The Lefschetz fixed point theorem and its converse have many generalizations. One of these generalizations is to endomorphisms of a space relative to a fixed subspace. In this paper we define relative Lefschetz numbers and Reidemeister traces using traces in bicategories with shadows. We use the functoriality of this trace to identify different forms of these invariants and to prove a relative ...

The fixed point index of topological fixed point theory is a well studied integer-valued algebraic invariant of a mapping which can be characterized by a small set of axioms. The coincidence index is an extension of the concept to topological (Nielsen) coincidence theory. We demonstrate that three natural axioms are sufficient to characterize the coincidence index in the setting of continuous m...

where τ, σ , and c are real constants with τ ≥ 0, σ ≥ 0, |c| < 1, g(t,x) is a T-periodic function for t > 0 and, for an arbitrary bounded domain D ⊂ R, g(t,x) is a Lipschitzfunction on [0,T]×D. Moreover, p ∈ C(R,R), p(t +T) = p(t) and ∫ T 0 p(t)dt = 0. They obtained sufficient conditions which guarantee the existence of at least one T-periodic solution for the above system. However, for the exi...

In this paper, we present necessary and sufficient conditions for the existence of positive C3[0, 1]∩C4(0, 1) solutions for the singular boundaryvalue problem x′′′′(t) = p(t)f(x(t)), t ∈ (0, 1); x(0) = x(1) = x′(0) = x′(1) = 0, where f(x) is either superlinear or sublinear, p : (0, 1) → [0,+∞) may be singular at both ends t = 0 and t = 1. For this goal, we use fixed-point index results.

We provide the solutions of the fixed point conditions of the Yukawa sector for a large class of N = 1 supersymmetric theories including the minimal and next-to-minimal supersymmetric standard models and their grand unified and other extensions. We also introduce a test which can discriminate between infra-red stable, infra-red unstable and saddle point solutions, and illustrate our methods wit...

Existence of strong (i.e. classical) solutions to the generalized inverse of the threedimensional quasi-geostrophic equations, describing the large-scale motion of the atmosphere and oceans, is proved for a finite time interval. Both the dissipative and the nondissipative cases are considered. The spatial domain considered is doubly periodic in the horizontal directions and is bounded above and...

Recursive domain equations have natural solutions. In particular there are domains defined by strictly positive induction. The class of countably based domains gives a computability theory for possibly non-countably based topological spaces. A qcb0 space is a topological space characterized by its strong representability over domains. In this paper, we study strictly positive inductive definiti...