The Quasilinearization Method on an Unbounded Domain

We apply a method of quasilinearization to a boundary value problem for an ordinary differential equation on an unbounded domain. A uniquely determined Green’s function, which is integrable and of fixed sign, is employed. The hypotheses to apply the quasilinearization method imply uniqueness of solutions. The quasilinearization method generates a bilateral iteration scheme in which the iterates converge monotonically and quadratically to the unique solution. In this paper, we shall apply a method of quasilinearization to the singular boundary value problem (BVP), x′′(t) + q(t)x(t) = f(t, x(t)), t ∈ R, (1) x(0) = x0, x(t) bounded on R, (2) where x0 is real, f : R × R → R is continuous, q : R → R− is continuous, and q(t) ≤ −c < 0, t ∈ R, for some c > 0. We model the singular BVP based on the work of Bebernes and Jackson [1]. The method of quasilinearization has recently been studied and extended extensively. It is generating a rich history beginning with the works by Bellman [2, 3]. Lakshmikantham, Leela, Vatsala, and many co-authors have extensively developed the method and have applied the method to a wide range of problems. We refer the reader to the recent work by Lakshmikantham and Vatsala [12] and the extensive bibliography found there. The method we produce here is modeled by the method developed by Lakshmikantham, Leela and McRae [11]; this method is referred to as the improved generalized quasilinearization method. Analogous methods have been applied to two-point boundary value problems for ordinary differential equations and we refer the reader to the papers, [15, 13, 14, 10, 6, 7, 4]. To our knowledge, this paper provides the first application of the quasilinearization method to singular boundary value problems on unbounded domains. Devi and Vatsala [5] have recently applied the method to a singular BVP on a bounded domain. The method of quasilinearization for BVPs employs a delicate balance of upper and lower solution methods with monotone methods. Recently, Eloe, Grimm and Received by the editors August 10, 2001 and, in revised form, December 11, 2001. 1991 Mathematics Subject Classification. Primary 34B40, 34A45.