A Boundary Value Problem for a Nonlinear Differential Equation with a Small Parameter
نویسندگان
چکیده
It is the purpose of this paper to prove Theorems 1 and 2 below which relate the existence, uniqueness, and general behavior of the solution, y(x, e), for small e>0, of the two-point boundary value problem 0, which includes the point (0, y0). (iii) There exists a constant k>0 such thatf(x, y)^k for (x, y) in R. Then, for all sufficiently small e>0, there exists in R a solution y(x) =y(x, e) of (2) ty" + f(x, y)y' + g(x, y) = 0 satisfying the boundary conditions y(0) = yo, y(l) = yi. Further, y(x, e)—>m(x), y'(x, e)—>m'(x), as e—>0, uniformly on any subinterval 0 < 5 g x ^ 1. Remarks. From the proof of Theorem 1 it will be seen that the Presented to the Society, October 27, 1951; received by the editors March 28, 1951. 73
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