By R.P. Agarwal, Said R. Grace, Donal O'Regan
ISBN-10: 9048160952
ISBN-13: 9789048160952
ISBN-10: 9401725152
ISBN-13: 9789401725156
In this monograph, the authors current a compact, thorough, systematic, and self-contained oscillation concept for linear, half-linear, superlinear, and sublinear second-order traditional differential equations. a huge function of this monograph is the representation of a number of effects with examples of present curiosity. This ebook will stimulate additional study into oscillation theory.
This e-book is written at a graduate point, and is meant for college libraries, graduate scholars, and researchers operating within the box of standard differential equations.
Read Online or Download Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations PDF
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Extra resources for Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations
Example text
Integrating the above inequality from Tl to u, we get In(l+ h~ a~2)~ls)dS) :::: In(:~;/) for u::::T1 . 20) in the above inequality, we obtain -w' (u) :::: w(Td(a(u)p(u)) for u:::: T 1 . Finally, integrating this last inequality, we find that w(u)----+ -00 as u ----+ 00, which contradicts the fact that w(t) > 0 for t:::: T. 19) holds. 18), it follows that v(u) ----+ v(oo) < 00. 19), v(oo) = O. , (ii) holds. (ii) => (iii). It is obvious. (iii) => (iv). 16). Define y(t) = (3(t) + JtOO v 2(s)j(a(s)p(s))ds.
Then, Iv(t)l:::: ly(t)l, and y' (t) y2(t) v 2 (t) - Q(t) - a(t)p(t) < - Q(t) - a(t)p(t)' 20 Chapter 2 Hence, (iv) holds. (iv) =} (i). 1) is nonoscillatory. This completes the proof. 1). 4. 1) is nonoscillatory. (ii) There exist T~ to and a function y(t) E C([T,oo),JR) such that y(t) ~ ~(t) + 1= a~;;;~~) y2(s)ds for t ~ T. 21) (iii) There exist T 2' to and a function z(t) E C([T, (0), JR) such that z(t) where = ~(t) + 1 /-* 1 00 t ~(t) = t] a(s);(s) z2(s)ds = t 2' T, f32(s) a(s)p(s) M[S, t]ds (1 and for t M[S, t] = exp 2 8 t f3(T) ) a(T)p(T) dT .
8 that we can establish the higher order iterated comparison theorems by using the nonoscillatory characterizations. 3. 1). 1. 1) is nonoscillatory if and only if there exist T ~ to and a function h(t) E C 1 ([T,oo),lR) satisfying q(t) + a(t)h 2(t) - (a(t)h(t»' ::; 0 for t ~ T. Proof. 1) such that x(t) -=/:- 0 for t ~ T ~ to. Define h(t) = -x'(t)/x(t) for t ~ T. 1) that q(t) + a(t)h 2(t) - (a(t)h(t»' () qt +a ( ) ( x'(t»)2 t x (t) x 2 (t) q(t) - q(t) x 2(t) = + (a(t)x'(t»' x(t) - aCt) (X'(t»2 x 2 (t) o.