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

and Applied Analysis 3 Lemma 7 (see [15]). LetC 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 󵄩󵄩󵄩γ (x − y) + δ (Tx − Ty) 󵄩󵄩󵄩 ≤ (γ + δ) 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 , ∀x, y ∈ C. (21) 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). Lemma9 (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),∑∞ n=1 γ n = ∞; (ii) lim sup n→∞ (δ n /γ n ) = 0 or ∑∞ n=1 |δ n | < ∞; then lim n→∞ α n = 0. 3. Main Results In this section, we prove the strong convergence results on the iterative algorithm for k-strict pseudo-contractions. 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) ̸ = 0. Let F : C 󳨃→ H be a tLipschitzian and η-strongly monotone mapping and A : C 󳨃→ C a γ-strongly positive bounded linear operator. For a given x 0 ∈ 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−μt/2)−τL > 0, γ ∈ (1, 2); (ii) lim n→∞ γ n = 0, lim n→∞ δ n = 0, ∑∞ n=0 γ n = ∞, ∑∞ n=0 δ n = ∞, (γ n + δ n )k ≤ γ n ; (iii) lim n→∞ (α n /(γ n + δ n )) = 0, ∑∞ n=1 |α n − α n−1 | < ∞, ∑ ∞ n=1 |γ n − γ n−1 | < ∞, ∑∞ n=1 |δ n − δ n−1 | < ∞. Then the sequence {x n } converges strongly to a fixed point x∗ of T, which solves the variational inequality ⟨(I − A) x ∗ , z − x ∗ ⟩ ≤ 0, ∀z ∈ Fix (T) . (22) 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): 󵄩󵄩󵄩Txn − Tp 󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩 1 1 − k (Jx n − kx n ) − 1 1 − k (Jp − kp) 󵄩󵄩󵄩󵄩󵄩 = 1 1 − k 󵄩󵄩󵄩(Jxn − Jp) − k (xn − p) 󵄩󵄩󵄩 ≤ 1 + k 1 − k 󵄩󵄩󵄩xn − p 󵄩󵄩󵄩 . (23) Thus we immediately get that T is a (1 + k)/(1 − k)Lipschitzian mapping. Then we estimate ‖y n − p‖: 󵄩󵄩󵄩yn − p 󵄩󵄩󵄩 = 󵄩󵄩󵄩PC [αnτSxn + (I − μαnF)Txn] − PCp 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩αnτSxn + (I − μαnF)Txn − p 󵄩󵄩󵄩 = 󵄩󵄩󵄩αn (τSxn − μFp) + (I − μαnF)Txn − (I − μα n F)Tp 󵄩󵄩󵄩