Boundary Value Problems For The Quasilinear Differential Equation And Equation Systems

In this paper, we study the existence of positive solutions for the quasilinear differential equations and quasilinear differential equation systems, subject to linear or nonlinear boundary conditions.In chapter one, we study the existence of positive solutions of the equation (φ(x'))' + a(t)f(x(t)) = 0, subject to linear boundary value conditions by a simple application of a fixed point index theorem in cones. We introduce a function φ : R → R is an increasing homeomor-phism and homomorphism and φ(0) = 0.In chapter two, we study the existence of multiple positive solutions of the equation (φ(x'))' + a(t)f(x(t)) = 0, subject to nonlinear boundary value conditions, where φ : R → R is an increasing homeomorphism and homomorphism and φ(0) = 0. We show the existence of at least two positive solutions by using a new three functionals fixed point theorem in cones.In chapter three, we consider the existence of positive solution of the following equation systemswhere φ₁, φ₂ : R → R is an increasing homeomorphism and homomorphism and φ₁(0) = 0, φ₂ (0) = 0. We show the sufficient conditions for the existence of positive solution by using the nome type cone expansion-expression fixed point theorem.In chapter four, when nonlinear term f contains x' , we consider the existence of positive solution of the quasilinear differential equationsubject to nonlinear boundary value conditions by using the nome type cone expansion-expression fixed point theorem, where φ : R → R is an increasing homeomorphism and homomorphism and φ(0) = 0.