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Methods of reduction for Lagrange systems on time scales with nabla derivatives

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Chinese Physics B 2017年第1期26卷 243-249页

作者： Shi-Xin Jin Yi Zhang School of Science Nanjing University of Science and Technology Nanjing 210009 China College of Civil Engineering Suzhou University of Science and Technology Suzhou 215011 China

The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are *** equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results.

关键词： reduction of dynamical system cyclic integral energy integral time scales

Positive Solutions for Second-Order m-Point Boundary Value Problems on time scales

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Acta Mathematica Sinica,English Series 2006年第6期22卷 1797-1804页

作者： Wan Tong LI Hong Rui SUN School of Mathematics and Statistics Lanzhou University Lanzhou 730000 P. R. China

Let T be a time scale such that 0, T ∈ T. By means of the Schauder fixed point theorem and analysis method, we establish some existence criteria for positive solutions of the m-point boundary value problem on time scales where α ∈ Ctd（（O,T,[0,∞））,f∈ Ckd（[0,∞）×[0,∞））,β,γ ∈[0,∞）,ξi ∈（0,p（T）.b,ai∈ （0,∞）（for i = 1,..., m - 2） are given constants satisfying some suitable hypotheses. We show that this problem has at least one positive solution for sufficiently small b 〉 0 and no solution for sufficiently large b. Our results are new even for the corresponding differential equation （T = R） and difference equation （T = Z）.

关键词： time scales positive solution Schauder fixed point theorem

Oscillation Criteria of Second Order Half Linear Delay Dynamic Equations on time scales

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Acta Mathematicae Applicatae Sinica 2017年第1期33卷 83-92页

作者： H.A. AGWO A.M.M. KHODIER HEBA A. HASSAN Department of Mathematics Faculty of Education Ain Shams University 11757 beside Tabary school RoxyCairo Egypt

In this paper, we establish some new oscillation criteria for a non autonomous second order delay dynamic equation （r（t）g（x△（t）））△＋p（t）f（x（τ（t）））=0 on a time scale T. Oscillation behavior of this equation is not studied before. Our results not only apply on differential equations when T=R, difference equations when T=N but can be applied on different types of time scales such as when T=N for q〉1 and also improve most previous results. Finally, we give some examples to illustrate our main results.

关键词： time scales oscillation second order delay dynamic equations generalized Riccati technique generalized exponential function

Existence and exponential stability of almost-periodic solutions for MAM neural network with distributed delays on time scales

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Applied Mathematics(A Journal of Chinese Universities) 2021年第1期36卷 70-82页

作者： GAO Jin WANG Qi-ru LIN Yuan School of Computer Sciences Shenzhen Institute of Information TechnologyShenzhen 518172China School of Mathematics Sun Yat-sen UniversityGuangzhou 510275China Department of Public Courses Shenzhen Institute of Information TechnologyShenzhen 518172China

This paper is concerned with multidirectional associative memory neural network with distributed delays on almost-periodic time *** sufficient conditions on the existence,uniqueness and the global exponential stability of almost-periodic solutions are *** example is presented to illustrate the feasibility and effectiveness of the obtained results.

关键词： multidirectional associative memory neural networks time scales almost-periodic solutions,exponential stability.

Multiple Positive Solutions for a Fourth-order p-Laplacian Sturm-Liouville Boundary Value Problems on time scales

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Acta Mathematicae Applicatae Sinica 2015年第2期31卷 287-296页

作者： You-zheng DING Zhong-li WEI School of Mathematics Shandong University Department of Mathematics Shandong Jianzhu University

In this paper, we investigate the existence of multiple positive solutions for the following fourthorder p-Laplacian Sturm-Liouville boundary value problems on time scales ﹛[φp（u^△△（t））]^△△= f（t, u（σ（t）））, t ∈ [a, b],α0 u（a）- β0u^△（a） = 0, γ0 u（σ（b））＋ δ0 u△（σ（b）） = 0,α0（φp（u^△△））（a）- β0（φp（u^△△））△（a） = 0,γ0（φp（u^△△））（σ（b））＋ δ0（φp（u^△△））△（σ（b）） = 0,where φp（s） is the p-Laplacian operator. Under growth conditions on the nonlinearity f some existence results of at least two and three positive solutions for the above problem are obtained by virtue of fixed point theorems on cone. In particular, the nonlinearity f may be both sublinear and superlinear.

关键词： time scales p-Laplacian equation positive solution fixed point theorem

Some integral inequalities on time scales

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Applied Mathematics and Mechanics(English Edition) 2008年第1期29卷 23-29页

作者： Adnan Tuna Servet Kutukcu Department of Mathematics Faculty of Science and Arts Gazi University Bepevler 065000 Ankara Turkey

In this article, we study the reverse HSlder type inequality and HSlder inequality in two dimensional case on time scales. We also obtain many integral inequalities by using HSlder inequalities on time scales which gi... 详细信息

关键词： integral inequalities Hoelder's inequalities Hardy's inequalities time scales reverse inequality

Oscillation Theorems for Second-Order Nonlinear Neutral Delay Dynamic Equations on time scales

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Acta Mathematica Sinica,English Series 2008年第9期24卷 5-1432,1410页

作者： Samir H.SAKER Donal O’REGAN Ravi P.AGARWAL Department of Mathematics College of Science King Saud University Riyadh 11451 Saudi Arabia Department of Mathematics National University of Ireland Galway Ireland Department of Mathematical Sciences Florida Institute of Technology Melbourne Florida 32901 USA

By employing the generalized Riccati transformation technique,we will establish some new oscillation criteria and study the asymptotic behavior of the nonoscillatory solutions of the second-order nonlinear neutral delay dynamic equation [r(t)[y(t)+p(t)y(■(t))]~Δ]~Δ+q(t)f(y((δ(t)))=0 on a time scale■.The results improve some oscillation results for neutral delay dynamic equations and in the special case when■our results cover and improve the oscillation results for second- order neutral delay differential equations established by Li and Liu[***.,48(1996), 871 886].When■,our results cover and improve the oscillation results for second order neutral delay difference equations established by Li and Yeh[***.,36(1998),123-132].When ■ ■our results are essentially *** examples illustrating our main results are given.

关键词： oscillation neutral delay dynamic equation generalized Riccati technique time scales

Oscillation of third-order functional dynamic equations on time scales

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Science China Mathematics 2011年第12期54卷 2597-2614页

作者： SAKER Samir H. College of Science Research Centre King Saud University P.O. Box 2455 Riyadh 11451 Saudi Arabia

We consider the nonlinear functional dynamic equation （p（t）[（r（t）x^△（t））^△]γ）^△＋q（t）f（x（τ（t））） =0, for t≥t0,on a time scale T, where γ〉 0 is the quotient of odd positive integers, p, r, τ- and q are positive rd-continuous functions defined on the time scale 1F, and lirut→∞ τ（t） = ∞. The main aim of this paper is to establish some new sufficient conditions which guarantee that the equation has oscillatory solutions or the solutions tend to zero as →∞ τ. The main investigation depends on the Riccati substitution and the analysis of the associated Riccati dynamic inequality. Our results extend, complement and improve some previously obtained ones. In particular, the results provided substantial improvement for those obtained by Yu and Wang [J Comput Appl Math, 225 （2009）, 531-540]. Some examples illustrating the main results are given.

关键词： oscillation third-order dynamic equations time scales

HYERS-ULAM STABILITY OF SECOND-ORDER LINEAR DYNAMIC EQUATIONS ON time scales

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Acta Mathematica Scientia 2021年第5期41卷 1809-1826页

作者： Douglas R.ANDERSON Masakazu ONITSUKA Department of Mathematics Concordia CollegeMoorheadMN 56562 USA Department of Applied Mathematics kayama University of ScienceOkayama700-0005Japan

We investigate the Hyers-Ulam stability(HUS)of certain second-order linear constant coefficient dynamic equations on time scales,building on recent results for first-order constant coefficient time-scale *** particular,for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale,we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two *** for instability are also given.

关键词： stability second order Hyers-Ulam time scales