Non-oscillation domains of differential equations with two parameters
This research monograph is an introduction to single linear differential equations (systems) with two parameters and extensions to difference equations and Stieltjes integral equations. The scope is a study of the values of the parameters for which the equation has one solution(s) having one (finitely many) zeros. The prototype is Hill's equation or Mathieu's equation. For the most part no periodicity assumptions are used and when such are made, more general notions such as almost periodic functions are introduced, extending many classical and introducing many new results. Many of the proofs in the first part are variational thus allowing for natural extensions to more general settings later. The book should be accessible to graduate students and researchers alike and the proofs are, for the most part, self-contained.
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Scalar VolterraStieltjes integral equations
Linear Vector Ordinary Differential Equations
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aA(x almost-periodic functions assume assumption Besicovitch Bohr almost-periodic bounded set changes sign class of potentials closed set coefficients constant contains continuous convex convex set Corollary defined Differential Equations DIPRO disconjugacy domain disconjugate Edited equal to zero Euler equation example fixed full ray Hence implies inf v(x interval of non-oscillation Lebesgue measure lemma Let A(x Let q lim inf linearly dependent matrix mean value equal Moreover necessary condition non-oscillation domain non-oscillatory non-trivial solution Note oscillating potential oscillatory at infinity oscillatory at oo parabola parabolic segment periodic functions permutation matrix positive Lebesgue measure positive linear functional Proceedings proof of theorem q E S Remark resp satisfy scalar sequence set of positive solution y(x Stepanov a.p. function Sturm comparison theorem Sturm-Liouville equation sufficiently large symmetric trigonometric polynomials unbounded vector Wintner-disconjugate X E R yn(x zn(x