We introduce the generalized split common fixed point problem (GSCFPP) and show that the GSCFPP for nonexpansive operators is equivalent to the common fixed point problem. Moreover, we introduce a new iterative algorithm for finding a solution of the GSCFPP and obtain some strong convergence theorems under suitable assumptions.

1. Introduction

Let H₁ and H₂ be real Hilbert spaces and let A : H₁ → H₂ be a bounded linear operator. Given intergers p, r ≥ 1, let us recall that the multiple-set split feasibility problem (MSSFP) was recently introduced [1] and is to find a point:

(1.1)

where

and

are nonempty closed convex subsets of H₁ and H₂, respectively. If p = r = 1, the MSSFP (1.1) becomes the so-called split feasibility problem (SFP) [2] which is to find a point:

(1.2)

where C and Q are nonempty closed convex subsets of H₁ and H₂, respectively. Recently, the SFP (1.2) and MSSFP (1.1) have been investigated by many researchers; see, [3–10].

Since every closed convex subset in a Hilbert space is looked as the fixed point set of its associating projection, the MSSFP (1.1) becomes a special case of the split common fixed point problem (SCFPP), which is to find a point:

(1.3)

where U_i : H₁ → H₁ (i = 1,2, …, p) and T_j : H₂ → H₂ (j = 1,2, …, r) are nonlinear operators. If p = r = 1, the problem (1.3) reduces to the so-called two-set SCFPP, which is to find a point:

(1.4)

Censor and Segal in [11] firstly introduced the concept of SCFPP in finite-dimensional Hilbert spaces and considered the following iterative algorithm for the two-set SCFPP (1.4) for Class-ℑ operators:

(1.5)

where x₀ ∈ H₁, 0 < γ < 2/∥A∥² and I is the identity operator. They proved the convergence of the algorithm (1.5) to a solution of problem (1.4). Moreover, they introduced a parallel iterative algorithm, which converges to a solution of the SCFPP (1.3). However, the parallel iterative algorithm does not include the algorithm (1.5) as a special case.

Very recently, Wang and Xu in [12] considered the SCFPP (1.3) for Class-ℑ operators and introduced the following iterative algorithm for solving the SCFPP (1.3):

(1.6)

Under some mild conditions, they proved some weak and strong convergence theorems. Their iterative algorithm (1.6) includes Censor and Segal’s algorithm (1.5) as a special case for the two-set SCFPP (1.4). Moreover, they prove that the SCFPP (1.3) for the Class-ℑ operators is equivalent to a common fixed point problem. This is also a classical method. Many problems eventually converted to a common fixed point problem; see [13–15].

Motivated and inspired by the aforementioned research works, we introduce a generalized split common fixed point problem (GSCFPP) which is to find a point:

(1.7)

Then, we show that the GSCFPP (1.7) for nonexpansive operators is equivalent to the following common fixed point problem:

(1.8)

where V_j = I − γA^*(I − T_j)A (0 < γ ≤ 1/∥A∥²) for every j ∈ ℕ. Moreover, we give a new iterative algorithm for solving the GSCFPP (1.7) for nonexpansive operators and obtain some strong convergence theorems.

2. Preliminaries

Throughout this paper, we write x_n⇀x and x_n → x to indicate that {x_n} converges weakly to x and converges strongly to x, respectively.

An operator T : H → H is said to be nonexpansive if ∥Tx − Ty∥≤∥x − y∥ for all x, y ∈ H. The set of fixed points of T is denoted by F(T). It is known that F(T) is closed and convex. An operator f : H → H is called contraction if there exists a constant ρ ∈ [0,1) such that ∥f(x) − f(y)∥≤ρ∥x − y∥ for all x, y ∈ H. Let C be a nonempty closed convex subset of H. For each x ∈ H, there exists a unique nearest point in C, denoted by P_Cx, such that ∥x − P_Cx∥≤∥x − y∥ for every y ∈ C. P_C is called a metric projection of H onto C. It is known that for each x ∈ H,

(2.1)

for all y ∈ C.

Let {T_n} be a sequence of operators of H into itself. The set of common fixed points of {T_n} is denoted by F({T_n}), that is,

. A sequence {T_n} is said to be strongly nonexpansive if each {T_n} is nonexpansive and

(2.2)

whenever {x_n} and {y_n} are sequences in C such that {x_n − y_n} is bounded and ∥x_n − y_n∥−∥T_nx_n − T_ny_n∥→0; see [16, 17]. A sequence {z_n} in H is said to be an approximate fixed point sequence of {T_n} if z_n − T_nz_n → 0. The set of all bounded approximate fixed point sequences of {T_n} is denoted by

; see [16, 17]. We know that if {T_n} has a common fixed point, then

is nonempty; that is, every bounded sequence in the common fixed point set is an approximate fixed point sequence. A sequence {T_n} with a common fixed point is said to satisfy the condition (Z) if every weak cluster point of {x_n} is a common fixed point whenever

. A sequence {T_n} of nonexpansive mappings of H into itself is said to satisfy the condition (R) if

(2.3)

for every nonempty bounded subset D of H; see [18].

In order to prove our main results, we collect the following lemmas in this section.

Lemma 2.1 (see [16].)Let C be a nonempty subset of a Hilbert space H. Let {T_n} be a sequence of nonexpansive mappings of C into H. Let {λ_n} be a sequence in [0,1] such that lim inf _n→∞λ_n > 0. Let {U_n} be a sequence of mappings of C into H defined by U_n = λ_nI + (1 − λ_n)T_n for n ∈ ℕ, where I is the identity mapping on C. Then {U_n} is a strongly nonexpansive sequence.

Lemma 2.2 (see [16].)Let H be a Hilbert space, C a nonempty subset of H, and {S_n} and {T_n} sequences of nonexpansive self-mappings of C. Suppose that {S_n} or {T_n} is a strongly nonexpansive sequence and is nonempty. Then .

Lemma 2.3 (see [17].)Let H be a Hilbert space, and C a nonempty subset of H. Both {S_n} and {T_n} satisfy the condition (R) and {T_ny : n ∈ ℕ, y ∈ D} is bounded for any bounded subset D of C. Then {S_nT_n} satisfies the condition (R).

Lemma 2.4 (see [19].)Let {x_n} and {y_n} be bounded sequences in a Banach space X and let {β_n} be a sequence in [0,1] with 0 < lim inf _n→∞β_n ≤ limsup _n→∞β_n < 1. Suppose x_n+1 = (1 − β_n) y_n + β_n x_n for all integers n ≥ 0 and

(2.4)

Then lim _n→∞∥y_n − x_n∥ = 0.

Lemma 2.5 (see [20].)Assume that {a_n} is a sequence of nonnegative real numbers such that

(2.5)

where {γ_n} is a sequence in (0,1) and {δ_n} is a sequence such that

(i)
,
(ii)
limsup _n→∞δ_n/γ_n ≤ 0 or .

Then lim _n→∞a_n = 0.

3. Main Results

Now we state and prove our main results of this paper.

Lemma 3.1. Let A : H₁ → H₂ be a given bounded linear operator and let T_n : H₂ → H₂ be a sequence of nonexpansive operators. Assume

(3.1)

For each constant γ > 0, V_n is defined by the following:

(3.2)

Then Fix ({V_n}) = A⁻¹(Fix ({T_n})). Moreover, for 0 < γ ≤ 1/∥A∥², V_n is nonexpansive on H₁ for n ∈ ℕ.

Proof. Since the inclusion A⁻¹(Fix ({T_n}))⊆Fix ({V_n}) is evident, now we only need to show the converse inclusion. If z ∈ Fix ({V_n}), then we have A^*(I − T_n)Az = 0. Since A⁻¹(Fix ({T_n})) ≠ ∅, we take an arbitrary p ∈ A⁻¹(Fix ({T_n})). Hence

(3.3)

It follows that (1/2)∥Az − T_nAz∥² ≤ 0, then Az = T_nAz for every n ∈ ℕ, hence z ∈ A⁻¹(Fix ({T_n})). Next we turn to show that V_n is a nonexpansive operator for n ∈ ℕ. Since T_n is nonexpansive, we have

(3.4)

Hence

(3.5)

For 0 < γ ≤ 1/∥A∥², we can immediately obtain that V_n is a nonexpansive operator for every n ∈ ℕ.

From Lemma 3.1, we can obtain that the solution set of GSCFPP (1.7) is identical to the solution set of problem (1.8).

Theorem 3.2. Let {U_n} and {V_n} be sequences of nonexpansive operators on Hilbert space H₁. Both {U_n} and {V_n} satisfy the conditions (R) and (Z). Let f : H₁ → H₁ be a contraction with coefficient ρ ∈ [0,1). Suppose Ω = Fix (U_n)⋂Fix (V_n) ≠ ∅. Take an initial guess x₁ ∈ H₁ and define a sequence {x_n} by the following algorithm:

(3.6)

where {α_n}, {β_n}, {γ_n}, and {λ_n} are sequences in [0,1]. If the following conditions are satisfied:

(i)
α_n + β_n + γ_n = 1, for all n ≥ 1;
(ii)
lim _n→∞α_n = 0 and ;
(iii)
0 < lim inf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(iv)
0 < lim inf _n→∞λ_n ≤ limsup _n→∞λ_n < 1;
(v)
lim _n→∞ | λ_n+1 − λ_n | = 0,

then {x_n} converges strongly to w ∈ Ω where w = P_Ωf(w).

Proof. We proceed with the following steps.

Step 1. First show that there exists w ∈ Ω such that w = P_Ωf(w).

In fact, since f is a contraction with coefficient ρ, we have

(3.7)

for every x, y. Hence P_Ωf is also a contraction. Therefore, there exists a unique w ∈ Ω such that w = P_Ωf(w).

Step 2. Now we show that {x_n} is bounded.

Let p ∈ Ω, then p ∈ Fix ({U_n}) and p ∈ Fix ({V_n}). Hence

(3.8)

Then

(3.9)

By induction on n,

(3.10)

for every n ∈ ℕ. This shows that {x_n} and {V_nx_n} are bounded, and hence, {U_ny_n}, {y_n}, and {f(x_n)} are also bounded.

Step 3. We claim that and , where A_n = λ_nI + (1 − λ_n)V_n.

We first show the former equality. Let {z_n} be a bounded sequence in H₁. If , then

(3.11)

Hence

. On the other hand, if

, combining (3.11) and limsup _n→∞λ_n < 1, we obtain that ∥V_nz_n − z_n∥→0. Hence

. Therefore,

Next, we show the latter equality. Using Lemma 2.1, we know that {A_n} is a strongly nonexpansive sequence. Thus, since , from Lemma 2.2 we have

(3.12)

Step 4. {S_n} satisfies the condition (R), where S_n = U_nA_n.

Let D be a nonempty bounded subset of H₁. From the definition of {A_n}, we have, for all y ∈ D,

(3.13)

It follows that

(3.14)

Since {V_n} satisfies the condition (R) and lim _n→∞ | λ_n+1 − λ_n | = 0, we have

(3.15)

that is, {A_n} satisfies the condition (R). Since {A_ny : n ∈ ℕ, y ∈ D} is bounded for any bounded subset D of H₁, by using Lemma 2.3, we have that {V_nA_n} satisfies the condition (R), that is, {S_n} satisfies the condition (R).

Step 5. We show ∥x_n+1 − x_n∥→0.

We can write (3.6) as x_n+1 = β_nx_n + (1 − β_n)z_n where z_n = (α_nf(x_n) + γ_nS_nx_n)/1 − β_n. It follows that

(3.16)

From Step 2, we may assume that {x_n} ⊂ D′, where D′ is a bounded set of H₁. Then from (3.16), we obtain

(3.17)

It follows that

(3.18)

Since {S_n} satisfies the condition (R), combining α_n → 0 as n → ∞, we have

(3.19)

Hence by Lemma 2.4, we get ∥z_n − x_n∥→0 as n → ∞. Consequently,

(3.20)

Step 6. We claim that .

From (3.6), we have

(3.21)

and hence

(3.22)

Since ∥x_n+1 − x_n∥→0, α_n → 0 and limsup _n→∞β_n < 1, we derive

(3.23)

Thus (3.23) and Steps 2 and 3 imply that

(3.24)

Step 7. Show limsup _n→∞〈f(w) − w, x_n − w〉≤0, where w = P_Ωf(w).

Since {x_n} is bounded, there exist a point v ∈ H₁ and a subsequence of {x_n} such that

(3.25)

and

. Since {U_n} and {V_n} satisfy the condition (Z), from Step 6, we have v ∈ F({U_n})∩F({V_n}). Using (2.1), we get

(3.26)

Step 8. Show x_n → w = P_Ωf(w).

Since w ∈ Ω, using (3.8), we have

(3.27)

which implies that

(3.28)

for every n ∈ ℕ. Consequently, according to Step 7, ρ ∈ [0,1), and Lemma 2.5, we deduce that {x_n} converges strongly to w = P_Ω(w). This completes the proof.

Combining Lemma 3.1 and Theorem 3.2, we can obtain the following strong convergence theorem for solving the GSCFPP (1.7).

Theorem 3.3. Let {U_n} and {T_n} be sequences of nonexpansive operators on Hilbert space H₁ and H₂, respectively. Both {U_n} and {T_n} satisfy the conditions (R) and (Z). Let f : H₁ → H₁ be a contraction with coefficient ρ ∈ [0,1). Suppose that the solution set Ω of GSCFPP (1.7) is nonempty. Take an initial guess x₁ ∈ H₁ and define a sequence {x_n} by the following algorithm:

(3.29)

where γ ∈ (0, 1/∥A∥²), and {α_n}, {β_n}, {γ_n}, {λ_n} are sequences in [0,1]. If the following conditions are satisfied:

(i)
α_n + β_n + γ_n = 1, for all n ≥ 1;
(ii)
lim _n→∞α_n = 0 and ;
(iii)
0 < liminf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(iv)
0 < liminf _n→∞λ_n ≤ limsup _n→∞λ_n < 1;
(v)
lim _n→∞ | λ_n+1 − λ_n | = 0,

then {x_n} converges strongly to w ∈ Ω where w = P_Ωf(w).

Proof. Set V_n = I − γA^*(I − T_n)A. By Lemma 3.1, V_n is a nonexpansive operator for every n ∈ ℕ. We can rewrite (3.29) as

(3.30)

We only need to prove that {V_n} satisfies the conditions (R) and (Z). Assume that D is a nonempty bounded subset of H₁. For every y ∈ D, we have

(3.31)

Since {T_n} satisfies the condition (R), and D′ = {Ay : y ∈ D} is bounded, it follows from (3.31) that

(3.32)

Therefore, {V_n} satisfies the condition (R).

Assume that x_n⇀z and x_n − V_nx_n → 0; we next show that V_nz = z. By using x_n − V_nx_n → 0, we have A^*(I − T_n)Ax_n → 0. Since A⁻¹(Fix ({T_n})) ≠ ∅, we choose an arbitrary point p ∈ A⁻¹(Fix ({T_n})); then for every n ∈ ℕ,

(3.33)

Hence

(3.34)

Then we get

. Since {T_n} satisfies the condition (Z) and Ax_n⇀Az, we have Az ∈ F({T_n}). From Lemma 3.1, we have z ∈ Fix ({V_n}).

Let T : H → H be a nonexpansive mapping with a fixed point, and define T_n = T for all n ∈ ℕ. Then {T_n} satisfies the conditions (R) and (Z). Thus, one obtains the algorithm for solving the two-set SCFPP (1.4).

Corollary 3.4. Let U and T be nonexpansive operators on Hilbert space H₁ and H₂, respectively. Let f : H₁ → H₁ be a contraction with coefficient ρ ∈ [0,1). Suppose that the solution set Ω of SCFPP (1.4) is nonempty. Take an initial guess x₁ ∈ H₁ and define a sequence {x_n} by the following algorithm in (3.29), where γ ∈ (0, 1/∥A∥²), and {α_n}, {β_n}, {γ_n}, {λ_n} are sequences in [0,1]. If the following conditions are satisfied:

(i)
α_n + β_n + γ_n = 1, for all n ≥ 1;
(ii)
lim _n→∞α_n = 0 and ;
(iii)
0 < lim inf _n→∞β_n ≤ limsup _n→∞β_n < 1;
(iv)
0 < lim inf _n→∞λ_n ≤ limsup _n→∞λ_n < 1;
(v)
lim _n→∞ | λ_n+1 − λ_n | = 0.

Then {x_n} converges strongly to w ∈ Ω where w = P_Ωf(w).

Remark 3.5. By adding more operators to the families {U_n} and {T_n} by setting U_i = I for i ≥ p + 1 and T_j = I for j ≥ r + 1, the SCFPP (1.3) can be viewed as a special case of the GSCFPP (1.7).

Acknowledgment

This research is supported by the science research foundation program in Civil Aviation University of China (07kys09), the Fundamental Research Funds for the Central Universities (Program No. ZXH2011D005), and the NSFC Tianyuan Youth Foundation of Mathematics of China (No. 11126136).

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Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

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Strong Convergence Theorems for the Generalized Split Common Fixed Point Problem

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