Libro

Descripción

Nonlinear differential equations and boundary value problems occupy a central place in modern analysis, both because of their theoretical richness and because of their wide range of applications in physical sciences, engineering, and mathematical modeling. Among these problems, those governed by the $p$-Laplacian operator form a particularly important class due to their quasilinear nature and their natural appearance in many real-world phenomena such as non-Newtonian fluid mechanics, combustion theory, porous media diffusion, nonlinear elasticity, and reaction-diffusion processes. This book," Quadrature Method for Boundary Value Problems: Existence, Multiplicity and Bifurcation of Positive Solutions via the Time Map", is devoted to the rigorous study of positive solutions of one-dimensional boundary value problems associated with the $p$-Laplacian operator. Its main objective is to systematically develop and apply the quadrature method, also known as the time-map method, in order to obtain exact results concerning existence, uniqueness, multiplicity, and bifurcation of positive solutions. The major strength of this approach lies in its ability to transform an infinite-dimensional nonlinear differential problem into the study of a single scalar equation involving one real variable. This elegant reduction not only simplifies the theoretical analysis but also allows a precise description of the bifurcation structure of solutions. The method therefore becomes a powerful tool for understanding the global behavior of positive solutions. This work is organized around three main directions. The first establishes the theoretical foundations of the quadrature method: the energy identity, the construction of the time map, the quadrature theorem, and the essential analytical properties of the method.