Let \(Q=(0,T)\times(0,1).\) This paper is concerned with the singularly perturbed problem \(T_\varepsilon\), of elliptic-hyperbolic type, in \(Q\), consisting of the system
\[ \begin{aligned} \varepsilon\frac{\partial^2u}{\partial t^2}- \dfrac{\partial u}{\partial t}&= \frac{\partial v}{\partial x}+F(x,u(t,x))-f(t,x)\\ \varepsilon\frac{\partial^2v}{\partial t^2} -\frac{\partial v}{\partial t}&= \frac{\partial u}{\partial x}+G(x,v(t,x))-g(t,x) \end{aligned} \tag{S}\(_\varepsilon\) \]
and the conditions:
\[ \begin{aligned} u(t,0)=u(t,1)=0,&\quad 0<t<T,\\ u(0,x)=a_1(x), \quad v(0,x)=a_2(x), &\quad 0<x<1 \end{aligned}\tag{1} \]
and
\[ u(T,x)=b_1(x),\quad v(T,x)=b_2(x),\quad 0<x<1.\tag{2} \]
In the context of Hilbert Sobolev spaces and variational methods, under suitable assumptions on the data, the limit problem \(T_o\), as \(\varepsilon\to 0\), is the hyperbolic telegraph system \((S_o)\) (\(\varepsilon=0\) in S\(_\varepsilon)\), together with conditions (1). The authors study existence, uniqueness, regularity of solutions \((u_\varepsilon,v_\varepsilon)\) to \(T_\varepsilon\), \((u,v)\) to \(T_o\), and the asymptotic behavior of \((u_\varepsilon,v_\varepsilon)\), in terms of \((u,v)\), as \(\varepsilon\to 0\). The main tool, in their investigations is the theory of maximal monotone operators.