Algorithms for a System of General Variational Inequalities in Banach Spaces

Remark 2.1. Evidently, the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, for instance, [6, 19, 20].

Let D be a subset of C, and let Π be a mapping of C into D. Then Π is said to be sunny if

whenever Π(x) + t(x − Π(x)) ∈ C for x ∈ C and t ≥ 0. A mapping Π of C into itself is called a retraction if Π² = Π. If a mapping Π of C into itself is a retraction, then Π(z) = z for every z ∈ R(Π), where R(Π) is the range of Π. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. Then following lemma concerns the sunny nonexpansive retraction.

Lemma 2.2 (see [21].)Let C be a closed convex subset of a smooth Banach space E, let D be a nonempty subset of C, and let Π be a retraction from C onto D. Then Π is sunny and nonexpansive if and only if

for all u ∈ C and y ∈ D.

Remark 2.3. (1) It is well known that if E is a Hilbert space, then a sunny nonexpansive retraction Π_C is coincident with the metric projection from E onto C.

(2) Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with the set F(T) ≠ ∅. Then the set F(T) is a sunny nonexpansive retract of C.

Lemma 2.4 (see [22].)Assume that {α_n} is a sequence of nonnegative real numbers such that

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

• (a)

  $$,

• (b)

  limsup⁡_n→∞(δ_n/γ_n) ≤ 0 or $$.

Then lim⁡_n→∞α_n = 0.

Lemma 2.5 (see [23].)Let X be a Banach space, {x_n}, {y_n} be two bounded sequences in X and {β_n} be a sequence in [0,1] satisfying

Suppose that x_n+1 = β_nx_n + (1 − β_n)y_n, for all n ≥ 1 and then lim⁡_n→∞∥y_n − x_n∥ = 0.

Lemma 2.6 (see [24].)Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:

Lemma 2.7 (see [25].)Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E, and let G be a nonexpansive mapping of C into itself. If {x_n} is a sequence of C such that x_n⇀x and x_n − Gx_n → 0, then x is a fixed point of G.

Lemma 2.8 (see [26].)Let C be a nonempty closed convex subset of a real Banach space E. Assume that the mapping F : C → E is accretive and weakly continuous along segments (i.e., F(x + ty)⇀F(x) as t → 0). Then the variational inequality

is equivalent to the dual variational inequality

Lemma 2.9. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let Π_C be a sunny nonexpansive retraction from E onto C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive. For given $$, where $$, then $$ is a solution of the problem (1.1) if and only if x^* is a fixed point of the mapping Q defined by

where λ_i  (i = 1,2, …, N) are real numbers.

Proof. We can rewrite (1.1) as

By Lemma 2.2, we can check (2.19) is equivalent to This completes the proof.

Lemma 2.10. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let Π_C be a sunny nonexpansive retraction from E onto C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive. Let Q be defined as Lemma 2.9. If 0 ≤ λ_i ≤ γ_i/K², then Q : C → C is nonexpansive.

Proof. First, we show that for all i ∈ {i, 2, …, N}, the mapping Π_C(I − λ_iA_i) is nonexpansive. Indeed, for all x, y ∈ C, from the condition λ_i ∈ [0, γ_i/K²] and Lemma 2.6, we have

which implies for all i ∈ {1,2, …, N}, the mapping Π_C(I − λ_iA_i) is nonexpansive, so is the mapping Q.

Algorithm 3.1. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Π_C be a sunny nonexpansive retraction from E to C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive. Let B : C → E be a strongly positive bounded linear operator with coefficient α > 0 and F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ (0, α). For any t ∈ (0,1), define a net {x_t} as follows:

where, for any i,   λ_i ∈ (0,   γ_i/K²) is a real number.

Remark 3.2. We notice that the net {x_t} defined by (3.1) is well defined. In fact, we can define a self-mapping W_t : C → C as follows:

From Lemma 2.10, we know that if, for any i,   λ_i ∈ (0, γ_i/K²), the mapping Π_C(I − λ₁A₁)Π_C(I − λ₂A₂) ⋯ Π_C(I − λ_NA_N) = Q is nonexpansive and | | I − tB|| ≤ 1 − tα. Then, for any x, y ∈ C, we have

This shows that the mapping W_t is contraction. By Banach contractive mapping principle, we immediately deduce that the net (3.1) is well defined.

Theorem 3.3. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Π_C be a sunny nonexpansive retraction from E to C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive. Let B : C → E be a strongly positive bounded linear operator with coefficient α > 0, and let F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ (0, α). Assume that Ω ≠ ∅ and λ_i ∈ (0,   γ_i/K²). Then the net {x_t} generated by the implicit method (3.1) converges in norm, as t → 0⁺ to the unique solution $$ of VI

Proof. We divide the proof of Theorem 3.3 into four steps.

• (I)

  Next we prove that the net {x_t} is bounded.

Take that x^* ∈ Ω, we have

It follows that Therefore, {x_t} is bounded. Hence, {y_t}, {By_t}, {A_ix_t}, and {F(y_t)} are also bounded. We observe that

From Lemma 2.10, we know that Q : C → C is nonexpansive. Thus, we have

Therefore,

(II) {x_t} is relatively norm-compact as t → 0⁺.

Let {t_n}⊂(0,1) be any subsequence such that t_n → 0⁺ as n → ∞. Then, there exists a positive integer n₀ such that 0 < t_n < 1/2, for all n ≥ n₀. Let $$. It follows from (3.9) that

We can rewrite (3.1) as

For any x^* ∈ Ω ⊂ C, by Lemma 2.2, we have

With this fact, we derive that

It turns out that

In particular,

Since {x_n} is bounded, without loss of generality, $$ can be assumed. Noticing (3.10), we can use Lemma 2.7 to get $$. Therefore, we can substitute $$ for x^* in (**) to get

Consequently, the weak convergence of {x_n} to $$ actually implies that $$ strongly. This has proved the relative norm compactness of the net {x_t} as t → 0⁺.

(III) Now, we prove that $$ solves the variational inequality (3.4). From (3.1), we have

For any z ∈ Ω, we obtain

Now we prove that 〈y_t − x_t, j(z − x_t)〉≥0. In fact, we can write y_t = Q(x_t). At the same time, we note that z = Q(z), so

Since I − Q is accretive (this is due to the nonexpansivity of Q), we can deduce immediately that Therefore, Since B, F is strongly positive, we have It follows that Combining (3.20) and (3.22), we get Now replacing t in (3.23) with t_n and letting n → ∞, noticing that $$, we obtain which is equivalent to its dual variational inequality (see Lemma 2.8) that is, $$ is a solution of (3.4).

(IV) Now we show that the solution set of (3.4) is singleton.

As a matter of fact, we assume that x^* ∈ Ω is also a solution of (3.4) Then, we have

From (3.25), we have So, Therefore, $$. In summary, we have shown that each cluster point of {x_t} (as t → 0) equals $$. Therefore, $$ as t → 0. This completes the proof.

Algorithm 3.4. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E. Let Π_C be a sunny nonexpansive retraction from E to C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive. Let B : C → E be a strongly positive bounded linear operator with coefficient α > 0, and let F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ (0, α). For arbitrarily given x₀ ∈ C, let the sequence {x_n} be generated iteratively by

where {α_n} and {β_n} are two sequences in [0,1] and, for any i,   λ_i ∈ (0, γ_i/K²) is a real number.

Theorem 3.5. Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space E, and let Π_C be a sunny nonexpansive retraction from E to C. Let {A_i : C → E,   i = 1,2, …, N} be a finite family of γ_i-inverse-strongly accretive.Let B : C → E be a strongly positive bounded linear operator with coefficient α > 0, and let F : C → E be a strongly positive bounded linear operator with coefficient ρ ∈ (0, α). Assume that Ω ≠ ∅. For given x₀ ∈ C, let {x_n} be generated iteratively by (3.29). Suppose the sequences {α_n} and {β_n} satisfy the following conditions:

• (1)

  lim⁡_n→∞α_n = 0 and $$,

• (2)

  0 < liminf⁡_n→∞β_n ≤ limsup⁡_n→∞β_n ≤ 1.

Then {x_n} converges strongly to $$ which solves the variational inequality (3.4).

Proof. Set y_n = Π_C(I − λ₁A₁)Π_C(I − λ₂A₂) ⋯ Π_C(I − λ_NA_N)x_n for all n ≥ 0. Then x_n+1 = β_nx_n + (1 − β_n)Π_C(α_nF + (I − α_nB))y_n for all n ≥ 0. Pick up x^* ∈ Ω.

From Lemma 2.10, we have

Hence, it follows that By induction, we deduce that Therefore, {x_n} is bounded. Hence, {A_ix_i}  (i = 1,2, …, N), {y_n}, {By_n}, and {F(y_n)} are also bounded. We observe that Set x_n+1 = β_nx_n + (1 − β_n)z_n for all n ≥ 0. Then z_n = Π_C(α_nF + (I − α_nB))y_n. It follows that This implies that Hence, by Lemma 2.5, we obtain lim⁡_n→∞∥z_n − x_n∥ = 0. Consequently, At the same time, we note that It follows that From Lemma 2.10, we know that Q : C → C is nonexpansive. Thus, we have Thus, lim⁡_n→∞∥x_n − Q(x_n)∥ = 0. We note that Next, we show that where $$ is the unique solution of VI(3.4).

To see this, we take a subsequence $$ of {z_n} such that

We may also assume that $$. Note that z ∈ Ω in virtue of Lemma 2.7 and (3.40). It follows from the variational inequality (3.4) that Since z_n = Π_C(α_nF + (I − α_nB))y_n, according to Lemma 2.2, we have From (3.44), we have It follows that Finally, we prove $$. From x_n+1 = β_nx_n + (1 − β_n)z_n and (3.46), we have We can apply Lemma 2.4 to the relation (3.47) and conclude that $$. This completes the proof.