Abstract
In the paper we consider the following Cauchy problem for the systems of the ordinary dierential equations { ψ′′(t) + K(t,ψ(t),ψ′(t)) = S(t), t ∈ (0,T), ψ(0) = ψ0, ψ′(0) = ψ1, (∗) where K and S are some vector-valued functions, ψ0,ψ1 ∈ Rd are some xed vectors. Such systems appear in various classical mechanics problems because they follow from Newton's second law of the dynamics. If we use the Faedo-Galerkin method for the seeking of the weak solution to the initial boundary value problem for the linear or nonlinear hyperbolic equations, then we also use the problems of type (*) for the construction of the approximation functions. In practice, the Cauchy problems of type (*) often are reduced to the following Cauchy problem of the higher dimension: ψ′(t) = θ(t), t ∈ (0,T), θ′(t) = S(t)−K(t,ψ(t),θ(t)), t ∈ (0,T), ψ(0) = ψ0, θ(0) = ψ1. (∗∗) Then the standard Peano/Caratheodory types existence theorems are used. But these are local theorems and they should be complemented by some extension theorems. On the other hand we can seek the global solutions of problems (*) or (**) directly. In the present paper we and the conditions of the global solvability of the Cauchy problem (*). Corresponding results for problem (**) we got in our previous paper. These conditions contain the Caratheodory existence conditions and the Lasalle extension conditions for (*). Finally, we use these facts to proving the existence theorem for the initial boundary value problem for some nonlinear hyperbolic equations with the variable exponent of the nonlinearity.
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