Repulsive Singularity Research Articles

Positive periodic solution of second-order difference equation with a repulsive singularity

A second-order difference equation with periodic boundary value conditions is studied in this article, the nonlinear term may be singular. A few conditions are given to guarantee the existence of positive periodic solution by fixed-point theorem in cones. Our results can be applied to both superlinearity and semilinearity in the case that the difference equation has no singularity, and can be applied to both strong singularity and weak singularity in the case that the difference equation has a repulsive singularity.

Multiplicity of positive periodic solutions to third-order variable coefficients singular dynamical systems

In this paper, by applying a nonlinear alternative principle of Leray–Schauder and Guo–Krasnosel’skii fixed point theorem on compression and expansion of cones, together with truncation technique, we study the existence of multiplicity noncollision periodic solutions to third-order singular dynamical systems. By combining the analysis of the sign of Green’s function for a linear equation, we consider the systems where the potential has a repulsive singularity at origin. The so-called strong force condition is not needed, and the nonlinearity may have sign changing behavior. Recent results in the literature, even in the scalar case, are generalized and improved.

On the positive periodic solutions of a class of Liénard equations with repulsive singularities in degenerate case

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h, it has been shown that for any M>1 there exists sM∈R such that the equation x″+f(x)x′+h(t,x)=s has two positive T-periodic solutions u1(⋅;s) and u2(⋅;s) satisfying min⁡{u1(t;s):t∈[0,T]}>M and min⁡{u2(t;s):t∈[0,T]}<1/M for every s<sM. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T-periodic solutions of indefinite differential equations.

Second-Order Differential Equation with Indefinite and Repulsive Singularities

Second-Order Differential Equation with Indefinite and Repulsive Singularities

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Spreading Equilibria Under Mildly Singular Potentials: Pancakes Versus Droplets

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of pi /2. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient for purely repulsive potentials. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.

Scalar perturbation of gravitating double-kink solutions

In this letter, a two-dimensional (2D) gravity-scalar model is studied. This model supports interesting double-kink solutions, and the corresponding metric solutions can be derived analytically. Depending on a tunable parameter c, the metric can be symmetric or asymmetric. The Schrödinger-like equation for normal modes of the physical linear perturbation is derived. As c varies, the effective potential can have one or two singular barriers. If c is larger than a critical value, the zero mode will be normalizable, despite of the appearance of a strong repulsive singularity. The double-kink solution is always stable against linear perturbations.

Periodic bouncing solutions of the Lazer–Solimini equation with weak repulsive singularity

We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

An exact expression of positive periodic solution for a first-order singular equation

The main purpose of this paper is to obtain an exact expression of the positive periodic solution for a first-order differential equation with attractive and repulsive singularities. Moreover, we prove the existence of at least one positive periodic solution for this equation with an indefinite singularity by applications of topological degree theorem, and give the upper and lower bounds of the positive periodic solution.

Applications of Schauder’s fixed point theorem to singular radially symmetric systems

We study the existence of periodic orbits of radially symmetric systems with a repulsive singularity. We show that such a system has a family of orbits rotating around the origin with small angular momentum. Our proofs are based on the use of topological degree theory and Schauder’s fixed point theorem.

Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities

In this paper, we investigate the existence of a positive periodic solution for the following fourth-order p-Laplacian generalized neutral differential equation with attractive and repulsive singularities: \t\t\t(φp(u(t)−c(t)u(t−δ(t)))″)″+f(u(t))u′(t)+g(t,u(t))=k(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\bigl(\\varphi_{p}\\bigl(u(t)-c(t)u\\bigl(t-\\delta(t)\\bigr) \\bigr)''\\bigr)''+f \\bigl(u(t)\\bigr)u'(t)+g\\bigl(t,u(t)\\bigr)=k(t), $$\\end{document} where g has a singularity at the origin. The novelty of the present article is that we show that attractive and repulsive singularities enable the achievement of a new existence criterion of a positive periodic solution through an application of coincidence degree theory. Recent results in the literature are generalized and significantly improved.

Periodic Orbits for Radially Symmetric Systems with Singularities and Semilinear Growth

We are concerned with non-autonomous radially symmetric systems, with certain strong repulsive singularities near the origin and with some semilinear growth near infinity. By use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T, and these solutions rotate exactly once around the origin in their period time. The result in this paper shows that both of the antiperiodic and the periodic eigenvalues play the same role.

Study on a kind of \u03d5-Laplacian Li\xe9nard equation with attractive and repulsive singularities

In this paper, by application of the Manasevich-Mawhin continuation theorem, we investigate the existence of a positive periodic solution for a kind of ϕ-Laplacian singular Liénard equation with attractive and repulsive singularities.

Twist periodic solutions for differential equations with a combined attractive–repulsive singularity

We study the existence of twist periodic solutions of second order differential equations with an attractive–repulsive singularity. Such twist periodic solutions are stable in the sense of Lyapunov. The proof is based on the third order approximation method in combination with some location information obtained by the averaging method and the method of upper and lower solutions on the reversed order.

Periodic Orbits of Radially Symmetric Keplerian-Like Systems with a Singularity

We study planar radially symmetric Keplerian-like systems with repulsive singularities near the origin and with some semilinear growth near infinity. By the use of topological degree theory, we prove the existence of two distinct families of periodic orbits; one rotates around the origin with small angular momentum, and the other one rotates around the origin with both large angular momentum and large amplitude.

Periodic orbits of a singular superlinear planar system

We present a planar system which depends periodically on time. If the force admits a repulsive singularity and a superlinear growth, we show that such a system has a family of periodic orbits rotating around the origin with angular momentum from zero to infinity.

A result on the quasi-periodic solutions of forced isochronous oscillators at resonance

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearity $$x'' + V'(x) + g(x) = e(t,x,x'),$$ where the assumptions on V, g and e are regular, described precisely in the introduction. Using a variant of Moser’s twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

Nontrivial periodic solutions of second order singular damped dynamical systems

Assuming that the linear equation $x'' + h(t)x' + a(t)x = 0$ has a positive Green's function, we study the existence of nontrivial periodic solutions of second order damped dynamical systems \[ x'' + h(t)x' + a(t)x = f(t, x) + e(t), \] where $h$, $a\in \C(\!(\R/T\Z),\R)$, $e\! =\! (e_1,\ldots, e_N\!)^T\!\! \in \C(\!(\R/T\Z),\R^N\!)$, $N \ge 1$, and the nonlinearity $f = (f_1,\ldots, f_N)^T\in\C((\R=T\Z)\times\R^N\setminus\{0\},\R^N)$ has a repulsive singularity at the origin. We consider a very general singularity and do not need any kind of strong force condition. The proof is based on a nonlinear alternative principle of Leray-Schauder. Recent results in the literature are generalized and improved.

Periodic, quasi-periodic and unbounded solutions of radially symmetric systems with repulsive singularities at resonance

In this paper, we are concerned with periodic solutions, quasi-periodic solutions and unbounded solutions for radially symmetric systems with singularities at resonance, which are 2π-periodic in time. The method is based on the qualitative analysis of Poincare map with action-angle variables. The existence of infinitely many periodic and quasi-periodic solutions or unbounded motions depends on the oscillatory properties of a certain function.