APPROXIMATION OF SOLUTIONS OF SPLIT INVERSE PROBLEM FOR MULTI-VALUED DEMI-CONTRACTIVE MAPPINGS IN HILBERT SPACES

Let H1 and H2 denote two Hilbert spaces, and let Aj: H1 → H2 be bounded linear operators. Additionally, let Ui: H1 → 2H1 and Tj: H2 → 2H2, where 1 ≤ i ≤ n and 1 ≤ j ≤ r, represent multi-valued demi-contractive operators with demi-contractive constants βi and µj, respectively. Consider Γ, which is defined as the set of elements x in C = ∩ n i=1 F(Ui) such that Ajx belongs to F(Tj) for all 1 ≤ j ≤ r. It is known that Γ is not empty.

Furthermore, assume that Ui(x) and Uj(y) are bounded for all x in H1, y in H2, and 1 ≤ i ≤ n, 1 ≤ j ≤ r. Additionally, Ui(p) = {p} for all p in F(Ui), where 1 ≤ i ≤ n, and Tj(p) = {p} for all p in F(Tj), where 1 ≤ j ≤ r.

Under these conditions, there exists an x0 in H1 such that the sequence {xk} defined by the following iterations: qk = xk + γ Pr j=1 A∗j (bj,k − Ajxk), where bj,k belongs to Tj(Ajxk) for all 1 ≤ j ≤ r, and xk+1 = (1 − αk)qk + αk nPn i=1 ui,k, where ui,k belongs to Ui(qk) for all 1 ≤ i ≤ n, converges weakly to x* in Γ.

Moreover, if there exists a non-zero σ in H1 such that (hui − q, σi ≥ 0 for all 1 ≤ i ≤ n, where ui belongs to Ui(q) and q belongs to H1, and hA∗j (bj − Ajy), σi ≥ 0 for all 1 ≤ j ≤ r, where bj belongs to Tj(Ajy) and y belongs to H1, then the sequence {xk} converges strongly to x* in Γ.