The problem is solved by using a technique of the sequential continuation of solution in the cyclic mode for each of the equation alternately.
The author uses continuation of solution with respect to a parameter using the implicit function theorem and hence requires the autonomous part of the nonlinear perturbations to be C 2 smooth.
These manifolds correspond to S a − and S r −, respectively, for ε > 0. The attracting and repelling manifolds of the perturbed system are calculated using the homotopy continuation of solution trajectories to a suitable boundary value problem.
Numerical techniques employed include fixed point (shooting) as well as polynomial approximation methods for the computation of periodic trajectories, an algorithm for the computation of invariant tori and pseudoarc length continuation of solution branches.
Uniqueness results play a significant role in the continuation of solutions and in the theory of autonomous systems.
The continuation of solutions for the two-component Camassa-Holm system after wave breaking is studied in this paper.
Guo [21, 22] studied the persistence properties and unique continuation of solutions to the 2-component Camassa-Holm system in the case λ = 0.
Differential inclusions are generalization of differential equations, therefore all problems considered for differential equations, that is, existence of solutions, continuation of solutions, dependence on initial conditions and parameters, are present in the theory of differential inclusions.
It is obvious that condition (5) can be invaluable in ordinary differential and integral equations for investigations of continuation of solutions, asymptotic properties of solutions, absolute stability, and instability of solutions. .
Therefore standard results on existence, uniqueness, and continuation of solutions of Volterra integral equations (cf., e.g., Burton [20], Gripenberg et al. [21], Miller [1]) ensure that there is a unique solution of (4.7) (up to a possible explosion time).
The oscillation of (1.5) has been studied by many authors, the interested reader is referred to the seminal books by Došlý and Řehák [3] and Agarwal et al. [4, 5], where in addition to mainly oscillation theory, the existence, uniqueness, and continuation of solutions are also discussed.
