Mixed problem for one-dimensional wave equation with dynamic boundary condition

utt − uxx +B1(ut) +B2(x, u) = f(t, x), x ∈ (0, 1), t > 0, (1.1) εutt(t, 0)− ux(t, 0) + b10 (ut(t, 0)) + b20 (u(t, 1)) = f0(t), t > 0, (1.2) δutt(t, 1) + ux(t, 1) + b11 (ut(t, 1)) + b21 (u(t, 1)) = f1(t), t > 0, (1.3) u(0, x) = φ(x), ut(0, x) = ψ(x), (1.4) where B1, B2, b10, b11, b20, b21, f , f0 and f1 are given real valued functions. The functions B1, b10 and b11 have the form B1(s) = μ|s|q−1s, b10(s) = μ0|s|0s and b11(s) = μ1|s|1s, respectively, where μ, μ0, μ1 and q, q0, q1 are real constants such that μ ≥ 0, μ0 ≥ 0, μ1 ≥ 0 and q > 1, q0 > 1, q1 > 1. (1.5)