Related papers: Index theory for linear self-adjoint operator equa…

We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making…

We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point…

Morse index theory provides an elegant and useful tool for describing several aspects of a Lagrangian system in terms of its variational properties. In the classical framework it provides an equality between the spectral properties of a…

We use Morse theory to study impulsive problems. First we consider asymptotically piecewise linear problems with superlinear impulses, and prove a new existence result for this class of problems using the saddle point theorem. Next we…

In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with reasonable self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems,…

In this sequence, we first prove an abstract Morse index theorem in a Hilbert space modeling a variational problem with constraints. Then, our abstract formulation is applied to study several optimization setups including closed CMC…

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type…

The aim of the present manuscript is to develop an index theory for singular Lagrangian systems, with a particular focus on the important class of singular operators given by Bessel type differential operators. The main motivation is to…

The goal of this article is to study closed connected sets of periodic solutions, of autonomous second order Hamiltonian systems, emanating from infinity. The main idea is to apply the degree for SO(2)-equivariant gradient operators defined…

The Maslov index is a powerful tool for computing spectra of selfadjoint, elliptic boundary value problems. This is done by counting intersections of a fixed Lagrangian subspace, which designates the boundary condition, with the set of…

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces.…

The classical Morse index theorem establishes a fundamental connection between the Morse index-the number of negative eigenvalues that characterize key spectral properties of linear self-adjoint differential operators-and the count of…

Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…

We develop the concept of operators in Hilbert spaces which are similar to their adjoints via antiunitary operators, the latter being not necessarily involutive. We discuss extension theory, refined polar and singular-value decompositions,…

We analyze the chain recurrent set of skew product semiflows obtained from nonautonomous differential equations -- ordinary differential equations or semilinear parabolic differential equations. For many gradient-like dynamical systems,…

We develop the local Morse theory for a class of non-twice continuously differentiable functionals on Hilbert spaces, including a new generalization of the Gromoll-Meyer's splitting theorem and a weaker Marino-Prodi perturbation type…

We prove an index theorem for families of linear periodic Hamiltonian systems, which is reminiscent of the Atiyah-Singer index theorem for selfadjoint elliptic operators. For the special case of one-parameter families, we compare our…

In recent years, we have established the iteration theory of the index for symplectic matrix paths and applied it to periodic solution problems of nonlinear Hamiltonian systems. This paper is a survey on these results.

A simple version of exact finite dimensional reduction for the variational setting of mechanical systems is presented. It is worked out by means of a thorough global version of the implicit function theorem for monotone operators. Moreover,…