Approximation of Solutions of Split Inverse Problem for Multi-valued Demi-Contractive Mappings in Hilbert Spaces

Abstract

Let H1 and H2 be two Hilbert spaces and Aj : H1 → H2 be bounded linear operators and Ui: H1 → 2H1, Tj : H2 → 2H2, 1 ≤ i ≤ n, 1 ≤ j ≤ r be two multi-valued demi-contractive operators with demi-contractive constants βi and µj , respectively, such that Γ = {x ∈ C = ∩ n i=1F(Ui) : Ajx ∈ F(Tj )} 6= ∅. Moreover, suppose Ui(x) and Uj (y) are bounded ∀x ∈ H1, y ∈ H2, 1 ≤ i ≤ n, 1 ≤ j ≤ r and such that Ui(p) = {p} ∀p ∈ F(Ui), 1 ≤ i ≤ n and Tj (p) = {p} ∀p ∈ F(Tj ), 1 ≤ j ≤ r. Then, for some x0 ∈ H1, the sequence {xk} defined by (qk = xk + γ Pr j=1 A∗ j (bj,k − Ajxk), where bj,k ∈ Tj (Ajxk) ∀1 ≤ j ≤ r, xk+1 = (1 − αk)qk + αk nPn i=1 ui,k, where ui,k ∈ Ui(qk) ∀ 1 ≤ i ≤ n, converges weakly to x ∗ ∈ Γ. Moreover, if there exists σ 6= 0 ∈ H1, such that( hui − q, σi ≥ 0 ∀ 1 ≤ i ≤ n, ui ∈ Ui(q) and q ∈ H1, hA∗ j (bj − Ajy), σi ≥ 0 ∀ 1 ≤ j ≤ r, bj ∈ Tj (Ajy) and y ∈ H1, then, the sequence {xk} converges strongly to x ∗ ∈ Γ.