Abstract
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 ε, 0>ε≪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.
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Communicated by Li Li
The project is supperted by National Natural Science Foundation of China
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Songlin, C. Singular perturbation for a nonlinear boundary value problem of first order system. Appl Math Mech 17, 1095–1100 (1996). https://doi.org/10.1007/BF00119958
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DOI: https://doi.org/10.1007/BF00119958