In what follows, we will divide the proof into four steps.
Firstly, we prove that the sequence xn is bounded.
From (18) and Lemmas 1 and 3, we have(19)yn−x∗2=xn−x∗−τnxn−PCxn+A∗I−PQAxn2=xn−x∗2−2τnxn−x∗,xn−PCxn+Axn−PQAxn,Axn−Ax∗+τn2xn−PCxn+A∗I−PQAxn2≤xn−x∗2−2τnxn−PCxn2+Axn−PQAxn2+τn21+A2xn−PCxn2+1+1A2Axn−PQAxn2≤xn−x∗2−τn2−τn1+A2xn−PCxn2+Axn−PQAxn2.
By condition (1), we have τn2−τn1+A2>0, so yn−x∗≤xn−x∗. Therefore, from (18), we obtain(20)xn+1−x∗=αnψxn−x∗+1−αnWnyn−x∗≤αnψxn−x∗+1−αnWnyn−x∗≤αnψxn−ψx∗+αnψx∗−x∗+1−αnWnyn−x∗≤αnδxn−x∗+αnψx∗−x∗+1−αnyn−x∗≤αnδxn−x∗+αnψx∗−x∗+1−αnxn−x∗=1−αn1−δxn−x∗+αn1−δψx∗−x∗1−δ≤maxxn−x∗,ψx∗−x∗1−δ.
By introduction, we obtain(21)xn+1−x∗≤maxx0−x∗,ψx∗−x∗1−δ,for all n≥0. The above inequality implies that the sequence xn is bounded. Combining with (18), we know that yn, Wnyn, and ψxn are also bounded.
Secondly, we show that the following inequality holds:(22)xn+1−x∗2≤1−αn˜xn−x∗2+αn˜δn,where αn˜=αn1−δ2 and(23)δn=2αnψx∗−x∗,xn+1−x∗−tnxn−PCxn2+Axn−PQAxn2αn1−δ2,with tn=τn1−αn2−τn1+A2.
From equations (18) and (19) and Lemma 3, we have(24)xn+1−x∗2=αnψxn−ψx∗+1−αnWnyn−x∗+αnψx∗−x∗2≤αnψxn−ψx∗+1−αnWnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnψxn−ψx∗2+1−αnWnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnδ2xn−x∗2+1−αnyn−x∗2+2αnψx∗−x∗,xn+1−x∗≤αnδ2xn−x∗2+1−αnxn−x∗2−τn1−αn2−τn1+A2×xn−PCxn2+Axn−PQAxn2+2αnψx∗−x∗,xn+1−x∗=1−αn1−δ2xn−x∗2+αn1−δ22ψx∗−x∗,xn+1−x∗1−δ2−τn1−αn2−τn1+A2xn−PCxn2+Axn−PQAxn2αn1−δ2.
So, inequality (22) holds.
Thirdly, we show that limsupn⟶∞δn is finite.
Since xn is bounded, we have(25)δn≤2ψx∗−x∗,xn+1−x∗1−δ2≤2ψx∗−x∗⋅xn+1−x∗1−δ2<∞.
This implies that limsupn⟶∞δn<∞. Next, we will show that lim supn⟶∞δn<−1 by contraction. If we assume that limsupn⟶∞δn<−1, then there exists n0∈N, such that δn≤−1 for all n≥n0. From (22), we obtain(26)xn+1−x∗2≤1−αn˜xn−x∗2+αn˜δn≤1−αn˜xn−x∗2−αn˜≤xn−x∗2−αn˜.
By introduction, we have(27)xn+1−x∗2≤xn0−x∗2−∑i=n0nαi˜.
Since ∑i=0∞αi=∞, ∑i=0∞αi˜=1−δ2∑i=0∞αi=∞, then there exists N>n0, such that ∑i=n0Nαi˜>xn0−x∗2. Combining with the last inequality, we have(28)xN+1−x∗2≤xn0−x∗2−∑i=n0Nαi˜<0,which is contradicted with the fact that xN+1−x∗2 is nonnegative. Thus, limsupn⟶∞δn≥−1. So, limsupn⟶∞δn is finite.
Lastly, we show that limsupn⟶∞δn≤0.
Since limsupn⟶∞δn is finite, there exists a subsequence nk such that(29)limsupn⟶∞δn=limk⟶∞δnk=limk⟶∞2αnkψx∗−x∗,xnk+1−x∗αnk1−δ2−tnkxnk−PCxnk2+Axnk−PQAxnk2αnk1−δ2.
Since ψx∗−x∗,xnk+1−x∗ is bounded, without loss of generality, we may assume the limit of ψx∗−x∗,xnk+1−x∗ exists. From (29), we may also assume the following limit exists:(30)limk⟶∞tnkxnk−PCxnk2+Axnk−PQAxnk2αnk1−δ2.
These conditions limn⟶∞αn=0,0<ɛ≤τnk≤2/1+A2−ɛ and tn=τn1−αn2−τn1+A2 imply tnk/αnk1−δ2⟶∞k⟶∞. So, we obtain(31)limk⟶∞xnk−PCxnk2+Axnk−PQAxnk2=0,that is,(32)limk⟶∞xnk−PCxnk=limk⟶∞Axnk−PQAxnk2=0.
Next, we prove that any weak cluster point of the sequence xnk is a solution of the SFP (1).
Since xnk is bounded, let x¯ be a weak cluster point of the sequence xnk; without loss of generality, we assume that xnk⇀x¯; then, we obtain Axnk⇀Ax¯. From the fact that PC and PQ are nonexpansive, Lemma 2 implies I−PC and I−PQ are demiclosed at zero; from (32), we obtain x¯=PCx¯ and Ax¯=PQAx¯, i.e., x¯∈C,Ax¯∈Q, hence x¯∈S.
Finally, we show that xnk+1−xnk⟶0k⟶∞.
From (18) and the definition of Wn, we know(33)Wnk−ynk=1−η1ynk+η1PCUnk,2ynk−ynk≤η1ynk−PCUnk,2ynk=ynk−PCUnk,2ynknk⟶0, k⟶∞,and, from (32), we have(34)ynk−xnk=τnkxnk−PCxnk+A∗I−PQAxnk≤τnkxnk−PCxnk+A⋅Axnk−PQAxnk⟶0, k⟶∞.
So,(35)xnk+1−xnk=αnkψxnk+1−αnkWnkynk−xnk≤αnkψxnk−xnk+1−αnkWnkynk−xnk+1−αnkynk−xnk⟶0, k⟶∞.
This implies that any weak cluster point of xnk+1 also belongs to S. Without loss of generality, we assume that xnk+1 converges weakly to x^∈S. Now, combing (29), Lemma 1, and the fact that x∗=PSψx∗, we can obtain(36)limsupn⟶∞δn≤limk⟶∞2ψx∗−x∗,xnk+1−x∗1−δ2=2ψx∗−x∗,x^−x∗1−δ2≤0.
From Lemma 5, we get limn⟶∞xn−x∗=0, which ends the proof.
From Theorem 1, we obtain the following subresult on the split feasibility problem (1).