In this paper, we focus on nonoscillatory solutions of two (2D) and three (3D) dimensional time scale systems and discuss nonexistence of such solutions

Akin, Elvan

Ozturk, Ozkan

Tiryaki, Ismail Ugur

Yeni, Gulsah

English

Missouri University of Science and Technology (Missouri S&T): Scholars' Mine

Missouri University of Science and Technology Scholars' Mine Mathematics and Statistics Faculty Research & Creative Works Mathematics and Statistics 01 Jan 2018 Mini Review: A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems Elvan Akin Missouri University of Science and Technology, akine@mst.edu Ozkan Ozturk Ismail Ugur Tiryaki Gulsah Yeni Follow this and additional works at: https://scholarsmine.mst.edu/math_stat_facwork  Part of the Mathematics Commons, and the Statistics and Probability Commons Recommended Citation E. Akin et al., "Mini Review: A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems," International Journal of Applied and Experimental Mathematics, vol. 3, no. 1, Graphy Publications, Jan 2018. This work is licensed under a Creative Commons Attribution 3.0 License. This Article - Journal is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Mathematics and Statistics Faculty Research & Creative Works by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact scholarsmine@mst.edu. AbstractIn this paper, we focus on nonoscillatory solutions of two (2D) and three (3D) dimensional time scale systems and discuss nonexistence of such solutions.A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale SystemsPublication History:Received: November 17, 2017Accepted: January 18, 2018Published: January 20, 2018Keywords: Paraconsistent logic, Linear-time temporal logic, Clinical reasoningverication, Paraconsistent modelMini Review Open AccessIntroductionThe motivation of studying dynamic equations on time scales is to unify continuous and discrete analysis and harmonize them in one comprehensive theory and eliminate obscurity from both. A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most well known examples for time scales are R (leading to dierential equations,  [10]), Z (leading to dierence equations, [11]), and                                         (leading to q-dierence equations,  [9]). In 1988, the theory of time scales was initiated by Stefan Hilger in his Ph.D. thesis, [8]. Since the calculus of time scales has been recently developed, we give a brief introduction to time scales calculus. Nevertheless, an excellent introduction can be found in [5, 6] by Bohner and Peterson.There are two jump operators: For t ε T, the forward jump operator σ: T → T is given by σ(t):=inf {s ε T: s>T} for all t ε  T while the backward jump operator ρ: T→T is dened by ρ(t):= sup{s ε T : s < t} for all t ε T. Finally, the graininess function: μ: T → [0,∞) is given by μ(t):= σ(t) - t for all t ε T.We dene inf 0 = sup T. There are four types of points in T. If σ(t)>t, hen t is called right - scattered, while if ρ(t)<t, t is called left - scattered. If t is right and left - scattered at the same time, then we say that t is isolated. If t < supT and σ(t)=t, then t is called right - dense, while if t > inf T and ρ(t)=t, we say t is left - dense. Also, if t is right and left - dense at the same time, then we say that t is dense.If supT < ∞, then TK=T\(ρ(sup T), sup T], and TK=T if sup T=∞. Suppose that f : T→R is a function. Then fσ: T→R is dened by fσ(t)=f(σ(t)) for all t ε T. For any ε > 0, if there exists a δ > 0 such that                                                               for all                                          ,then f is called delta (or Hilger) dierentiable on Tk and f∆ is called delta derivative of f. Let f : T→R be a function with t ε TK. If f is dierentiable at t, f is continuous at t. If f is continuous at t and t is right-scattered, then f is dierentiable at t andIf t is right dense, then f is dierentiable at t if*Corresponding Author: Dr. Elvan Akin, Department of Mathematics and Statistics, Missouri University of Science and Technology, 400 W 12th Street, Rolla, Missouri 65409 USA, E-mail: akine@mst.eduCitation: Akn E, Ozturk O, Tiryaki IU, Yeni G  (2018) A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems. Int J Appl Exp Math 3: 126. doi:  https://doi.org/10.15344/2456-8155/2018/126Copyright: © 2018 Akn et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.International Journal ofApplied & Experimental MathematicsElvan Akın*, Ozkan Ozturk, Ismail U. Tiryaki and Gulsah YeniDepartment of Mathematics and Statistics, Missouri University of Science and Technology, 400 W 12th Street, Rolla, Missouri 65409 USA Int J Appl Exp Math                                                                                                                                                                                                IJAEM, an open access journal ISSN: 2456-8155                                                                                                                                                                                                      Volume 3. 2018. 126 exists as a fnite number. If f is dierentiable at t, then fσ(t)=f(t)+μ(t)f∆(t) holds for all type of points in T. Let f, g : T→R be dierentiable at t ε TK. Then we have the product and quotient rules for derivatives as follows:1. 2. If g(t)g(σ(t))≠0, then     is dierentaible at t withf : T→R is called right dense continuous (rd-continuous), denoted by Crd,Crd(T); or Crd(T,R), if it is continuous at right dense points in T and its left sided limits exist as a nite number at left dense points in T. Throughout we denote continuous functions by C. Also, the Cauchy integral is dened byEvery rd-continuous function has an antiderivative. Moreover, F given byis an antiderivative of f.Higher Dimensional Time Scale SystemsThe study of higher dimensional time scale systems in nature and society has been motivated by their applications such as population dynamics, genomic and neuron dynamics and epidemiology in biological sciences,  [1,16] and in astrophysics, gas dynamics and fuid mechanics, relativistic mechanics, nuclear physics, and chemically reacting systems, [4,7,12,17].                                               Akn et al., Int J Appl Exp Math 2018, 3: 126                                               https://doi.org/10.15344/2456-8155/2018/1260 2: {1, , ,...}, 1q q q qΝ = >| ( ) ( ) ( )( ( ) ) | | ( ) |f t f s f t t s t sσ σ ε σ∆− − − ≤ − ( , )s t tδ δ∈ − + ∩Τ( ) ( )( )   ( )f t f tf ttσµ∆ −=( ) ( )( ) lim   s tf t f sf tt sσ∆→−=−( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )fg t f t g t f t g t f t g tσ σ∆ ∆ ∆ ∆= + +fg( ) ( ) ( ) ( )( )( ) ( )f f t g t f t g ttg g t g tσ∆ ∆ ∆  −=  ( ) ( ) ( ) for all ,baf t t F b F a a b∆ = − ∈Τ∫0( ) ( )  for ttF t f s s t= ∆ ∈Τ∫Int J Appl Exp Math                                                                                                                                                                                                IJAEM, an open access journal ISSN: 2456-8155                                                                                                                                                                                                      Volume 3. 2018. 126 Recently, nonoscillatory solutions of 2D and 3D time scale systems have been considered in [13-15] and [2,3], respectively. One of our main goals in these articles is to nd integral conditions which eliminate nonoscillatory solutions of given systems. Depending on which dimension we deal with one naturally attempt to nd single, double or triple integral conditions. A question we have here is whether we can eliminate all types of nonoscillatory solutions or not. In this contribution, we only satisfy single integral conditions for both time scale systems and other integral conditions can be found in references above. Our approach is based on the sign of components of nonoscillatory solutions of systems and we assume without loss of generality that the rst component of such solutions is always positive.We rst start with the following 2D time scale system                                                                                                     (2.1)where a, b ε Crd ([t0,∞)T, R+) and f, g ε C(R,R) satisfying uf(u) > 0, ug(u) > 0 for u≠0 and                                                                                                     (2.2)where F and G are positive constants and Φα, and Φβ are old power functions, i.e., Φp(u)=|u|psgn u, p > 0 and p ε {α, β}.Throughout this article, we assume that T is unbounded above. We call (x, y) a proper solution if it is dened on [t0, ∞)T and sup{|x(s)|, |y(s)| : s ε [t, ∞)T} > 0 for t ≥t0. By t ≥t1, we mean t ε [t1, ∞)T := [t1, ∞)   T. A solution (x, y) of (2.1) is said to be nonoscillatory if the component functions x and y are both nonoscillatory, i.e., either eventually positive or eventually negative. Otherwise, it is said to be oscillatory. Denitions above can be modied for higher dimensional time scale systems.Assume that (x, y) is a nonoscillatory solution of system (2.1) such that x oscillates but y is eventually positive. Then the rst equation of system (2.1) yields x∆(t)=a(t)f(y(t))>0 eventually one sign for all large t ≥ t0, a contradiction. The case where y is eventually negative is similar. Therefore, we have that the component functions x and y are themselves nonoscillatory. In other words, any nonoscillatory solution (x, y) of system (2.1) is one of the following types:                              Type (a): sgn x(t) = sgn y(t);                              Type (b): sgn x(t) ≠ sgn y(t).For convenience, let us set                                                                                                      (2.3)TheoremAny nonoscillatory solution of system (2.1) cannot be of1. Type (a) if A(t0) < 1 and B(t0) = 1.2. Type (b) if A(t0) = 1 and B(t0) < 1.Secondly, let's consider 3D time scale systems of the form                                                                                                  (2.4)where a, b, c ε Crd([t0,∞)T,R+ amd f, g, h ε C(R, R) satisfying uf(u) > 0, ug(u) > 0, and uh(u) > 0 for u ≠ 0 and                                                                                                   (2.5)where F, G, and H are positive constants and Φp(u)=|u|psgn u, p>0 and p ε {α, β, γ}. By a similar argument, we can conclude that any nonoscillatory solution (x, y, z) of (2.4) is one of following types:                             Type (a): sgn x(t) = sgn y(t) = sgn z(t),                             Type (b): sgn x(t) ≠ sgn y(t) = sgn z(t),                             Type (c): sgn x(t) = sgn y(t) ≠ sgn z(t),                             Type (d): sgn x(t) = sgn z(t) ≠ sgn y(t). For convenience let's set LemmaAny nonoscillatory solution of system (2.4) can not be of9. Type (a) if C(t0) = 1.10. Type (c) if A(t0) = 1;11. Type (d) if B(t0) = 1.Not only from Lemma 2.2 but also from other results in above references one can observe that Type (b) solution of system (2.1) is not eliminated. In fact, it is worth to emphasize that components of Type (b) solutions have nite limits. Note that when we deal with 2D time scale systems such type does not occur. Here is the question comes to our mind: Is it related with dimensions of given systems? When we focus on four dimensional (4D) time scale systems, we realize that nonoscillatory solutions whose components have nite limits do not occur either. Moreover, we have already known that all types of nonoscillatory solutions of 4D time scale systems can be eliminated by some integral conditions. Therefore, we would like to emphasize that this is a matter of considering odd or even dimensional time scale systems.Competing InterestsThe authors declare that there is no competing interest regarding the publication of this article. References1. Agarwal RP, O'Regan D, Saker SH (2014) Oscillation and Stability of Delay Models in Biology. Springer. 2. Ekin-Bohner E, Dosla Z, Lawrence B (2012) Almost oscillatory Three Dimensional Dynamic Systems. Adv. Difference Equ 2012: 46.3. Akinn E, Ozturk O (2016) Nonoscillation Criteria for Three Dimensional Time-Scale Systems.4. Arthur AM, Robinson PD (1969) Complementary variational principle for ∇2 = f(Φ) with applications to the Thomas - Fermaind Liouville equations. Proc Cambridge Philos Soc 65: 535-542.5. Bohner M, Peterson A (2001) Dynamic Equations on Time Scales: An Introduction with Applications. Birkhauser, Boston. 6. Bohner M, Peterson A (2003) Advances in Dynamic Equations on Time Scales. Birkhauser, Boston. 7. Davis HT (1960) Introduction to Nonlinear Differential Integral Equations U.S. Atomic Energy Commission, Washington DC.8. Hilger S (1988) Ein Mabkettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten [thesis]. Universitat Wurzburg.Citation: Akn E, Ozturk O, Tiryaki IU, Yeni G  (2018) A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems. Int J Appl Exp Math 3: 126. doi:  https://doi.org/10.15344/2456-8155/2018/126       Page 2 of 3( ) ( ) ( ( ))( ) ( ) ( ( )),x t a t f y ty t b t g x t∆∆ == −rd 0, C ([ , ) ,R )a b t+Τ∈ ∞∩0 00 0 0( ) ( )  and ( ) ( ) ,  t tA t a t t B t b t t t∞ ∞= ∆ = ∆ ∈Τ∫ ∫( ) ( ) ( ( ))( ) ( ) ( ( ))( ) ( ) ( ( ))x t a t f y ty t b t g z tz t c t h x t∆∆∆ = = = −( ) ( ) ( ), ,  for all 0,( ) ( ) ( )f u g u h uF G H uu u uα β γ≥ ≥ ≥ ≠Φ Φ Φ00 0( ) ( ) ,  tc t c t t t∞= ∆ ∈Τ∫Int J Appl Exp Math                                                                                                                                                                                                IJAEM, an open access journal ISSN: 2456-8155                                                                                                                                                                                                      Volume 3. 2018. 126 9. Kac V, Cheung P (2002) Quantum calculus. Universitext, Springer-Verlag, New York. 10. Kelley WG, Peterson AC (2010) The Theory of Differential Equations: Classical and Qualitative, Springer.11. Kelley WG, Peterson AC (2001) Difference Equations, Second Edition: An Introduction with Applications. Academic Press.12. Nehari Z (1963) On a nonlinear differential equatioan rising in nuclear physics. Proc. Roy. Irish Academy Sect A.13. Ozturk O, Akin E (2016) Nonoscillation Criteria for Two Dimensional Time-Scale Systems. Nonauton Dyn Syst 3: 1-13.14. Ozturk O, Akin E (2016) On Nonoscillatory Solutions of Two Dimensional Nonlinear Delay Dynamical Systems. Opuscula Math 36: 651-669.15. Ozturk O, Akin E, Tiryaki IU (2015) On Nonoscillatory Solutions of Emden-Fowler Dynamic Systems on Time Scales. Filomat.16. Ruan S, Wolkowicz GSK, Wu J (1999) Differential Equations with Applications to Biology. AMS.17. Shevyelo VN (1965) Problems methods and fundamental results in the theory of oscillation of solutions of nonlinear nonautonomous ordinary differential equations. Proc. 2nd All-Union Conf. on Theoretical and Appl Mech, Moscow.Citation: Akn E, Ozturk O, Tiryaki IU, Yeni G  (2018) A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems. Int J Appl Exp Math 3: 126. doi:  https://doi.org/10.15344/2456-8155/2018/126       Page 3 of 3

Mini Review: A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems

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Mini Review: A Note on Nonoscillatory Solutions for Higher Dimensional Time Scale Systems

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