A basic theory for initial value problems of first order ordinary differential equations with Lp-Carathéodory functions and applications

Abstract

A basic theory on the existence and uniqueness of solutions and local solutions for initial value problems (IVPs) of first order ordinary differential equations is established, where the nonlinearities are Lp-Carathéodory functions. The basic theory contains three parts: (i) the existence and uniqueness of solutions in C([a,b],[c,d]); (ii) the existence and uniqueness of local solutions in C([a,a+h1(p)];[c,d]), where the expression of the number h1(p) will be given and is a computable constant, and (iii) the existence and uniqueness of solutions and nonnegative solutions by establishing a new Bihari inequality. The classical results dealt with the above part (ii) when f is continuous or an L1-Carathéodory function, and only obtained the existence of a number h such that the IVPs had local solutions in C[a,a+h]. Our methodology is to use the results obtained on (i) to study the results mentioned in (ii) when p∈[1,p] while the classical results used the Schauder fixed point theorems directly to obtain the existence and uniqueness of local solutions when f is continuous or p=1. Our results in parts (i) and (iii) are new. The results in part (ii) are new when p∈(1,∞] and improve the classical results when p=1. Our results will enrich the classical basic theory. As applications of the basic theory, the existence of solutions and nonnegative local solutions for logistic type population models with heterogeneous environments is studied, where the local rates of changes in the population density are Lp-Carathéodory functions.