Boundary value problems for higher-order nonlinear ordinary differential equations

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The aim of this research is to study the existence, the uniqueness, the number of zeros and the distribution of zeros of solutions of boundary value problems for higher-order ordinary differential equations, and to obtain the detailed information for the set of solutions of higher-order elliptic differential equations on the base of the results for ordinary differential equations.The new results are as follows :1.For a fourth-order or even-order quasilinear ordinary differential equation, necessary and sufficient conditions for the existence of a positive solution are obtained. For a 2-system of the second-order ordinary differential equations and a 2-system of the second-order elliptic equations, the existence of a positive solution is discussed.2.A degenerate elliptic equation with arbitrary nonlinearity is considered on exterior domain, and necessary and sufficient conditions for all solutions to be oscillatory are established. A degenerate elliptic equation is considered on strip-like domain, and the nonexistence of a positive solution is discussed.3.For an initial value problem for the heat equation, stationary isothermic surfaces and uniformly dense domains are discussed, and an interaction between degenerate diffusion and shape of domain is discussed.4.For a singular eigenvalue problem to a higher-order ordinary differential equation on an infinite interval, it is shown that there is a countable sequence of eigenfunctions having exactly n zeros.5.For a fourth-order nonlinear elliptic differential equation including the poly-harmonic operator, the existence results and nonexistence results are obtained.