A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

Definition 1. A mapping T : C ↦ C is said to be L-Lipschitzian, if there exists a constant L > 0 such that

It is clear that a Lipschitzian map is a contractive map when 0 < L < 1 and is a nonexpansive map when L = 1. If k = 0; then a k-strict pseudo-contraction map is a nonexpansive map.

Definition 2. A mapping P_C : H ↦ C is said to be the metric projection, if for any x ∈ H, there exists a unique nearest point in C denoted by P_Cx such that

Lemma 3 (see [13].)Let x ∈ H and z ∈ C be any points. There holds

Lemma 4 (see [9], Demiclosedness princple.)Let C be a nonempty closed convex subset of a real Hilbert space H and let T : C ↦ C be a nonexpansive mapping with F(T) ≠ ∅. If {x_n} is a sequence in C weakly converging to x and if {(I − T)x_n} converges strongly to y, then (I − T)x = y; in particular if y = 0, then x ∈ F(T).

Lemma 5 (see [14].)Let λ be a number in [0,1] and μ ≥ 0. Let F : H ↦ H be a t-Lipschitzian and η-strongly monotone operator on a Hilbert space. Associate with a nonexpansive mapping T : H ↦ H and define the mapping T^λ : H ↦ H by

Lemma 6 (see [4].)Assume that A is a strongly positive bounded linear operator on a Hilbert space H with coefficient $$ and 0 < ρ ≤ ∥A∥⁻¹; then $$.

Lemma 7 (see [15].)Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C ↦ C be a k-strict pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (γ + δ)k ≤ γ; then

Lemma 8 (see [16].)Let H be a Hilbert space and C a nonempty convex subset of H. Let T : C ↦ H be a k-strict pseudo-contractive mapping. Define a mapping Jx = δx + (1 − δ)Tx for all x ∈ C. Then as δ ∈ [k, 1), J is a nonexpansive mapping such that F(J) = F(T).

Lemma 9 (see [17].)Let {α_n} be a sequence of nonnegative real numbers satisfying the following relation: α_n+1 ≤ (1 − γ_n)α_n + δ_n, where (i) {γ_n}⊂(0,1), $$; (ii) limsup _n→∞(δ_n/γ_n) = 0 or $$; then lim _n→∞α_n = 0.

Theorem 10. Let C be a nonempty closed convex subset of a real Hilbert space H, S : C ↦ H a non-self-L-Lipschitzian mapping, and T : C ↦ C a k-strict pseudo-contractive mapping such that Fix (T) ≠ ∅. Let F : C ↦ H be a t-Lipschitzian and η-strongly monotone mapping and A : C ↦ C a $$-strongly positive bounded linear operator. For a given x₀ ∈ C, let the sequences {x_n} and {y_n} generated by (11), where {α_n}, {γ_n}, {δ_n}∈[0,1], satisfy the following conditions:

• (i)

  [1 − μ(η − μt²/2)]((1 + k)/(1 − k)) ≤ 1, μ(η − μt²/2) − τL > 0, $$;

• (ii)

  lim _n→∞γ_n = 0, lim _n→∞δ_n = 0, $$, $$, (γ_n + δ_n)k ≤ γ_n;

• (iii)

  lim _n→∞(α_n/(γ_n + δ_n)) = 0, $$, $$, $$.

Proof. The proof is divided into five steps.

Step  1. We first show that the sequences {x_n}, {y_n} are bounded. Take p ∈ Fix (T), own to T : C ↦ C be a k-strict pseudo-contractive mapping, and define Jx = kx + (1 − k)Tx. By Lemma 8  J is nonexpansive and Fix (J) = Fix (T); therefore Tx = (1/(1 − k))(Jx − kx):

Thus we immediately get that T is a (1 + k)/(1 − k)-Lipschitzian mapping. Then we estimate ∥y_n − p∥:

On the other hand, notice that lim _n→∞γ_n = 0, lim _n→∞δ_n = 0; without loss of generality, we may assume that γ_n + δ_n ≤ ∥A∥⁻¹; thus

Step  2. Now we prove that ∥x_n+1 − x_n∥→0 as n → ∞. Denote ξ = μ(η − (μt²/2)):

Step  3. Now we prove that ∥x_n+1 − Tx_n∥→0 as n → ∞:

Step  4. Now we show that limsup _n→∞〈(I − A)x^*, x_n − x^*〉≤0, where x^* ∈ Fix (T) is the unique solution of the variational inequality. Take a subsequence $$ of {x_n} such that

Observe that the sequence {x_n} is bounded; without loss of generality we may assume that $$. By Lemma 4, we get x^′ ∈ Fix (T). Therefore by Lemma 3, we have

Step  5. Next we prove that ∥x_n+1 − x^*∥→0 as n → ∞:

Notice that

Remark 11. The iterative algorithm in Theorem 10 here is a new approximating method, and Lemma 7 plays a key role in the proof of the main results which makes the proof simple.

Remark 12. The results in this paper improve and extend some recent related results. For example, Theorem 10 here improves and extends Theorem  3.2 in [8] in the following ways:

• (i)

  the nonexpansive mapping T : C ↦ C in [8] is extended to the case of k-strict pseudo-contractions T : C ↦ C;

• (ii)

  the self-contraction f : C ↦ C in [8] is extended to the case of a (possiblly non-self) Lipschitzian mapping S : C ↦ H.