The p-regularity theory and poincaré’s small parameter method

Abstract

Some applications of p -regularity theory in various areas of mathematics were described in [2, 4, 7]. In this paper, p -regularity theory is applied to singular nonlinear ordinary differential equations (ODEs), including ODEs with a small parameter. We formulate an implicit function theorem in the singular case, which is used to analyze the existence of the solution to a boundary value problem for nonlinear ODEs. Additionally, a new method is proposed for solving ODEs in a close-to-resonance case. The method is a modification of the wellknown Poincare small parameter method and the p -factor method. Recall the definitions of a p -factor operator and a p -regular mapping. Consider a mapping F : X → Z , where F ∈ C p + 1 ( X , Z ) and X and Z are Banach spaces. Denote by x * the solution to the equation F ( x ) = 0. It is assumed that Im F '( x *) ≠ Z , which means that the mapping F is singular at the point x * . Without loss of generality, we assume that Z can be decomposed into a direct sum of closed subspaces: Z = Z 1 ⊕ Z 2 ⊕ … ⊕ Z p . In this paper, the p -factor operator is used as a tool for deriving various results in the singular case. We propose two different methods for defining Z 1 , Z 2 , …, Z p and the corresponding p -factor operator. First method. Define Z 1 = cl (Im F '( x *)) ; Z i = cl(span I m F ( i ) ( x *)[ · ] i ) for i = 2, 3, …, p ‐ 1; and Z p = W p , where W i is the closed complement of ( Z 1 ⊕ Z 2 ⊕ … ⊕ Z i – 1 ) to Z and : Z → W i is the projector onto W i along ( Z 1 ⊕ Z 2 ⊕ … ⊕ Z i – 1 ). Following [6], we PW i