By Qingkai Kong

This article is a rigorous remedy of the fundamental qualitative thought of normal differential equations, first and foremost graduate point. Designed as a versatile one-semester path yet supplying sufficient fabric for 2 semesters, a quick path covers center subject matters equivalent to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré—Bendixson theorem, and bifurcation thought, and second-order themes together with oscillation thought, boundary worth difficulties, and Sturm—Liouville difficulties. The presentation is apparent and easy-to-understand, with figures and copious examples illustrating the which means of and motivation in the back of definitions, hypotheses, and common theorems. A thoughtfully conceived choice of routines including solutions and tricks toughen the reader's knowing of the fabric. necessities are restricted to complex calculus and the user-friendly conception of differential equations and linear algebra, making the textual content compatible for senior undergraduates in addition.

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**Example text**

R m ), TJ = IjO, j = 1 , . . , m; moreover, \ \ v p ( x ' ) , B^(Gm)\\ ^ c||w(or), Wp(G)||, where the constant c > 0 is independent o f u ( x ) . 8 can be found in O. V. Besov, V. P. Il'in, and S. M. Nikol'skii [1], S. M. Nikol'skii [1], S. V. Uspenskii, G. V. Demidenko, and V. G. Perepelkin [1] § 9. Boundary-Value Problems for Ordinary Differential Equations on the Half-Axis Consider the differential operator with constant coefficients d\ dm dm~l We assume that the characteristic polynomial satisfies the following condition: the equation L m (A) = 0 has no purely imaginary roots.

3. Let u(x) G Wlp(G), 1 ^ p < oo, /? n), /% £ 0 6e integers, j = 1 , . . , n . If /3a < 1 — |a|/p, £/ien f/ze wea& derivative D@u(x) can be changed in a set of measure zero in such a way that D%u(x) G C(G); moreover, sup \D%u(x)\ ^ c||u(x), iy'(G)||, wAere i/ie r£G constant c > 0 z's independent of u ( x ) . 3 generalize the embedding theorems due to S. L. Sobolev [2, 3] to the isotropic case. 4 (on extension). Let a domain G satisfy the strong l-horn condition. Then there exists a linear continuous extension operator IT : , Kp

1 satisfy the assumptions of the Lizorkin PROOF. The functions /•**(£) theorem. 2 below. 1. J+(s,xn)\ <: c ID* Df J_ (s,-* n )| ^ 16 1. Preliminaries n-l where (s)2 = ]T] s 2 / a j , c, 8 > 0 are constants. j=i PROOF. For the sake of definiteness, we estimate the functions J+(s, xn). By the homogeneity of L(ca is,cani\) = c L ( i s , i X ) , c > 0, the roots of the equation L ( i s , i X ) = 0 are also homogeneous: \k(ca s) = can\k(s). Therefore, there exist constants 6, A > 0 such that s = s(s)-a> , A ^ |At(S)| £ I m A t ( S ) £ 2 < f >0, A ^ |Afc(s)| ^ -Im Aj(s) ^ 2 J > 0, j = l , .