the notation of higher order differential equations

... existence theorem for single and multiple positive periodic solutions of (1.3) Our aim of this article is to study the existence of periodic solutions for the higherorder difference equations ... the existence of periodic solutions for the higherorder difference equation x(n + k) = g(x(n)) − f (n, x(n − τ (n)) We show the existence of periodic solutions by using Schauder’s fixed point theorem, ... fixed point theorem 1 Introduction and main results Let R denote the set of the real numbers, Z the integers and N the positive integers In this article, we investigate the existence of periodic...

Ngày tải lên: 18/06/2014, 22:20

... 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 ... [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 ... for a class of even -order differential equations Qi-Ming Zhang∗1 and Xiaofei He2 College of Science, Hunan University of Technology, Zhuzhou, Hunan 412000, P.R China College of Mathematics and...

Ngày tải lên: 20/06/2014, 21:20

... and they give rise to many applications see Not only the new theory of the so-called “dynamic equations unifies the theories of differential equations and difference equations but also extends these ... G ut p τ Δτ ≤ G u T 2.53 P T The rest of the proof is similar to that of Theorem 2.8, and the details are omitted The proof is completed Theorem 2.10 Assume that the function f t, u0 , , un−1 ... discrete analysis The cases when a time scale is equal to the real numbers or to the integers represent the classical theories of differential and of difference equations Many other interesting...

Ngày tải lên: 21/06/2014, 05:20

... for the existence of positive solutions The purpose of this paper is to investigate further the singular BVP for delay higherorder dynamic equation 1.1 By the use of the fixed point theorem of ... on the existence of positive solutions to the BVP 1.1 are established The paper is organized as follows In Section 2, we give some lemmas, which will be required in the proof of our main theorem ... Tk T − {t2 } ; otherwise, set Tk T Tk T By an interval a, b we always mean the intersection of the real interval a, b with the given time scale, that is, a, b ∩ T Other types of intervals are...

Ngày tải lên: 21/06/2014, 20:20

... (3:8) then (1.1) is oscillatory or tends to zero as t ® ∞ Proof The proof of the corollary is similar to the proof of Corollary 3.3 and so it is omitted From the above results on the oscillation of ... (2:13) then every solution of (1.1) is oscillatory or tends to zero as t ® ∞ Proof The proof of the corollary is similar to the proof of Corollary 2.8 and so it is omitted Example 2.11 Consider the ... Oscillation of solutions of non-linear neutral delay differential equations of higher order for p(t) = ± Arch Math 40, 359–366 (2004) Tang, S, Li, T, Thandapani, E: Oscillation of higher- order half-linear...

Ngày tải lên: 20/06/2014, 22:20

... aspects of these equations are yet to be explored For some recent work on equations of fractional order with deviated arguments, see [46-48] and the references therein In this article, we consider the ... t∈[η,1] β(1 − η) , η} − βη Proof Note that u“ (t) ≤ 0, by applying the concavity of u, the proof is easy So we omit it where γ = min{βη, Proofs of main theorems Define the operator T : C[0, 1] ® ... Difference Equations 2011, 2011:2 http://www.advancesindifferenceequations.com/content/2011/1/2 Page of 11 as an important area of investigation For the general theory and applications of integer order...

Ngày tải lên: 21/06/2014, 03:20

... to one of the conditions of the lemma, admits estimate 2.18 Therefore, estimate 2.31 is valid 10 Boundary Value Problems Proof of the main results Proof of Theorem 1.1 Without loss of generality, ... Therefore, all the conditions of Theorem 1.1 are fulfilled which guarantee the existence of at least one ω-periodic solution of 1.4 I Kiguradze et al 13 Proof of Corollary 1.4 Without loss of ... Then in view of 1.21 , 1.22 from 3.14 , inequalities 1.13 and 1.14 hold, where w pk k s ds 0, , n − k 3.21 On the other hand, according to 1.23 , the numbers fore, all the conditions of Theorem...

Ngày tải lên: 21/06/2014, 22:20

... third order differential equations, ” Bulletin of the Institute of Mathematics Academia Sinica, vol 13, no 4, pp 367– 384, 1985 20 N Parhi and S Parhi, “On the behaviour of solutions of the differential ... functionaldifferential equations of third order, ” Bulletin of the Calcutta Mathematical Society, vol 86, no 3, pp 253–266, 1994 15 N Parhi and P Das, “On nonoscillation of third order differential equations, ” ... completes the proof of Theorem 3.5 Remark 3.6 If f ≡ in Theorem 3.5, then p ≡ and hence the theorem is not applicable to homogeneous equation: r t y t q t k y t p t h y g t 3.18 Acknowledgment The...

Ngày tải lên: 22/06/2014, 02:20

... neutral differential equations The main results of this paper are the comparison theorems contained in the next section where we relate the oscillation of solutions of (1.1) to nonexistence of eventually ... view of our assumption (c) and the first part of Lemma 3.1 The following theorems are obtained in a similar manner by applying the theorems in the the previous section, respectively The proofs ... eventually positive solution w of degree of inequality (2.14) contradicts (c) The proof is complete The proof of the next theorem is similar, and hence we omit it Theorem 2.2 Let δ = 1, α < 0,...

Ngày tải lên: 22/06/2014, 18:20

... G j with period 2n The rest a ss umpt ions f or F a n d for the characteristic roots are the same as in the first paragraph But here we consider the case when there exist the following relations ... proportional coefficients It is easily seen that with these values of q f p 5M the coefficients o f W in (2.8) are equal to zero Therefore, equating the coefficients jqp o f sin < 5, co s in (2.8) ... at the term s corresp o n d in g qj = qjsịlì p v = = PvsK 0' = 1* 2, Qisfll /, V = 1, 2, + ••• r), fo r w hich qisfll + + Pi ^ + ••• Prsflr - + ± ^ + Xs/I • O n the other hand, follow in g the...

Ngày tải lên: 08/04/2015, 15:36

Ngày tải lên: 14/03/2014, 13:20

... numbers of higher order, ” Advances in Difference Equations, vol 2010, Article ID 309480, pages, 2010 V Kurt, “A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of ... Difference Equations r r In the special case x 0, Gn,χ,q Gn,χ,q are called the nth generalized q-Genocchi numbers of order r attached to χ Therefore, we obtain the following theorem Theorem 2.1 ... to χ The purpose of this paper is to present a systemic study of some families of higher- order generalized q-Genocchi numbers and polynomials attached to χ by using the generating function of those...

Ngày tải lên: 21/06/2014, 05:20

... systemic study of some formulas of the twisted q-extension of the generalized Euler numbers and polynomials of order k attached to χ On the Twisted q-Extension of the Generalized Euler Polynomials ... order k attached to χ 4 Advances in Difference Equations We firstly consider the twisted q-extension of the generalized Euler polynomials of higher order as follows: ∞ tn n! En,χ,ζ,q x n ex y qt ... are called the nth generalized h, q -Euler polynomials of order r attached r to χ In the special case x 0, En,χ,ζ,q r En,χ,ζ,q are called the nth generalized h, r -Euler r numbers of order r By...

Ngày tải lên: 21/06/2014, 16:20

... polynomials and the generalized higher- order Euler polynomials The main purpose of this paper is to give the symmetry identities for the twisted generalized Euler polynomials of higher order using the symmetric ... integral on Zp From these symmetric properties, we derive the symmetry identities for the twisted generalized Euler polynomials of higher order In 14 , Kim gave the relation between the power sum polynomials ... Journal of Inequalities and Applications Comparing the coefficients on both sides of 2.17 and 2.18 , we obtain the following theorem which shows the relationship between the power sums and the twisted...

Ngày tải lên: 21/06/2014, 18:20

... question about the values of the first derivative at the end points of u the interval of integration, t and t According to the theory, it holds that u 0 Therefore, we approximate the values of the first ... This change of the problem setting is useful for obtaining of a priori estimates necessary for the application of the Fredholm-type existence theorem Lemma 5.5 during the proof Theorem 5.3 existence ... u1 ≡ u2 , and the result In the following theorem we formulate sufficient conditions for the existence of at least one solution of problem 5.1a and 5.1b with a > In the proof of this theorem, we...

Ngày tải lên: 21/06/2014, 20:20

... means of properties of the associated vector field u t , p t u t together with the Kneser’s property of the cross sections of the solutions’ funnel, the authors provide conditions which guarantee the ... damped solution of problem 1.1 , 1.10 if and only if u is a damped solution of problem 2.13 , 1.10 Therefore, we can borrow the arguments of 12 in the proofs of this section Theorem 3.3 If u ... If u has other zeros, we use the previous arguments and get that either u has a finite number of zeros and then fulfils 2.21 or u is oscillatory Remark 3.4 According to the proof of Theorem 3.3,...

Ngày tải lên: 21/06/2014, 20:20

... n! 1.11 k Then the values of En,ζ x at x are called the ζ-Euler numbers of order k When k 1, the polynomials or numbers are called the ζ-Euler polynomials or numbers The purpose of this paper ... of symmetry for the ζ-Euler polynomials of higher order By using our identities of symmetry for the ζ-Euler polynomials of higher order, we can obtain many identities related to the Frobenius-Euler ... to the Frobenius-Euler polynomials of higher order On the Symmetry for the ζ-Euler Polynomials of Higher Order Let w1 , w2 ∈ N with w1 ≡ mod and w2 ≡ mod Then we set R m Zm p w1 , w2 e w1 x1...

Ngày tải lên: 21/06/2014, 20:20

... polynomials attached to χ of higher order Symmetric Properties for the Generalized Bernoulli Polynomials of Higher Order Let χ be the Dirichlet’s character with conductor d ∈ N Then we note that t ... Bn,χ x n tn n! 2.12 k Then the values of Bn,χ x at x are called the generalized Bernoulli numbers attached to χ of order k When k 1, the polynomials of numbers are called the generalized Bernoulli ... of the Jangjeon Mathematical Society, vol 11, no 2, pp 191–198, 2008 ¨ 23 A Tekcan, A Ozkoc, B Gezer, and O Bizim, “Some relations involving the sums of Fibonacci ¸ numbers,” Proceedings of the...

Ngày tải lên: 22/06/2014, 02:20

... respectively, the ring of p-adic rational integers, the field of p-adic rational numbers and the p-adic completion of the algebraic closure of Qp The p-adic absolute value in Cp is 1/p When one talks of ... 1.16 The purpose of this paper is to give the distribution of extended higher order q-Euler and q-Genocchi polynomials In 24 , Choi-Anderson-Srivastava have studied the q-extension of the Apostol-Euler ... polynomials of order n, and the multiple Hurwitz zeta functions see 24 Actually, their results and definitions are not new see 18, 20 and the definition of the Apostol-Bernoulli numbers in their paper...

Ngày tải lên: 22/06/2014, 03:20