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In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R k →Y is a C 2-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R 1 that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ0)∈X×R k and we describe the solution set of the studied equation in a small neighbourhood of this point.

In this work we consider a class of planar second order Hamiltonian systems: $\ddot{q}+\nabla V(q)=0$, where a potential $V$ has a singularity at a point $\xi \in {𝐑}^2$: $V(q)\to -\mathrm{\infty }$, as $q\to \xi$ and the unique global maximum $0\in 𝐑$ that is achieved at two distinct points $a,b\in {𝐑}^2\setminus \{\xi \}$. For a class of potentials that satisfy a strong force condition introduced by W. B. Gordon [Trans. Amer. Math. Soc. 204 (1975)], via minimization of action integrals, we establish the existence of at least two solutions which wind around $\xi$ and join $\{a,b\}$ to $\{a,b\}$. One of them, $Q$, is a heteroclinic...

We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ ℝ² with two positive parameters α and β. The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map F of index zero (to be defined later) that depends on the parameters α and β. Applying the implicit function...

We shall be concerned with the existence of almost homoclinic solutions for a class of second order functional differential equations of mixed type: $q̈\left(t\right)+V_q(t,q\left(t\right))+u(t,q\left(t\right),q(t-T),q(t+T))=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ and T>0 is a fixed positive number. By an almost homoclinic solution (to 0) we mean one that joins 0 to itself and q ≡ 0 may not be a stationary point. We assume that V and u are T-periodic with respect to the time variable, V is C¹-smooth and u is continuous. Moreover, f is non-zero, bounded, continuous and square-integrable....

We consider a planar autonomous Hamiltonian system :q+∇V(q) = 0, where the potential V: ℝ2 {ζ→ ℝ has a single well of infinite depth at some point ζ and a strict global maximum 0at two distinct points a and b. Under a strong force condition around the singularity ζ we will prove a lemma on the existence and multiplicity of heteroclinic and homoclinic orbits - the shadowing chain lemma - via minimization of action integrals and using simple geometrical arguments.

We consider a conservative second order Hamiltonian system $$\ddot{q}+\nabla V\left(q\right)=0$$ in ℝ3 with a potential V having a global maximum at the origin and a line l ∩ 0 = ϑ as a set of singular points. Under a certain compactness condition on V at infinity and a strong force condition at singular points we study, by the use of variational methods and geometrical arguments, the existence of homoclinic solutions of the system.

A special case of G-equivariant degree is defined, where G = ℤ₂, and the action is determined by an involution $T:{ℝ}^p\oplus {ℝ}^q\to {ℝ}^p\oplus {ℝ}^q$ given by T(u,v) = (u,-v). The presented construction is self-contained. It is also shown that two T-equivariant gradient maps $f,g:(ℝⁿ,S^{n-1})\to (ℝⁿ,ℝⁿ\setminus 0)$ are T-homotopic iff they are gradient T-homotopic. This is an equivariant generalization of the result due to Parusiński.

In this work we will consider a class of second order perturbed Hamiltonian systems of the form $q̈+V_q(t,q)=f\left(t\right)$, where t ∈ ℝ, q ∈ ℝⁿ, with a superquadratic growth condition on a time periodic potential V: ℝ × ℝⁿ → ℝ and a small aperiodic forcing term f: ℝ → ℝⁿ. To get an almost homoclinic solution we approximate the original system by time periodic ones with larger and larger time periods. These approximative systems admit periodic solutions, and an almost homoclinic solution for the original system is obtained...