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1528 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 Oscillation Criteria for Second Order Nonlinear Neutral Perturbed Dynamic Equations on Time Scales Xiuping Yu Department of Mathematics and Physics, Hebei Institute of Architecture and Civil Engineering, Zhangjiakou, China Email: xiuping66@163.com Hua Du Information Science and Engineering College, Hebei North University, Zhangjiakou, China Email: dhhappy88@126.com Hongyu Yang Department of Mechanic and Engineering, Zhangjiakou Vocational Technology Institute, Zhangjiakou, China Email: yanghy88@sina.com Abstract—To investigate the oscillatory and asymptotic behavior for a certain class of second order nonlinear neutral perturbed dynamic equations on time scales. By employing the time scales theory and some necessary analytic techniques, and introducing the class of parameter functions and generalized Riccati transformation, some new sufficient conditions for oscillation of such dynamic equations on time scales were established. The results not only improve and extend some known results in the literature, but also unify the oscillation of second order nonlinear neutral perturbed differential equations and second order nonlinear neutral perturbed difference equations. In particular, the results are essentially new under the relaxed conditions for the parameter function. Some examples are given to illustrate the main results. Dynamic equations on time scales are widely used in many fields such as computer, electrical engineering, population dynamics, and neural network, etc. Index Terms—oscillation, nonlinear neutral perturbed dynamic equation, time scales, Riccati transformation. I. INTRODUCTION The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Higher in his Ph.D. thesis [1] in 1988 in order to unify continuous and discrete analysis. Not only can this theory of so-called “dynamic equations” unify the theories of differential equations and of difference equations, but also it is able to extend these classical cases “in between”, e.g., to socalled q-difference equations. Several authors have expounded on various aspects of this new theory, see the survey paper by Agarwal [2] and references cited therein. A book on the subject of time scales by Bohner and Peterson [3] summarizes and organizes much of the time scale calculus. A time scales T is an arbitrary nonempty closed subset of the real numbers . There are many interesting time scales and they give rise to plenty of applications, the cases when the time scale is equal to reals or the integers represent the classical theories of differential and of difference equ-ations. Another useful +∞ time scale a time scale P , n 0[( na b), na ( b) a] ab∪ = + + + is wi-dely used to study population in biological communities, electric circuit and so on [3]. In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of some dynamic equations on time scales, and we refer the reader to the papers [4-17] and references cited therein. Regarding neutral dynamic equations, Argarwal et al [6] considered the second order neutral delay dynamic quation Δ { ( t)[( xt ( ) ctxt ( ) ( )) ] γ Δ α + − τ } + ftxt ( , ( − δ)) = 0. (1) where γ > 0 is an odd positive integer, τ andδ are position constants, α Δ () t > 0, and proved that the oscillation of (1) is equivalent to the oscillation of a first order delay dynamic inequality. Saker [7] considered (1) where γ ≥ 1 , is an odd positive integer, the condition α Δ () t > 0is abolished and established some new sufficient conditions for oscillation of (1). However the results established in [6-7] are only valid for the time scales , , or h , q , where q = { t: t = q k , k∈ , q > 1} . Sahiner et al [8] considered the general equation Δ γ Δ { α( t)[( xt ( ) + ctx ( ) ( τ( t)) ] } + ftx ( , ( δ( t))) = 0 . (2) on a time scale T , where γ ≥ 1 and τ () t ≤t, δ () t ≤ t, and followed the argument in [6-7] by reducing the oscillation of (2) to the oscillation of a first order delay dynamic inequality and established some sufficient conditions for the oscillation. However one can easily see that the two © 2013 ACADEMY PUBLISHER doi:10.4304/jcp.8.6.1528-1535
JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013 1529 examples presented in [8] to illustrate the main results are valid only when T= and cannot be applied when T = . Agarwal, O′Regan and Saker [3]considered (2) where γ ≥ 1 is an odd positive integer and α Δ () t > 0, and established some new oscillation criteria by employing the Riccati transformation technique which can be applied on any time scale T and improved the results in [6, 8]. Bohner and Saker [9] considered perturbed nonlinear dynadynamic equation Δ { ( t)(( x( t)) ) γ Δ } F( t, x σ ) G( t, x σ Δ α + = , x ). (3) on a time scales T . Where γ > 0 is an odd positive integer, using Riccati transformation techniques, they obtained some sufficient conditions for the solution to be oscillatory or converge to zero. Following this trend, we shall study the oscillation for the second-order neutral nonlinear perturbed dynamic equations of the form and Δ γ { α( t)(( x( t) + c( t) x( τ( t))) ) } Δ + F( tx , ( δ( t))) = Gtx ( , ( δ( t)), x), Δ γ { α( t)(( x( t) − c( t) x( τ( t))) ) } Δ + F( tx , ( δ( t))) = Gtx ( , ( δ( t)), x). on an arbitrary time scales T , where γ is a quotient of positive odd integer, α, c is a positive real-valued rdcontinuous function defined on a time scales T and the following conditions are satisfied: +∞ 1 γ (H1) 0 ≤ct ( ) ≤ c < 1, t ( α( t)) 0 ∫ Δ =∞, for all t ∈ T ; t 0 (H2) τ , δ : T → T satisfies τ () t ≤ t, for all t ∈ T , either δ () t ≥ t or δ () t ≤ t for all suffici-ently large t , and lim τ ( t) = lim δ ( t) =∞; t→∞ t→∞ (H3) pq , : T → are rd-continuous function, such that qt () − pt () > 0, for all t ∈ T ; 2 (H4) F : T× → and G : T× → are functions such that uF(, t u ) > 0and uG(, t u, v ) > 0, for all u ∈ − {0} , v ∈ , t ∈ T ; (H5) F(, tu) u γ ≥ qt (), and Gtuv (, , ) u γ ≤ pt () for all uv∈ , −{0} , t ∈ T . We note that in all the above results the conditions 0 ≤ ct ( ) < 1, γ ≥ 1 and δ () t ≤ t are required. And some authors utilized the kernel function ( t− s ) m Δ Δ (4) (5) or the general class of functions H (, ts) and obtained some oscillation Δs criteria, but the condition H ( ts , ) ≤ 0 is required. In this paper the study is free of these restrictions and contains the cases when 0< γ < 1, δ ( t) ≥t, and − 1 < ct ( ) ≤ 0 . In particular, by utilizing the general class of functions H (, ts ), we shall derive some sufficient conditions for the solutions of (4) and (5) to be oscillatory or converge Δs to zero when the condition H (, t s) ≤ 0is relaxed. Our results are different from the existing results for neutral equations on time scales that were established in [6-11, 13-17]. Also, we give some examples to illustrate the main results. Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T = ∞ , and define the time scale interval [ t 0 , ∞) T by [ t , ∞ ) : = [ t , ∞) ∩ T . By a solution of (4), we mean a 0 T 0 nontrivial real-valued function x (t) satisfying (4) for t ≥ t . A solution x (t) of (4) is said to be oscillatory if 0 it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. Equation (4) is said to be oscillatory if all its solutions are oscillatory. Our attention is restricted to those solutions of (4) which exist on some half line[ t 0 , ∞) and satisfy sup{| xt ( ) |: t≥ t x } > 0 , for any t ≥ t . x 0 The paper is organized as follows. In next section, we present some basic formula and lemma concerning the calculus on time scales. In Section 3, we will use Riccati transformation techniques and the general class of functions H (, ts) and give some sufficient conditions for the oscillatory behavior of solutions of (4) and (5). In last section, we give some examples to illustrate our main results. Through this paper, we let γ d () t = max[0, d()], t Q() t = ( q() t − p())(1 t −c( δ ())), t + ∫ Δs α () s d () t = max[0, − d()], t ρ(, t u): = , − () s and for sufficiently largeT ∗ , δ () t 1 γ u t 1 γ ∫ Δs α u 1, δ ( t) t, ∗ ⎧ ≥ β (, tT ) = ⎨ γ ∗ ⎩ρ (, tT ), δ() t ≤ t. II. SOME PRELIMINARIES ON TIME SCALES A time scales T is an arbitrary nonempty closed subset of the real numbers . In this paper, we only consider time scales interval of form [ t 0 , ∞) T , on T we define the forward jump operatorσ and the graininess μ by { } σ (): t = inf s∈ T : s > t and μ(): t = σ () t − t. A point t ∈ T with σ () t = tis called right-dense, while t is referred to as being right-scattered if σ () t > t . A function f : T → is said to be rd-continuous if it is continu-ous at each right-dense point and if there exists a left limit in all left-dense points. The ( Δ derivative) f Δ of f is defined by Δ f ( σ ( t)) − f( s) f () t = lim , where Ut () = T \{ σ ()} t . σ () t − s s→t sU ∈ () t © 2013 ACADEMY PUBLISHER
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Journal of Computers ISSN 1796-203X
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Corn Moisture Measurement using a C
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Call for Papers and Special Issues
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(Contents Continued from Back Cover
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