Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The sufficiency of the Matkowsky condition in the problem of resonance


Author: Ching Her Lin
Journal: Trans. Amer. Math. Soc. 278 (1983), 647-670
MSC: Primary 34E15
DOI: https://doi.org/10.1090/S0002-9947-1983-0701516-5
MathSciNet review: 701516
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the sufficiency of the Matkowsky condition concerning the differential equation $ \varepsilon y'' + f(x,\varepsilon )y' + g(x,\varepsilon )y = 0\;( - a \leqslant x \leqslant b)$ under the assumption that $ f(0,\varepsilon ) = 0$ identically in $ \varepsilon ,{f_x}(0,\varepsilon ) \ne 0$ with $ f > 0$ for $ x < 0$ and $ f < 0$ for $ x > 0$. Y. Sibuya proved that the Matkowsky condition implies resonance in the sense of N. Kopell if $ f$ and $ g$ are convergent power series for $ \vert\varepsilon \vert < \rho \;(\rho > 0),f(x,0)=-2x$ and the interval $ [ - a,b]$ is contained in a disc $ D$ with center at 0. The main problem in this work is to remove from Sibuya's result the assumption that $ D$ is a disc.


References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun, Eds., Handbook of mathematical functions, Nat. Bur. Standards Appl. Math. Ser. 55, U. S. Government Printing Office, Washington, D.C., 1964. MR 0167642 (29:4914)
  • [2] R. C. Ackerberg and R. E. O'Malley, Boundary layer problems exhibiting resonance, Stud. Appl. Math. 49 (1970), 61-73. MR 0269940 (42:4833)
  • [3] L. P. Cook and W. Eckhaus, Resonance in a boundary value problem of singular perturbation type, Stud. Appl. Math. 52 (1973), 129-139. MR 0342799 (49:7543)
  • [4] P. F. Hsieh and Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Anal. Appl. 16 (1966), 84-103. MR 0200512 (34:403)
  • [5] N. Kopell, A geometric approach to boundary layer problems exhibiting resonance, SIAM J. Appl. Math. 37 (1979), 436-458. MR 543963 (82d:34079)
  • [6] H. O. Kreiss and S. V. Parter, Remarks on singular perturbations with turning points, SIAM J. Math. Anal. 5 (1974), 230-251. MR 0348212 (50:710)
  • [7] C-H. Lin, Phragmen-Lindelof theorem in a cohomological form, Univ. of Minnesota Math. Report 81-151.
  • [8] -, The sufficiency of Matkowsky-condition in the problem of resonance, Ph.D. Thesis, Univ. of Minnesota, June 1982.
  • [9] B. J. Matkowsky, On boundary layer problems exhibiting resonance, SIAM Rev. 17 (1975), 82-100. MR 0358004 (50:10469)
  • [10] F. W. Olver, Sufficient conditions for Ackerber-O'Malley resonance, SIAM J. Math. Anal. 9 (1978), 328-355. MR 0470383 (57:10137)
  • [11] R. E. O'Malley, Introduction to singular perturbation, Academic Press, New York, 1974.
  • [12] Y. Sibuya, Uniform simplification in a full neighborhood of a turning point, Mem. Amer. Math. Soc. No. 149 (1974). MR 0440140 (55:13020)
  • [13] -, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, North-Holland, Amsterdam, 1975.
  • [14] -, A theorem concerning uniform simplification at a transition point and the problem of resonance, SIAM J. Math. Anal. 12 (1981), 653-668. MR 625824 (82m:34060)
  • [15] W. Wasow, Asymptotic expansions for ordinary differential equations, Interscience, New York, 1965. MR 0203188 (34:3041)
  • [16] A. M. Watts, A singular perturbation problem with a turning point, Bull. Austral. Math. Soc. 5 (1971), 61-73. MR 0296444 (45:5504)
  • [17] E. T. Whittaker and G. H. Watson, A course of modern analysis, 4th ed., Cambridge Univ. Press, Cambridge, 1952.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34E15

Retrieve articles in all journals with MSC: 34E15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0701516-5
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society