Nonlinear boundary value problems for ordinary differential equations

Andrzej Granas ; Ronald Guenther ; John Lee

GDML_Books, (1985), p.

Résumé

CommentsThis tract is intended to be accessible to a broad spectrum of readers. Those with out much previous experience with differential equations might find it profitable (when the need arises) to consult one of the following standard texts: Coddington-Levinson [17], Hale [35], Hartman [38], Mawhin-Rouche [61]. The bibliography given below is restricted mostly to the problems discussed in the tract or closely related topics. A small number of additional references are included however in order to provide a guide to further study; most of these contain extensive bibliographies for the material they cover. The following references include some of the recent surveys and monographs that are related to the subject matter of this tract in a substantial way: Bailey-Shampine-Waltman [7], Bernfeld-Lakshmikantahm [11], Cesari [15], Eloe-Henderson [21], Gaines-Mawhin [25], Gudkov-Klokow-Lepin-Ponomarov [34], Jackson [43], Keller [47], Lefschetz [57], Mawhin [60], Protter-Weinberger [69].

CONTENTSComments............................................................................................................................5CHAPTER IIntroduction§ 1. Elementary theory of second order differential equations...........................................12§ 2. Topological preliminaries.............................................................................................14§ 3. The maximum principle................................................................................................16§ 4. Existence and a priori bounds-examples.....................................................................19§ 5. Problems with other boundary conditions....................................................................25CHAPTER IIThe Bernstein theory of the equation y" = f(t, y, y')§ 1. The homogeneous Dirichlet, Neumann, and periodic problems...................................28§ 2. The homogeneous Sturm-Liouville problem................................................................34§ 3. Inhomogeneous boundary conditions..........................................................................35§ 4. Examples and remarks................................................................................................39§ 5. Bernstein-Nagumo growth conditions..........................................................................44§ 6. Nonlinear boundary conditions....................................................................................50§ 7. Uniqueness..................................................................................................................52CHAPTER IIIApplications§ 1. Steady-state temperature distributions........................................................................56§ 2. The Thomas-Fermi problem........................................................................................59§ 3. Singular boundary value problems..............................................................................62§ 4. Osmotic flow.................................................................................................................64§ 5. Positive solutions to diffusion equations......................................................................70CHAPTER IVOther second order boundary value problems§ 1. Periodic solutions to differential equations of Nirenberg type......................................76§ 2. The Dirichlet problem for y" = f(y') and the Neumann problem for y" = f(t,y,y').............85§ 3. Upper and lower solutions...........................................................................................94CHAPTER VEven order systems and higher order equations§ 1. General existence theorems........................................................................................99§ 2. Second order systems...............................................................................................102§ 3. Third and fourth order problems................................................................................108§ 4. Higher even order equations......................................................................................111CHAPTER VINumerical solution of boundary value problems§ 1. Newton’s method........................................................................................................113§ 2. The shooting method for the Dirichlet problem..........................................................115§ 3. The shooting method for the Neumann problem........................................................120§ 4. Quasilinearization for boundary value problems........................................................121References.......................................................................................................................125

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Zbl 0615.34010

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