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...

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.

By using the fixed-point index theory and Leggett-Williams fixed-point theorem,we study the existence of multiple solutions to the three-point boundary value problem u′′′ t a t f t, u t , u′ t 0, 0 < t < 1; u 0 u′ 0 0; u′ 1 − αu′ η λ, where η ∈ 0, 1/2 , α ∈ 1/2η, 1/η are constants, λ ∈ 0,∞ is a parameter, and a, f are given functions. New existence theorems are obtained, which extend and comple...

generalized geraghty type fuzzy mappings oncomplete metric spaces are introduced and a fixed point theorem thatgeneralizes some recent comparable results for fuzzy mappings incontemporary literature is obtained. example is provided to show thevalidity of obtained results over comparable classical results for fuzzymappings in fixed point theory. as an application, existence of coincidencefuzzy p...