Journal of Fixed Point Theory and Applications最新文献

In this paper, we are concerned with the bifurcation from infinity and multiplicity of solutions of the semilinear elliptic system

which can be considered as the stationary problem of reaction–diffusion equations. We treat this problem in the framework of dynamical systems, and deal with it via the approach of a pure dynamical nature, which is different from those in the literature. By using the Shape theory of attractors and the Poincaré–Lefschetz duality theory of Conley index, we establish some new multiplicity results of solutions of the system on bifurcations from infinity under an appropriate Landesman–Lazer type condition, improving the earlier works in the literature.

We study the Dirichlet problem for an elliptic system derived from FitzHugh–Nagummo model as follows:

where (Omega ) represents a bounded smooth domain in (mathbb {R}^2) and (varepsilon , gamma ) are positive constants. The parameter (delta _{varepsilon }>0) is a constant dependent on (varepsilon ), and the nonlinear term f(u) is defined as (u(u-a)(1-u)). Here, a is a function in (C^2(Omega )cap C^1({overline{Omega }})) with its range confined to ((0,frac{1}{2})). Our research focuses on this spatially inhomogeneous scenario whereas the scenario that a is spatially constant has been studied extensively by many other mathematicians. Specifically, in dimension two, we utilize the Lyapunov–Schmidt reduction method to establish the existence of a single interior peak solution. This is contingent upon a mild condition on a, which acts as an indicator of a location-dependent activation threshold for excitable neurons in the biological environment.

We study the determinant of the second variation of an optimal control problem for general boundary conditions. Generically, these operators are not trace class and the determinant is defined as a principal value limit. We provide a formula to compute this determinant in terms of the linearisation of the extrenal flow. We illustrate the procedure in some special cases, proving some Hill-type formulas.

Abstract

In this paper, we consider the coupled elliptic system with critical exponent and logarithmic terms: $$begin{aligned} {left{ begin{array}{ll} -Delta u=lambda _{1}u+ mu _1|u|^{2p-2}u+beta |u|^{p-2}|v|^{p}u+theta _1 ulog u^2, &{} quad xin Omega , -Delta v=lambda _{2}v+ mu _2|v|^{2p-2}v+beta |u|^{p}|v|^{p-2}v+theta _2 vlog v^2, &{}quad xin Omega , u=v=0, &{}quad x in partial Omega , end{array}right. } end{aligned}$$ where (Omega subset {mathbb R}^N) is a bounded smooth domain, (2p=2^*=frac{2N}{N-2}) is the Sobolev critical exponent. When (N ge 5) , for different ranges of (beta ,lambda _{i},mu _i,theta _{i}) , (i=1,2) , we obtain existence and nonexistence results of positive solutions via variational methods. The special case (N=4 ) was studied by Hajaiej et al. (Positive solution for an elliptic system with critical exponent and logarithmic terms, arXiv:2304.13822, 2023). Note that for (Nge 5) , the critical exponent is given by (2pin left( 2,4right) ) ; whereas for (N=4) , it is (2p=4) . In the higher-dimensional cases (Nge 5) brings new difficulties, and requires new ideas. Besides, we also study the Brézis–Nirenberg problem with logarithmic perturbation $$begin{aligned} -Delta u=lambda u+mu |u|^{2p-2}u+theta u log u^2 quad text { in }Omega , end{aligned}$$ where (mu >0, theta <0) , (lambda in {mathbb R}) , and obtain the existence of positive local minimum and least energy solution under some certain assumptions.

This work is devoted to the study of singular strongly non-linear integro-differential equations of the type

where f is a Carathéodory function, (Phi ) is a strictly increasing homeomorphism, and k is a non-negative integrable function, which is allowed to vanish on a set of zero Lebesgue measure, such that (1/k in L^p_textrm{loc}({mathbb {R}}^{+}_0)) for a certain (p>1). By considering a suitable set of assumptions, including a Nagumo–Wintner growth condition, we prove existence and non-existence results for boundary value problems associated with the non-linear integro-differential equation of our interest in the sub-critical regime on the real half line.

In this work, we use techniques from Stieltjes calculus and fixed point index theory to show the existence and multiplicity of solution of a first order non-linear boundary value problem with linear boundary conditions that extend the periodic case. We also provide the Green’s function associated to the problem as well as an example of application.

Abstract

Fix a suitable link on the disk. Recently, F. Morabito associates each Hamiltonian symplecticmorphism preserving the link to a braid type. Based on this construction, Morabito defines a family of pseudometrics on the braid groups by using the Hofer metric. In this paper, we show that two Hamiltonian symplecticmorphisms define the same braid type provided that their Hofer distance is sufficiently small. As a corollary, the pseudometrics defined by Morabito are nondegenerate.

In this paper, we use truncation argument combined with method of minimization, argument of comparison, topological degree arguments and sub-supersolutions method to show existence of multiple positive solutions (which are ordered in the (C(overline{Omega }))-norm) for the following class of problems:

where (Omega ) is a bounded smooth domain of (mathbb {R}^N) ((Nge 1), kappa ,mu ,lambda > 0,qge 1) are parameters, the nonlinearity (f: mathbb {R}rightarrow mathbb {R}) is a continuous function that can change sign and satisfies an area condition and (h: mathbb {R}rightarrow mathbb {R}) is a general nonlinearity.

We classify, up to fiberwise symplectomorphisms, a saturated neighborhood of a singular fiber of an integrable system (which is proper onto its image and has connected fibers) containing (k geqslant 1) focus-focus critical points. Our proof recovers the classification for (k=1) which was known prior to this paper. Our result shows that there is a one-to-one correspondence between such neighborhoods and k formal power series, up to a ((mathbb {Z}_2 times D_k))-action, where (D_k) is the kth dihedral group. The k formal power series determine the dynamical behavior of the Hamiltonian vector fields associated to the components of the momentum map on the symplectic manifold ((M,omega )) near the singular fiber containing the k focus-focus critical points. This proves a conjecture of San Vũ Ngọc from 2003.

We construct finite energy foliations and transverse foliations of neighbourhoods of the circular orbits in the rotating Kepler problem for all negative energies. This paper would be a first step towards our ultimate goal that is to recover and refine McGehee’s results on homoclinics [23] and to establish a theoretical foundation to the numerical demonstration of the existence of a homoclinic–heteroclinic chain in the planar circular restricted three-body problem [20], using pseudoholomorphic curves.