Journal of Fixed Point Theory and Applications最新文献

We study time-periodic and spatially localised solutions (breathers) in general infinite conservative and dissipative nonlinear Klein–Gordon lattices. First, in the time-reversible (conservative) case, we give a concise proof of the existence of breathers not using the concept of the anticontinuous limit. The existence problem is converted into an operator equation for time-reversal initial conditions generating breather solutions. A nontrivial solution of this operator equation is established facilitating Schauder’s fixed point theorem. Afterwards, we prove the existence and uniqueness of breather solutions in damped and forced infinite nonlinear Klein–Gordon lattice systems utilising the contraction mapping principle.

In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of (b^m)-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for (b^{2m})-symplectic manifolds depending only on the topology of the manifold. Moreover, for (b^m)-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions.

We introduce a notion of strong closing property of contact forms, inspired by the (C^infty ) closing lemma for Reeb flows in dimension three. We then prove a sufficient criterion for strong closing property, which is formulated by considering a monoidal functor from a category of manifolds with contact forms to a category of filtered vector spaces. As a potential application of this criterion, we propose a conjecture which says that a standard contact form on the boundary of any symplectic ellipsoid satisfies strong closing property.

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.