Publications of Khatibzadeh, Hadi

Asymptotically periodic solutions to some second order evolution and difference equations

In this paper we examine the asymptotic periodicity of the solutions of some second-order differential inclusions on [0,∞)associated with maximal monotone operators in a Hilbert space H, whose forcing terms are periodic functions perturbed by functions from L^1(0,∞; H; tdt). It is worth pointing out that strong solutions do not exist in general, so we need to consider weak solutions for this class of evolution inclusions. Similar second-order difference inclusions are also addressed. Our main results on asymptotic periodicity represent significant extensions of the previous theorems proved by R.E. Bruck (1980) and B. Djafari Rouhani (2012).

Strong and weak solutions to second order differential inclusions governed by monotone operators

In this paper we introduce the concept of a weak solution for second order differential inclusions of the form u″(t) ∈ Au(t) + f(t), where A is a maximal monotone operator in a Hilbert space H. We prove existence and uniqueness of weak solutions to two point boundary value problems associated with such kind of equations. Furthermore, existence of (strong and weak) solutions to the equation above which are bounded on the positive half axis is proved under the optimal condition tf(t) ∈ L^ 1(0, ∞; H), thus solving a long-standing open problem (for details, see our comments in Section 3 of the paper). Our treatment regarding weak solutions is similar to the corresponding theory related to the first order differential inclusions of the form f(t) ∈ u′(t) + Au(t) which has already been well developed.