SINGULAR PERTURBATION FOR A NONLINEAR BOUNDARY VALUE PROBLEM OF FIRST ORDER SYSTEM

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Singular perturbation for a nonlinear boundary value problem of first order system

In this paper, we study the following perturbed nonlinear boundary value problem of the form:% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaH1o% qzceWG4bGbauaacqGH9aqpcaWGMbGaaiikaiaadshacaGGSaGaamiE% aiaacYcacaWG5bGaaiilaiabew7aLjaacMcaaeaacqaH1oqzceWG5b% GbauaacqGH9aqpcaWGNbGaaiikaiaadshacaGGSaGaamiEaiaacYca% caWG5bGaaiilaiabew7aLjaacMcaaeaacaWG4bGaaiikaiaaicdaca% GGPaGaeyypa0JaamyqaiaacIcacqaH+oaEdaWgaaWcbaGaaGymaaqa% baGccaGGSaGaeqOVdG3aaSbaaSqaaiaaikdaaeqaaOGaaiilaiaadI% hacaGGOaGaaGymaiaacMcacqGHsislcaWG4bGaaiikaiaaigdacaGG% PaGaeyOeI0IaamiEaiaacIcacaaIWaGaaiykaiaacYcacaWG5bGaai% ikaiaaigdacaGGPaGaeyOeI0IaamyEaiaacIcacaaIWaGaaiykaiaa% cYcacqaH1oqzcaGGPaaabaGaamyEaiaacIcacaaIWaGaaiykaiabg2% da9iaadkeacaGGOaGaeqOVdG3aaSbaaSqaaiaaigdaaeqaaOGaaiil% aiabe67a4naaBaaaleaacaaIYaaabeaakiaacYcacaWG4bGaaiikai% aaigdacaGGPaGaeyOeI0IaamiEaiaacIcacaaIXaGaaiykaiabgkHi% TiaadIhacaGGOaGaaGimaiaacMcacaGGSaGaamyEaiaacIcacaaIXa% GaaiykaiabgkHiTiaadMhacaGGOaGaaGimaiaacMcacaGGSaGaeqyT% duMaaiykaaaaaa!9385![begin{gathered} varepsilon x' = f(t,x,y,varepsilon ) hfill varepsilon y' = g(t,x,y,varepsilon ) hfill x(0) = A(xi _1 ,xi _2 ,x(1) - x(1) - x(0),y(1) - y(0),varepsilon ) hfill y(0) = B(xi _1 ,xi _2 ,x(1) - x(1) - x(0),y(1) - y(0),varepsilon ) hfill end{gathered} ]where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdG3aaS% baaSqaaiaaigdaaeqaaOGaaiilaiaabccacqaH+oaEdaWgaaWcbaGa% aGOmaaqabaaaaa!3C9E![xi _1 ,{text{ }}xi _2 ] are functions of , 0>1. Under some suitable conditions, we give the asymptotic expansion of solution of any order, and obtain the estimation of remainder term by using the comparison theorem.The project is supperted by National Natural Science Foundation of China