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.
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Barbu, V.: A class of boundary value problems for second order abstract differential equations. J. Fac. Sci. Tokyo Sect. I 19, 295–319 (1972)
Barbu, V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff, Leyden (2002)
Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les especes de Hilbert. North-Holland, Amsterdam (1973)
Bruck, R.E.: Periodic forcing of solutions of a boundary value problem for a second order differential equation in Hilbert space. J. Math. Anal. Appl. 76, 159–173 (1980)
Bruck, R.E.: On the weak asymptotic almost-periodicity of bounded solutions of u″ ∈ Au + f for monotone operators. J. Diff. Equ. 37, 309–317 (1980)
Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of bounded solutions to a nonhomogeneous second order evolution equation of monotone type. Nonlinear Anal. 71, e147–e152 (2009)
Djafari Rouhani, B., Khatibzadeh, H.: Asymptotic behavior of bounded solutions to some second order evolution systems. Rocky Mountain J. Math. 40, 1289–1311 (2010)
Djafari Rouhani, B., Khatibzadeh, H.: A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation. J. Math. Anal. Appl. 363, 648–654 (2010)
Mitidieri, E.: Some remarks on the asymptotic behaviour of the solutions of second order evolution equations. J. Math. Anal. Appl. 107, 211–221 (1985)
Moroşanu, G.: Nonlinear Evolution Equations and Applications. D. Reidel, Dordrecht (1988)
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Khatibzadeh, H., Moroşanu, G. Strong and Weak Solutions to Second Order Differential Inclusions Governed by Monotone Operators. Set-Valued Var. Anal 22, 521–531 (2014). https://doi.org/10.1007/s11228-013-0270-3
- Differential inclusion
- Maximal monotone operator
- Boundary value problem
- Strong solution
- Weak solution
- Bounded solution
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