Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^YA-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ? R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ? 0 for all u in T(x) with $\text{∥ x ∥ ? r > 0}\text{and some}\text{K: X → Y}{}^1$, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) ? T(y) ?A(x ? y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Ne?as, Petryshyn, and Babu?ka.     

Covering mappings and their applications to differential equations unsolved for the derivative

We continue to study the properties of covering mappings of metric spaces and present their applications to differential equations. To extend the applications of covering mappings, we introduce the notion of conditionally covering mapping. We prove that the solvability and the estimates for solutions of equations with conditionally covering mappings are preserved under small Lipschitz perturbations. These assertions are used in the solvability analysis of differential equations unsolved for the derivative.     

Fredholm alternatives for nonlinear A-proper mappings with applications to nonlinear elliptic boundary value problems

Let X and Y be real Banach spaces with a projectionally complete scheme Γ = {X_n, P_n; Y_n, Q_n} and let T: X → Y be an asymptotically linear mapping which is A-proper with respect to Γ and whose asymptotic derivative T_∞?L(X, Y) is also A-proper with respect to Γ. Necessary and sufficient conditions are given in order that the equation $\text{T(x) = ?}$ be solvable for a given ? in Y. Under certain additional conditions it is shown that solutions can be constructed as strong limits of finite dimensional Galerkin type approximates x_n?X_n. Theorems 1 and 2 include as special cases the recent results of Kachurovskii, Hess, Ne?as, and the author. The abstract results for A-proper mappings are then applied to the (constructive) solvability of boundary value problems for quasilinear elliptic equations of order 2m with asymptotically linear terms of order 2m ? 1.     

Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.     

Coincidence points of mappings in vector metric spaces with applications to differential equations and control systems

We prove a theorem on the coincidence points of two mappings acting on spaces equipped with a vector metric. By way of application, we obtain sufficient conditions for the existence of a solution of an ordinary differential equation unsolved for the derivative of the unknown function and local solvability conditions for a control system with mixed constraints.     

Global Hopf bifurcation for differential equations with state-dependent delay

We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of S¹-equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay.     

Nonautonomous bifurcation patterns for one-dimensional differential equations

Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained.     

Hopf bifurcation for neutral functional differential equations

In this paper, we extend the computation of the properties of Hopf bifurcation, such as the direction of bifurcation and stability of bifurcating periodic solutions, of DDE introduced by Kazarinoff et al. [N.D. Kazarinoff, P. van den Driessche, Y.H. Wan, Hopf bifurcation and stability of periodic solutions of differential–difference and integro-differential equations, J. Inst. Math. Appl. 21 (1978) 461–477] to a kind of neutral functional differential equation (NFDE). As an example, a neutral delay logistic differential equation is considered, and the explicit formulas for determining the direction of bifurcation and the stability of bifurcating periodic solutions are derived. Finally, some numerical simulations are carried out to support the analytic results.     

Divisibility of multilinear mappings with applications to affine differential geometry

For real finite-dimensional vector spaces V, W we call a bilinear symmetric mapping h?:?V?×?V?→?W non-degenerate if the components of h with respect to a certain basis are linearly independent and non-degenerate. We say that a symmetric trilinear mapping C?:?V?×?V?×?V?→?W is divisible by h if there exists a linear form α such that C(v,?v,?v)?=?α(v)h(v,?v) for every v?∈?V. In affine differential geometry of affine immersions h is the second fundamental form and C – the cubic form of the immersion. The immersion has pointwise planar normal sections if h(v,?v)?∧?C(v,?v,?v)?=?0 for every tangent vector v. We prove that it implies that C is divisible by h if h is non-degenerate and the codimension is greater than two. For immersions with Wiehe's or Sasaki's choice of transversal bundles divisibility of C by h implies vanishing of C.     

Normal forms and Hopf bifurcation for partial differential equations with delays

The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.