On Variational Inclusion and Common Fixed Point Problems in q-Uniformly Smooth Banach Spaces

Proposition 1.1 (see [1].)Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let Q : C → D be a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:

• (a)

  Q is sunny and nonexpansive,

• (b)

  ∥Qx−Qy∥² ≤ 〈x − y, J(Qx − Qy)〉, for all x, y ∈ C,

• (c)

  〈x − Qx, J(y − Qx)〉≤0, for all x ∈ C, y ∈ D.

Lemma 2.1 (see [16].)Let C be a closed convex subset of a strictly convex Banach space E. Let T₁ and T₂ be two nonexpansive mappings from C into itself with F(T₁)⋂ F(T₂) ≠ ∅. Define a mapping S by

where λ is a constant in (0,1). Then S is nonexpansive and F(S) = F(T₁)⋂ F(T₂).

Lemma 2.2 (see [30].)Let {α_n} be a sequence of nonnegative numbers satisfying the property:

where {γ_n}, {b_n}, and {c_n} satisfy the restrictions:

• (i)

  lim _n→∞γ_n = 0, $$,

• (ii)

  $$,

• (iii)

  limsup _n→∞c_n ≤ 0.

Then, lim _n→∞α_n = 0.

Lemma 2.3 (see [31], page 63.)Let q > 1. Then the following inequality holds:

for arbitrary positive real numbers a, b.

Lemma 2.4 (see [17].)Let E be a real q-uniformly smooth Banach space, then there exists a constant C_q > 0 such that

In particular, if E is a real 2-uniformly smooth Banach space, then there exists a best smooth constant K > 0 such that

Lemma 2.5 (see [20].)Let C be a nonempty convex subset of a real q-uniformly smooth Banach space E and let T : C → C be a λ-strict pseudocontraction. For α ∈ (0,1), one defines T_αx = (1 − α)x + αTx. Then, as α ∈ (0, μ], μ = min {1, {qλ/C_q} ^1/(q−1)}, T_α : C → C is nonexpansive such that F(T_α) = F(T).

Lemma 2.6 (see [21].)Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E which admits weakly sequentially continuous generalized duality mapping j_q from E into E^*. Let T : C → C be a nonexpansive mapping. Then, for all {x_n} ⊂ C, if x_n⇀x and x_n − Tx_n → 0, then x = Tx.

Lemma 2.7 (see [21].)Let C be a nonempty, closed, and convex subset of a real q-uniformly smooth Banach space E. Let V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0. Let 0 < μ < (qη/C_qk^q) ^1/(q−1) and τ = μ(η − C_qμ^q−1k^q/q). Then for each t ∈ (0, min {1, 1/τ}), the mapping S : C → E defined by S : = (I − tμV) is a contraction with a constant 1 − tτ.

Lemma 2.8 (see [21].)Let C be a nonempty, closed and convex subset of a real q-uniformly smooth Banach space E. Let Q_C be a sunny nonexpansive retraction from E onto C. Let V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E a L-Lipschitzian mapping with constant L ≥ 0, and T : C → C a nonexpansive mapping such that F(T) ≠ ∅. Let 0 < μ < (qη/C_qk^q) ^1/(q−1) and 0 ≤ γL < τ, where τ = μ(η − C_qμ^q−1k^q/q). Then {x_t} defined by

Has the following properties:

• (i)

  {x_t} is bounded for each t ∈ (0, min {1, 1/τ}),

• (ii)

  lim _t→0∥x_t − Tx_t∥ = 0,

• (iii)

  {x_t} defines a continuous curve from (0, min {1, 1/τ}) into C.

Lemma 2.9. Let C be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C. Let Q : C → D be a retraction and let j, j_q be the normalized duality mapping and generalized duality mapping on E, respectively. Then the following are equivalent:

• (a)

  Q is sunny and nonexpansive,

• (b)

  ∥Qx−Qy∥² ≤ 〈x − y, j(Qx − Qy)〉, for all x, y ∈ C,

• (c)

  〈x − Qx, j(y − Qx)〉≤0, for all x ∈ C, y ∈ D,

• (d)

  〈x − Qx, j_q(y − Qx)〉≤0, for all x ∈ C, y ∈ D.

Proof. From Proposition 1.1, we have a⇔b⇔c. We need only to prove c⇔d.

Indeed, if y − Qx ≠ 0, then 〈x − Qx, j(y − Qx)〉≤0⇔〈x − Qx, j_q(y − Qx)〉≤0, for all x ∈ C, y ∈ D (by the fact that j_q(x) = ∥x∥^q−2j(x), ∀x ≠ 0).

If y − Qx = 0, then 〈x − Qx, j(y − Qx)〉 = 〈x − Qx, j_q(y − Qx)〉 = 0, for all x ∈ C, y ∈ D. This completes the proof.

Lemma 2.10. Let C be a nonempty, closed, and convex subset of a q-uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping j_q from E into E^*. Let Q_C be a sunny nonexpansive retraction from E onto C. Let V : C → E be a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E a L-Lipschitzian with constant L ≥ 0, and T : C → C a nonexpansive mapping such that F(T) ≠ ∅. Let 0 < μ < (qη/C_qk^q) ^1/(q−1) and 0 ≤ γL < τ, where τ = μ(η − C_qμ^q−1k^q/q). For each t ∈ (0, min {1, 1/τ}), let {x_t} be defined by (2.6), then {x_t} converges strongly to x^* ∈ F(T) as t → 0, which is the unique solution of the following variational inequality:

Proof. We first show the uniqueness of a solution of the variational inequality (2.7). Suppose both $$ and x^* ∈ F(T) are solutions of (2.7). It follows that

Adding up (2.8), we have On the other hand, we have that It is a contradiction. Therefore, $$ and the uniqueness is proved. Below we use x^* to denote the unique solution of (2.7).

Next, we prove that x_t → x^* as t → 0.

Since E is reflexive and {x_t} is bounded due to Lemma 2.8 (i), there exists a subsequence $$ of {x_t} and some point $$ such that $$. By Lemma 2.8(ii), we have $$. Taken together with Lemma 2.6, we can get that $$. Setting y_t = tγfx_t + (I − tμV)Tx_t, where t ∈ (0, min {1, 1/τ}), then we can rewrite (2.6) as x_t = Q_Cy_t.

We claim $$.

From Lemma 2.9, we have

It follows from (2.11) and Lemma 2.7 that It follows that Therefore, we get Using that the duality map j_q is weakly sequentially continuous from E to E^* and noticing (2.14), we get that

We prove that $$ solves the variational inequality (2.7). Since

we derive that For all z ∈ F(T), note that It follows from Lemma 2.9 and (2.18) that where $$.

Now replacing t in (2.19) with t_n and letting n → ∞, from (2.15) and Lemma 2.8 (ii), we obtain $$, that is, $$ is a solution of (2.7). Hence $$ by uniqueness. Therefore, $$ as n → ∞. And consequently, x_t → x^* as t → 0.

Lemma 2.11. Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Let the mapping A : C → E be a α-inverse-strongly accretive operator. Then the following inequality holds:

In particular, if 0 < λ ≤ (qα/C_q) ^1/(q−1), then I − λA is nonexpansive.

Proof. Indeed, for all x, y ∈ C, it follows from Lemma 2.4 that

It is clear that if 0 < λ ≤ (qα/C_q) ^1/(q−1), then I − λA is nonexpansive. This completes the proof.

Lemma 2.12. Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Suppose M₁, M₂ : C → 2^E are two m-accretive mappings and ρ₁, ρ₂ are two arbitrary positive constants. Let A, B : C → E be α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let G : C → C be a mapping defined by

If 0 < ρ₁ ≤ (qα/C_q) ^1/(q−1) and 0 < ρ₂ ≤ (qβ/C_q) ^1/(q−1), then G : C → C is nonexpansive.

Proof. For all x, y ∈ C, by Lemma 2.11, we have

which implies that G : C → C is nonexpansive. This completes the proof.

Lemma 2.13. Let C be a nonempty closed convex subset of a real q-uniformly smooth Banach space E. Suppose A, B : C → E are two inverse strongly accretive operators, M₁, M₂ : C → 2^E are two m-accretive mappings, and ρ₁, ρ₂ are two arbitrary positive constants. Then (x^*, y^*) ∈ C × C is a solution of general system (1.15) if and only if x^* = Gx^*, where G is defined by Lemma 2.12.

Proof. Note that

This completes the proof.

Lemma 2.14 (see [18].)Let E be a q-uniformly smooth Banach space and C a nonempty convex subset of E. Assume for each i ≥ 0, T_i : C → E is a λ_i-strict pseudocontraction with λ_i ∈ (0,1). Assume inf {λ_i : i ≥ 1} = λ > 0 and $$. Let $$ be a positive sequence such that $$, then $$ is a λ-strict pseudocontraction and $$.

Remark 2.15. Under the assumptions of Lemma 2.14, if for each i ≥ 1 the mapping T_i : C → E is replaced by T_i : C → C, respectively, where C is a nonempty closed convex subset of E, then noticing the fact

by Lemma 2.14, we deduce that $$ is a λ-strict pseudocontraction with λ = inf {λ_i : i ≥ 1} and $$.

Theorem 3.1. Let C be a nonempty closed convex subset of a strictly convex, and uniformly smooth Banach space E which admits a weakly sequentially continuous generalized duality mapping j_q : E → E^*. Let Q_C be a sunny nonexpansive retraction from E onto C. Assume the mappings A, B : C → E are α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let M₁, M₂ : C → 2^E two m-accretive mappings and ρ₁, ρ₂ be two arbitrary positive constants. Suppose V : C → E is k-Lipschitz and η-strongly accretive with constants k, η > 0, f : C → E being L-Lipschitz with constant L ≥ 0. Let $$ be an infinite family of λ_i-strict pseudocontractions with {λ_i}⊂(0,1) and inf   {λ_i : i ≥ 0} = λ > 0. Let 0 < μ < (qη/C_qk^q) ^1/(q−1), 0 < ρ₁ < (qα/C_q) ^1/(q−1), 0 < ρ₂ < (qβ/C_q) ^1/(q−1), 0 ≤ γL < τ,  0 < σ ≤ d, where τ = μ(η − C_qμ^q−1k^q/q) and d = min   {1, {qλ/C_q} ^1/(q−1)}. Assume {ξ_i}⊂(0,1) and $$. Define a mapping $$, for all x ∈ C. For arbitrarily given x₀ ∈ C and δ ∈ (0,1), let {x_n} be the sequence generated iteratively by

Assume that {α_n}, {β_n}, and {γ_n} are three sequences in (0,1) satisfying the following conditions:

• (i)

  $$,

• (ii)

  $$,

• (iii)

  $$.

Suppose in addition that $$. Then {x_n} converges strongly to some point x^* ∈ F, which is the unique solution of the following variational inequality:

Proof. We divide the proof into several steps.

Step  1. First, we show that sequences {x_n} are bounded. From lim _n→∞α_n = 0 and 0 < liminf _n→∞γ_n ≤ limsup _n→∞γ_n < 1, there exist some a, b ∈ (0,1) such that {γ_n}⊂[a, b]. We may assume, without loss of generality, that {α_n}⊂(0, (1 − b)min {1, 1/τ}). From Lemma 2.7, we deduce that

Taking x^* ∈ F, it follows from Lemma 2.13 that Putting $$, then we can deduce that $$. By Lemma 2.11, we obtain It follows from (3.5) that In view of Remark 2.15, let S : C → C be the mapping defined by $$ for all x ∈ C, then we can deduce that S : C → C is a λ-strict pseudocontraction and $$. By virtue of Lemma 2.5 and 0 < σ ≤ d, where d = min {1, {qλ/C_q} ^1/(q−1)}, we can get that T : C → C is nonexpansive and $$. Putting l_n = δTx_n + (1 − δ)y_n, it follows that It follows from (3.7) that Hence, {x_n} is bounded, so are {y_n}, {k_n}, {z_n}, and {l_n}.

Step  2. In this part, we will claim that ∥x_n+1 − x_n∥ → 0, as n → ∞.

We observe that

It follows from (3.9) that Again from (3.1), we have It follows from (3.10) that Substituting (3.12) into (3.11), we have where M′ = sup _n≥0{μ∥Vl_n∥ + γ∥fx_n∥, ∥l_n − x_n∥, ∥k_n − x_n∥} < ∞. From (i), (ii), (iii), (3.13), and Lemma 2.2, we deduce that We observe that which implies that Noticing conditions (i) and (ii) and (3.14), we have Let In view of Lemma 2.1, we see that W : C → C is nonexpansive such that Noticing that one has In view of (3.17), (iii) and (3.21), we deduce that We define x_t = Q_C[tγfx_t + (I − tμV)Wx_t], then it follows from Lemma 2.10 that {x_t} converges strongly to some point x^* ∈ F(W) = F, which is the unique solution of the variational inequality (3.2).

Step  3. We show that

where x^* is the solution of the variational inequality of (3.2). To show this, we take a subsequence $$ of {x_n} such that Without loss of generality, we may further assume that $$ for some point z ∈ C due to reflexivity of the Banach space E and boundness of {x_n}, it follows from (3.22) and Lemma 2.6 that z ∈ F(W) = F. Since the Banach space E has a weakly sequentially continuous generalized duality mapping j_p : E → E^*, we obtain that Step  4. We prove that lim _n→∞∥x_n − x^*∥. Setting h_n = α_nγfx_n + γ_nx_n + [(1 − γ_n)I − α_nμV]l_n, for all n ≥ 0. It follows from (3.1) that x_n+1 = Q_Ch_n. In view of Lemmas 2.3, 2.7, and 2.9, we have which implies Put a_n = α_n(τ − γL) and c_n = q〈γfx^* − μVx^*, j_q(x_n+1 − x^*)〉/[1 + (q − 1)(τ − γL)α_n](τ − γL). Apply Lemma 2.2 to (3.27) to obtain x_n → x^* ∈ F as n → ∞. This completes the proof.

Remark 3.2. Compared with the known results in the literature, our results are very different from those in the following aspects.

• (i)

  The results in this paper improve and extend corresponding results in [6–13]. Especially, our result extends their results from 2-uniformly smooth Banach space or Hilbert space to more general q-uniformly smooth Banach space.

• (ii)

  Our Theorem 3.1 extends one nonexpansive mapping in [6, Theorem 2.1], one λ-strict pseudocontraction in [8, Theorem 3.1], and an infinite family of nonexpansive mappings in [10, Theorem 3.1] to an infinite family of λ_i-strict pseudocontractions. And our Theorem 3.1 gets a common element of the common fixed-point set of an infinite family of λ_i-strict pseudocontractions and the solution set of the general system of variational inclusions for two inverse strongly accretive mappings in a q-uniformly smooth Banach space.

• (iii)

  We by f(x_n) replace the u which is a fixed element in iterative scheme (1.16), where f is a L-Lipschitzian. And we also add a Lipschitz and strong accretive operator V in our scheme (3.1). In particular, whenever C = E, f = u, V = I, $$ and q = 2, our scheme (3.1) reduces to (1.16).

• (iv)

  It is worth noting that the Banach space E does not have to be uniformly convex in our Theorem 3.1. However, it is very necessary in Theorem 3.1 of Qin et al. [8] and many other literature.

Corollary 3.3. Let C be a nonempty closed convex subset of a strictly convex, and 2-reflexive E which admits a weakly sequentially continuous normalized duality mapping j : E → E^*. Let Q_C be a sunny nonexpansive retraction from E onto C. Assume the mappings A, B : C → E are α-inverse strongly accretive and β-inverse strongly accretive, respectively. Let M₁, M₂ : C → 2^E be two m-accretive operators and ρ₁, ρ₂ two arbitrary positive constants. Suppose V : C → E is a k-Lipschitzian and η-strongly accretive operator with constants k, η > 0, f : C → E being a L-Lipschitzian with constant L ≥ 0. Let 0 < μ < η/K²k², 0 < ρ₁ < α/K², 0 < ρ₂ < β/K² and 0 ≤ γL < τ, where τ = μ(η − K²μk²). Let T : C → C be a nonexpansive with F = F(T)⋂ F(G) ≠ ∅. For arbitrarily given δ ∈ (0,1) and x₀ ∈ C, let {x_n} be the sequence generated iteratively by

Assume that {α_n}, {β_n}, and {γ_n} are three sequences in (0,1) satisfying the following conditions:

• (i)

  $$, lim _n→∞α_n = 0, $$,

• (ii)

  0 < lim  inf _n→∞γ_n ≤ lim  sup _n→∞γ_n < 1, $$,

• (iii)

  $$, lim _n→∞β_n = β > 0.

Then {x_n} converges strongly to x^* ∈ F, which is the unique solution of the following variational inequality: