"continuation solution" is a correct and usable phrase in written English.
It is usually used to connote a solution that resolves a problem or a situation and ensures that it continues in some way. For example, "The company resorted to a continuation solution to ensure the factory remained open."
The aim of this article, is to present a unique continuation result for solutions of a differential inequalities of the form: ‖ P ( x, D ) u ( x ) ‖ E ≤ ‖ V ( x ) u ( x ) ‖ E, (1).
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.
In this paper we prove a global continuation result for periodic solutions of the following retarded functional differential equation (RFDE for short) on a manifold, depending on a parameter λ ≥ 0 : x ′ ( t ) = λ f ( t, x t ).
Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations.
Roughly speaking, p ∈ M is a bifurcation point if any of its neighborhoods in M contains T-periodic orbits corresponding to arbitrarily small values of λ > 0. The main outcome of this paper, Theorem 3.3 below, is a global continuation result for T-periodic solutions of equation (1.1).
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. .
The problem is solved by using a technique of the sequential continuation of solution in the cyclic mode for each of the equation alternately.
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.
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.
