Regularization for evolution equations in Hilbert spaces involving monotone operators via the semi-flows method

Abstract

In a Hilbert space H consider the equation d/dt x(t) + T(t)x(t) + α(t)x(t) = ƒ(t), t ≥ 0, where the family of operators T(t), t ≥ 0 converges in a certain sense to a monotone operator S, the function α vanishes at infinity and the function ƒ converges to a point h. In this paper we provide sufficient conditions that guarantee the fact that full limiting functions of any solution of the equation are points of the orthogonality set O(h; S) of S at h, namely the set of all x ∈ H such that ⟨Sx - h, x - z⟩ = 0, for all z ∈ S-1(h). If the set O(h; S) is a singleton, then the original solution converges to a solution of the algebraic equation Sz = h. Our problem is faced by using the semi-flow theory and it extends to various directions the works [1, 12].