Bounded solutions of delay dynamic equations on time scales

Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 http://www.advancesindifferenceequations.com/content/2012/1/183 R E S E A R C H Open Access Bounded solutions of delay dynamic equations on time scales Josef Diblík1,2* and Jiˇ rí Vítovec1 *Correspondence: [email protected].cz; [email protected] 1Department of Mathematics, Faculty of Electrical Engineering and Communications, Brno University of Technology, Brno, Czech Republic 2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic Abstract In this paper we discuss the asymptotic behavior of solutions of the delay dynamic equation y(t)=f(t,y(τ(t))), where f:T×R→R,τ:T→Tis a delay function and Tis a time scale. We formulate the principle which gives the guarantee that the graph of at least one solution of the above mentioned equation stays in the prescribed domain. This principle uses the idea of the retraction method and is a suitable tool for investigating the asymptotic behavior of solutions of dynamic equations. This is illustrated by an example. 1 Introduction 1.1 Time scale calculus We assume that the reader is familiar with the notion of time scales. Thus, note just that T,[a,b]T:= [a,b]∩T(resp. (a,b)T:= (a,b)∩Tor similarly, we define any combination of right and left open or closed interval), [a,∞)T:= [a,∞)∩T,σ,ρ,μand fstand for the time scale, a finite time scale interval, an infinite time scale interval, a forward jump operator, a backward jump operator, graininess and a -derivative of f.Further,the symbols C(T), Crd(T)andC rd(T) stand for the class of continuous, rd-continuous and rd-continuous -derivative functions. See [], which is the initiating paper of the time scale theory, the thesis []and[] containing a lot of information on time scale calculus. Now, we remind further aspects of time scale calculus, which will be needed later; see, e.g.,[]. Definition  Let Tbe a time scale. A function f:T×R→Ris called (i) rd-continuous,ifgdefined by g(t):=f(t,y(t)) is rd-continuous for any rd-continuous function y:T→R; (ii) bounded on a set S⊂T×R, if there exists a constant M>such that f(t,y)≤Mfor all (t,y)∈S; (iii) Lipschitz continuous on a set S⊂T×R, if there exists a constant L>such that f(t,y)–f(t,y)≤L|y–y|for all (t,y), (t,y)∈S. ©2012 Diblík and Vítovec; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 2 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 1.2 Delay dynamic equations on time scales Let τ:T→Tbe an increasing rd-continuous function satisfying τ(t)≤tfor all t∈T.Let the function f:T×R→Rbe rd-continuous. We consider the delay dynamic equation y(t)=ft,yτ(t) () on time scales T. (Note that y(τ(t)) ∈Crd(T) in view of Definition .) For a given t∈T, a function y:[τ(t),∞)T→Ris said to be a solution of ()on [τ(t),∞)Tprovided y∈Crd([τ(t),∞)T), y∈C rd([t,∞)T)andysatisfies () for all t∈[t,∞)T. If, moreover, an initial function ϕ∈C([τ(t),t]T)begivenand y(t)=ϕ(t), t∈τ(t),tT,() then we say that yis a solution of the initial value problem (IVP) ()and(). 1.3 Existence and uniqueness of solutions of delay dynamic equations For the next study, it is important to verify the existence and uniqueness of solutions of IVP ()and(). However, the following theorem (in a more general form) can be found in [, Theorem .]. Theorem  (Picard-Lindelöf theorem) Let t∈T,t>t,m>.Let Ym:= y∈R:y–ϕ(t)≤mforallt∈τ(t),tT, where the properties of ϕare described in previous Section ..Assume that f ∈Crd([t, t]T×Ym)is bounded on [t,t]T×Ym,with bound M >,and Lipschitz continuous on [t,t]T×Ym.Then the initial value problem () and () has a unique solution y on the interval [τ(t), σ(ξ)]T⊂[τ(t),t]T,where ξ:= max[t,t+δ]T and δ:= min{t–t,m/M}. Carefully tracing the proof of Theorem in [], it easy to verify that if Theorem holds, then the solution of the IVP ()and() depends continuously on the initial data. 2 Problem under consideration Let b,c:T→Rbe rd-continuous functions such that b(t)<c(t) for all t∈[τ(t),∞]Tand b(t)<ϕ(t)<c(t) for all t∈τ(t),tT.() We define a set ⊂T×Ras := (t,y):t∈τ(t),∞T,b(t)<y<c(t). Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 3 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 Then the closure equals := (t,y):t∈τ(t),∞T,b(t)≤y≤c(t) and the boundary ∂ =∂B∪∂C,where ∂B:= (t,y):t∈τ(t),∞T,y=b(t) and ∂C:= (t,y):t∈τ(t),∞T,y=c(t). Consider the delay dynamic equation () and the initial value problem (). Let b∗:= minb(t):t∈τ(t), tT and c∗:= maxc(t):t∈τ(t),tT. Let t∈T,t>t. Throughout, we will assume that a function fis bounded and Lipschitz continuous on a domain S=S(t,y)⊂T×Rand τ(t),tT×b∗,c∗∩⊂S. This condition says that by Theorem  every initial value problem ()and()withϕsatisfying () has exactly one solution on the interval [τ(t), σ(ξ)]T,σ(ξ)>t.Itisalsoeasyto show that this solution depends continuously on the initial function ϕ. Our aim is to establish sufficient conditions for the right-hand side of equation ()in order to guarantee the existence of at least one solution y(t)of()definedon[τ(t),∞]T such that (t,y(t)) ∈for each t∈[τ(t),∞]T. The main result generalizes some previous results of the first author (and his co-authors) concerning the asymptotic behavior of solutions of discrete equations; see, e.g.,[–]. In papers [, ], in our best knowledge, the retract principle is extended to discrete equations (see [–] as well). In [, , ,] delayed discrete equations are considered by the retract technique, and in []theretract principle is given for discrete time scales. Papers [, ] are devoted to the extension of theretractprincipletodynamicequations.Inthepresentpaper,wegiveanattemptto enlarge the retract principle to delayed dynamic equations. For further consideration, it is convenient to establish the following concept. Definition  ApointM=(t,b(t)) ∈∂B,t≥t, is called the point of strict egress for the set with respect to equation ()if ft,ψτ(t)<b(t), () where ψ:[τ(t),t]T→Ris an arbitrary rd-continuous function such that b(s)<ψ(s)<c(s) for every s∈[τ(t), t)Tand ψ(t)=b(t). Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 4 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 ApointM=(t,c(t)) ∈∂C,t≥t, is called the point of strict egress for the set with respect to equation ()if ft,ψτ(t)>c(t), () where ψ:[τ(t),t]T→Ris an arbitrary rd-continuous function such that b(s)<ψ(s)<c(s) for every s∈[τ(t), t)Tand ψ(t)=c(t). Remark  The geometrical meaning of the point of strict egress is evident. If a point (t∗,b(t∗)) ∈∂Bis a point of strict egress for the set with respect to (), and y(t)isa (unique) solution of ()satisfyingy(t∗)=b(t∗), then due to (), y(t∗)–b(t∗)=ft∗,ψτ(t∗)–b(t∗)<. From the definition of a -derivative, we get y(t)–b(t)<(or(t,y(τ(t))) /∈)fort∈ (t∗,t∗+δ)Twith a small positive δif t∗is a right-dense point and for t=σ(t∗)ift∗is rightscattered. By analogy, if (t∗,c(t∗)) ∈∂Cis a point of strict egress for the set with respect to (), and y(t) is a (unique) solution of ()satisfyingy(t∗)=c(t∗), then due to (), yt∗–ct∗=ft∗,ψτt∗–ct∗>. From the definition of a -derivative, we get y(t)–c(t)>(or(t,y(t)) /∈)fort∈(t∗,t∗+ δ)Twith a small positive δif t∗is a right-dense point and for t=σ(t∗)ift∗is right-scattered. Definition  ([]) If A⊂Bare subsets of a topological space and π:B→Ais a continuous mapping from Bonto Asuch that π(p)=pfor every p∈A,thenπis said to be a retraction of Bonto A. When a retraction of Bonto Aexists, Ais called a retract of B. 3 Existence theorem The proof of the following theorem is based on the retract method, which is well known for ordinary differential equations and goes back to Ważewski []. Below we will assume that the function f, except for the indicated conditions, satisfies all the assumptions given in Section . Theorem  Let f :T×R→R.Let b,c:T→Rbe delta differentiable functions such that b(t)<c(t)for each t ∈[τ(t),∞)T.If,moreover,every point M ∈∂B∪∂Cis the point of strict egress for the set with respect to equation (), then there exists an rd-continuous initial function ϕ∗:[τ(t),t]T→Rsatisfying b(t)<ϕ∗(t)<c(t)for all t ∈τ(t), tT, such that the initial problem y(t)=ϕ∗(t), t∈τ(t),tT() Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 5 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 defines a solution y of () on the interval [τ(t), ∞)Tsatisfying b(t)<y(t)<c(t)for all t ∈[τ(t), ∞)T.() Proof The idea of the proof is simple. We suppose that the statement of the theorem is not valid. Then it is possible to prove that there exists a retraction of a segment B:= [α,β]with α<βonto a two-point set A:= {α,β}. But it is well known that the boundary of a nonempty (closed) interval cannot be its retract (see []). So, in our case, such a retractive mapping cannot exist because it is incompatible with continuity. Without any special comment, throughout the proof, we use the property that the initial value problem in question has a unique solution and the property of continuous dependence of solutions on their initial data. Suppose now that ϕ∗satisfying the inequality b(t)<ϕ∗(t)<c(t) for all t∈τ(t),tT and generating the solution y=y(t) which satisfies () for any t∈[τ(t),∞)Tdoes not exist. This means that for any rd-continuous initial function ϕsatisfying the inequality b(t)<ϕ(t)<c(t) for all t∈τ(t),tT,() there exists a t∈T,t>tsuch that for a corresponding solution y=y(t) of the initial problem y(t)=ϕ(t), t∈τ(t),tT, we have t,yt /∈ and, t,y(t)∈for all t∈[t,t)T. Let us define auxiliary mappings P,Pand P. First, define the mapping P:B→T×R,where B=t,ϕ(t)∈T×R:b(t)≤ϕ(t)≤c(t), such that (i) for ϕ(t),b(t)<ϕ(t)<c(t),t∈[τ(t),t]T, we define P:t,ϕ(t)→t,yt; (ii) for ϕ(t)satisfying b(t)<ϕ(t)<c(t),t∈[τ(t), t)Tand ϕ(t)=b(t),weput t=tand define P:t,ϕ(t)→t,yt=t,b(t); Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 6 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 (iii) for ϕ(t),b(t)<ϕ(t)<c(t),t∈[τ(t), t)Tand ϕ(t)=c(t),weputt=tand define P:t,ϕ(t)→t,yt=t,c(t). Second, we define the mapping P:B→T×R,where B=P(B)=t,y(t)∈T×R:y(t)≤btor y(t)≥ct, as P:t,yt→⎧ ⎨ ⎩ (t,c(t)) if y(t)≥c(t), (t,b(t)) if y(t)≤b(t). Third, we define the mapping P:B→T×R,where B=P(B)=t,˜ y∈T×R:˜ y=btor ˜ y=ct, as P:t,˜ y→⎧ ⎨ ⎩ (t,c(t)) if ˜ y=c(t), (t,b(t)) if ˜ y=b(t). We will show that the composite mapping P:= P◦P◦P,P:B→A, where A=t,b(t),t,c(t), is continuous with respect to the second coordinate ϕ(t)ofthepoint(t,ϕ(t)) ∈B. The definition of the mapping Pimplies that only two resulting points are possible, namely either P(B)=(t,c(t)) or P(B)=(t,b(t)). (I) We consider the first possibility, i.e.,P(B)=(t,c(t)). Let b(t)<ϕ(t)<c(t) for all t∈[τ(t),t)T.Then Pt,ϕ(t)=t,yt,t,yt /∈and yt≥ct. Let y(t)>c(t). Then ρ(t)<tand the continuity of the mapping Pis obvious. Indeed, if ϕ,ε(t), t∈τ(t),t,b(t)<ϕ,ε(t)<c(t), t∈τ(t), t() is the initial problem defining the solution yε(t), εis a sufficiently small number and ϕ,ε(t)–ϕ(t)<ε,t∈τ(t),t, Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 7 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 then due to the property of continuous dependence of solutions on their initial data, t,ε= tand Pt,ϕ,ε(t)=t,ε,yεt,ε=t,yεt with yεt>ct. Consequently, P(t,ϕ,ε(t)) = (t,c(t)). Let y(t)=c(t). By the assumption of the theorem, every boundary point of ∂ is the point of strict egress for the set with respect to equation (). Then for the solution yε(t) defined by (), we have Pt,ϕ,ε(t)=t,ε,yεt,ε either with yε(t,ε)>c(t,ε)orwitht,ε=t,yε(t,ε)=c(t). (We do not describe all the possibilities for the occurrence of the first or of the second alternative.) In both alternatives we get P(t,ϕ,ε(t)) = (t,c(t)) again. Hence, the mapping Pis continuous in the considered case. (II) We proceed analogously with the case P(B)=(t,b(t)). The continuity of the mapping Pwas proved for initial functions ϕsatisfying b(t)< ϕ(t)<c(t) for all t∈[τ(t), t)Tand b(t)<ϕ(t)<c(t). The desired retraction Prcan be defined as a mapping of the second coordinates realized by P. Then the mapping b(t),c(t)Pr –→b(t),c(t) is continuous and b(t)Pr –→b(t),c(t)Pr –→c(t), i.e.,thepoints{b(t)},{c(t)}are stationary. In this situation we proved that there exists a retraction Prof the set B:= [b(t), c(t)] onto the two-point set A:= {b(t), c(t)}(see Definition ). In regard to the above mentioned fact, this is impossible. Our supposition is false, and there exists the initial problem () such that the corresponding solution y=y∗(t) satisfies the inequalities ()forevery t∈[τ(t),∞]T. The theorem is proved.  4Example Let us consider the delay dynamic equation of the type () y=ft,yτ(t):=  t·yτ(t)+sin(y(τ(t))) +t+cos(y(τ(t))) +t() defined for each t∈[a,∞)Twith a∈T,τ(a)>andμ(t)=O(t) (which means that there exists d>suchthatμ(t)≤dt for each t∈T). Note that we have no further requirements for the function τ(except for those mentioned in Section .). Moreover, let t∈T,t>a be sufficiently large. With the aid of Theorem , we will show that there exists an initial function ϕ∗(t)∈–t–,t–,t∈τ(t),tT,() Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 8 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 which defines a solution y(t) for all t∈[τ(t),∞)Tof the dynamic equation ()satisfying y(t)<t–.() We define -differentiable functions b,c:[τ(t),∞)T→Rsatisfying b(t)<c(t)foreach t∈Tas b(t):=–t–,c(t):=t–. We will verify that every point M∈∂ =∂B∪∂C,where ∂B:= (t,y):t∈τ(t),∞T,y=–t–, ∂C:= (t,y):t∈τ(t),∞T,y=t–, is a point of strict egress for the set := (t,y):t∈τ(t),∞T,–t– <y<t– with respect to the dynamic equation (). For an arbitrary function ψ:[τ(t),t]T→R,t∈[t,∞)Tsuch that b(s)<ψ(s)<c(s), s∈[τ(t),t)Tand ψ(t)=b(t), we need (see ())  t·ψτ(t)+sin(ψ(τ(t))) +t+cos(ψ(τ(t))) +t<– t , () and analogously, for an arbitrary function ψ:[τ(t),t]T→R,t∈[t,∞)Tsuch that b(s)< ψ(s)<c(s), s∈[τ(t),t)Tand ψ(t)=c(t), we need (see ())  t·ψτ(t)+sin(ψ(τ(t))) +t+cos(ψ(τ(t))) +t> t .() Inequalities ()and() will be valid if  t+ +t< t( + d). Indeed, (we underline that for tsufficiently large the last inequality holds) we have   t·ψτ(t)+sin(ψ(τ(t))) +t+cos(ψ(τ(t))) +t < t+ +t< t( + d) < t = t(t+μ(t)). Hence, in view of Definition ,everypointM∈∂ is a point of strict egress for the set . Therefore, all the assumptions of Theorem  hold and there exists an initial value function ϕ∗with the property () such that the initial problem y(t)=ϕ∗(t) defines a solution yon the interval T=[τ(t), ∞)Tof () satisfying the inequality ()foreveryt∈[τ(t),∞)T. Note that this solution (due to ()) tends to zero as t→∞. Diblík and Vítovec Advances in Difference Equations 2012, 2012:183 Page 9 of 9 http://www.advancesindifferenceequations.com/content/2012/1/183 Remark  In our example of equation () with bounded and vanishing solution, the graininess μ(t) plays an important role. Roughly speaking, ‘the bigger’ the graininess will be, ‘the harder’ it will be to construct an example of equation () with a bounded solution. This follows from the formulas ()and(), where the function fis between -derivatives of two functions, and these derivatives decrease to zero if μ(t) goes to infinity. However, in the cases (as is, e.g., the example above) the graininess satisfying μ(t)=O(t)is‘sufficiently big’ and covers every well-known case of time scales (e.g.,T=R,T=Z,T=hZ,h>,and T=qN,q>). Competing interests The authors declare that they have no competing interests. Authors’ contributions The authors have made the same contribution. All authors read and approved the final manuscript. Acknowledgements The research was supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague), by ‘Operational Programme Research and Development for Innovations’, No. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS, and by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. Received: 21 July 2012 Accepted: 9 October 2012 Published: 24 October 2012 References 1. Hilger, S: Analysis on measure chains - a unified approach to continuous and discrete calculus. Results Math. 18, 18-56 (1990) 2. 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PWN, Warsaw (1967) doi:10.1186/1687-1847-2012-183 Cite this article as: Diblík and Vítovec: Bounded solutions of delay dynamic equations on time scales. Advances in Difference Equations 2012 2012:183.