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Articles 1 - 15 of 15
Full-Text Articles in Physical Sciences and Mathematics
Generalization Of Statistical Limit-Cluster Points And The Concepts Of Statistical Limit Inferior-Superior On Time Scales By Using Regular Integral Transformations, Ceylan Yalçin
Turkish Journal of Mathematics
With the aid of regular integral operators, we will be able to generalize statistical limit-cluster points and statistical limit inferior-superior ideas on time scales in this work. These two topics, which have previously been researched separately from one another sometimes only in the discrete case and other times in the continuous case, will be studied at in a single study. We will investigate the relations of these concepts with each other and come to a number of new conclusions. On some well-known time scales, we shall analyze these ideas using examples.
An Improved Oscillation Criteria For First Order Dynamic Equations, Özkan Öcalan
An Improved Oscillation Criteria For First Order Dynamic Equations, Özkan Öcalan
Turkish Journal of Mathematics
In this work, we consider the first-order dynamic equations \begin{equation*} x^{\Delta }(t)+p(t)x\left( \tau (t)\right) =0,\text{ }t\in \lbrack t_{0},\infty )_{\mathbb{T}} \end{equation*} where $p\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) , $ $\tau \in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ and $\tau (t)\leq t,\ \lim_{t\rightarrow \infty }\tau (t)=\infty $. When the delay term $\tau (t)$ is not necessarily monotone, we present a new sufficient condition for the oscillation of first-order delay dynamic equations on time scales.
Existence Of Positive Solutions For Nonlinear Multipoint P-Laplacian Dynamic Equations On Time Scales, Abdülkadi̇r Doğan
Existence Of Positive Solutions For Nonlinear Multipoint P-Laplacian Dynamic Equations On Time Scales, Abdülkadi̇r Doğan
Turkish Journal of Mathematics
In this paper, we investigate the existence of positive solutions for nonlinear multipoint boundary value problems for p-Laplacian dynamic equations on time scales with the delta derivative of the nonlinear term. Sufficient assumptions are obtained for existence of at least twin or arbitrary even positive solutions to some boundary value problems. Our results are achieved by appealing to the fixed point theorems of Avery-Henderson. As an application, an example to demonstrate our results is given.
On Nonoscillatory Solutions Of Three Dimensional Time-Scale Systems, Elvan Akin, Taher S. Hassan, Özkan Öztürk, İsmai̇l Uğur Ti̇ryaki̇
On Nonoscillatory Solutions Of Three Dimensional Time-Scale Systems, Elvan Akin, Taher S. Hassan, Özkan Öztürk, İsmai̇l Uğur Ti̇ryaki̇
Turkish Journal of Mathematics
In this article, we classify nonoscillatory solutions of a system of three-dimensional time scale systems. We use the method of considering the sign of components of such solutions. Examples are given to highlight some of our results. Moreover, the existence of such solutions is obtained by Knaster's fixed point theorem.
Solutions To Nonlinear Second-Order Three-Point Boundary Value Problems Of Dynamic Equations On Time Scales, Abdülkadi̇r Doğan
Solutions To Nonlinear Second-Order Three-Point Boundary Value Problems Of Dynamic Equations On Time Scales, Abdülkadi̇r Doğan
Turkish Journal of Mathematics
In this paper, we consider existence criteria of three positive solutions of three-point boundary value problems for $p$-Laplacian dynamic equations on time scales. To show our main results, we apply the well-known Leggett-Williams fixed point theorem. Moreover, we present some results for the existence of single and multiple positive solutions for boundary value problems on time scales, by applying fixed point theorems in cones. The conditions we used in the paper are different from those in [Dogan A. On the existence of positive solutions for the one-dimensional $ p $-Laplacian boundary value problems on time scales. Dynam Syst Appl 2015; …
A Generalization Of The Alexander Polynomial As An Application Of The Delta Derivative, İsmet Altintaş, Kemal Taşköprü
A Generalization Of The Alexander Polynomial As An Application Of The Delta Derivative, İsmet Altintaş, Kemal Taşköprü
Turkish Journal of Mathematics
In this paper, we define the delta derivative in the integer group ring and we show that the delta derivative is well defined on the free groups. We also define a polynomial invariant of oriented knot and link by carrying the delta derivative to the link group. Since the delta derivative is a generalization of the free derivative, this polynomial invariant called the delta polynomial is a generalization of the Alexander polynomial. In addition, we present a new polynomial called the difference polynomial of oriented knot and link, which is similar to the Alexander polynomial and is a special case …
Stability Of Abstract Dynamic Equations On Time Scales By Lyapunov's Second Method, Alaa Hamza, Karima Oraby
Stability Of Abstract Dynamic Equations On Time Scales By Lyapunov's Second Method, Alaa Hamza, Karima Oraby
Turkish Journal of Mathematics
In this paper, we use the Lyapunov's second method to obtain new sufficient conditions for many types of stability like exponential stability, uniform exponential stability, $h$-stability, and uniform $h$-stability of the nonlinear dynamic equation \begin{equation*} x^{\Delta}(t)=A(t)x(t)+f(t,x),\;t\in \mathbb{T}^+_\tau:=[\tau,\infty)_{\mathbb T}, \end{equation*} on a time scale $\mathbb T$, where $A\in C_{rd}(\mathbb T,L(X))$ and $f:\mathbb T\times X\to X$ is rd-continuous in the first argument with $f(t,0)=0.$ Here $X$ is a Banach space. We also establish sufficient conditions for the nonhomogeneous particular dynamic equation \begin{equation*} x^{\Delta}(t)=A(t)x(t)+f(t),\,t\in\mathbb{T}^+_{\tau}, \end{equation*} to be uniformly exponentially stable or uniformly $h$-stable, where $f\in C_{rd}(\mathbb T,X)$, the space of rd-continuous functions from …
On The Existence And Uniqueness Of Solutions To Dynamic Equations, Başak Karpuz
On The Existence And Uniqueness Of Solutions To Dynamic Equations, Başak Karpuz
Turkish Journal of Mathematics
In this paper, we prove the well-known Cauchy-Peano theorem for existence of solutions to dynamic equations on time scales. Some simple examples are given to show that there may exist more than a single solution for dynamic initial value problems. Under some certain conditions, it is also shown that there exists only one solution.
Limit Behaviors Of Nonoscillatory Solutions Of Three-Dimensional Time Scale Systems, Özkan Öztürk, Raegan Higgins
Limit Behaviors Of Nonoscillatory Solutions Of Three-Dimensional Time Scale Systems, Özkan Öztürk, Raegan Higgins
Turkish Journal of Mathematics
In this article, we investigate the oscillatory behavior of a three-dimensional system of dynamic equations on an unbounded time scale. A time scale $\T$ is a nonempty closed subset of real numbers. An example is given to illustrate some of the results.
Dynamic Shum Inequalities, Ravi Agarwal, Martin Bohner, Donal O'Regan, Samir Saker
Dynamic Shum Inequalities, Ravi Agarwal, Martin Bohner, Donal O'Regan, Samir Saker
Turkish Journal of Mathematics
Recently, various forms and improvements of Opial dynamic inequalities have been given in the literature. In this paper, we give refinements of Opial inequalities on time scales that reduce in the continuous case to classical inequalities named after Beesack and Shum. These refinements are new in the important discrete case.
New Oscillation Tests And Some Refinements For First-Order Delay Dynamic Equations, Başak Karpuz, Özkan Öcalan
New Oscillation Tests And Some Refinements For First-Order Delay Dynamic Equations, Başak Karpuz, Özkan Öcalan
Turkish Journal of Mathematics
In this paper, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales. We also present some examples to which none of the previous results in the literature can apply.
Multiple Positive Solutions Of Nonlinear $M$-Point Dynamic Equations For $P$-Laplacian On Time Scales, Abdülkadi̇r Doğan
Multiple Positive Solutions Of Nonlinear $M$-Point Dynamic Equations For $P$-Laplacian On Time Scales, Abdülkadi̇r Doğan
Turkish Journal of Mathematics
In this paper, we study the existence of positive solutions of a nonlinear $ m $-point $p$-Laplacian dynamic equation $$(\phi_p(x^\Delta(t)))^\nabla+w(t)f(t,x(t),x^\Delta(t))=0,\hspace{2cm} t_1< t 1.$ Sufficient conditions for the existence of at least three positive solutions of the problem are obtained by using a fixed point theorem. The interesting point is the nonlinear term $f$ is involved with the first order derivative explicitly. As an application, an example is given to illustrate the result.
Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur
Stability Of Perturbed Dynamic System On Time Scales With Initial Time Difference, Coşkun Yakar, Bülent Oğur
Turkish Journal of Mathematics
The behavior of solutions of a perturbed dynamic system with respect to an original unperturbed dynamic system, which have initial time difference, are investigated on arbitrary time scales. Notions of stability, asymptotic stability, and instability with initial time difference are introduced. Sufficient conditions of stability properties are given with the help of Lyapunov-like functions.
Lebesgue-Stieltjes Measure On Time Scales, Asli Deni̇z, Ünal Ufuktepe
Lebesgue-Stieltjes Measure On Time Scales, Asli Deni̇z, Ünal Ufuktepe
Turkish Journal of Mathematics
The theory of time scales was introduced by Stefan Hilger in his Ph. D. thesis in 1988, supervised by Bernd Auldbach, in order to unify continuous and discrete analysis [5]. Measure theory on time scales was first constructed by Guseinov [4], then further studies were made by Guseinov-Bohner [1], Cabada-Vivero [2] and Rzezuchowski [6]. In this article, we adapt the concept of Lebesgue-Stieltjes measure to time scales. We define Lebesgue-Stieltjes \Delta and \nabla-measures and by using these measures, we define an integral adapted to time scales, specifically Lebesgue-Stieltjes \Delta-integral. We also establish the relation between Lebesgue-Stieltjes measure and Lebesgue-Stieltjes \Delta-measure, …
Self-Adjoint Boundary Value Problems On Time Scales And Symmetric Green's Functions, Gusein Sh. Guseinov
Self-Adjoint Boundary Value Problems On Time Scales And Symmetric Green's Functions, Gusein Sh. Guseinov
Turkish Journal of Mathematics
In this note, higher order self-adjoint differential expressions on time scales, and associated with them self-adjoint boundary conditions, are discussed. The symmetry peoperty of the corresponding Green's functions is emphasized.