OSCILLATION AND NONOSCILLATION FOR IMPULSIVE DYNAMIC EQUATIONS ON CERTAIN TIME SCALES

where T is an unbounded-above time scale with 0∈ T f : JT×R→R is a given function, Ik ∈ C(R,R), tk ∈ T, 0 = t0 < t1 < ··· < tm < tm+1 < ··· <∞, y(t k ) = limh→0+ y(tk + h) and y(t− k ) = limh→0+ y(tk − h) represent the right and left limits of y(t) at t = tk in the sense of the time scale; that is, in terms of h > 0 for which tk + h, tk − h ∈ [t0,∞)∩T, whereas if tk is left-scattered (resp., right-scattered), we interpret y(t− k ) = y(tk) (resp., y(t k )= y(tk)). Impulsive differential equations have become important in recent years in mathematical models of real processes and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There have been significant developments in impulse theory also in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs of Bainov and Simeonov [5], Lakshmikantham et al. [22], Samoı̆lenko and Perestyuk [25], and the references therein. In recent years, dynamic equations on times scales have received much attention.


Introduction
This paper is concerned with the existence of oscillatory and nonoscillatory solutions of first-order impulsive dynamic equations on certain time scales. We consider the problem y Δ (t) = f t, y(t) , t ∈ J T := [0,∞) ∩ T, t = t k , k = 1,..., y t + k = I k y t − k , k = 1,..., (1.1) where T is an unbounded-above time scale with 0 ∈ T f : J T × R → R is a given function, I k ∈ C(R,R), t k ∈ T, 0 = t 0 < t 1 < ··· < t m < t m+1 < ··· < ∞, y(t + k ) = lim h→0 + y(t k + h) and y(t − k ) = lim h→0 + y(t k − h) represent the right and left limits of y(t) at t = t k in the sense of the time scale; that is, in terms of h > 0 for which t k + h, t k − h ∈ [t 0 ,∞) ∩ T, whereas if t k is left-scattered (resp., right-scattered), we interpret y(t − k ) = y(t k ) (resp., y(t + k ) = y(t k )). Impulsive differential equations have become important in recent years in mathematical models of real processes and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There have been significant developments in impulse theory also in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs of Bainov and Simeonov [5], Lakshmikantham et al. [22], Samoȋlenko and Perestyuk [25], and the references therein. In recent years, dynamic equations on times scales have received much attention.
We refer the reader to the books by Bohner and Peterson [10,11], Lakshmikantham et al. [23], and the references therein. The time scale calculus has tremendous potential for applications in mathematical models of real processes, for example, in physics, chemical technology, population dynamics, biotechnology and economics, neural networks, social sciences; see the monographs of Aulbach and Hilger [4], Bohner and Peterson [10,11], Lakshmikantham et al. [23], and the references therein. The existence of solutions of boundary value problem on a measure chain (i.e., time scale) was recently studied by Henderson [20] and Henderson and Tisdell [21]. The question of existence of solutions to some classes of impulsive dynamic equations on time scales was treated very recently by Henderson [19] and Benchohra et al. in [1,7,8]. The aim of this paper is to initiate the study of oscillatory and nonoscillatory solutions to impulsive dynamic equations on time scales. For oscillation and nonoscillation of impulsive differential equations, see, for instance, the monograph of Bainov and Simonov [5] and the papers of Graef et al. [16,17]. The purpose of this paper is to give some sufficient conditions for existence of oscillatory and nonoscillatory solutions of the first-order dynamic impulsive problem (1.1) on time scales. There has been, in fact, a good deal of research already devoted to oscillation questions for dynamic equations on time scales; see, for example, [2,9,12,14,15,24]. For the purposes of this paper, we will rely on the nonlinear alternative of Leray-Schauder type combined with a lower and upper solutions method. Our results can be considered as contributions to this emerging field.

Preliminaries
We will briefly recall some basic definitions and facts from time scale calculus that we will use in the sequel.
A time scale T is an closed subset of R. It follows that the jump operators σ,ρ : T → T defined by σ(t) = inf{s ∈ T : s > t}, ρ(t) = sup{s ∈ T : s < t} (2.1) (supplemented by inf ∅ := supT and sup∅ := inf T) are well defined. The point t ∈ T is left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) > t, respectively. If T has a right-scattered minimum m, ,[a,b), and so on will denote time scales intervals where a,b ∈ T with a < ρ(b).
Definition 2.1. Let X be a Banach space. The function g : T → X will be called rd− continuous provided it is continuous at each right-dense point and has a left-sided limit at each point, and write g ∈ C rd (T) = C rd (T,X). For t ∈ T k , the Δ derivative of g at t, denoted by g Δ (t), is the number (provided it exists) such that for all ε > 0, there exists a neighborhood U of t such that for all s ∈ U. A function F is called an antiderivative of g : T → X provided where μ(t) = σ(t) − t which is called the graininess function. The set of all rd−continuous functions g that satisfy 1 + μ(t)g(t) > 0 for all t ∈ T will be denoted by + . The generalized exponential function e p is defined as the unique solution y(t) = e p (t,a) of the initial value problem y Δ = p(t)y, y(a) = 1, where p is a regressive function. An explicit formula for e p (t,a) is given by For more details, see [10]. Clearly, e p (t,s) never vanishes.

Main result
We will assume for the remainder of the paper that, for each k = 1,..., the points of impulse t k are right-dense. In order to define the solution of (1.1), we will consider the space Remark 3.1. In light of the right-density assumption on each impulse point, we observe that this restriction precludes certain time scales. For example, time scales that are excluded from this work include discrete time scales, time scales associated with q-differences, harmonic numbers time scales, and so forth. We observe further that, in the context of impulsive problems on time scales, such restrictions on impulse points are not uncommon; see, for example, [13,19].
Let us start by defining what we mean by a solution of problem (1.1).
... For the study of this problem, we first list the following hypotheses: Theorem 3.4. Assume that hypotheses (H1)-(H4) hold. Then the problem (1.1) has at least one solution y such that Proof. The proof will be given in several steps.
Step 1. Consider the problem Transform the problem (3.5) into a fixed point problem. Consider the following modified problem: (ii) By the definition of τ it is clear that In order to apply the nonlinear alternative of Leray-Schauder type, we first show that N is continuous and completely continuous.  (3.14) 6 Oscillation and nonoscillation By (H1) and Remark 3.5 we have, for each t ∈ J 1 , (3.15) Thus N(y) ∞ ≤ .
Claim 3. N maps bounded set into equicontinuous sets of PC.
Let u 1 ,u 2 ∈ J 1 , u 1 < u 2 and B q be a bounded set of PC as in Claim 2. Let y ∈ B q . Then (3.16) As u 2 → u 1 , the right-hand side of the above inequality tends to zero. As a consequence of Claims 1 to 3 together with the Arzela-Ascoli theorem, we can conclude that N : C([t 0 ,t 1 ],R) → C([t 0 ,t 1 ],R) is continuous and completely continuous. This implies by Remark 3.5 that for each t ∈ J 1 we have Thus From the choice of U there is no y ∈ ∂U such that y = λN(y) for some λ ∈ (0,1). As a consequence of the nonlinear alternative of Leray-Schauder type [18], we deduce that N has a fixed point y in U which is a solution of the problem (3.6).

Claim 5.
The solution y of (3.6) satisfies Let y be the above solution to (3.6). We prove that Suppose not. Then there exist e 1 ,e 2 ∈ J 1 , e 1 < e 2 such that α(e 1 ) = y(e 1 ) and y(t) < α(t) ∀t ∈ e 1 ,e 2 . Using the fact that α is a lower solution to (3.6), the above inequality yields which is a contradiction. Analogously, we can prove that This shows that the problem (3.6) has a solution in the interval [α,β] which is solution of (3.5). Denote this solution by y 0 .

Oscillation and nonoscillation
Step 2. Consider the following problem: (3.30) Consider the following modified problem: (3.32) A solution to (3.31)-(3.32) is a fixed point of the operator N 1 : that is, Using the same reasoning as that used for problem (3.5), we can conclude the existence of at least one solution y to (3.32)-(3.41). We now show that this solution satisfies Let y be the above solution to (3.32)-(3.41). We show that Assume this is false. Then since y(t + 1 ) ≥ α(t + 1 ), there exist e 3 ,e 4 ∈ J 2 with e 3 < e 4 such that α(e 3 ) = y(e 3 ) and which is a contradiction. Analogously, we can prove that This shows that the problem (3.32)-(3.41) has a solution in the interval [α,β] which is a solution of (3.30). Denote this solution by y 1 .
Step 3. We continue this process and take into account that y m := y| [tm−1,tm] is a solution to the problem Consider the following modified problem: . . .

(3.45)
The proof is complete.
The following theorem gives sufficient conditions to ensure the nonoscillation of solutions of problem (1.1).
The following theorem discusses the oscillation of solutions of problem (1.1).
Proof. Suppose on the contrary that y is a nonoscillatory solution of (1.1). Then there exists T y > 0 such that y(t) > 0 for all t > T y , or y(t) < 0 for all t > T y . In the case y(t) > 0 for all t > T y , we have β(t k ) > 0 for all t k > T y , k = 1,..., which is a contradiction, since β(t k ) is an oscillatory upper solution. Analogously, in the case y(t) < 0 for all t > T y , we have α(t k ) < 0 for all t k > T y , k = 1,..., which is also a contradiction, since α(t k ) is an oscillatory lower solution.

An example
As an application of our results, we consider the following impulsive dynamic equation where f : J T × R → R. Assume that there exist g 1 (·),g 2 (·) ∈ C(J T ,R) such that and, for each t ∈ J T ,   Since all the conditions of Theorem 3.4 are satisfied, the problem (4.1) has at least one solution y on J T with α ≤ y ≤ β. If g 1 (t) > 0, then α is positive and nondecreasing, thus y(t) is nonoscillatory. If g 2 (t) < 0, then β is negative and nonincreasing, thus y(t) is nonoscillatory. If the sequences α(t k ) and β(t k ) are both oscillatory, then y(t) is oscillatory.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation