ITERATIVE METHODS FOR SOLVING THE MULTIPLE-SETS SPLIT FEASIBILITY PROBLEM

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Introduction
Let E n and E m be two real Euclidian spaces, n, m be positive integers, This problem was first introduced by Censor and Elfving in 1994 [5] for modeling inverse problems that arise from phase retrievals and in image reconstruction [3], [4]. Recently, the MSSFP can also be used to model the intensity-modulated radiation therapy [7]- [10] and references therein. Denote by  the set of solution for (1.1). Throughout, this paper, we assume that  ≠ 0.
For solving the split convex feasibility problem, that is (1.1) with N = M = 1, Byrne [3], [4] introduced a well-known iterative method, named CQ-method and defined by with a fixed real number 2 0; 2 / , A  where C P and Q P denote the metric projections on the sets C and Q, respectively, and A T is the transpose of A.
In the case that n = m and A = E the MSSFP deduces to the convex feasibility problem (CFP), that is to find a point p C  . To solve the CFP, Censor et al. [6] proposed a string-averaged algorithmic scheme in which the endpoints of strings of sequential projections onto the constraints are averaged.
Recently, Nguyen Buong [1], [2] used properties of metric projections instead of the proximity function to construct a general scheme, 1 1 2 where the mappings P1 and P2 are defined by one of the following cases: In the present article, we propose a iterative algorithmic scheme which is given with a self adaptive step-size. We also give a relaxed variant of this scheme by using projections onto half-spaces instead of those onto the original convex sets.

Preliminaries
In this section, we introduce some definitions and lemmas which can be used in the proof of our main result.
Definitions 1.1. A mapping T from a subset K of E n into E m is called: where is a positive number, and firmly nonexpansive if, in addition, for some fixed (0;1)  and a nonexpansive mapping U, and we say T isaveraged.
For a closed convex subset K of E n , there exists a mapping PK from E n onto K such that PK is called the metric projection on K. We know that PK is firmly nonexpansive [10] (hence, nonexpansive) and 1/2-averaged [5]. Moreover, 2 2 2 , , . ...  ( )

Main result
Let the string    Step 0: Let x 1 and 1 be any point in E n and any positive real number, respectively, and set k:=1; Step 1: Assume that the k th iterate x k has been constructed. If Step 2: Set k: = k + 1 and go to Step 1.
where, the parameter k and k , for all k ≥ 1, satisfy,  z  if and only if converges to a solution of (1.1) as k   .
Proof. We consider only the case when the algorithm does not terminate in a finite number of iterations. First, we prove that {x k } is bounded. Take a point p  . Then, since 1 is nonexpansive and  we can write that where y k =k A T (E -2 )Ax k  0 as k   , that is followed from (3.4) again, (3.2) and the property of . k So, x = P1 , x and hence, x  . Then, i.e., all the sequence {x k } converges to x as k   . The proof is completed.

Remark 1
In the case that S2 = M and ( ) 1 that is the proximity function, introduced by Xu [15]. By taking we obtain that the upper bound for k equal to 4.
In algorithmic schemes 1, we assume that all the projections i C P and j Q P can be easily calculated, but in practice they are sometime difficult to compute or even impossible. In this case, one can turn to relaxed method, proposed by Yang [16] and studied in [11], [14] with the proximity function q(x) defined in the previous section. Now, we give a relaxed variant for algorithmic scheme 1. First, we assume that the convex subsets Ci and Qj in this part satisfy the following assumptions: We define the following half-spaces: . It is difficult to confirm that z is a solution of (1.1). So, we consider the following relaxed algorithmic scheme.
Algorithmic scheme 2 Step 0: Let x 1 and 1 be any point in E n and any positive real number, respectively, and set k = 1; Step 1: The kth iterate x k is constructed by The following Lemma is essential in proving convergence.

Lemma 3.3 [12]
Suppose h is a convex function on E n , then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded subsets of E n .  Proof.
Take a point p. Since , Indeed, as in the proof of Theorem 3.1, we get that

Numerical examples
In this section, we present some preliminary numerical results, calculated by several methods of algorithmic schemes 1 and 2. The methods, used in computations, are (1.3) and new ones with a selfadaptive step size. In the first example, the sets Ci and Qj are defined by   3  2  1  1  2  3  1  2  3   3  2  2  1  2  3  1  2  3   2  2  2  3  3  1  2  3  1  2  3 , , :   The computational results by method (3.6) with the same data as the above and new 1 k and 2 k are given in the following numerical table.
Clearly, the numerical results in Table 7 show that new method (3.6)-(3.7) with is a little faster than the first one, that is usually called the relaxed simultaneous method.

Conclusion
In this paper, we proposed a general approach to construct iterative methods for solving the multiplesets split feasibility problem (MSSFP), that is stringaveraged algorithmic schemes.