Fixed Point Index Research Articles

Sunflower model: time-dependent coefficients and topology of the periodic solutions set

We investigate the structure of the set of periodic solutions of a time-dependent generalized version of the sunflower equation (in fact of the delayed Lienard equation), where the coefficients can vary periodically, thus allowing for environmental oscillations. Our result stems from a more general analysis, based on fixed point index and degree-theoretic methods, of the set of T-periodic solutions of T-periodically perturbed coupled delay differential equations on differentiable manifolds.

A Landesman–Lazer type result for periodic parabolic problems on [formula omitted] at resonance

We are concerned with T-periodic solutions of nonautonomous parabolic problem of the form ut=Δu+V(x)u+f(t,x,u), t>0, x∈RN, with V∈L∞(RN)+Lp(RN), where 2<p<∞ if N=1,2 and N≤p<∞ for N≥3 and T-periodic continuous perturbation f:RN×R→R. The so-called resonant case is considered, i.e. when N:=Ker(Δ+V)≠{0} and f is bounded. We derive a formula for the fixed point index of the associated translation along trajectories operator in terms of the Brouwer topological degree of the time averaging of the mapping fˆ:N→N, fˆ being the restriction of f to N. By use of the formula and continuation techniques we show that Landesman–Lazer type conditions imply the existence of T-periodic solutions.

RETRACTED ARTICLE: Sharp geometrical properties of a-rarefied sets via fixed point index for the Schrödinger operator equations

In this paper, we use the theory of fixed point index for the Schrodinger operator equations to obtain a geometrical property of a-rarefied sets at infinity on cones. Meanwhile, we give an example to show that the reverse of this property is not true.

Positive doubly periodic solutions of telegraph equations with delays

This paper deals with the existence of positive doubly periodic solutions for the nonlinear telegraph equation with delays $\mathcal{L}u=f( t, x, u(t-\tau_{1}, x),\ldots,u(t-\tau_{n}, x) )$ , $(t, x)\in\mathbb{R}^{2}$ , where $\mathcal{L}u:=u_{tt}-u_{xx}+c u_{t}+a(t, x) u $ is a linear telegraph operator acting on function $u: \mathbb{R}^{2}\to \mathbb{R}$ , $c>0$ is a constant, $a\in C(\mathbb{R}^{2}, (0, \infty))$ is 2π-periodic in t and x, $f\in C(\mathbb{R}^{2}\times[0, \infty)^{n}, [0, \infty))$ is 2π-periodic in t and x, and $\tau_{1}, \ldots, \tau_{n}\in[0, \infty)$ are constants. Some existence results of positive doubly 2π-periodic weak solutions are obtained under that $f(t, x, \eta_{1}, \ldots, \eta_{n})$ satisfies some superlinear or sublinear growth conditions on $\eta_{1}, \ldots, \eta_{n}$ . The discussion is based on the fixed point index theory in cones.

On the equivalent classification of three-dimensional competitive Atkinson/Allen models relative to the boundary fixed points

We study the fixed point index on the carrying simplex of the competitive map. The sum of the indices of the fixed points on the carrying simplex for the three-dimensional competitive map is unit. Based on that, we analyze the asymptotic behavior of the three-dimensional competitive Atkinson/Allen model. We present all the equivalence classes relative to the boundary of the carrying simplex of the low-dimensional (two or three) map, depending upon relationship among the model coefficients. For the two-dimensional case, there are only three dynamic scenarios, and every orbit converges to some fixed point. For the three-dimensional case, there are total $33$ stable equivalence classes, and in $18$ of them all the compact limit sets are fixed points. Further, we focus on the analysis of the dynamics of the other $15$ cases. Hopf bifurcation is studied and a necessary condition for it occurring is given, which implies that the classes $19$-$25$, $28$, $30$ and $32$ do not have any Hopf bifurcation. However, the class $26$ and class $27$ do admit Hopf bifurcations, which means that these two classes may have isolated invariant closed curves in their carrying simplex, and such an invariant closed curve corresponds to either a subharmonic or a quasiperiodic solution in continuous time systems. Each system in class $27$ has a heteroclinic cycle and the numerical simulation also reveals that there exist systems having May-Leonard phenomenon: the existence of nonquasiperiodic oscillation.

Positive solutions for a second order boundary value problem on time scales

In this paper using fixed point index theory we study the existence of positive solutions for a second order boundary value problem on time scales. Our existence theorems will be expressed under some conditions concerning the first eigenvalue of a relevant linear operator.

Positive solutions of sub-superlinear Sturm–Liouville problems

In this paper, we introduce the notion of a strict lower/upper solution to nonlinear Sturm–Liouville boundary value problems. Based on the maximum principles, we establish a result of Leray–Schauder degree on the ordered intervals induced by the pairs of strict lower and upper solutions. Applying the result and the fixed point index theory in cones, we obtain the global existence results of positive solutions for sub-superlinear Sturm–Liouville problems.

Nontrivial solutions of boundary value problems for second-order functional differential equations

In this paper we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.

Nonzero radial solutions for a class of elliptic systems with nonlocal BCs on annular domains

We provide new results on the existence, non-existence, localization and multiplicity of nontrivial solutions for systems of Hammerstein integral equations. Some of the criteria involve a comparison with the spectral radii of some associated linear operators. We apply our results to prove the existence of multiple nonzero radial solutions for some systems of elliptic boundary value problems subject to nonlocal boundary conditions. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.

Viable periodic solutions in state-dependent impulsive problems

Periodic solutions of state-dependent impulsive ODEs in a prescribed set of constraints are examined. The so-called impulsive index is introduced, as a topological tool, and its properties are studied. Some sufficient conditions for its homotopy property are discussed in detail. In a construction of the impulsive index the fixed point index on ANRs is applied to an induced discrete semidynamical system on a barrier where jumps occur. Several illustrative examples are added.

Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate

The paper is concerned with a predator-prey diffusive dynamics subject to homogeneous Dirichlet boundary conditions, in which the growth rate of the the predator is nonlinear. Taking $m$ as the main parameter, we show the existence, stability and exact number of positive solution when $m$ is large, and some numerical simulations are done to complement the analytical results. The main tools used here include the fixed point index theory, the super-sub solution method, the bifurcation theory and the perturbation technique.

Fixed point indices of central configurations

Central configurations of n point particles in \({E \approx \mathbb{R}^d}\) with respect to a potential function U are shown to be the same as the fixed points of the normalized gradient map \({F = -\nabla_{M}U / ||\nabla_{M}U||_{M}}\) , which is an SO(d)-equivariant self-map defined on the inertia ellipsoid. We show that the SO(d)-orbits of fixed points of F are all fixed points of the map induced on the quotient by SO(d), and we give a formula relating their indices (as fixed points) with their Morse indices (as critical points). At the end, we give an example of a nonplanar relative equilibrium which is not a central configuration.

Compression fixed point theorems of operator type

This paper is concerned with new fixed point theorems of operator type that are proved by fixed point index theory and are generalizations of the Leggett–Williams fixed point theorems. With additional structures on the inward and outward boundaries of a conical region, the theorems can be used to prove the existence of solutions of boundary value problems for nonlinear differential equations with dependence on higher-order derivatives, and they can provide nonlocal upper and lower bounds on the solutions. An application is given to demonstrate these advantages.

Lefschetz indices for n-valued maps

We give a systematic exposition of the construction and properties of Lefschetz fixed-point indices for the n-valued maps studied by Schirmer, Brown and others, using techniques introduced by Dold. In particular, we discuss Dold’s fixed-point transfer [Math. Z. 148 (1976), 215–244] and a homotopy version of that transfer constructed by Ponto and Shulman [Algebr. Geom. Topol. 14 (2014), 1275–1306].

Qualitative Analysis of a Predator-Prey Model with Predator Saturation and Competition

In this paper, we consider positive solutions of a predator-prey model with predator saturation and competition under Dirichlet boundary conditions. First, we establish the existence of positive solutions to the system mainly using the fixed point index theory in cone. Second, the uniqueness of positive solution is obtained under certain conditions. We show that there is a unique and asymptotically stable positive solution for c?(?,c?(a)) when β goes to ? or b goes to 0. Finally, we discuss the extinction and persistence results of time-dependent positive solutions to the system.

Multiple positive solutions of integral BVPs for high-order nonlinear fractional differential equations with impulses and distributed delays

In this article, we study the existence of multiple solutions of the integral boundary value problems for high-order nonlinear fractional differential equations with impulses and distributed delays. Some sufficient criteria will be established by the fixed point index theorem in cones. As application, one example is given to demonstrate the validity of our main results.

The spectral analysis for a singular fractional differential equation with a signed measure

In this paper, by using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain some properties of the first eigenvalue of a fractional differential equation. Based on these properties, the fixed point index of the nonlinear operator is calculated explicitly and some sufficient conditions for the existence of positive solutions are established.

Fixed point indices of planar continuous maps

We characterize the sequences of fixed point indices $\{i(f^n, p)\}_{n\ge 1}$ of fixed pointsthat are isolated as an invariant set for a continuous map $f$ in the plane. In particular, we provethat the sequence is periodic and $i(f^n, p) \le 1$ for every $n \ge 0$. This characterization allows us tocompute effectively the Lefschetz zeta functions for a wide class of continuous maps in the \(2\)-sphere, to obtainnew results of existence of infinite periodic orbits inspired on previous articles of J. Franks and to give a partial answer to a problem of M. Shub about the growth of the number of periodic orbits of degree--\(d\) maps in the 2-sphere.

Existence of Positive Solutions for a Fourth-Orderm-Point Boundary Value Problem

By using Krasnosel’skii’s fixed point theorem and the fixed point index theorem in the special function space, we obtain some sufficient conditions for the existence of positive solutions of fourth-order boundary value problem with multipoint boundary conditions. Applications of our results to some special problems are also discussed.

The Existence and Multiplicity of Solutions for Singular Boundary Value Systems with p-Laplacian

This paper presents sufficient conditions for the existence of positive solutions for the fourth-order boundary value problem system with p-Laplacian operator. The existence of single or multiple positive solutions for the system is showed through the fixed point index theory in cones under some assumptions.