A Unified Iterative Treatment for Solutions of Problems of Split Feasibility and Equilibrium in Hilbert Spaces
We at first raise the so called split feasibility fixed point problem which covers the problems of split feasibility, convex feasibility, and equilibrium as special cases and then give two types of algorithms for finding solutions of this problem and establish the corresponding strong convergence theorems for the sequences generated by our algorithms. As a consequence, we apply them to study the split feasibility problem, the zero point problem of maximal monotone operators, and the equilibrium problem and to show that the unique minimum norm solutions of these problems can be obtained through our algorithms. Since the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, each type of our algorithms can be considered as a generalized methodology for solving the aforementioned problems.
Throughout this paper, denotes a real Hilbert space with inner product and the norm , the identity mapping on , the set of all natural numbers, and the set of all real numbers. For a self-mapping on , denotes the set of all fixed points of . If is a set-valued mapping; then denotes its domain, that is, .
Let and be nonempty closed convex subsets of two Hilbert spaces and , respectively, and let be a bounded linear mapping. The split feasibility problem (SFP) is the problem of finding a point with the property The SFP was first introduced by Censor and Elfving  for modeling inverse problems which arise from phase retrievals and medical image reconstruction. Recently, it has been found that the SFP can also be used to model the intensity-modulated radiation therapy. For details, the readers are referred to Xu  and the references therein.
Start with any and generate a sequence through the iteration where , the adjoint of , and and are the metric projections onto and , respectively.
The sequence generated by the algorithm (2) converges weakly to a solution of SFP(1), cf. [2–4]. Under the assumption that SFP(1) has a solution, it is known that a point solves SFP(1) if and only if is a fixed point of the operator cf. , where Xu also proposed the regularized method, and proved that the sequence converges strongly to the minimum norm solution of SFP(1) provided that the parameters and verify some suitable conditions. This regularized method was further investigated by Yao et al.  and Yao et al. .
Putting and , SFP(1) is of the forms: As a metric projection is firmly nonexpansive, it is reasonable to require the and in (5) to be firmly nonexpansive only and call it a split feasibility fixed point problem (SFFP). Many interesting problems in the literature can be described as SFFP.
(i) A set-valued map is called monotone if for all and for any , . is said to be maximal monotone if its graph is not properly contained in the graph of any other monotone operator. A point is called a zero point of a maximal monotone operator if . The set of all zero points of is denoted by , which is equal to for any , where denotes the resolvent of a monotone operator ; that is, for any . It is known that for any , is firmly nonexpansive. Now, let and be two maximal monotone operators on and , respectively. Replacing and with and , respectively, in (1), the SFP becomes a SFFP: Putting , the previous SFFP is reduced to the common zero point problem of two maximal monotone operators:
(ii) Let . An equilibrium problem is the problem of finding such that whose solution set is denoted by . For solving an equilibrium problem, we usually assume that the function satisfies the following conditions:(A1) , for all ;(A2) is monotone, that is, , for all ;(A3) for all , ;(A4) for all is convex and lower semicontinuous.Blum and Oettli  and Aoyama et al.  showed that there exists a unique such that Moreover, For , define by for all . Combettes and Hirstoaga  showed that there hold(a) is single-valued;(b) is firmly nonexpansive;(c);(d) is closed and convex.
Now, let and be two functions satisfying conditions (A1)–(A4). Replacing and with and , respectively, in (1), the SFP becomes a SFFP: Putting , , and , the previous SFFP is reduced to the common equilibrium problem:
(iii) When , and the bounded linear operator is the identity mapping, SFP(1) is reduced to the convex feasibility problem (CFP): which in turn can be described as a SFFP: where and . Although SFFP(5) contains CFP as a special case, it cannot cover the multiple-set split convex feasibility problem described in .
Let and be two families of firmly nonexpansive self-mappings on and , respectively, so that and and obtain the following main result.
Let , , , and be sequences in with and for all . Let be a sequence in and let be a bounded sequence in . Suppose that the solution set of SFFP(16) is nonempty. For any , start with an arbitrary and define a sequence byThen, the sequence converges strongly to provided that the following conditions are satisfied: (i), , ;(ii), ;(iii)there are two nonnegative real-valued functions and on with
Based on the concept of using contractions to approximate nonexpansive mappings, another type of algorithms for SFFP(5) is also introduced, and the corresponding strong convergence theorem for the sequence generated by such algorithm is given too.
In Section 5, since resolvents of monotone operators are firmly nonexpansive, we replace the sequences and of firmly nonexpansive mappings in the previous condition (iii) by two sequences of resolvents of maximal monotone operators. Then, the proposed algorithm becomes a scheme to approach the minimum norm solution of zero point problem of maximal monotone operators and the equilibrium problem. It is worth noting that as Blum and Oettli  showed that the variational inequalities, convex differentiable optimization, and Nash equilibria in noncooperative games can be formulated as equilibrium problems, the proposed algorithm can be considered as a generalized methodology for solving all aforementioned problems.
In order to facilitate our investigation in this paper, we recall some basic facts. A mapping is said to be(i)nonexpansive if (ii)firmly nonexpansive if (iii)-averaged by if for some and some nonexpansive mapping ;(iv)-inverse strongly monotone (-ism), with , if If is nonexpansive, then the fixed point set of is closed and convex, cf. . If is averaged, then is nonexpansive with . It is well known that is firmly nonexpansive if and only if it is -averaged, cf. , and so is . Here we would like to mention that the term “averaged mapping” originated in [12, 13]. In , Baillon el at. showed that if is a -averaged mapping by on a nonempty closed convex subset of a uniformly convex Banach space, then if and only if for all in . Moreover, in , Bruck and Reich showed that if the above satisfies and is odd, then converges strongly to a fixed point of .
Let be a nonempty closed convex subset of . The metric projection from onto is the mapping that assigns each the unique point in with the property It is known that is firmly nonexpansive and characterized by the inequality: for any ,
We need some lemmas that will be quoted in the sequel.
Lemma 1 (see ). If is a self-mapping on , then the following assertions hold.(a) is nonexpansive if and only if the complement is -averaged.(b)If is -ism and , then is ()-ism.(c) is -averaged if and only if is ()-ism.(d)If is -averaged and is -averaged, then the composite is -averaged.(e)If and are averaged on so that , then
For , the resolvent of a maximal monotone operator on has the following properties.
Lemma 2. Let be a maximal monotone operator on . Then,(a) is single-valued and firmly nonexpansive;(b) and ;(c) (the resolvent identity) for , the following identity holds:
Lemma 3. Let . Then,(a);(b)for any , (c)for with ,
Lemma 4 (see  (demiclosedness principle)). Let be a nonexpansive self-mapping on and suppose that is a sequence in such that converges weakly to some and . Then, .
Lemma 5 (see ). Let be a sequence of nonnegative real numbers satisfying where , , and verify the following conditions:(i), ;(ii);(iii) and .Then, .
Lemma 6 (see ). Let be a sequence in that does not decrease at infinity in the sense that there exists a subsequence such that For any , define . Then, as and , for all .
3. Weak Convergence Theorems
In this section, we at first transform SFFP(5) into a fixed point problem for the operator , where is any positive real number. And then use fixed point algorithms to solve SFFP(5). From now on until the end of this paper, unless we state specifically, and , (resp., and , ) are firmly nonexpansive self-mappings on (resp., ), and denotes a bounded linear operator from into with adjoint .
Lemma 7. Let be a firmly nonexpansive self-mapping on with . Then, for any , one has
Proof. Since and is firmly nonexpansive, we have and hence,
Although , for all is similar to the characterization inequality (24) for the metric projection , as needs not to be in , it is in general different from . For example, let for all , which is obviously firmly nonexpansive with . Thus, for all , while for all .
Proposition 8. For any , the operator is -averaged and is -averaged.
Proof. Using the fact that is firmly nonexpansive, it is routine to show that is -ism, and so is -ism by Lemma 1(b). Thus, Lemma 1(c) shows that is -averaged. As is firmly nonexpansive, it is -averaged. Therefore, is -averaged by Lemma 1(d).
Proposition 9. Let be the solution set of SFFP(5); that is, . For any , let . Suppose that . Then, .
Proof. If solves SFFP(5), we have and . Now, note that implies
which means that . Consequently, . This shows that .
For the inverse inclusion, let be any member of and pick . It is readily seen from that Since , an application of Lemma 7 yields which together with (36) implies that This comes to conclude that , and hence once we note that . Finally, since by assumption, we have . Thus, follows from Lemma 1(e).
Proposition 12 (see ). Let be a -averaged self-mapping on with and assume that is a sequence in such that Then, for any , the sequence generated by the Mann's algorithm converges weakly to a fixed point of .
4. Strong Convergence Theorems
Lemma 14. For any and all , one has
Proof. Since is firmly nonexpansive, so is . Hence, for all , we have Consequently,
Theorem 15. Let , , , and be sequences in with and for all . Let be a sequence in and let be a bounded sequence in . Suppose that the solution set of SFFP(16) is nonempty. For any , start with any and define a sequence by Then the sequence converges strongly to provided that the following conditions are satisfied: (i), , ;(ii), ;(iii)there are two nonnegative real-valued functions and on with
Proof. Put . For simplicity, put . In view of Proposition 9, is nonexpansive, so
from which follows that is a bounded sequence. Taking into account Lemma 3, we get
Meanwhile, we have by Lemma 14 that
Therefore, we deduce that
We now carry on with the proof by considering the following two cases: (I) is eventually decreasing and (II) is not eventually decreasing.
Case I. Suppose that is eventually decreasing; that is, there is such that is decreasing. In this case, exists in . From inequality (52) we have which together with the boundedness of and conditions (i) and (ii) implies Then, an application of condition (iii) follows that for all , Since is bounded, it has a subsequence such that converges weakly to some and where the last inequality follows from (24) since by Lemma 4. Choose so that . From (52) we have Accordingly, because of (56) and condition (i), we can apply Lemma 5 to inequality (57) with , , , and to conclude that
Case II. Suppose that is not eventually decreasing. In this case, by Lemma 6, there exists a nondecreasing sequence in such that and Then, it follows from (52) and (59) that Therefore, and then proceeding just as in the proof in Case I, we obtain which in conjunction with condition (iii) shows that for all and then follows that From (60) we have and thus, Letting and using (64) and condition (i), we obtain Also, since which together with (62) and condition (i) implies that , and so by virtue of (67). Consequently, we conclude that via (59) and (69). This completes the proof.
This theorem says that the sequence converges strongly to a point of which is the nearest to . In particular, if is taken to be , then the limit point of the sequence is the unique minimum solution of SFFP(16).
Corollary 16. Let , , and be sequences in with and for all . Let be a sequence in and let be a bounded sequence in . Suppose that the solution set of SFFP(16) is nonempty. For any , start with any and define a sequence by Then, the sequence converges strongly to provided that the following conditions are satisfied: (i), , ;(ii), ;(iii)there are two nonnegative real-valued functions and on with(iv)either or .
If the sequence (resp., ) of firmly nonexpansive mappings consists of a single mapping (resp., ), then and obviously verify condition (iii), and hence, we have the following corollary.
Corollary 17. Let , , , and be sequences in with and for all . Let be a sequence in and let be a bounded sequences in . Assume that the solution set of SFFP(5) is nonempty. For any , start with an arbitrary and define the sequence by Then, converges strongly to provided that the following conditions are satisfied: (i), , ;(ii), .
When the sequence is taken to be a constant , then because is an averaged mapping, we can apply Corollary 3.4 of Huang and Hong  to obtain the following result.
Theorem 18. Let , , , and be sequences in with and for all . Suppose that and suppose that is a bounded sequence in . Assume that the solution set of SFFP(5) is nonempty. For any , start with an arbitrary and define the sequence by Then, converges strongly to provided that the following conditions are satisfied: (i), , ;(ii), .
Theorem 19. Let be a sequence in . Suppose that and assume that the solution set of SFFP(5) is nonempty. Start with any and define a sequence by Then, the sequence converges strongly to the minimum norm solution of SFFP(5) provided that the following conditions are satisfied: (i);(ii);(iii)either or .
Proof. Put , , and for all . Then,
Take . From Proposition 9, we have
from which follows that is bounded and so is . Choose so that for all . We have
In view of conditions (i), (ii), and (iii), we can apply Lemma 5 to (82) to get
and then from
we see that
Consequently, the demiclosedness principle ensures that each weak limit point of is a fixed point of the averaged mapping . And then we conclude from Proposition 9 that each weak limit point of lies in .
Let be the minimum norm element of ; that is, . We shall show that converges strongly to . To see this, we compute as follows: If , then an application of Lemma 5 to (86) yields that . Hence, to complete the proof, it suffices to show that . For this, taking into account Proposition 8, we can write for some and some nonexpansive mapping . Then, from we obtain which ensures that , and hence, once we note that . Since is bounded, we can take a subsequence so that it is weakly convergent to and where the last inequality comes from the characterization inequality of a metric projection. Now, applying (89) and (90) to the equality we obtain and thus,