Topological Degree Research Articles

Homotopy continuation methods for coupled-cluster theory in quantum chemistry

Homotopy methods have proven to be a powerful tool for understanding the multitude of solutions provided by the coupled-cluster polynomial equations. This endeavour has been pioneered by quantum chemists that have undertaken both elaborate numerical as well as mathematical investigations. Recently, from the perspective of applied mathematics, new interest in these approaches has emerged using both topological degree theory and algebraically oriented tools. This article provides an overview of describing the latter development.

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

Rotating periodic solutions for p-Laplacian differential systems

In this paper, we study existence of rotating periodic solutions for p-Laplacian differential systems. We first build a new continuation theorem by topological degree, and then obtain the existence of rotating periodic solutions for two kinds of p-Laplacian differential systems via this continuation theorem, extend some existing relevant results.

Moser's theorem with frequency-preserving

This paper mainly concerns the KAM persistence of the mapping $\mathscr {F}:\mathbb {T}^{n}\times E\rightarrow \mathbb {T}^{n}\times \mathbb {R}^{n}$ with intersection property, where $E\subset \mathbb {R}^{n}$ is a connected closed bounded domain with interior points. By assuming that the frequency mapping satisfies certain topological degree condition and weak convexity condition, we prove some Moser-type results about the invariant torus of mapping $\mathscr {F}$ with frequency-preserving under small perturbations. To our knowledge, this is the first approach to Moser's theorem with frequency-preserving. Moreover, given perturbed mappings over $\mathbb {T}^n$, it is shown that such persistence still holds when the frequency mapping and perturbations are only continuous about parameter beyond Lipschitz or even Hölder type. We also touch the parameter without dimension limitation problem under such settings.

Ambrosetti–Prodi Alternative for Coupled and Independent Systems of Second-Order Differential Equations

This paper deals with two types of systems of second-order differential equations with parameters: coupled systems with the boundary conditions of the Sturm–Liouville type and classical systems with Dirichlet boundary conditions. We discuss an Ambosetti–Prodi alternative for each system. For the first type of system, we present sufficient conditions for the existence and non-existence of its solutions, and for the second type of system, we present sufficient conditions for the existence and non-existence of a multiplicity of its solutions. Our arguments apply the lower and upper solutions method together with the properties of the Leary–Schauder topological degree theory. To the best of our knowledge, the present study is the first time that the Ambrosetti–Prodi alternative has been obtained for such systems with different parameters.

Existence Result for a Double Phase Problem Involving the (p(x), q(x))-Laplacian Operator

ABSTRACT The Dirichlet boundary value problem for elliptic equations involving the (p(x), q(x))-Laplacian operator with a reaction term depending on the gradient and on two real parameters is considered in this paper. Using the topological degree theory for a class of demicontinuous operators of generalized (S +) and the theory of the variable exponent Sobolev spaces, we prove the existence of at least one weak solution of such problem.

An infinite dimensional version of the intermediate value theorem

Let mathfrak {f}= I-k be a compact vector field of class C^1 on a real Hilbert space mathbb {H}. In the spirit of Bolzano’s Theorem on the existence of zeros in a bounded real interval, as well as the extensions due to Cauchy (in mathbb {R}^2) and Kronecker (in mathbb {R}^k), we prove an existence result for the zeros of mathfrak {f} in the open unit ball mathbb {B} of mathbb {H}. Similarly to the classical finite dimensional results, the existence of zeros is deduced exclusively from the restriction mathfrak {f}|_mathbb {S} of mathfrak {f} to the boundary mathbb {S} of mathbb {B}. As an extension of this, but not as a consequence, we obtain as well an Intermediate Value Theorem whose statement needs the topological degree. Such a result implies the following easily comprehensible, nontrivial, generalization of the classical Intermediate Value Theorem: If a half-line with extremeq notin mathfrak {f}(mathbb {S})intersects transversally the functionmathfrak {f}|_mathbb {S}for only one point of mathbb {S}, then any value of the connected component ofmathbb {H}{setminus }mathfrak {f}(mathbb {S})containingqis assumed bymathfrak {f}inmathbb {B}. In particular, such a component is bounded.

Existence and approximate solutions of a nonlinear model for the Antarctic circumpolar current

We consider a nonlinear Neumann boundary value problem which is derived for the Antarctic Circumpolar Current. By the theory of topological degree, we prove the existence results for the problem with semilinear oceanic vorticity term. We also construct the approximate solutions for such a nonlinear model.

Results on nonlocal controllability for impulsive fractional functional integro-differential equations via degree theory

This manuscript mainly focuses on the nonlocal controllability analysis for the impulsive fractional functional integro-differential equation (IFrFIDE) in n-dimensional Euclidean space. To attain the solution representation of the given system, we use the Laplace transform technique and the Mittag-Leffler function. Initially, we obtained the existence and uniqueness of a solution by using the topological degree method, contraction mapping and Gronwall’s inequality. Moreover, we also explore the nonlocal controllability results for our given problem. Furthermore, two numerical examples and a filter system are provided for our given dynamical system. It is helpful to demonstrate the efficacy of our findings.

Parametric Topological Entropy for Multivalued Maps and Differential Inclusions with Nonautonomous Impulses

The main purpose of this paper is to investigate a parametric topological entropy for impulsive differential inclusions on tori. In this way, besides other matters, we would like to extend our recent results concerning impulsive differential equations as well as those on “nonparametric” topological entropy to impulsive differential inclusions. Parametric topological entropy, which is usually called a topological entropy for nonautonomous dynamical systems, is considered here via the compositions of associated multivalued Poincaré translation operators with the single-valued time-dependent impulsive maps. On compact polyhedra and, in particular on tori, parametric topological entropy for families of admissible multivalued maps can be estimated from below by means Ivanov-type inequality in terms of the asymptotic Nielsen and Lefschetz numbers which are, unlike the topological entropy, homotopy invariants. In the scalar case, an effective criterion for a positive parametric topological entropy can be given by topological degree arguments for equi-continuous impulsive maps. In a single-valued nonparametric case, a positive topological entropy usually signifies topological chaos. Some simple illustrative examples are provided.

Solvability of a Boundary Value Problem Involving Fractional Difference Equations

In this current work, we apply the topological degree and fixed point theorems to investigate the existence, uniqueness, and multiplicity of solutions for a boundary value problem associated with a fractional-order difference equation. Moreover, we provide some appropriate examples to verify our main conclusions.

Solitons in the Gauged Skyrme-Maxwell Model

We consider soliton solutions of the U(1) gauged Skyrme model with the pion mass term. The domain of existence of gauged Skyrmions is restricted from above by the value of the pion mass. Concentrating on the solutions of topological degree one, we find that coupling to the electromagnetic field breaks the symmetry of the configurations, the Skyrmions carrying both an electric charge and a magnetic flux, with an induced dipole magnetic moment. The Skyrmions also possess an angular momentum, which is quantized in the units of the electric charge. The mass of the gauged Skyrmions monotonically decreases with increase of the gauge coupling.

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Existence of Weak Solution for $p(x)$-Kirchhoff Type Problem Involving the $p(x)$-Laplacian-like Operator by Topological Degree

Regularity of the equilibrium measure for meromorphic correspondences

Let f be a meromorphic correspondence on a compact Kähler manifold X of dimension k. Assume that its topological degree is larger than the dynamical degree of order $$k-1$$ . We obtain a quantitative regularity of the equilibrium measure of f in terms of its super-potentials.

Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with p-Laplacian Operator

The purpose of this paper is to investigate the initial value problem of Hadamard-type fractional relativistic oscillator equation with p-Laplacian operator. By overcoming the perturbation of singularity to fractional relativistic oscillator equation, the multiplicity of positive solutions to the problem were proved via the methods of reducing and topological degree in cone, which extend and enrich some previous results.

Robust feasibility of systems of quadratic equations using topological degree theory

Robust feasibility of systems of quadratic equations using topological degree theory

Existence of solution for impulsive fractional differential equations with nonlocal conditions by topological degree theory

This article aims to investigate the existence and uniqueness of solutions to impulsive fractional differential equations. Firstly, we show a formula for solutions to an impulsive fractional problem involving a generalization of the classical Caputo derivative with a nonlocal condition in a Banach space. Secondly, some sufficient conditions for the existence of solutions are established by applying fixed point methods. Furthermore, we reconsider some of operators for deriving the existence and uniqueness of the solution to impulsive fractional problem using topological degree techniques. The results support this in one example.

Existence of weak solution for p-Kirchoff type problem via topological degree

Existence of weak solution for p-Kirchoff type problem via topological degree

Periodic solutions to superlinear indefinite planar systems: A topological degree approach

We deal with a planar differential system of the form{u′=h(t,v),v′=−λa(t)g(u), where h is T-periodic in the first variable and strictly increasing in the second variable, λ>0, a is a sign-changing T-periodic weight function and g is superlinear. Based on the coincidence degree theory, in dependence of λ, we prove the existence of T-periodic solutions (u,v) such that u(t)>0 for all t∈R. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or ϕ-Laplacian-type differential operators).

Response solutions of 2-dimensional degenerate systems under quasi-periodic perturbations

This paper considers quasi-periodic perturbations of 2-dimensional degenerate systems. It is proved that if the equilibrium point of the unperturbed system is hyperbolic-type degenerate, then the perturbed system has a small response solution. The proof is based on the topological degree theory and some KAM techniques developed in [3].