Acta mathematica scientia,Series B ›› 1988, Vol. 8 ›› Issue (4): 389-398.

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ON UNIFORMLY VALID ESTIMATE OF SOLUTIONS TO SINGULAR PERTURBATION ROBIN BOUNDARY VALUE PROBLEM

Wang Huaizhong Zhou Qinde   

  1. Dept. of Math., Jilin University, Changchun, China
  • Received:1986-02-23 Online:1988-12-25 Published:1988-12-25

Abstract: In this paper we study the Robin boundary value problem with a small parameter
εy"=f(t, y, ω(ε)y', ε),
a0y(0) +b0y'(0)=ξ(ε), a1y(1)+b1y'(1)=η(ε),
where the function ω(ε) is continuous on ε ≥ 0 with ω(0)=0. Assuming all known functions are suitably smooth, f satisfies Nagumo's condition, fy>0, ai2-bi2≠0, (-1)iaibi ≤ 0 (i=0, 1) and the reduced equation 0=f(t, y, 0, 0) has a solution y(t) (0 ≤ t ≤ 1), we prove the existence and the uniqueness of the solution for the boundary value problem and givo an asymptotic expansion of the solution in the power ε1/2 which is uniformly valid on 0 ≤ t ≤ 1.

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