the solution of second order differential equation

... expressions of x 2n ,x 2n1 ,y 2n ,ωt, and xt are important in the process of studying the spectrum containment of the almost periodic solution of 1.1. Before giving the main theorem, we list the ... B}. We postpone the proof of this theorem to the next section. 3. The Proof of Main Theorem To show the Main Theorem, we need some more lemmas. Lemma 3.1. Let f ∈APR,thenσ b f i 2n ,σ b f i 2n1 ,σ b h 2n ,σ b h 2n1  ... ∈ R, which guarantees the uniqueness of solution of 1.1 and cannot be omitted. To study the spectrum of almost periodic solution of 1.1, we firstly study the solution of 1.1.Let f 1 n   n1 n  s n f  σ  dσ...

Ngày tải lên: 21/06/2014, 20:20

Ngày tải lên: 21/06/2014, 20:20

Ngày tải lên: 21/06/2014, 20:20

... interval symmetrizes about the origin. Then, similar to the proof of (2.9), we can verify the condition f  (α) = f  (β), and the other conditions in Lemma 2.2 are satisfied all the way. Hence, it also ... contributions QZ carried out the theoretical proof and drafted the manuscript. XH participated in the design and coordination. Both of the two authors read and approved the final manuscript. References [1] ... [α, β], too. Then, by a similar method to the proof of (2.3) together with Lemma 2.2, we can obtain (2.4) immediately. For the other ordinary cases, i.e., a = 0, we only need to move the interval...

Ngày tải lên: 20/06/2014, 21:20

... and the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory. Motivated by the above works, we consider the singular third -order ... “Existence of solutions for a class of third -order nonlinear boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 104–112, 2004. 14 D. Guo, Semi-Ordered ... problem,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 413–425, 2006. 12 Y. Sun, “Positive solutions of singular third -order three-point boundary value problem,” Journal of Mathematical...

Ngày tải lên: 21/06/2014, 05:20

... multiple of k. On the other hand, in the remaining two cases, the study of the regular variation of the solutions gives the additional information that the positive solutions, even if they are ... M  RV  2 M  ∞,0 . The above theorem shows how the study of the regular variation of the solutions and the M-classification supplement each other to give an asymptotic description of nonoscillatory solutions. ... general equations. The aim of this section is to analyze the relations between the classification of the eventually positive solutions according to their regularly varying behavior, and the M-classification....

Ngày tải lên: 21/06/2014, 11:20

... economics. The theory of impulsive differential equations has become an important area of investigation in the recent years and is much richer than the corresponding theory of differential equations. ... mentioning the works by Guo 31.In31, Guo investigated the minimal nonnegative solution of the following initial value problem for a second order nonlinear impulsive integrodifferential equation of Volterra ... we will use the cone theory and monotone iterative technique to investigate the existence of minimal nonnegative solution for a class of second- order nonlinear impulsive differential equations on...

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Ngày tải lên: 13/08/2014, 16:20

... 1+hν 2 ⎤ ⎥ ⎦ . (2:7) Further , let N be the number of positive roots of the function in (2.6), and W be the number of sign changes in its coefficients. Because the radius of convergence of this series is ∞, then ... verified that in the vicinity of zero, the function g(z) is of order O (z ν ). By virtue of this asymptotic and because g(z) is an odd function, the integral along the left-hand side of the contour ... δ >0. Then, we can state the following theorem. Theorem 3.2. Let the conditions of Theorem 2.1, (3.6) and (3.7) hold. If the operator- value function q(t) has properties 1-3, then the following...

Ngày tải lên: 21/06/2014, 02:20

... fractional -order models have proved to be more accurate than integer- order models, i.e., there are more degrees of freedom in the fractional -order models. In consequence, the subject of fractional differential ... as an important area of investigation. For the general theory and applications of integer order differential equations wit h deviat ed arguments, we refer the reader to the refer- ences [39-45]. As ... mentioning that the condit ions of our theorems are easily to verify, so they are applicable to a variety of problems, see Examples 4.1 and 4.2. The proof of our main resul ts is based upon the following...

Ngày tải lên: 21/06/2014, 03:20

... ε k , 3.7 and therefore,  Tz k−1 − z k  ≤ ε k 3.8 as announced. The next result is used in order to establish the fact that the sequence defined in Theorem 3.1 approximates the solution of the nonlinear ... in order to approximate the solution of the nonlinear Volterra integro-differential equation. We define, following a fixed-point approach, a sequence of functions which approximate the solution of ... estimate the rate of convergence of the sequence of projections. For this purpose, consider the dense subset {t i } i≥1 of distinct points in 0, 1 and let T n be the set {t 1 , ,t n } ordered...

Ngày tải lên: 21/06/2014, 10:20

... completed the estimate of the error bounds for asymptotic solutions to second order linear difference equations in the first case. For the second case, we leave it to the second part of this paper: ... still open. The purpose of this and the next paper Error bounds for asymptotic solutions of second- order linear difference equations II: the second case is to estimate error bounds for solutions ... finite. Equation 3.10 is a inhomogeneous second- order linear difference equation; its solution takes the form of a particular solution added to an arbitrary linear combination of solutions to the...

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