Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 28, pp. 1–20. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLOCAL PROBLEMS FOR HYPERBOLIC EQUATIONS FROM THE VIEWPOINT OF STRONGLY REGULAR BOUNDARY CONDITIONS LUDMILA S. PULKINA Abstract. In this article, we consider a nonlocal problem for hyperbolic equa- tion with integral conditions and show their close connection with the notion of strongly regular boundary conditions. This has an important bearing on the method of the study of solvability. We propose also a new approach which enables us to prove a unique solvability of the nonlocal problem with integral condition. 1. Introduction In this article, we consider the nonlocal problem for hyperbolic equations Lu ≡ u tt − (a(x, t)u x ) x + c(x, t)u = f (x, t). (1.1) The question is to find a solution of (1.1) in Q T = (0,l) × (0,T ), with l,T < ∞, satisfying the initial conditions u(x, 0) = 0, u t (x, 0) = 0 (1.2) and nonlocal conditions  l 0 K i (x)u(x, t)dx =0, i =1, 2. (1.3) Various phenomena of modern natural science often lead to nonlocal problems on mathematical modeling, and nonlocal models turn out to be often more precise that local conditions; see [5]. Nonlocal problems form a relatively new division of differential equations theory and generate a need in developing some new methods of research [30]. Nowadays various nonlocal problems for partial differential equations are actively studied and one can find a lot of papers dealing with them; see [2, 9, 14, 13, 18] and references therein. We focus our attention on nonlocal problems with integral conditions for hyperbolic equations which have been studied in [1, 3, 4, 6, 12, 9, 25, 17, 19, 23, 27, 28]. Systematic studies of nonlocal problems with integral conditions originated with the papers by Cannon [10] and Kamynin [16]. These and further investigations of nonlocal problems show that classical methods most widely used to prove solvability of initial-boundary problems break down when applied to 2010 Mathematics Subject Classification. 35L10, 35L20, 35L99, 35D30, 34B10. Key words and phrases. Nonlocal problem; hyperbolic equation; nonlocal integral conditions; dynamical boundary conditions; strongly regular boundary conditions; weak solution. c 2020 Texas State University. Submitted December 17, 2019. Published April 6, 2020. 1