Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

Definition 1. Let A : C → X be a mapping of C into X. Then A is said to be

• (i)

  accretive if for each x, y ∈ C there exists j(x − y) ∈ J(x − y) such that

  $$ ()
  where J is the normalized duality mapping;

• (ii)

  α-strongly accretive if for each x, y ∈ C there exists j(x − y) ∈ J(x − y) such that

  $$ ()
  for some α ∈ (0,1);

• (iii)

  β-inverse strongly accretive if for each x, y ∈ C there exists j(x − y) ∈ J(x − y) such that

  $$ ()
  for some β > 0;

• (iv)

  λ-strictly pseudocontractive if for each x, y ∈ C there exists j(x − y) ∈ J(x − y) such that

  $$ ()
  for some λ ∈ (0,1).

Definition 2. Let C be a nonempty convex subset of a real Banach space X. Let $$ be a finite family of nonexpansive mappings of C into itself and let λ₁, …, λ_N be real numbers such that 0 ≤ λ_i ≤ 1 for every i = 1, …, N. Define a mapping K : C → C as follows:

Such a mapping K is called the K-mapping generated by T₁, …, T_N and λ₁, …, λ_N.

Lemma 3 (see [4].)Let C be a nonempty closed convex subset of a strictly convex Banach space. Let $$ be a finite family of nonexpansive mappings of C into itself with $$ and let λ₁, …, λ_N be real numbers such that 0 < λ_i < 1 for every i = 1, …, N − 1 and 0 < λ_N ≤ 1. Let K be the K-mapping generated by T₁, …, T_N and λ₁, …, λ_N. Then $$.

Theorem 4 (see [5], Theorem 3.2.)Let X be a real smooth Banach space. Let T, F : X → CB(X), and A : D(A) ⊂ X → 2^X be three multivalued mappings, let g : X → D(A) be a single-valued mapping, and let N(·, ·) : X × X → X be a single-valued continuous mapping satisfying the following conditions:

• (C1)

  A∘g : X → 2^X is m-accretive and H-uniformly continuous;

• (C2)

  T : X → CB(X) is H-uniformly continuous;

• (C3)

  F : X → CB(X) is H-uniformly continuous;

• (C4)

  the mapping x ↦ N(x, y) is ϕ-strongly accretive and μ-H-Lipschitz with respect to the mapping T, where ϕ : [0, ∞)→[0, ∞) is a strictly increasing function with ϕ(0) = 0;

• (C5)

  the mapping y ↦ N(x, y) is accretive and ξ-H-Lipschitz with respect to the mapping F.

For arbitrary x₀ ∈ D(A), define the sequence {x_n} iteratively by

where {u_n} is defined by for any w_n ∈ Tx_n, k_n ∈ Fx_n, and some ɛ > 0, where {σ_n} is a positive real sequence such that $$. Then, there exists $$ such that, for $$ and for all n ≥ 0, {x_n} converges strongly to $$, and, for any $$ and $$, $$ is a solution of the MVVI (16).

Proposition 5 (see [12].)Let {λ_n} and {b_n} be sequences of nonnegative numbers and {α_n}⊂(0,1) a sequence satisfying the conditions that {λ_n} is bounded, $$, and b_n → 0, as n → ∞. Let the recursive inequality

be given where ψ : [0, ∞)→[0, ∞) is a strictly increasing function such that it is positive on (0, ∞) and ψ(0) = 0. Then λ_n → 0, as n → ∞.

Proposition 6 (see [13].)Let X be a real smooth Banach space. Let T, and F : X → 2^X be two multivalued mappings, and let N(·, ·) : X × X → X be a nonlinear mapping satisfying the following conditions:

• (i)

  the mapping  x ↦ N(x, y)is  ϕ-strongly accretive with respect to the mapping  T;

• (ii)

  the mapping  y ↦ N(x, y)2009  is accretive with respect to the mapping  F.

Proposition 7 (see [14].)Let X be a real Banach space and let S : X → 2^X∖{∅} be a lower semicontinuous and ϕ-strongly accretive mapping; then, for any x ∈ X, Sx is a one-point set; that is, S is a single-valued mapping.

Lemma 8. Let {s_n} be a sequence of nonnegative real numbers satisfying

where {α_n}, {β_n}, and {γ_n} satisfy the following conditions:

• (i)

  {α_n}⊂[0,1] and $$;

• (ii)

  limsup _n→∞β_n ≤ 0;

• (iii)

  γ_n ≥ 0, for all n ≥ 0, and $$.

Then limsup _n→∞s_n = 0.

Lemma 9. In a smooth Banach space X, there holds the inequality

Lemma 10 (see [1].)Let C be a nonempty closed convex subset of a smooth Banach space X. Let Π_C be a sunny nonexpansive retraction from X onto C and let A be an accretive operator of C into X. Then, for all λ > 0,

Lemma 11 (see [16].)Let C be a nonempty closed convex subset of a real smooth Banach space X. Let D be a nonempty subset of C. Let Π be a retraction of C onto D. Then the following are equivalent:

• (i)

  Π is sunny and nonexpansive;

• (ii)

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

• (iii)

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

Lemma 12 (see [17].)Let X be a uniformly convex Banach space and $$, r > 0. Then there exists a continuous, strictly increasing, and convex function φ : [0, ∞]→[0, ∞], φ(0) = 0 such that

for all $$ and all α, β, γ ∈ [0,1] with α + β + γ = 1.

Lemma 13 (see [18].)Let C be a nonempty closed convex subset of a Banach space X. Let S₀, S₁, … be a sequence of mappings of C into itself. Suppose that $$. Then for each y ∈ C, {S_ny} converges strongly to some point of C. Moreover, let S be a mapping of C into itself defined by Sy = lim _n→∞S_ny for all y ∈ C. Then lim _n→∞sup {∥Sx − S_nx∥ : x ∈ C} = 0.

Lemma 14 (see [19].)Let X be a uniformly smooth Banach space or a reflexive and strictly convex Banach space with a uniformly Gáteaux differentiable norm. Let C be a nonempty closed convex subset of X, let T : C → C be a nonexpansive mapping with Fix (T) ≠ ∅, and let f ∈ Ξ_C. Then the net {x_t} defined by x_t = tf(x_t)+(1 − t)Tx_t converges strongly to a point in Fix (T). If one defines a mapping Q : Ξ_C → Fix (T) by Q(f): = s − lim _t→0x_t, for all f ∈ Ξ_C, then Q(f) solves the VIP as follows:

Lemma 15 (see [20].)Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let $$ be a sequence of nonexpansive mappings on C. Suppose $$ is nonempty. Let {λ_n} be a sequence of positive numbers with $$. Then a mapping S on C defined by $$ for x ∈ C is defined well and nonexpansive, and $$ holds.

Lemma 16 (see [21].)Given a number r > 0. A real Banach space X is uniformly convex if and only if there exists a continuous strictly increasing function φ : [0, ∞)→[0, ∞),   φ(0) = 0, such that

for all λ ∈ [0,1] and x, y ∈ X such that ∥x∥ ≤ r and ∥y∥ ≤ r.

Lemma 17. Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let A : C → X be an α-inverse strongly accretive mapping. Then, one has

where λ > 0. In particular, if 0 < λ ≤ α/κ², then I − λA is nonexpansive.

Theorem 18. Let X be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let Π_C be a sunny nonexpansive retraction from X onto C. Let T, F : X → CB(X), and A : C → 2^C be three multivalued mappings, let g : X → C be a single-valued mapping, and let N(·, ·) : X × X → C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that

(C6) N(Tx, Fx) + A(g(x)) : X → 2^C∖{∅} is ζ-inverse strongly accretive with ζ ≥ κ².

Proof. First of all, by Lemma 17 we know that I − λ_iA_i is a nonexpansive mapping, where λ_i ∈ (0, α_i/κ²) for each i = 1, …, N. Hence, from the nonexpansivity of Π_C, it follows that G_i is a nonexpansive mapping for each i = 1, …, N. Since B : C → C is the K-mapping generated by G₁, …, G_N and ρ₁, …, ρ_N, by Lemma 3, we deduce that $$. Utilizing Lemma 10, and the definition of G_i, we get Fix (G_i) = VI (C, A_i) for each i = 1, …, N. Thus, we have

Now, let us show that for any v ∈ C, λ > 0, there exists a point $$ such that $$ is a solution of the MVVI (15), for any $$ and $$. Indeed, following the argument idea in the proof of Chidume et al. [5, Theorem 3.1], we put Vx : = N(Tx, Fx) for all x ∈ X. Then by Proposition 6, V is ϕ-strongly accretive. Since T and F are H-uniformly continuous and N(·, ·) is continuous, Vx is continuous and hence lower semicontinuous. Thus, by Proposition 7,  Vx is single-valued. Moreover, since V is ϕ-strongly accretive and by assumption A∘g : X → 2^C is m-accretive, we have that V + λA∘g is an m-accretive and ϕ-strongly accretive mapping, and hence by Cioranescu [22, page 184], for any x ∈ X, we have that (V + λA∘g)(x) is closed and bounded. Therefore, by Morales [23], V + λA∘g is surjective. Hence, for any v ∈ X and λ > 0, there exists $$ such that $$, where $$ and $$. In addition, in terms of Proposition 7, we know that V + λA∘g is a single-valued mapping. Assume that N(Tx, Fx) + λA(g(x)) : X → C is ζ-inverse strongly accretive with ζ ≥ κ². Then by Lemma 17, we conclude that the mapping x ↦ x − (N(Tx, Fx) + λA(g(x))) is nonexpansive.

Without loss of generality, we may assume that v = 0 and λ = 1. Let p ∈ Δ and let r (≥∥f(p) − p∥/(1 − ρ)) be sufficiently large such that $$. Then p ∈ D(A) = C such that 0 ∈ N(w, k) + A∘g(p) for any w ∈ Tp and k ∈ Fp. Let M : = sup {∥u∥ : u ∈ N(w, k) + A(g(x)), x ∈ B, w ∈ Tx, k ∈ Fx}. Then as A∘g, T, and F are H-uniformly continuous on X, for ɛ₁ : = ϕ(r)/8(1 + ɛ), ɛ₂ : = ϕ(r)/8μ(1 + ɛ), and ɛ₃ : = ϕ(r)/8ξ(1 + ɛ), there exist δ₁, δ₂, δ₃ > 0 such that for any x, y ∈ X, ∥x − y∥ < δ₁, ∥x − y∥ < δ₂ and ∥x − y∥ < δ₃ imply H(A∘g(x), A∘g(y)) < ɛ₁, H(Tx, Ty) < ɛ₂ and H(Fx, Fy) < ɛ₃, respectively.

Let us show that x_n ∈ B for all n ≥ 0. We show this by induction. First, x₀ ∈ B by construction. Assume that x_n ∈ B. We show that x_n+1 ∈ B. If possible we assume that x_n+1 ∉ B, then ∥x_n+1 − p∥ > r. Further from (35) it follows that

and hence

which immediately yields

Since N(·, ·) is ϕ-strongly accretive with respect to T and A(g(·)) is accretive, we deduce from (41) that

Again from (35), we have that

Also, from Proposition 7, Vx = N(Tx, Fx) is a single-valued mapping; that is, for any k, k^′ ∈ Fx and w, w^′ ∈ Tx, we have N(w, k) = N(w, k^′) and N(w, k) = N(w^′, k). On the other hand, it follows from Nadler [24] that, for k_n+1 ∈ Fx_n+1 and w_n+1 ∈ Tx_n+1, there exist $$ and $$ such that

respectively. Therefore, from (42) and (36), we have

So, we get ∥x_n+1 − p∥ ≤ r, a contradiction. Therefore, {x_n} is bounded.

Let us show that lim _n→∞∥x_n − x_n+1∥ = 0 and lim _n→∞∥x_n − y_n∥ = 0.

Indeed, we define G : C → C by Gx : = x − (N(Tx, Fx) + A(g(x))) for all x ∈ C. Then, G is a nonexpansive mapping and the iterative scheme (35) can be rewritten as follows:

Taking into account condition (iv), we may assume that {β_n}⊂[a, b] for some a, b ∈ (0,1). From (47), we can rewrite y_n by

where z_n = (α_nf(x_n) + γ_nBx_n + δ_nS_nx_n)/(1 − β_n). Now, we have where 1/(1 − b) ²sup _n≥0{∥f(x_n)∥ + ∥Bx_n∥ + ∥S_nx_n∥}   ≤ M₀ for some M₀ > 0. By simple calculation, we have

So, from (49), we get

Also, for convenience, we write

By simple calculation, we get

From (51) and (53), we deduce that

and hence where $$ for some M₁ > 0. Utilizing Lemma 17, we conclude from (55), conditions (i), (ii), and (vi), and the assumption on {S_n} that

Furthermore, utilizing Lemma 16, we obtain from (39) and (47) that

which immediately yields So, from (56) and conditions (ii), (v), and (vi), we get which together with the properties of φ and φ₁ implies that

Note that

Hence, from (60), it follows that

Let us show that lim _n→∞∥x_n − Bx_n∥ = 0 and lim _n→∞∥x_n − Sx_n∥ = 0.

Indeed, from the definition of y_n, we can rewrite y_n by

where e_n = γ_n + δ_n and $$.

Utilizing Lemma 12, from (63) we have

which implies that From (62) and conditions (ii), (iii), and (iv), we have From the properties of φ₂, we have By Lemma 16, we deduce from the definition of $$ the following which implies that From (67) and condition (iii), we have From the properties of φ₃, we have From the definition of y_n, we can rewrite y_n by where d_n = α_n + δ_n and $$.

Utilizing Lemma 12, from (72) and the convexity of ∥·∥², we have

which implies that From (62), (74), and conditions (ii), (iii), and (iv), we have

By the properties of φ₄, we have

From (71), (76), and

we have

Observe that

Utilizing Lemma 13, we conclude from (78) that

Define a mapping Wx = (1 − θ₁ − θ₂)Bx + θ₁Sx + θ₂Gx, where θ₁, θ₂ ∈ (0,1) are two constants with θ₁ + θ₂ < 1. Then by Lemma 15, we have Fix (W) = Fix (B)∩Fix (S)∩Fix (G) = Δ. We observe that

From (60), (76), and (80), we obtain

Now, we claim that

where q = s − lim _t→0x_t with x_t being the fixed point of the contraction Then x_t solves the fixed point equation x_t = tf(x_t)+(1 − t)Wx_t. Thus we have By Lemma 9, we conclude that where It follows from (86) that

Letting n → ∞ in (88) and noticing (87), we derive

where M₂ > 0 is a constant such that $$ for all t ∈ (0,1) and n ≥ 0. Taking t → 0 in (89), we have

On the other hand, we have

It follows that

Taking into account that x_t → q as t → 0, we have

Since X has a uniformly Fréchet differentiable norm, the duality mapping J is norm-to-norm uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence (83) holds. Noticing that J is norm-to-norm uniformly continuous on bounded subsets of X, we deduce from (62) that

Finally, let us show that x_n → q as n → ∞. Indeed, utilizing Lemma 9, we obtain from (47) that

and hence

Applying Lemma 8 to (96), we conclude from conditions (ii) and (vi) and (94) that x_n → q as n → ∞. This completes the proof.

Corollary 19. Let X be a uniformly convex and 2-uniformly smooth Banach space and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let Π_C be a sunny nonexpansive retraction from X onto C. Let T, F : X → CB(X), and A : C → 2^C be three multivalued mappings, let g : X → C be a single-valued mapping, and let N(·, ·) : X × X → C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (C6) N(Tx, Fx) + A(g(x)) : X → 2^C∖{∅} is ζ-inverse strongly accretive with ζ ≥ κ².

Proof. Since T_i is a η_i-strictly pseudocontractive mapping for each i = 1, …, N, it is known that A_i : = I − T_i is η_i-inverse strongly accretive for each i = 1, …, N. In Theorem 18, we put G_i = Π_C(I − λ_iA_i) for i = 1, …, N, where λ_i ∈ (0, η_i/κ²). It is not hard to see that Fix (T_i) = VI (C, A_i). As a matter of fact, we have, for λ_i > 0,

Accordingly, we conclude that $$  $$. Therefore, the desired result follows from Theorem 18.

Remark 20. Theorem 18 improves, extends, supplements, and develops [5, Theorem 3.2] and [25, Theorem 3.1] in the following aspects.

Lemma 21. Let C be a nonempty closed convex subset of a smooth Banach space X and let A : C → X be a ξ-strictly pseudocontractive and ν-strongly accretive mapping with ξ + ν ≥ 1. Then, for λ ∈ (0,1], one has

In particular, if $$, then I − λA is nonexpansive.

Theorem 22. Let X be a nonempty closed convex subset of a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let Π_C be a sunny nonexpansive retraction from X onto C. Let T, F : X → CB(X), and A : C → 2^C be three multivalued mappings, let g : X → C be a single-valued mapping, and let N(·, ·) : X × X → C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4. Consider that

(H6) N(Tx, Fx) + A(g(x)) : X → C is ξ₀-strictly pseudocontractive and ν₀-strongly accretive with ξ₀ + ν₀ ≥ 1.

Proof. First of all, by Lemma 21, we know that I − λ_iA_i is a nonexpansive mapping, where $$ for each i = 1, …, N. Hence, from the nonexpansivity of Π_C, it follows that G_i is a nonexpansive mapping for each i = 1, …, N. Since B : C → C is the K-mapping generated by G₁, …, G_N and ρ₁, …, ρ_N, by Lemma 3, we deduce that $$. Utilizing Lemma 10 and the definition of G_i, we get Fix (G_i) = VI (C, A_i) for each i = 1, …, N. Thus, we have

Repeating the same arguments as those in the proof of Theorem 18, we can prove that for any v ∈ C, λ > 0, there exists a point $$ such that $$ is a solution of the MVVI (15), for any $$ and $$. In addition, in terms of Proposition 7, we know that V + λA∘g is a single-valued mapping due to the fact that V + λA∘g is ϕ-strongly accretive. Assume that N(Tx, Fx) + A(g(x)) : X → C is ξ₀-strictly pseudocontractive and ν₀-strongly accretive with ξ₀ + ν₀ ≥ 1. Then by Lemma 21, we conclude that the mapping x ↦ x − (N(Tx, Fx) + λA(g(x))) is nonexpansive.

Without loss of generality, we may assume that v = 0 and λ = 1. Let p ∈ Δ and let r(≥∥f(p) − p∥/(1 − ρ)) be sufficiently large such that $$. Observe that

Utilizing (106) and repeating the same arguments as those in the proof of Theorem 18, we can derive x_n ∈ B for all n ≥ 0. Hence {x_n} is bounded.

Let us show that lim _n→∞∥x_n − x_n+1∥ = 0 and lim _n→∞∥x_n − y_n∥ = 0.

Indeed, we define G : C → C by Gx : = x − (N(Tx, Fx) + A(g(x))) for all x ∈ C. Then, G is a nonexpansive mapping and the iterative scheme (102) can be rewritten as follows:

Repeating the same arguments as those of (56), (60), (62), (76), and (80) in the proof of Theorem 18, we can obtain that

Define a mapping Wx = (1 − θ₁ − θ₂)Bx + θ₁Sx + θ₂Gx, where θ₁, θ₂ ∈ (0,1) are two constants with θ₁ + θ₂ < 1. Then by Lemma 15, we have that Fix (W) = Fix (B)∩Fix (S)∩Fix (G) = Δ. We observe that

From (109), we obtain

Now, we claim that

where q = s − lim _t→0x_t with x_t being the fixed point of the contraction Then x_t solves the fixed point equation x_t = tf(x_t)+(1 − t)Wx_t. Repeating the same arguments as those of (93) in the proof of Theorem 18, we can deduce that Since X has a uniformly Gáteaux differentiable norm, the duality mapping J is norm-to-weak  ^* uniformly continuous on bounded subsets of X. Consequently, the two limits are interchangeable and hence (112) holds. Noticing that J is norm-to-weak  ^* uniformly continuous on bounded subsets of X, we conclude from (108) that

Finally, let us show that x_n → q as n → ∞. Indeed, repeating the same arguments as those (96) in the proof of Theorem 18, we can deduce from (107) that

Applying Lemma 8 to (116), we infer from conditions (ii) and (vi) and (115) that x_n → q as n → ∞. This completes the proof.

Corollary 23. Let X be a uniformly convex Banach space which has a uniformly Gáteaux differentiable norm and let C be a nonempty closed convex subset of X such that C ± C ⊂ C. Let Π_C be a sunny nonexpansive retraction from X onto C. Let T, F : X → CB(X), and A : C → 2^C be three multivalued mappings, let g : X → C be a single-valued mapping, and let N(·, ·) : X × X → C be a single-valued continuous mapping satisfying conditions (C1)–(C5) in Theorem 4 and (H6) N(Tx, Fx) + A(g(x)) : X → C is ξ₀-strictly pseudocontractive and ν₀-strongly accretive with ξ₀ + ν₀ ≥ 1.