An inertial-type algorithm for approximation of solutions of Hammerstein integral inclusions in Hilbert spaces

Abstract

Let H be a real Hilbert space. Let $F:H\rightarrow 2^{H}$ F : H → 2 H and $K:H\rightarrow 2^{H}$ K : H → 2 H be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion $0\in u+KFu$ 0 ∈ u + K F u has a solution. We construct an inertial-type algorithm and show its strong convergence to a solution of the inclusion. As far as we know, this is the first inertial-type algorithm for Hammerstein inclusions in Hilbert spaces. We also give numerical examples to compare the new algorithm with some existing ones in the literature.

"Solutions of Split Equality Hammerstein Type Equation Problems in Reflexive Real Banach Spaces"

“…A strong convergence theorem is established under the assumption that the associated mappings are uniformly continuous and monotone. The convergence of the method does not require the existence of a constant γ 0 , unlike the results in Chidume and Zegeye [15], Uba et al [36] and Bello et al [3]. A numerical example is also provided to clearly exhibit the behavior of the convergence of the proposed method.…”

“…In 2021, Bello et al [3] introduced an inertial type algorithm for solving Hammerstein type equations in real Hilbert spaces: Let H be a real Hilbert space and let F, K : H → H be maximal monotone and bounded mappings. For arbitrary u 1 , v 1 , u 2 , v 2 ∈ H, define the sequences {h n } , {p n } , {u n } , and {v n } by…”