Abstract
Let H be a real Hilbert space. Let F:Hrightarrow 2^{H} and K:Hrightarrow 2^{H} be two maximal monotone and bounded operators. Suppose the Hammerstein inclusion 0in u+KFu 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.
Highlights
Let be a measurable bounded subset of Rn
We introduce an inertial algorithm for approximating solution of Hammerstein inclusion in Hilbert spaces
6 Conclusion In this paper, we introduced a novel inertial algorithm for approximating solutions of Hammerstein inclusions 0 ∈ u + KFu in Hilbert spaces
Summary
Let be a measurable bounded subset of Rn. A nonlinear integral equation of Hammerstein type is of the form u(x) + k(x, y)f y, u(y) dy = w(x),. Brézis and Browder [5] proved the following theorem for the approximation of solutions of Hammerstein equations with angle-bounded operators using a suitably defined Galerkin method. The first satisfactory result for approximating solution of Hammerstein equation was given by Chidume and Zegeye [26,27,28] They considered the product space E = H × H and defined the auxiliary operator T : E → E by. Theorem 4.1 (Chidume and Shehu (CS12) [24]) Let H be a real Hilbert space, and let F, K : H → H be bounded, continuous, and monotone mappings. Theorem 4.3 (Shehu (S14) [41]) Let H be a real Hilbert space, and let F : H → H be a bounded, coercive, and maximal monotone mapping. ∞, and Suppose that is a KFu converges strongly to u∗
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