Fixed points for planar maps with multiple twists, with application to nonlinear equations with indefinite weight

In this paper, we investigate the dynamical properties associated with planar maps which can be represented as a composition of twist maps together with expansive–contractive homeomorphisms. The class of maps we consider present some common features both with those arising in the context of the Poincaré–Birkhoff theorem and those studied in the theory of topological horseshoes. In our main theorems, we show that the multiplicity results of fixed points and periodic points typical of the Poincaré–Birkhoff theorem can be recovered and improved in our setting. In particular, we can avoid assuming area-preserving conditions and we also obtain higher multiplicity results in the case of multiple twists. Applications are given to periodic solutions for planar systems of non-autonomous ODEs with sign-indefinite weights, including the non-Hamiltonian case. The presence of complex dynamics is also discussed. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

A mathematical study of diffusive logistic equations with mixed type boundary conditions

<p style='text-indent:20px;'>The purpose of this paper is to provide a careful and accessible exposition of static bifurcation theory for a class of mixed type boundary value problems for diffusive logistic equations with indefinite weights, which model population dynamics in environments with spatial heterogeneity. We discuss the changes that occur in the structure of the positive solutions as a parameter varies near the first eigenvalue of the linearized problem, and prove that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) located some distance away from the boundary of the environment. A biological interpretation of main theorem is that an initial population will grow exponentially until limited by lack of available resources if the diffusion rate is below some critical value; this idea is generally credited to the English economist T. R. Malthus. On the other hand, if the diffusion rate is above this critical value, then the model obeys the logistic equation introduced by the Belgian mathematical biologist P. F. Verhulst. The approach in this paper is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in partial differential equations.</p>