The present paper is devoted to the investigation of lossless transmission lines terminated by a nonlinear Josephson junction circuit. Such lines are described by a first order hyperbolic system of partial differential equations with sine-nonlinearity. We formulate a mixed problem for this system with nonlinear boundary conditions generated by a nonlinear resistive circuit. In contrast of our previous result  here we cannot reduce the mixed problem to an initial value one on the boundary because the hyperbolic system is not linear one. We extend results from  and present in an operator form the mixed problem in question. Then we cut off the domain and show that operator defined is contractive one. Its unique fixed point is an approximated solution of the mixed problem.