Recursos de colección

Project Euclid (Hosted at Cornell University Library) (202.070 recursos)

Topological Methods in Nonlinear Analysis

1. Topological shadowing and the Grobman-Hartman theorem

Zgliczyński, Piotr
We give geometric proofs for the Grobman-Hartman theorem for diffeomorphisms and ODEs. Proofs use covering relations and cone conditions for maps and isolating segments and cone conditions for ODEs. We establish topological versions of the Grobman-Hartman theorem as the existence of some semiconjugaces.

2. A note on Conley index and some parabolic problems with locally large diffusion

Carbinatto, Maria C.; Rybakowski, Krzysztof P.
We prove singular Conley index continuation results for a class of scalar parabolic equations with locally large diffusion considered by Fusco [On the explicit construction of an ODE which has the same dynamics as scalar parabolic PDE, J. Differential Equations 69 (1987), 85-110] and Carvalho and Pereira [A scalar parabolic equation whose asymptotic behavior is dictated by a system of ordinary differential equations, J. Differential Equations 112 (1994), 81-130].

3. On a class of cocycles having attractors which consist of singletons

Guzik, Grzegorz
We give a new simple sufficient condition for existence of the global pullback attractor which consists of singletons for general cocycle mappings on an arbitrary complete metric space. In particular, we need not have any structure on a parameter space, so the criterion can be applied in both cases: nonautonomous as well as random dynamical systems. Our considerations lead us also to new large class of iterated function systems with point-fibred attractors.

4. An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators

Feltrin, Guglielmo; Zanolin, Fabio
Using Mawhin's coincidence degree theory, we obtain some new continuation theorems which are designed to have as a natural application the study of the periodic problem for cyclic feedback type systems. We also discuss some examples of vector ordinary differential equations with a $\phi$-Laplacian operator where our results can be applied. Our main contribution in this direction is to obtain a continuation theorem for the periodic problem associated with $(\phi(u'))' + \lambda k(t,u,u') = 0$, under the only assumption that $\phi$ is a homeomorphism.

5. Properties of unique positive solution for a class of nonlocal semilinear elliptic equation

Jiang, Ruiting; Zhai, Chengbo
We study a class of nonlocal elliptic equations $$-M\bigg(\int_{\Omega}|u|^{\gamma}dx\bigg)\Delta u=\lambda f(x,u)$$ with the Dirichlet boundary conditions in bounded domain. Under suitable assumptions on $M$ and the nonlinear term $f$, the existence and new properties of a unique positive solutions are obtained via a monotone operator method and a mixed monotone operator method.

6. Mayer-Vietoris property of the fixed point index

Barge, Héctor; Wójcik, Klaudiusz
We study a Mayer-Vietoris kind formula for the fixed point index of maps of ENR triplets $f\colon (X;X_1,X_2)\to (X;X_1,X_2)$ having compact fixed point set. We prove it under some suitable conditions. For instance when $(X;X_1,X_2)=(E^n;E^n_+,E^n_-)$. We use these results to generalize the Poincaré-Bendixson index formula for vector fields to continuous maps having a sectorial decomposition, to study the fixed point index $i(f,0)$ of orientation preserving homeomorphisms of $E^2_+$ and $(E^3;E^3_+,E^3_-)$ and the fixed point index in the invariant subspace.

7. Concentration of ground state solutions for fractional Hamiltonian systems

Torres, César; Zhang, Ziheng
We are concerned with the existence of ground states solutions to the following fractional Hamiltonian systems: $$\begin{cases} - _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\ u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n), \end{cases} \tag*{(\mbox{FHS})_\lambda}$$ where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda> 0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix for all $t\in \mathbb{R}$, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$ and $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$. Assuming that $L(t)$ is a positive semi-definite symmetric matrix for all $t\in \mathbb{R}$, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$, $W(t,u)$ satisfies the Ambrosetti-Rabinowitz condition and some other reasonable...

8. Global and local structures of oscillatory bifurcation curves with application to inverse bifurcation problem

Shibata, Tetsutaro
We consider the bifurcation problem $$-u''(t) = \lambda (u(t) + g(u(t))), \quad u(t) > 0, \quad t \in I := (-1,1), \quad u(\pm 1) = 0,$$ where $g(u) = g_1(u) := \sin \sqrt{u}$ and $g_2(u) := \sin u^2 (= \sin (u^2))$, and $\lambda > 0$ is a bifurcation parameter. It is known that $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(g,\alpha)$. When $g(u) = g_1(u)$, this problem has been proposed in Cheng [On an open problem of Ambrosetti, Brezis and Cerami,...

9. Trajectory attractor and global attractor for Keller-Segel-Stokes model with arbitrary porous medium diffusion

Sun, Wenlong; Li, Yeping
We investigate long-time behavior of weak solutions for the Keller-Segel-Stokes model with arbitrary porous medium diffusion in 2D bounded domains. We first prove the existence of the trajectory attractor $\mathcal{A}^{{\rm tr}}$ for the translation semigroup in the trajectory space. Further, we construct the global attractor $\mathcal{A}$ in a generalized sense. The results are shown by the definition of trajectory attractor and global attractor, and energy estimates.

10. Singularly perturbed $N$-Laplacian problems with a nonlinearity in the critical growth range

Zhang, Jianjun; do Ó, João Marcos; Miyagaki, Olímpio H.
We consider the following singularly perturbed problem: $$-\varepsilon^N\Delta_N u+V(x)|u|^{N-2}u= f(u),\quad u(x)>0\quad \mbox{in } \mathbb R^N,$$ where $N\ge 2$ and $\Delta_N u$ is the $N$-Laplacian operator. In this paper, we construct a solution $u_\varepsilon$ which concentrates around any given isolated positive local minimum component of $V$, as $\varepsilon\rightarrow 0$, in the Trudinger-Moser type of subcritical or critical case. In the subcritical case, we only impose on $f$ the Berestycki and Lions conditions. In the critical case, a global condition on the nonlinearity $f$ is imposed. However, any monotonicity of $f(t)/t^{N-1}$ or Ambrosetti-Rabinowitz type conditions are not required.

11. Existence of multiple solutions for a quasilinear elliptic problem

Cossio, Jorge; Herrón, Sigifredo; Vélez, Carlos
In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the $p$-derivative at zero and the $p$-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the $p$-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.

12. CQ method for approximating fixed points of nonexpansive semigroups and strictly pseudo-contractive mappings

Piri, Hossein; Rahrovi, Samira
We use the CQ method for approximating a common fixed point of a left amenable semigroup of nonexpansive mappings, an infinite family of strictly pseudo-contraction mappings and the set of solutions of variational inequalities for monotone, Lipschitz-continuous mappings in a real Hilbert space. Our results are a generalization of a result announced by Nadezhkina and Takahashi [N. Nadezhkina and W. Takahashi, Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz-continuous monotone mappings, SIAM J. Optim. 16 (2006), 1230-1241] and some other recent results.

13. On some applications of convolution to linear differential equations with Levitan almost periodic coefficients

We investigate some properties of Levitan almost periodic functions with particular emphasis on their behavior under convolution. These considerations allow us to establish the main result concerning Levitan almost periodic solutions to linear differential equations of the first order. In particular, we state a condition, which guarantees that a special linear equation possesses a Levitan almost periodic solution. We also compare the class of Levitan almost periodic functions and the class of almost periodic functions with respect to the Lebesgue measure, and simultaneously, give an answer to the open question posed by Basit and Günzler in the paper [Difference property...

14. Existence theory for quasilinear elliptic equations via a regularization approach

Liu, Jiaquan; Liu, Xiangqing; Wang, Zhi-Qiang
In this paper, we further develop a regularization approach initiated in our earlier work for the study of solution structure of quasilinear elliptic equations containing several special cases of mathematical models.

15. Rigorous numerics for fast-slow systems with one-dimensional slow variable: topological shadowing approach

Matsue, Kaname
We provide a rigorous numerical computational method to validate periodic, homoclinic and heteroclinic orbits as the continuation of singular limit orbits for the fast-slow system \begin{equation*} \begin{cases} x' = f(x,y,\varepsilon), & \\ y' =\varepsilon g(x,y,\varepsilon) & \end{cases} \end{equation*} with one-dimensional slow variable $y$. Our validation procedure is based on topological tools called isolating blocks, cone conditions and covering relations. Such tools provide us with existence theorems of global orbits which shadow singular orbits in terms of a new concept, the covering-exchange. Additional techniques called slow shadowing and $m$-cones are also developed. These techniques give us not only generalized topological verification...

16. Critical Brezis-Nirenberg problem for nonlocal systems

Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

17. Critical Brezis-Nirenberg problem for nonlocal systems

Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.

18. Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

Teng, Kaimin; Agarwal, Ravi P.
We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

19. Infinitely many solutions for a class of quasilinear equation with a combination of convex and concave terms

Teng, Kaimin; Agarwal, Ravi P.
We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

20. A noncommutative version of Farber's topological complexity

Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.