Nonuniqueness Theorem for a Singular Cauchy-nicoletti Problem

The nonuniqueness of a regular or singular Cauchy problem for ordinary differential equations is studied in several papers such as [3, 4, 5, 13, 14, 15, 16, 17]. Most of these results can also be found in the monograph [1]. The uniqueness of solutions of Cauchy initial value problem for ordinary differential equations with singularity is investigated in [7, 8, 9, 12]. The topological structure of solution sets to a large class of boundary value problems for ordinary differential equations is studied in [2]. First results on the nonuniqueness for a singular Cauchy-Nicoletti boundary value problem are given in [10, 11, 12] by Kiguradze, where sufficient conditions for the nonuniqueness are written in the form of one-sided inequalities for the components in the right-hand side f (t,x1, . . . ,xn) of the corresponding equation. An expression for the estimation of the jth component f j(t,x1, . . . ,xn) of f depends on t and xj and is linear in |xj|. In [6], we studied the nonuniqueness for a singular Cauchy problem. Our criteria involve vector Lyapunov functions and the estimations need not be linear. The present paper deals with the nonuniqueness of the singular Cauchy-Nicoletti problem and extends the results of [6] to this more general problem. Supposing −∞ ≤ a < A ≤ ∞, b > 0, we will use the following notations throughout the paper: Rk and R+ denote k-dimensional real Euclidean space and the interval [0,∞), respectively. | · | is used for the notation of Hölder’s 1-norm (the sum of the absolute values of components). x = (x1, . . . ,xn) denotes a variable vector from Rn with components x1, . . . ,xn, while x0 = (x01, . . . ,x0n) stands for a fixed vector from Rn with components x01, . . . ,x0n.N is equal to the set {1, . . . ,n}. l denotes a fixed number from the set {1, . . . ,n}.