Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems

In this article, zero points of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated. A hybrid projection iterative algorithm is considered for analyzing the convergence of the iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions. Some applications of the main results are also provided.

AMS Classification: 47H05; 47H09; 47J25; 90C33.

The theory of monotone operators has emerged as an effective and powerful tool for studying a wide class of unrelated problems arising in various branches of social, engineering, and pure sciences in unified and general framework. Two notions related to monotone operators have turned out to be very useful in the study of various problems involving such operators. The first one, which is inspired by the notion of subdifferential of a convex function, is the concept of enlargement of a given operator; see [1-3] and the references therein. It allows to make a quantitative analysis in different problems involving monotone operators, like for example variational inequalities, inclusions, etc. The second notion is the one of generalized sum of two monotone operators; see [4,5] and the references therein. In recent years, much attention has been given to develop efficient numerical methods for treating zero point problems of monotone operators and fixed point problems of mappings which are Lipschitz continuous; see [6-28] and the references therein. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [29-31] and the references therein. It has recently applied to solve split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [32-35] and the reference therein.

In this article, zero points of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated based on a hybrid iterative method.

The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a hybrid iterative method is proposed and analyzed. Strong convergence theorems for common elements in the zero point set of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, the solution set of a monotone variational inequality, and the fixed point set of a strict pseudocontraction are established in the framework of real Hilbert spaces without any compact assumptions. In Section 4, applications of the main results are discussed.

In what follows, we always assume that H is a real Hilbert space with inner product 〈· , ·〉 and norm || · ||. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonlinear mapping. F(S) stands for the fixed point set of S; that is, F(S):= {x ∈ C : x = Tx}.

Recall that S is said to be nonexpansive iff

If C is a bounded, closed, and convex subset of H, then F(S) is not empty, closed, and convex; see [36].

S is said to be κ-strictly pseudocontractive iff there exists a constant κ ∈ [0, 1) such that

It is clear that the class of κ-strictly pseudocontractive mappings includes the class of non-

expansive mappings.

Let A : C → H be a mapping. A is said to be monotone iff

A is said to be inverse-strongly monotone iff there exists a constant α > 0 such that

For such a case, A is also said to be α-inverse-strongly monotone.

A is said to be Lipschitz continuous iff there exists a positive constant L such that

Recall that the classical variational inequality is to find an x ∈ C such that

It is known that x ∈ C is a solution to (2.1) if and only if x is a fixed point of the mapping Proj_C(I - rA), where r > 0 is a constant, I stands for the identity mapping, and Proj_C stands for the metric projection from H onto C. If A is α-inverse-strongly monotone and r ∈ (0, 2α], then the mapping Proj_C(I - rA) is nonexpansive; see [37] for more details. It follows that V I(C, A), where V I(C, A) stands for the solution set of (2.1), is closed and convex.

A set-valued mapping R : H ⇉ H is said to be monotone iff, for all x, y ∈ H, f ∈ Rx and g ∈ Ry imply 〈x - y, f - g〉 > 0. A monotone mapping R : H ⇉ H is maximal iff the graph G(R) of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping R is maximal if and only if, for any (x, f) ∈ H × H, 〈x - y, f - g〉 ≥ 0, for all (y, g) ∈ G(R) implies f ∈ Rx.

The class of monotone operators is one of the most important classes of operators. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of zero points for maximal monotone operators; see [38-45] and the references therein. For a maximal monotone operator M on H and r > 0, we may define the single-valued resolvent J_r : H → D(M ), where D(M ) denotes the domain of M. It is known that J_r is firmly nonexpansive and M ^-1(0) = F(J_r), where F (J_r):= {x ∈ D(M ): x = J_rx}, and M ^-1(0): {x ∈ H : 0 ∈ Mx}.

In this article, zero points of the sums of a maximal monotone operator and an inverse-strongly monotone mapping, solutions of a monotone variational inequality, and fixed points of a strict pseudocontraction are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.

In order to prove our main results, we also need the following definitions and lemmas.

Lemma 2.1 [46]. Let C be a nonempty, closed, and convex subset of H, and S : C → C a κ-strict pseudocontraction. Define a mapping S_αx = βx + (1 - β)Sx for all x ∈ C. If β ∈ [κ, 1), then the mapping S_β is a nonexpansive mapping such that F (S_β) = F (S).

Lemma 2.2 [47]. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping. Then the mapping I - S is demiclosed at zero, that is, if {x_n} is a sequence in C such that and x_n - Sx_n → 0, then .

Lemma 2.3. Let C be a nonempty, closed, and convex subset of H, B : C → H a mapping, and M : H ⇉ H a maximal monotone operator. Then F(J_r(I - sB)) = (B + M)^-1(0).

Proof. Notice that

This completes the proof.

Lemma 2.4 [48]. Let C be a nonempty, closed, and convex subset of H, A : C → H a Lipschitz monotone mapping, and N_Cx the normal cone to C at x ∈ C; that is, N_Cx = {y ∈ H : 〈x - u, y〉, ∀u ∈ C}. Define

Then W is maximal monotone and 0 ∈ Wx if and only if x ∈ V I(C, A).

Now, we are in a position to give our main results.

Theorem 3.1. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a κ-strict pseudocontraction with a nonempty fixed point set, A : C → H an α-inverse-strongly monotone mapping, and B : C → H a β-inverse-strongly monotone mapping. Let M : H ⇉ H be a maximal monotone operator such that D(M) ⊂ C. Assume that is not empty. Let {x_n} be a sequence generated by the following iterative process:

where , {r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, 2β), and {α_n} and {β_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1, κ ≤ β_n ≤ b < 1;

(b) 0 < r ≤ r_n ≤ r' < 2α;

(c) 0 < s ≤ s_n ≤ s' < 2β,

where a, b, r, r', s, and s' are real constants. Then the sequence {x_n} converges strongly to .

Proof. First, we show that C_n is closed and convex for each n ≥ 1. From the assumption, we see that C₁ = C is closed and convex. Suppose that C_m is closed and convex for some m ≥ 1. We show that C_m+1 is closed and convex for the same m. Let v₁, v₂ ∈ C_m+1 and v = tv₁ + (1 - t)v₂, where t ∈ (0, 1). Notice that

is equivalent to

It is clearly to see that v ∈ C_m+1. This shows that C_n is closed and convex for each n ≥ 1. Put

and

where S_n is defined by

We see from Lemma 2.1 that S_n is nonexpansive with F (S_n) = F (S). Since A is α-inverse-strongly monotone, and B is β-inverse-strongly monotone, we see from the restriction (b) that

and

Now, we show that for each n ≥ 1. Notice that . Suppose that for some m ≥ 1. For any , we see from (3.2), and (3.3) that

This shows that p ∈ C_m+1. This proves that . Note that . For each , we have || x₁ - x_n || ≤ || x₁ - p ||. Since B is inverse-strongly monotone, we see from Lemma 2.3 that (B + M)^-1(0) is closed, and convex. Since A is Lipschitz continuous, we find that VI(C, A) is close, and convex. In view of Lemma 2.2, we obtain F(S) is closed, and convex. This proves that is closed and convex. It follows that

This implies that {x_n} is bounded. Since and , we have

It follows that

This proves that lim_n→∞ || x_n - x₁ || exists. Notice that

It follows that

In view of , we see that

This implies that

We, therefore, obtain from (3.6) that

On the other hand, we see from (3.3) that

It follows that

In view of the restrictions (a), and (c), we find from (3.7) that

Since is firmly nonexpansive, we find that

This finds that

It follows from (3.1) that

which in turn implies that

In view of the restriction (a), we see from (3.7), and (3.8) that

On the other hand, we see from (3.2) that

It follows that

In view of the restrictions (a), and (b), we find from (3.7) that

Since Proj_C is firmly nonexpansive, we arrive at

which finds that

This implies that

It follows that

In view of the restriction (a), we see from (3.7), and (3.11) that

On the other hand, we have

In view of (3.7), we see from the restriction (a) that

Note that

It follows from (3.10) and (3.13) that

In view of

we see from (3.14) and (3.15) that

Note that

which yields that

In view of the restriction (b), we conclude from (3.16) that

Since {x_n} is bounded, there exists a subsequence of {x_n} such that . In view of Lemma 2.2, we obtain from (3.17) that q ∈ F(S). In view of (3.10), and (3.15), we see that , and , respectively. Now, we are in a position to show that q ∈ VI(C, A).

Define

Then W is maximal monotone. Let (x, y) ∈ G(W). Since y - Ax ∈ N_Cx and z_n ∈ C, we have

On the other hand, we have from z_n = Proj_C(I - r_nA₁)v_n that

and hence

It follows that

In view of the restriction (b), we obtain from (3.13) that 〈x - q, y〉 ≥ 0. We have q ∈ A^-10 and hence q ∈ VI(C, A).

Next, we prove that q ∈ (B + M)^-1(0). Notice that

that is,

Let µ ∈ ν. Since M is monotone, we find from (3.18) that

In view of the restriction (c), we see from (3.10) that

This implies that -Bq ∈ Mq, that is, q ∈ (B + M)^-1(0). This completes . Assume that there exists another subsequence of {x_n} weak converges weakly to . We can easily conclude from Opial's condition (see [49]) that q = q'.

Finally, we show that and {x_n} converges strongly to q. This completes the proof of Theorem 3.1. In view of the weak lower semicontinuity of the norm, we obtain from (3.5) that

which yields that . It follows that {x_n} converges strongly to . This completes the proof.

We conclude from Theorem 3.1 the following results on nonexpansive mappings.

Corollary 3.2. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set, A : C → H be an α-inverse-strongly monotone mapping, and B : C → H be a β-inverse-strongly monotone mapping. Let M : H ⇉ H be a maximal monotone operator such that D(M) ⊂ C. Assume that is not empty. Let {x_n} be a sequence generated by the following iterative process:

where , {r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, 2β), and { α_n} is a sequence in (0, 1). Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1;

(b) 0 < r ≤ r_n ≤ r' < 2α;

(c) 0 < s ≤ s_n ≤ s' < 2β,

where a, r, r', s, and s' are real constants. Then the sequence {x_n} converges strongly to .

If A = 0, then Corollary 3.2 is reduced to the following.

Corollary 3.3. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set, and B : C → H be a β-inverse-strongly monotone mapping. Let M : H ⇉ H be a maximal monotone operator such that D(M) ⊂ C. Assume that is not empty. Let {x_n} be a sequence generated by the following iterative process:

where {s_n} is a sequence in (0, 2β), and {α_n} is a sequence in (0, 1).

Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1;

(b) 0 < s ≤ s_n ≤ s' < 2β,

where a, s, and s' are real constants. Then the sequence {x_n} converges strongly to .

If B = 0, then Corollary 3.2 is reduced to the following.

Corollary 3.4. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a nonexpansive mapping with a nonempty fixed point set, A : C → H a α-inverse-strongly monotone mapping. Let M : H ⇉ H be a maximal monotone operator such that D(M) ⊂ C. Assume that is not empty. Let {x_n} be a sequence generated by the following iterative process:

where ,{r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, +∞), and {α_n} is a sequence in (0, 1). Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1;

(b) 0 < r ≤ r_n ≤ r' < 2α;

(c) 0 < s ≤ s_n < ∞,

where a, r, r', and s are real constants. Then the sequence {x_n} converges strongly to .

Let f : H → (-∞, +∞] be a proper convex lower semicontinuous function. Then the subdifferential ∂ of f is defined as follows

From Rockafellar [50], we know that ∂f is maximal monotone. It is not hard to verify that 0 ∈ ∂ f (x) if and only if .

Let I_C be the indicator function of C, i.e.,

Since I_C is a proper lower semicontinuous convex function on H, we see that the subdifferential ∂I_C of I_C is a maximal monotone operator. It is clearly that J_sx = Proj_Cx, ∀x ∈ H. Notice that (B + ∂I_C)^{- 1}(0) = V I(C, B). Indeed,

In view of Theorem 3.1, we have the following.

Corollary 3.5. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be aα κ -strict pseudocontraction with a nonempty fixed point set, A : C → H be an α-inverse-strongly monotone mapping, and B : C → H be a β-inverse-strongly monotone mapping. Assume hat is not empty. Let {x_n} be a sequence generated by he following iterative process:

where {r_n} is a sequence in (0, 2α), {s_n} is a sequence in (0, 2β), and {α_n} and {β_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1, κ ≤ β_n ≤ b < 1;

(b) 0 < r ≤ r_n ≤ r' < 2α;

(c) 0 < s ≤ s_n ≤ s' < 2β,

where a, b, r, r', s, and s' are real constants. Then the sequence {x_n} converges strongly to .

Let F be a bifunction of C × C into ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [51] (see also Fan [52]).

To study the equilibrium problem (4.1), we may assume that F satisfies the following conditions:

(A1) F(x, x) = 0 for all x ∈ C;

(A2) F is monotone, i.e., F(x, y) + F(y, x) = 0 for all x, y ∈ C;

(A3) for each x, y, z ∈ C,

(A4) for each x∈C,y↦F(x,y) is convex and lower semi-continuous.

Putting F(x, y) = 〈Ax, y - x〉 for every x, y ∈ C, we see that the equilibrium problem (4.1) is reduced to the variational inequality (2.1).

The following lemma can be found in [51,53].

Lemma 4.1. Let C be a nonempty, closed, and convex subset of H and F:CxC→ℝ a bifunction satisfying (A1)-(A4). Then, for any s > 0 and x ∈ H, there exists z ∈ C such that

Further, define

for all s > 0 and × ∈ H. Then, the following hold:

(a) T_s is single-valued;

(b) T_s is firmly nonexpansive; that is,

(c) F(T_s) = EP (F );

(d) EP(F) is closed and convex.

Lemma 4.2 [8] . Let C be a nonempty, closed, and convex subset of H, F a bifunction from C×C to ℝ which satisfies (A1)-(A4), and A_F a multivalued mapping of H into itself defined by

Then A_F is a maximal monotone operator with the domain D(A_F ) ⊂ C, , where FP(F) stands for the solution set of (4.1), and

where T_s is defined as in (4.2).

In this section, we consider the problem of approximating a solution of the equilibrium problem.

Theorem 4.3. Let C be a nonempty, closed, and convex subset of H. Let S : C → C be a κ-strict pseudocontraction with a nonempty fixed point set, and F:C×C→ℝ a bifunction satisfying (A1)-(A4). Assume that is not empty. Let {x_n} be a sequence generated by the following iterative process:

where A_F is defined by (4.3), {s_n} is a positive sequence, and {α_n} and {β_n} are sequences in (0, 1). Assume that the following restrictions are satisfied

(a) 0 ≤ α_n ≤ a < 1, κ ≤ β_n ≤ b < 1;

(b) 0 < s ≤ s_n ≤ s' < ∞,

where a, b, s, and s' are real constants. Then the sequence {x_n} converges strongly to .

Proof. Putting A = B = 0, we immediately conclude from Lemmas 4.1 and 4.2 the desired conclusion.

The authors declare that they have no competing interests.

CW designed and performed all the steps of proof in this research and also wrote the paper. AL participated in the design of the study. All authors read and approved the final manuscript.

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