We are interested in finding solutions of nonlinear differential equations describing the behaviour of one-dimensional viscoelastic medium with implicit constitutive relations. We focus on a subclass of such models known as the strain-limiting models. To describe the response of viscoelastic solids we assume a nonlinear relationship among the linearized strain, the strain rate and the Cauchy stress. We consider the corresponding Cauchy problem for the stress variable. Under the monotonicity assumption of the nonlinear constitutive function, we convert the problem to a new form for the strain variable and prove local-in-time existence of solutions.