By Miroslav Bartusek, Zuzana Dosla, John R. Graef
ISBN-10: 0817635629
ISBN-13: 9780817635626
ISBN-10: 081768218X
ISBN-13: 9780817682187
First posed via Hermann Weyl in 1910, the limit–point/limit–circle challenge has encouraged, over the past century, a number of new advancements within the asymptotic research of nonlinear differential equations. This self-contained monograph strains the evolution of this challenge from its inception to its modern day extensions to the research of deficiency indices and analogous houses for nonlinear equations.
The booklet opens with a dialogue of the matter within the linear case, as Weyl initially acknowledged it, after which proceeds to a generalization for nonlinear higher-order equations. En course, the authors distill the classical theorems for moment and higher-order linear equations, and thoroughly map the development to nonlinear limit–point effects. the connection among the limit–point/limit–circle homes and the boundedness, oscillation, and convergence of strategies is explored, and within the ultimate bankruptcy, the relationship among limit–point/limit–circle difficulties and spectral idea is tested in detail.
With over a hundred and twenty references, many open difficulties, and illustrative examples, this paintings should be important to graduate scholars and researchers in differential equations, useful research, operator concept, and comparable fields.
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Example text
7. 62) 1 for any tl ::: O. 3 The Sublinear Equation imply II : : [II I [a(u)y' (u) y(u)r/ (u) / r 2(u) ]du I I {a(u)[y'(u)f /r(U)}du]! [II {a(u)l(u)[r'(u)]2 /r (u ) }d u ::: K 1 [II If y (t) is not eventually monotonic, let {t j} ---+ of zeros of y'(t). 62) implies where ]! I 3 r I (a(u)[y'(u)]2/ r(u)}du 00 be an increasing sequence 1 1 H(t) = {a(u)[y'(u)f /r(u)}du II and K 2 > 0 is a constant. 61) holds. If y(t) is eventually monotonic, then y(t)y/(t) ::: 0 for t ::: tl for large tl ::: O. 61) holds.
27) . 20) holds. 1 . That the converse relationship does not hold can be seen from the following results. 8. 2) hold. r ::/= 0 be continuous on IR+. f(x\ , .. , x n) I ~ g(IXll) on 1R" . 3. 3 Extension ofthe LPiLC Properties to Singular Solutions Proof. 28), J(t) == lyln-lI(t) - s [I yln-ll(tY)1 = I[I y1nl(s) dsl I)' Ir(s)llf(yIOI, ... , yln-ll(s))1 ds I)' ~M 00. 29) [I g (I:(; I) ds , o( ) I) for t E J where M = max Ir(s)l. I)'~s~b Since y is a singular solution, yIn-II is unbounded, and so J is unbounded on J as well.
A(u)r(u)] (tl- cx )/2du < 00 , and l OO [ I / (a (u )r (u )) tl- CX ]d U < 00. 4) is of the nonlinear limit-eircle type. (t)X 2k12k = a p (t )x z. 12) Now 1 1 [xz] = IPZ(t)xzI/PZ(t) + Z212]/ p! (t) s [p(t)(x 2k12k + K 1) + Z212]1 p! (t) ::: V(s)1 p : (t) + K I p : (t) ::: [P(t)x 2/2 I for some constant K 1 I O. Thus , we have ::: Since jJ(t) = p'(t)a/\t)lra(t), if we let r(s) denote the inverse function of s(t), we have L' {\p(r(v»11 p! = (r(v»} dv 1 1 {I{a(u)r (u»/ [a" (u )r a+ 1 (u) l'1/[a(u)r (u) ](,8-a)/2} du.