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the author's book on gewshnliche differentialgleichungen (ordinary differential equations) was published in 1972. the present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. the german book is widely used as a textbook for a first course in ordinary differential equations. this is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, peano's existence theorem. it is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. let me remark here that in the german system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations.
preface
note to the reader
introduction
chapter i. first order equations: some integrable cases
1. explicit first order equations
2. the linear differential equation. related equations
supplement: the generalized logistic equation
3. differential equations for families of curves. exact equations
4. implicit first order differential equations
chapter ii: theory of first order differential equations
5. tools from functional analysis
6. an existence and uniqueness theorem
supplement: singular initial value problems
7. the peano existence theorem
supplement: methods of functional analysis
8. complex differential equations. power series expansions
9. upper and lower solutions. maximal and minimal integrals
supplement: the separatrix
chapter iii: first order systems. equations of higher order
.10. the initial value problem for a system of first order
supplement i: differential inequalities and invariance
supplement ii: differential equations in the sense
of caratheodory
11. initial value problems for equations of higher order
supplement: second order differential inequalities
12. continuous dependence of solutions
supplement: general uniqueness and dependence theorems
13. dependence of solutions on initial values and parameters
chapter iv: linear differential equations
14. linear systems
15. homogeneous linear systems
16. inhomogeneous systems
supplement: l1-estimation of c-solutions
17. systems with constant coefficients
18. matrix functions. inhomogeneous systems
supplement: floquet theory
19. linear differential equations of order n
20. linear equations of order n with constant coefficients
supplement: linear differential equations with
periodic coefficients
chapter v: complex linear systems
21. homogeneous linear systems in the regular case
22. isolated singularities
23. weakly singular points. equations of fuchsian type
24. series expansion of solutions
25. second order linear equations
chapter vi: boundary value and eigenvalue problems
26. boundary value problems
supplement i: maximum and minimum principles
supplement ii: nonlinear boundary value problems
27. the sturm-liouville eigenvalue problem
supplement: rotation-symmetric elliptic problems
28. compact self-adjoint operators in hilbert space
chapter vii: stability and asymptotic behavior
29. stability
30. the method of lyapunov
appendix
a. topology
b. real analysis
c. complex analysis
d. functional analysis
solutions and hints for selected exercises
literature
index
notation
the author's book on gewshnliche differentialgleichungen (ordinary differential equations) was published in 1972. the present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. the german book is widely used as a textbook for a first course in ordinary differential equations. this is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, peano's existence theorem. it is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. let me remark here that in the german system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations.
prerequisites. in order to understand the main text, it suffices that the reader have a sound knowledge of calculus and be familiar with basic notions from linear algebra. for complex differential equations, some facts about hmomorphic functions and their integrals are required. these are summarized at the beginning of 8 and more fully described and partly proved in part c of the appendix. functional analysis is developed in the text when needed. in several places there are sections denoted as supplements, where more special subjects are treated or the theory is extended. more advanced tools such as lebesgue's theory of integration or schauder's fixed point theorem are occasionally used in those sections. the supplements and also 13 can be omitted in a first reading.
outline of contents. the book treats significantly more topics than can be covered in a one-semester course. it also contains material that is seldom found in textbooks and--what is perhaps more important--it uses new proofs for basic theorems. this aspect of the book calls for a closer look at contents and methods with emphasis on those places where we depart from the mainstream.
the first chapter treats classical cases of first order equations that can be solved explicitly. by means of a number of examples the student encounters the essential features of the initial value problem such as uniqueness and nonuniqueness, maximal solutions in the case of nonuniqueness, and continuous dependence on initial values in the small, but not in the large; see 1.vi-viii. the phase plane and phase portraits are explained in 3.vi-viii.
the theory proper starts with chapter ii. in this and the following chapter the initial value problem is treated first for one equation and then for systems of equations. the repetition caused by this separation of cases is minimal since all proofs carry over, while the student has the benefit that the reasoning is not burdened by technicalities about vector functions. the complex case, where the solutions are holomorphic functions, is treated in 8; the proofs follow the pattern set in 6 for the real case. the theory of differential inequalities in 9 is one-dimensional by its very nature. an extension to n dimensions leads to new phenomena that are treated in supplement i of 10.
chapter iv is devoted to linear systems and linear differential equations of higher order. in a supplement to 18 the floquet theory for systems with periodic coefficients is presented.
linear systems in the complex domain is the topic of chapter v. the main properties of systems with isolated singularities are developed in a novel way (see below). equations of mathematical physics are discussed in 25.
the main subject of chapter vi is the sturm-liouville theory of boundary value and eigenvalue problems. nonlinear boundary value problems and corresponding existence, uniqueness, and comparison theorems are also treated. in 28 the eigenvalue theory for compact self-adjoint operators in hilbert space is developed and applied to the sturm-liouville eigenvalue problem.
the last chapter deals with stability and asymptotic behavior of solutions. the linearization theorem of grobman-hartman is given without proof (the author is still looking for a really good proof). the method of lyapunov is developed and applied in 30.
an appendix consisting of four parts a (topology), b (real analysis), c (complex analysis), and d (functional analysis) contains notions and theorems that are used in the text or can lead to a deeper understanding of the subject. the fixed point theorems of brouwer and schauder are proved in b.v and d.xii.
in closing this overview, we point out that applications, mostly from mechanics and mathematical biology, are found in many places. exercises, which range from routine to demanding, are dispersed throughout the text, some with an outline of the solution. solutions of selected exercises are found at the end of the book.
special features. two general themes exercise a profound influence throughout the book: functional analysis and differential inequalities.
functional analysis. the contraction principle, that is, the fixed point theorem for contractive mappings in a banach space, is at the center. this theorem has all necessary properties to make it a fundamental principle of analysis: it is elementary, widely applicable, and far-reaching.2 its flexibility in connection with our subject comes to light when appropriate weighted maximum norms are used. a first example is found in the dissertation of morgenstern (1952); references to later authors in the literature axe historically unjustified. in linear complex systems, the weighted maximum norm in 21.ii leads to global existence without using analytic continuation and the monodromy theorem. moreover, this proof gives the growth properties of solutions that are needed in the treatment of singular points. the theorems on continuous dependence on initial values and parameters and on holomorphy with regard to complex parameters follow directly from the contraction principle, a fact which is still little known. differentiability with respect to real parameters requires ostrowski's theorem on approximate iteration 13.iv.
in the treatment of linear systems with weakly singular points, the crucial convergence proofs are also reduced to the contraction principle in a suitable banach space.3 for holomorphic solutions, i.e., power series expansions, this method was discovered by harris, sibuya, and weinberg (1969). the logarithmic case can also be treated along these lines. this approach leads also to theorems of lettenmeyer and others, which are beyond the scope of this book; cf. the original work cited above.
a theorem in appendix d.vii, which is partly due to holmes (1968), establishes a relation between the norm of a linear operator and its spectral radius. as explained in section d.ix, this result gives a better insight into the role of weighted maximum norms.
differential inequalities. the author, who also wrote the first monograph on differential inequalities (1964, 1970), has encountered many instances where authors are unaware of basic theorems on differential inequalities that would have made their reasoning much simpler and stronger. the distinction between weak and strong inequalities is a matter of fundamental importance. in partial differential equations this is common knowledge: weak maximum or comparison principles versus strong principles of this type. not so in ordinary differential equations. theorem 9.ix is a strong comparison principle that prescribes precisely the occurrence of strict inequalities, while most (all?) textbooks are content with the weak "less than or equal" statement. this principle is essential for our treatment of the sturm-liouville theory via prufer transformation. its usefulness in nonlinear sturm theory can be seen from a recent paper, walter (1997).
supplement i in 10 brings the two basic theorems on systems of differential inequalities, (i) the comparison theorem for quasimonotone systems, and (ii) max miiller's theorem for the general case. both were found in the mid twenties. quasimonotonicity is a necessary and sufficient condition for extending the classical theory (including maximal and minimal solutions) from one equation to systems of equations. more recently, both theorems (i) and (ii) have been applied to population dynamics, but it is not generally known that results on invariant rectangles are special cases of miiller's theorem. theorem 10.xii is the strong version of (i); it contains m. hirsch's theorem on strongly monotone flows, cf. hirsch (1985) and walter (1997).
a supplement to 26 describes a new approach to minimum principles for boundary value problems of sturmian type that applies also to nonlinear differential operators; cf. walter (1995). the strong minimum principle is generalized in 26.xix, so that it includes now the first eigenvalue case.
in supplement ii of 26 on nonlinear boundary value problems the method of upper and lower solutions for existence and serrin's sweeping principle for uniqueness are presented.
. miscellaneous topics. differential equations in the sense of caratheodory. the initial value problem is treated in supplement ii of 10 and a sturmliouville theory under caratheodory assumptions in 26.xxiv and 27.xxi. as a rule, the earlier proofs for the classical case carry over. this applies in particular to the strong comparison theorem 10.xv and the strong minimum principle in 26.xxv.
radial solutions of elliptic equations. this subject plays an active role in recent research on nonlinear elliptic problems. the radial a-operator is an operator of sturm-liouville type with a singularity at 0. the corresponding initial value problem is treated in a supplement of 6, and the eigenvalue problem and nonlinear boundary value problems for the unit ball in rn (for radial solutions) in a supplement to 27.
separatrices is the theme of a supplement in 9. differential inequalites are essential for proving existence and uniqueness.
special applications. we mention the generalized logistic equation in a supplement to 2, general predator-prey models in 3.vii, delay-differential equations in 7.xiv-xv, invariant sets in 10.xvi and the rubber band as a model for nonlinear oscillations in a nonsymmetric mechanical system in 11.x.
exact numerics. we give examples in which a combination of a numerical procedure and a sup-superfunction technique allows a mathematically exact computation of special values. the numerical part is based on an algorithm, developed by rudolf lohner (1987, 1988), that computes exact enclosures for the solutions of an initial value problem. in blow-up problems one obtains rather sharp enclosures for the location of the asymptote of the solutions; cf. 9.v. a different kind of sub-and supersolutions is used to compute a separatrix; in general, a separatrix is an unstable solution.
acknowledgments. it is a pleasure to thank all those who have contributed to the making of this volume. ttie translator, professor russell thompson, worked with expertise and patience in the face of changes and additions during the translation and furnished beautiful figures. he also suggested an improved division into chapters. irene redheffer acted as a mediator between author and translator with exceptional care and insight and translated the solutions section. her help and advice and that of professor ray redheffer were indispensable. my sincere thanks go to all of them and also to other helping hands and minds.
karlsruhe, august 1997
wolfgang walter
the author's book on gewshnliche differentialgleichungen (ordinary differential equations) was published in 1972. the present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. the german book is widely used as a textbook for a first course in ordinary differential equations. this is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, peano's existence theorem. it is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. let me remark here that in the german system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations.
preface
note to the reader
introduction
chapter i. first order equations: some integrable cases
1. explicit first order equations
2. the linear differential equation. related equations
supplement: the generalized logistic equation
3. differential equations for families of curves. exact equations
4. implicit first order differential equations
chapter ii: theory of first order differential equations
5. tools from functional analysis
6. an existence and uniqueness theorem
supplement: singular initial value problems
7. the peano existence theorem
supplement: methods of functional analysis
8. complex differential equations. power series expansions
9. upper and lower solutions. maximal and minimal integrals
supplement: the separatrix
chapter iii: first order systems. equations of higher order
.10. the initial value problem for a system of first order
supplement i: differential inequalities and invariance
supplement ii: differential equations in the sense
of caratheodory
11. initial value problems for equations of higher order
supplement: second order differential inequalities
12. continuous dependence of solutions
supplement: general uniqueness and dependence theorems
13. dependence of solutions on initial values and parameters
chapter iv: linear differential equations
14. linear systems
15. homogeneous linear systems
16. inhomogeneous systems
supplement: l1-estimation of c-solutions
17. systems with constant coefficients
18. matrix functions. inhomogeneous systems
supplement: floquet theory
19. linear differential equations of order n
20. linear equations of order n with constant coefficients
supplement: linear differential equations with
periodic coefficients
chapter v: complex linear systems
21. homogeneous linear systems in the regular case
22. isolated singularities
23. weakly singular points. equations of fuchsian type
24. series expansion of solutions
25. second order linear equations
chapter vi: boundary value and eigenvalue problems
26. boundary value problems
supplement i: maximum and minimum principles
supplement ii: nonlinear boundary value problems
27. the sturm-liouville eigenvalue problem
supplement: rotation-symmetric elliptic problems
28. compact self-adjoint operators in hilbert space
chapter vii: stability and asymptotic behavior
29. stability
30. the method of lyapunov
appendix
a. topology
b. real analysis
c. complex analysis
d. functional analysis
solutions and hints for selected exercises
literature
index
notation
the author's book on gewshnliche differentialgleichungen (ordinary differential equations) was published in 1972. the present book is based on a translation of the latest, 6th, edition, which appeared in 1996, but it also treats some important subjects that are not found there. the german book is widely used as a textbook for a first course in ordinary differential equations. this is a rigorous course, and it contains some material that is more difficult than that usually found in a first course textbook; such as, for example, peano's existence theorem. it is addressed to students of mathematics, physics, and computer science and is usually taken in the third semester. let me remark here that in the german system the student learns calculus of one variable at the gymnasium1 and begins at the university with a two-semester course on real analysis which is usually followed by ordinary differential equations.
prerequisites. in order to understand the main text, it suffices that the reader have a sound knowledge of calculus and be familiar with basic notions from linear algebra. for complex differential equations, some facts about hmomorphic functions and their integrals are required. these are summarized at the beginning of 8 and more fully described and partly proved in part c of the appendix. functional analysis is developed in the text when needed. in several places there are sections denoted as supplements, where more special subjects are treated or the theory is extended. more advanced tools such as lebesgue's theory of integration or schauder's fixed point theorem are occasionally used in those sections. the supplements and also 13 can be omitted in a first reading.
outline of contents. the book treats significantly more topics than can be covered in a one-semester course. it also contains material that is seldom found in textbooks and--what is perhaps more important--it uses new proofs for basic theorems. this aspect of the book calls for a closer look at contents and methods with emphasis on those places where we depart from the mainstream.
the first chapter treats classical cases of first order equations that can be solved explicitly. by means of a number of examples the student encounters the essential features of the initial value problem such as uniqueness and nonuniqueness, maximal solutions in the case of nonuniqueness, and continuous dependence on initial values in the small, but not in the large; see 1.vi-viii. the phase plane and phase portraits are explained in 3.vi-viii.
the theory proper starts with chapter ii. in this and the following chapter the initial value problem is treated first for one equation and then for systems of equations. the repetition caused by this separation of cases is minimal since all proofs carry over, while the student has the benefit that the reasoning is not burdened by technicalities about vector functions. the complex case, where the solutions are holomorphic functions, is treated in 8; the proofs follow the pattern set in 6 for the real case. the theory of differential inequalities in 9 is one-dimensional by its very nature. an extension to n dimensions leads to new phenomena that are treated in supplement i of 10.
chapter iv is devoted to linear systems and linear differential equations of higher order. in a supplement to 18 the floquet theory for systems with periodic coefficients is presented.
linear systems in the complex domain is the topic of chapter v. the main properties of systems with isolated singularities are developed in a novel way (see below). equations of mathematical physics are discussed in 25.
the main subject of chapter vi is the sturm-liouville theory of boundary value and eigenvalue problems. nonlinear boundary value problems and corresponding existence, uniqueness, and comparison theorems are also treated. in 28 the eigenvalue theory for compact self-adjoint operators in hilbert space is developed and applied to the sturm-liouville eigenvalue problem.
the last chapter deals with stability and asymptotic behavior of solutions. the linearization theorem of grobman-hartman is given without proof (the author is still looking for a really good proof). the method of lyapunov is developed and applied in 30.
an appendix consisting of four parts a (topology), b (real analysis), c (complex analysis), and d (functional analysis) contains notions and theorems that are used in the text or can lead to a deeper understanding of the subject. the fixed point theorems of brouwer and schauder are proved in b.v and d.xii.
in closing this overview, we point out that applications, mostly from mechanics and mathematical biology, are found in many places. exercises, which range from routine to demanding, are dispersed throughout the text, some with an outline of the solution. solutions of selected exercises are found at the end of the book.
special features. two general themes exercise a profound influence throughout the book: functional analysis and differential inequalities.
functional analysis. the contraction principle, that is, the fixed point theorem for contractive mappings in a banach space, is at the center. this theorem has all necessary properties to make it a fundamental principle of analysis: it is elementary, widely applicable, and far-reaching.2 its flexibility in connection with our subject comes to light when appropriate weighted maximum norms are used. a first example is found in the dissertation of morgenstern (1952); references to later authors in the literature axe historically unjustified. in linear complex systems, the weighted maximum norm in 21.ii leads to global existence without using analytic continuation and the monodromy theorem. moreover, this proof gives the growth properties of solutions that are needed in the treatment of singular points. the theorems on continuous dependence on initial values and parameters and on holomorphy with regard to complex parameters follow directly from the contraction principle, a fact which is still little known. differentiability with respect to real parameters requires ostrowski's theorem on approximate iteration 13.iv.
in the treatment of linear systems with weakly singular points, the crucial convergence proofs are also reduced to the contraction principle in a suitable banach space.3 for holomorphic solutions, i.e., power series expansions, this method was discovered by harris, sibuya, and weinberg (1969). the logarithmic case can also be treated along these lines. this approach leads also to theorems of lettenmeyer and others, which are beyond the scope of this book; cf. the original work cited above.
a theorem in appendix d.vii, which is partly due to holmes (1968), establishes a relation between the norm of a linear operator and its spectral radius. as explained in section d.ix, this result gives a better insight into the role of weighted maximum norms.
differential inequalities. the author, who also wrote the first monograph on differential inequalities (1964, 1970), has encountered many instances where authors are unaware of basic theorems on differential inequalities that would have made their reasoning much simpler and stronger. the distinction between weak and strong inequalities is a matter of fundamental importance. in partial differential equations this is common knowledge: weak maximum or comparison principles versus strong principles of this type. not so in ordinary differential equations. theorem 9.ix is a strong comparison principle that prescribes precisely the occurrence of strict inequalities, while most (all?) textbooks are content with the weak "less than or equal" statement. this principle is essential for our treatment of the sturm-liouville theory via prufer transformation. its usefulness in nonlinear sturm theory can be seen from a recent paper, walter (1997).
supplement i in 10 brings the two basic theorems on systems of differential inequalities, (i) the comparison theorem for quasimonotone systems, and (ii) max miiller's theorem for the general case. both were found in the mid twenties. quasimonotonicity is a necessary and sufficient condition for extending the classical theory (including maximal and minimal solutions) from one equation to systems of equations. more recently, both theorems (i) and (ii) have been applied to population dynamics, but it is not generally known that results on invariant rectangles are special cases of miiller's theorem. theorem 10.xii is the strong version of (i); it contains m. hirsch's theorem on strongly monotone flows, cf. hirsch (1985) and walter (1997).
a supplement to 26 describes a new approach to minimum principles for boundary value problems of sturmian type that applies also to nonlinear differential operators; cf. walter (1995). the strong minimum principle is generalized in 26.xix, so that it includes now the first eigenvalue case.
in supplement ii of 26 on nonlinear boundary value problems the method of upper and lower solutions for existence and serrin's sweeping principle for uniqueness are presented.
. miscellaneous topics. differential equations in the sense of caratheodory. the initial value problem is treated in supplement ii of 10 and a sturmliouville theory under caratheodory assumptions in 26.xxiv and 27.xxi. as a rule, the earlier proofs for the classical case carry over. this applies in particular to the strong comparison theorem 10.xv and the strong minimum principle in 26.xxv.
radial solutions of elliptic equations. this subject plays an active role in recent research on nonlinear elliptic problems. the radial a-operator is an operator of sturm-liouville type with a singularity at 0. the corresponding initial value problem is treated in a supplement of 6, and the eigenvalue problem and nonlinear boundary value problems for the unit ball in rn (for radial solutions) in a supplement to 27.
separatrices is the theme of a supplement in 9. differential inequalites are essential for proving existence and uniqueness.
special applications. we mention the generalized logistic equation in a supplement to 2, general predator-prey models in 3.vii, delay-differential equations in 7.xiv-xv, invariant sets in 10.xvi and the rubber band as a model for nonlinear oscillations in a nonsymmetric mechanical system in 11.x.
exact numerics. we give examples in which a combination of a numerical procedure and a sup-superfunction technique allows a mathematically exact computation of special values. the numerical part is based on an algorithm, developed by rudolf lohner (1987, 1988), that computes exact enclosures for the solutions of an initial value problem. in blow-up problems one obtains rather sharp enclosures for the location of the asymptote of the solutions; cf. 9.v. a different kind of sub-and supersolutions is used to compute a separatrix; in general, a separatrix is an unstable solution.
acknowledgments. it is a pleasure to thank all those who have contributed to the making of this volume. ttie translator, professor russell thompson, worked with expertise and patience in the face of changes and additions during the translation and furnished beautiful figures. he also suggested an improved division into chapters. irene redheffer acted as a mediator between author and translator with exceptional care and insight and translated the solutions section. her help and advice and that of professor ray redheffer were indispensable. my sincere thanks go to all of them and also to other helping hands and minds.
karlsruhe, august 1997
wolfgang walter
