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We apply the topological degree theory for condensing maps to study approximation of solutions to a fractional-order semilinear differential equation in a Banach space. We assume that the linear part of the equation is a closed unbounded generator of a $C_{0}$ -semigroup. We also suppose that the nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to obtain a result on the approximation of solutions to the problem. [ABSTRACT FROM AUTHOR]

If x is an equilibrium of an autonomous differential equation $\dot x=f(x)$ and det∥ f ( x )∥≠0, then x persists under autonomous perturbations and x transforms into a T-periodic solution under non-autonomous T-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form $-\dot u\in N_{B}(u)+f_{0}(u)$ , $u\in \mathbb {R}^{2}, $ i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set B of the reduced system is a non-moving unit ball of $\mathbb {R}^{2}$ . We show that an equilibrium ∥ u ∥=1 persists under periodic perturbations, if the projection $\overline {f}:\partial B\to \mathbb {R}^{2}$ of f on the tangent to the boundary ∂ B is nonsingular at u . [ABSTRACT FROM AUTHOR]

We consider a nonsmooth bifurcation equation depending on a small parameter $${\varepsilon > 0}$$ . In Theorem 1 we provide conditions ensuring the existence of branches of solutions, smoothly depending on $${\varepsilon}$$ , emanating from a curve of solutions of the bifurcation equation when $${\varepsilon = 0.}$$ Several examples will illustrate the different types of bifurcation that occur in the present nonsmooth case. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a new method to prove bifurcation of a branch of asymptotically stable periodic solutions of a T-periodically perturbed autonomous system from a T-periodic limit cycle of the autonomous unperturbed system. The method is based on a linear scaling of the state variables to convert, under suitable conditions, the singular Poincaré map (with two singularity conditions) associated to the perturbed autonomous system into an equivalent non-singular equation to which the classical implicit function theorem applies directly. As a result we obtain the existence of a unique branch of T-periodic solutions (usually found for bifurcations of co-dimension 2) as well as a relevant property of the spectrum of their derivatives. Finally, by a suitable representation formula of the classical Malkin bifurcation function, we show that our conditions are equivalent to the existence of a non-degenerate simple zero of the Malkin function. The novelty of the method is that it permits to solve the problem without explicit reduction of the dimension of the state space as it is usually done in the literature by the Lyapunov-Schmidt method. [ABSTRACT FROM AUTHOR]

We use the method of the topological degree, the theory of fractional powers of positive operators, and the Grisvard formula together with results proved by G. Raugel and G. R. Sell to study the periodic solutions of the incompressible Navier–Stokes equations in a thin three-dimensional domain. [ABSTRACT FROM AUTHOR]

For a nonlinear autonomous damped wave equation in a thin domain we provide conditions ensuring the existence of periodic solutions in time. Our approach uses both methods developed by Hale and Raugel and methods based on the topological degree theory together with some results on the functionalization of parameter. [ABSTRACT FROM AUTHOR]