aa r X i v : . [ m a t h . C A ] F e b AMBARZUMYAN TYPE THEOREMS ON A TIME SCALE
A. SINAN OZKAN
Abstract.
In this paper, we consider a Sturm–Liouville dynamic equationwith Robin boundary conditions on time scale and investigate the conditionswhich guarantee that the potential function is specified. Introduction
Time scale theory was introduced by Hilger in order to unify continuous anddiscrete analysis [16]. From then on this approach has received a lot of attentionand has applied quickly to various area in mathematics. Sturm–Liouville theoryon time scales was studied first by Erbe and Hilger [11] in 1993. Some importantresults on the properties of eigenvalues and eigenfunctions of a Sturm–Liouvilleproblem on time scales were given in various publications (see e.g. [2], [3], [4],[8]-[10], [12], [14]-[22] and the references therein).Inverse spectral problems consist in recovering the coefficients of an operatorfrom their spectral characteristics. Althouhgh there are vast literature for inverseSturm–Liouville problems on a continuous interval, there are no study on the gen-eral time scales. For Sturm-Liouville operator on a continuous interval, the studywhich starts inverse spectral theory, was published by Ambarzumyan [1] in 1929. Heprove that: if q is continuous function on (0 ,
1) and the eigenvalues of the problem − y ′′ + q ( t ) y = λy, t ∈ (0 , y ′ (0) = y ′ (1) = 0are given as λ n = n π , n ≥ q ≡ . Freiling and Yurko [13] generalized this result as λ = R q ( t ) dt implies q ≡ λ . The goal of this paper to prove an Ambarzumyan type theorem on a generaltime scale and to apply it on the a special time scale. In our main result, Theorem1, we generalize the results of Freiling and Yurko for Sturm-Liouville operator withmore general boundary conditions on a time scale.2.
Preliminaries and Main Results If T is a closed subset of R it called as a time scale. The jump operators σ , ρ and graininess operator on T are defined as follow: σ : T → T , σ ( t ) = inf { s ∈ T : s > t } if t = sup T , ρ : T → T , ρ ( t ) = sup { s ∈ T : s < t } if t = inf T , σ (sup T ) = sup T , ρ (inf T ) = inf T , µ : T → [0 , ∞ ) µ ( t ) = σ ( t ) − t . Mathematics Subject Classification.
Key words and phrases.
Ambarzumyan Theorem, Time scale, Sturm-Liouville equation; in-verse problem; dynamic equations.
A point of T is called as left-dense, left-scattered, right-dense, right-scattered andisolated if ρ ( t ) = t , ρ ( t ) < t , σ ( t ) = t , σ ( t ) > t and ρ ( t ) < t < σ ( t ) , respectively.A function f : T → R is called rd-continuous on T if it is continuous at allright-dense points and has left-sided limits at all left-dense points in T . The set ofrd-continuous functions on T is denoted by C rd ( T ) or C rd .Put T k := (cid:26) T − { sup T } , sup T is left-scattered T , the other cases , T k := (cid:0) T k (cid:1) k . Let t ∈ T k . Suppose that for given any ε > , there exist a neighborhood U = ( t − δ, t + δ ) ∩ T such that (cid:12)(cid:12) [ f ( σ ( t )) − f ( s )] − f ∆ ( t ) [ σ ( t ) − s ] (cid:12)(cid:12) ≤ ε | σ ( t ) − s | for all s ∈ U then, f is called differentiable at t ∈ T k . We call f ∆ ( t ) the deltaderivative of f at t. A function F : T → R defined as F ∆ ( t ) = f ( t ) for all t ∈ T k iscalled an antiderivative of f on T . In this case the Cauchy integral of f is definedby b Z a f ( t ) △ t = F ( b ) − F ( a ) , for a, b ∈ T .Some important relations whose proofs appear in [6], chapter1 will be needed. Wecollect them in the following lemma. Lemma 1.
Let f : T → R , g : T → R be two functions and t ∈ T k . i) lf f ∆ ( t ) exists, then f is continuous at t ;ii) if t is right-scattered and f is continuous at t , then f is differentiable at t and f ∆ ( t ) = f σ ( t ) − f ( t ) σ ( t ) − t , where f σ ( t ) = f ( σ ( t )) ; iii) if f ∆ ( t ) exists, then f σ ( t ) = f ( t ) + µ ( t ) f ∆ ( t ) ; iv) if f ∆ ( t ) , g ∆ ( t ) exist and ( f g )( t ) is defined, then ( f g ) ∆ ( t ) = (cid:0) f ∆ g + f σ g ∆ (cid:1) ( t ) and if ( gg σ ) ( t ) = 0 , then (cid:18) fg (cid:19) ∆ ( t ) = (cid:18) f ∆ g − f g ∆ gg σ (cid:19) ( t ) ;v) if f ∈ C rd ( T ) , then it has an antiderivative on T ;vi) if T consists of only isolated points and a, b ∈ T with a < b, then b R a f ( t ) △ t = P t ∈ [ a,b ) ∩ T µ ( t ) f ( t ); vii) if f ( t ) ≥ for all t ∈ [ a, b ] ∩ T and b R a f ( t ) △ t = 0 , then f ( t ) ≡ . Throughout this paper we assume that T is a bounded time scale, a = inf T and b = sup T . Consider the boundary value problem L = L ( q, h a , h b ) generated by theSturm–Liouville dynamic equation(1) ℓy := − y ∆∆ ( t ) + q ( t ) y σ ( t ) = λy σ ( t ) , t ∈ T k subject to the boundary conditions y ∆ ( a ) − h a y ( a ) = 0(2) y ∆ ( ρ ( b )) − h b y ( ρ ( b )) = 0(3)where q ( t ) is real valued continiuous function on T , h a , h b ∈ R and λ is the spec-tral parameter. Additionally, we assume that a = ρ ( b ), 1 + h a µ ( a ) = 0 and1 + h b µ ( ρ ( b )) = 0 . Definition 1.
The values of the parameter for which the equation (1) has nonzerosolutions satisfy (2) and (3), are called eigenvalues and the corresponding nontrivialsolutions are called eigenfunctions.
MBARTSUMIAN TYPE THEOREMS 3
It is proven in [6] that all eigenvalues of the problem (1)-(3) are real numbers.
Definition 2.
A solution y of (1) is said to have a zero at t ∈ T if y ( t ) = 0 , andit has a node between t and σ ( t ) if y ( t ) y ( σ ( t )) < . A generalized zero of y is thendefined as a zero or a node. Lemma 2 ([2]) . The eigenvalues of (1)-(3) may be arranged as −∞ < λ < λ <λ < ... and an eigenfunction corresponding to λ k +1 has exactly k generalized zerosin the open interval ( a, b ) . Lemma 3. If y ( t ) is an eigenfunction of the problem (1)-(3) then y σ ( a ) = 0 and y σ ( ρ ( b )) = 0 . Proof.
It is clear from Lemma 1 that y σ ( a ) = y ( a ) + µ ( a ) y ∆ ( a ) = y ( a ) [1 + h a µ ( a )]and y σ ( ρ ( b )) = y ( ρ ( b )) + µ ( ρ ( b )) y ∆ ( ρ ( b )) = y ( ρ ( b )) [1 + h b µ ( ρ ( b ))]. We claim that y ( a ) = 0 and y ( ρ ( b )) = 0. Otherwise, from (2) and (3) y ∆ ( a ) = 0 or y ∆ ( ρ ( b )) = 0hold, then by the uniqueness theorem of the solution of initial value problems y ( t )is identically vanish which contradicts that it is the eigenfunction. Therefore theproof is completed from the assumption 1 + h a µ ( a ) = 0 , h a µ ( ρ ( b )) = 0 . (cid:3) Theorem 1.
Let λ be the first eigenvalue of (1)-(3). If λ ≥ ρ ( b ) − a { h a − h b + ρ ( b ) Z a q ( t )∆ t } , then q ( t ) ≡ λ . Proof.
Let y ( t ) be the corresponding eigenfunction to λ . From eq(1) and Lemma2 we can write on T k y ∆∆1 ( t ) y σ ( t ) = q ( t ) − λ . It is from the relation y ∆∆1 ( t ) y σ ( t ) = (cid:2) y ∆1 ( t ) (cid:3) y σ ( t ) y ( t ) + (cid:20) y ∆1 ( t ) y ( t ) (cid:21) ∆ that (cid:20) y ∆1 ( t ) y ( t ) (cid:21) ∆ = q ( t ) − λ − (cid:2) y ∆1 ( t ) (cid:3) y σ ( t ) y ( t ) . From Lemma 3 we can integration of both sides from a to ρ ( b ). Therefore thefollowing equality is obtained ρ ( b ) Z a (cid:2) y ∆1 ( t ) (cid:3) y σ ( t ) y ( t ) ∆ t = y ∆1 ( a ) y ( a ) − y ∆1 ( ρ ( b )) y ( ρ ( b )) + ρ ( b ) Z a [ q ( t ) − λ ] ∆ t = h a − h b + ρ ( b ) Z a q ( t )∆ t − λ ( ρ ( b ) − a ) . It can be seen from Lemma 2 and our hypothesis that the right side of the lastequality is negative and the left side is non-negative. Thus y ∆1 ( t ) ≡ y ( t )is constant. Substituting y ( t ) is constant into equation (1), it is concluded that q ( t ) ≡ λ . (cid:3) A. SINAN OZKAN
Corollary 1.
The first eigenvalue of the problem − y ∆∆ + q ( t ) y σ = λy σ , y ∆ ( a ) = y ∆ ( ρ ( b )) = 0 is λ = ρ ( b ) − a ρ ( b ) R a q ( t )∆ t then, q ( t ) ≡ λ . This corollary is a generalization of the results of Freiling and Yurko [13] ontothe time scale.
Corollary 2.
Under the hypothesis ρ ( b ) R a q ( t )∆ t = 0 ; if q ( t ) = 0 , then the problem − y ∆∆ + q ( t ) y σ = λy σ ,y ∆ ( a ) = y ∆ ( ρ ( b )) = 0 has at least one negative eigenvalue. We conclude this paper with specializing our first result for a particular timescale which consists of only isolated points.
Remark 1.
Consider the time scale T = { x k ∈ R : a = x < x < x < ... < x n = b } and the following problem − y ∆∆ + q ( t ) y σ = λy σ ,y ∆ ( a ) = y ∆ ( ρ ( b )) = 0 . If the first eigenvalue of the problem satisfy λ ≥ max { q ( t ) : t ∈ T } , then q ( t ) ≡ λ . References [1] V.A. Ambarzumyan, ¨Uber eine Frage der Eigenwerttheorie, Z. Phys. 53 (1929), pp. 690–695.[2] R.P. Agarwal, M. Bohner, and P.J.Y. Wong, Sturm-Liouville eigenvalue problems on timescales, Appl. Math. Comput. 99 (1999), pp. 153–166.[3] P. Amster, P. De Na´poli, and J.P. Pinasco, Eigenvalue distribution of second-order dynamicequations on time scales considered as fractals, J. Math. Anal. Appl. 343 (2008), pp. 573–584.[4] P. Amster, P. De Na´poli, and J.P. Pinasco, Detailed asymptotic of eigenvalues on timescales, J. Differ. Equ. Appl. 15 (2009), pp. 225–231.[5] F. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York, 1964.[6] M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkha¨user, Boston, MA,2001.[7] M. Bohner and A. Peterson (eds.), Advances in Dynamic Equations on Time Scales,Birkha¨user, Boston, MA, 2003.[8] F.A. Davidson and B.P. Rynne, Global bifurcation on time scales, J. Math. Anal. Appl. 267(2002), pp. 345–360.[9] F.A. Davidson and B.P. Rynne, Self-adjoint boundary value problems on time scales, Elec-tron. J. Differ. Equ. 2007(175) (2007), pp. 1–10.[10] F.A. Davidson and B.P. Rynne, Eigenfunction expansions in L2 spaces for boundary valueproblems on time-scales, J. Math. Anal. Appl. 335 (2007), pp. 1038–1051.[11] L. Erbe and S. Hilger, Sturmian theory on measure chains, Differ. Equ. Dyn. Syst. 1 (1993),pp. 223–244.[12] L. Erbe and A. Peterson, Eigenvalue conditions and positive solutions, J. Differ. Equ. Appl.6 (2000), pp. 165–191.[13] G. Freiling and V.A. Yutko, Inverse Sturm–Liouville Problems and Their Applications, NovaScience, Huntington, NY, 2001.[14] G.S. Guseinov, Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int.J. Differ. Equ. 2 (2007), pp. 93–104.[15] G.S. Guseinov, An expansion theorem for a Sturm-Liouville operator on semi-unboundedtime scales, Adv. Dyn. Syst. Appl. 3 (2008), pp. 147–160.[16] S. Hilger, Analysis on measure chains – a unified approach to continuous and discrete calculus,Results in Math. 18 (1990), 18–56.[17] A. Huseynov, Limit point and limit circle cases for dynamic equations on time scales, Hacet.J. Math. Stat. 39 (2010), pp. 379–392.
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