It is well-known that condition H is crucial in obtaining a Picone’s formula in the case when **impulses** are absent. If H fails to hold, then Wirtinger, Leighton, and Sturm-Picone type results require employing a so-called “device of Picard.” We will show how this is possible for impulsive diﬀerential **equations** as well.

52 Read more

In this paper we study nonlinear diﬀerential **equations** subject to random **impulses** oc- curring at random moments. Randomness is introduced both through the time between **impulses**, which is distributed exponentially, and through the amount of **impulses**. The p-moment exponential stability of the solution is studied by employing appropriate gen- eralized Lyapunov’s functions. In the literature many authors study the stability of impul- sive systems with deterministic moments of **impulses** (see [, , ] and the references cited therein), the exponential stability of impulsive delay diﬀerential **equations** with determin- istic moments of **impulses** [–] and the p-moment stability of stochastic diﬀerential **equations** with or without **impulses** [, , , ]. The behavior of solutions of stochastic **equations** is totally diﬀerent from the behavior of **ordinary** diﬀerential **equations**, so the

12 Read more

x = b we write simply E . Consider the space (B, (·, ·) B ) is a seminormed linear space of functions mapping (–∞, ] × (–∞, ] into E, and satisfying the following fundamental ax- ioms which were adapted from those introduced by Hale and Kato for **ordinary** diﬀerential functional **equations**:

22 Read more

The theory of impulsive systems was developed not long ago as an independent area of mathematical analysis. The development arose out of curiosity to develop a mathematical framework that truly describes physical and biological processes as they occur in nature. Prior to this noble development, scientists had often made an underlying assumption that the behaviour of physical and biological systems described by **ordinary** **differential** **equations** is continuous and integrable in some sense. It was observed that the state of a system is susceptible to changes, and in some processes, these changes are often characterized by short- time perturbations (**impulses**) whose durations are negligible when compared with the total duration of their entry time evolution [15, 3, 16-18].

The problems of stability of solutions of diﬀerential **equations** via Lyapunov functions have been successfully investigated in the past. One type of stability, very useful in real world problems, is the so-called Lipschitz stability. Dannan and Elaydi [] introduced the notion of Lipschitz stability for **ordinary** diﬀerential **equations**. As is mentioned in [] this type of stability is important only for nonlinear problems, since it coincides with uniform stability in linear systems.

13 Read more

where p t ( ) ∈ PC 1 ([ , t 0 +∞ ), ) R , r t ( ) is a positive continuous function defined on [ , t 0 +∞ ) , δ τ ∈ +∞ , (0, ) , σ = max{ , } δ τ and ϕ ϕ , ′ : [ t 0 − σ , ] t 0 → R . They established adequate impulse controls under which the system remained oscillatory after undergoing controlled abrupt perturbations (called **impulses**). Abasiekwere and Moffat [37] examined the oscillations of a class of second order linear neutral impulsive **ordinary** **differential** **equations** with variable coefficients and constant retarded arguments and obtained sufficient conditions ensuring the oscillation of all solutions. Again, Abasiekwere, et al [38] closely considered a certain type of second order delay **differential** **equations** with constant impulsive jumps and obtained sufficient conditions for the oscillation of all its bounded solutions.

There is already a large body of research on the notion of impulsive diﬀerential equa- tions in the literature. The ﬁndings of most of these studies are mainly achieved through some such theories as ﬁxed point theory, topological degree theory (including continu- ation method and coincidence degree theory) and comparison method (including upper and lower solutions method and monotone iterative method) (see, for example, [–] and references therein). Recently, the existence and multiplicity of solutions for impul- sive problems have been thoroughly investigated by [–] using variational methods and the critical point theory, the whole ﬁndings of which can be considered as nothing but generalizations of the corresponding ones for the second-order **ordinary** diﬀerential **equations**. Put diﬀerently, the aforementioned achievements can be applied to impulsive systems in the absence of the **impulses** and still give the existence of solutions in this sit- uation. This is, somehow, to say that the nonlinear term V u functions more signiﬁcantly

16 Read more

Our result is closely related to those of Thompson [] and of Kongson et al. []. In [] and [], the authors established existence results for systems of second-order **ordinary** dif- ferential **equations** in more general bounding sets and subject to general boundary condi- tions () but not subject to **impulses**. Moreover, the proof in [] is incomplete as it fails to establish the required derivative bounds; these appear to require more assumptions on the Hartman-Nagumo growth bound than we assume here. Although our bounding sets are more restrictive than those in [], our proof is much simpler than theirs. In particular, the ideas introduced in our proof oﬀer a fresh starting point for further work aimed at iden- tifying the natural and most general concept of a bounding set and with this the natural and most general existence results possible for system () subject to nonlinear boundary conditions ().

18 Read more

There are signiﬁcant developments in the theory of **impulses** especially in the area of impulsive diﬀerential **equations** with ﬁxed moments, which provided a natural description of observed evolution processes, regarding as important tools for better understanding several real word phenomena in applied sciences [, , –]. In addition, motivated by the better formula of solutions cited by the work of Zhou et al. [, , ], the aim of this work is to discuss a boundary value problem for a coupled system of impulsive fractional diﬀerential equation. Exactly, this paper deals with the m-point boundary value problem of the following coupled system of impulsive fractional diﬀerential **equations** at resonance:

23 Read more

have their discontinuities at diﬀerent points from (a, b). Our aim in this paper is to prove the existence of a solution of problem (.), (.) satisfying the general linear bound- ary condition (.). To do this, we need a suitable linear space containing S. Due to state-dependent **impulses**, the Banach space of piece-wise continuous functions on [a, b] with the sup-norm cannot be used here. Therefore we choose the Banach space G L [a, b].

18 Read more

the conditions of the second result in []. Then (.) has a positive T -periodic solution if and only if e ¯ < . This means that (.) under (H) and (H)(i) does not have a T -periodic weak solution. However, if the **impulses** happen, i.e. if (H)(ii) is fulﬁlled for this singular equation (.), there may exist a positive T-periodic weak solution. Such a solution is called a periodic solution generated by **impulses** as pointed out in [].

11 Read more

(Kerr, 1988) developed some basic Solutions on **Differential** **Equations** using fractional powers of the Fourier Transform on the Schwartz space (Bailey and Swartzrauber, 1991) compute the Fractional Fourier transforms using Fast Fourier Transforms algorithm and discuss some properties of Discrete FrFT. (Almeida, 1994) suggested use of FrFT in the area of Communication in the time frequency plane. (Lin, 1999) investigated the use of FrFT in optics. (Ozaktas, kittay & Mendlovic, 1999) discussed the order fractional Fourier Transform which is a generalized from **Ordinary** Fourier transforms. (Bracewell, 2000) studied the Fourier analysis and the concerned are discussed. (Ozaktas, Zalevsky & kuttay, 2001) provide the concept of Fractional Fourier domains its properties and its relation to space-frequency representations. (Narayanan, & Prabhu, 2003) proposed some techniques in the field of signal restoration and noise removal by using FrFT. (Luchko, Martinez and Trujilo, 2008) introduced the new definition of Fractional Fourier Transforms using parameter ψ. (Roopkumar, 2016) used quaternion Fractional Fourier Transform techniques to prove Convolution and Product Theorem.

16 Read more

The coeﬃcients α, β and δ in (.) are positive ω-periodic functions, the kernel K satisﬁes the condition ∞ K(τ )dτ = . The change of variable t – τ = s and then the substitution of the type (.) in the case of the kernel (.) reduces integro-diﬀerential equation (.) to the system of **ordinary** diﬀerential **equations**

17 Read more

Korhan G¨ unel graduated from Ege University as a mathematician. He received his one of the M.Sc. degrees in computer engineering from Dokuz Eylul University, and received the other one in applied mathematics from Adnan Menderes University. He completed his Ph.D. degree in computer science at the Department of Mathematics in Ege University. His interests revolve around natural language processing, artificial intelligence applied to education, global optimization and machine learning for solving **differential** **equations**. Currently he works as Assistant Professor at the Department of Mathematics in Adnan Menderes University.

12 Read more

[10] Z. Fan and G. Li, “Existence Results for Semilinear Dif- ferential **Equations** with Nonlocal and Impulsive Condi- tions,” Journal of Functional Analysis, Vol. 258, No. 5, 2010, pp. 1709-1727. doi:10.1016/j.jfa.2009.10.023 [11] E. Hernandez, M. Pierri and G. Goncalves, “Existence

In this section, the modified mathematical model of Ngwa and Shu’s model for the evolution of the malaria in a population formulated in [1] is presented in (1) – (9). The **equations** are based on both human and mosquito population, infection rates of human and mosquito per unit time, rescaling of critical population of each class by total species population and bifurcation analysis. Following the basic ideas and structure of mathematical modeling in epidemiology, the model for the malaria disease was developed under the terms in tables 1 and 2 (see [1] for detailed analysis).

of the corresponding **equations** cannot be controlled in a single Hilbert or Banach space (in contrast to spin systems on a regular lattice, which have been well- studied, see e.g. [14] and modern developments in [19], and references therein). However, under mild conditions on the density of γ (holding for e.g. Poisson and Gibbs point processes in R n ), it is possible to apply the approach discussed above and construct a solution in the scale of Hilbert spaces S γ

21 Read more

Note that to solve example (1) y = 3, we could have simply integrated both sides. This follows from the very basic idea that anytime we are given two things that are equal, then as long as we do the same thing to one side of an equation as to the other then equality still holds. For an obvious example of this principle in action, if x = 2, then x + 4 = 2 + 4 and 3x = 6. So solving the **differential** equation y = 3, is pretty straightforward, we just have to integrate both sides: y ' dx 3 dx . The fundamental theorem of calculus then tells us that the integral of the derivative o f a function is just the function itself up to a constant, i.e. that y ' dx y c 1 , and also that 3 dx 3 x c 2 where we represent different constants by writing c 1 and c 2 , to distinguish them from

11 Read more

In this paper we have investigated the robust stability for the linear time-varying implicit dynamic **equations** on time scale. Some characterizations for robust stability of IDEs subjected to Lipschitz perturbations are derived. Many previous results for robust stability of the time-varying **ordinary** **differential** and difference **equations**, the time-varying **differential** algebraic **equations** and the time- varying implicit difference **equations** are also unified and extended .

11 Read more

Sabbavarapu Nageswara Rao has received M.Sc. from Andhra University, M. Tech (CSE) from Jawaharlal Nehru Technological University, and Ph.D from Andhra University under the esteemed guidance of Prof. K. Rajendra Prasad. He served in the cadre of Assistant professor and Associate professor in Sriprakash College of Engineering and Professor in AITAM, Tekkali. Presently, Dr. Rao is working in the Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia. His major research interest includes **ordinary** **differential** **equations**, difference **equations**, dynamic **equations** on time scales, p-Laplacian, fractional order **differential** **equations** and boundary value problems. He published several research papers on the above topics in various national and international journals.

12 Read more