Rolle's and Generalized Mean Value Theorems on ...

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Rolle's and Generalized Mean Value Theorems on Time Scales S. Gulsan Topal

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Department of Mathematics , Ege University , Bornova, Izmir, 35100, Turkey Published online: 17 Sep 2010.

To cite this article: S. Gulsan Topal (2002) Rolle's and Generalized Mean Value Theorems on Time Scales, Journal of Difference Equations and Applications, 8:4, 333-344, DOI: 10.1080/102619029001 To link to this article: http://dx.doi.org/10.1080/102619029001

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Journal of Difference Equations and Applications, 2002 Vol. 8 (4), pp. 333–343

Rolle’s and Generalized Mean Value Theorems on Time Scales S. GULSAN TOPAL Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey (Received 19 September 2001; Revised 19 September 2001; In final form 19 September 2001)

This Paper is Dedicated to Lynn Erbe In this paper, some basic rules of calculus on R are presented for arbitrary time scales. This allows us to prove Rolle’s and mean value theorems on a time scale. Keywords: Time scales; Delta and nabla derivatives; Rolle’s theorem; Mean value theorem AMS Subject Classification: 39A 10; 39A 11

INTRODUCTION We are concerned with proving Generalized Lagrange Mean Value Theorem on time scales. To understand the calculus on time scales, we need some preliminary definitions. Definition 1.1 Let T be a nonempty closed subset of real numbers R and define the forward jump operator s(t ) at t for t , sup T by

sðtÞ U inf{s . t : s [ T} ISSN 1023-6198 q 2002 Taylor & Francis Ltd DOI: 10.1080/10236190290017397

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and the backward jump operator rðtÞ at t for t . inf T by

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rðtÞ U sup{s , t : s [ T} for all t [ T: We assume throughout that T has the topology that it inherits from the standard topology on the real numbers R. If sðtÞ . t; we say t is right scattered, while if rðtÞ , t; we say t is left scattered. If sðtÞ ¼ t; we say t is right dense, while if rðtÞ ¼ t; we say t is left dense. We are concerned with calculus on time scales which is a unified approach to continuous and discrete calculus. In [5,11], Aulbach and Hilger have initiated the development of this calculus. Other papers in this area include Agarwal and Bohner [1], Erbe and Hilger [7], Erbe and Peterson [8 – 10]. The books of Kaymakc¸alan, Laksmikantham and Sivasundaram, [12] and Bohner and Peterson [6] are good references. The plan of this paper is as follows: the second section is devoted to a presentation of the basic results concerning the D and 7 derivatives of functions. In the third section, we state and give outline for the proof of Hilger’s Mean Value Theorem for pre 7 differentiable functions. The fourth section contains our main results which are Rolle’s and Generalized Mean Value Theorems on time scales and some examples.

CALCULUS ON TIME SCALES We introduce the sets Tk, Tk and T* which are derived from the time scale T as follows. If T has a left scattered maximum t1, then Tk ¼ T={t1 }; otherwise Tk ¼ T: If T has a right scattered minimum t2, then Tk ¼ T={t2 }; otherwise Tk ¼ T: Finally, T* ¼ Tk > Tk : If f : T ! C is a function and t [ T; then we define f D ðtÞ to be the real number (provided it exists) with the property that given any 1 . 0; there is a neighborhood U of t such that j f ðsðtÞÞ 2 f ðsÞ 2 f D ðtÞ½sðtÞ 2 sŠj # 1jsðtÞ 2 sj;

for all s [ U:

If t [ Tk ; then we define f 7 ðtÞ to be the real number (provided it exists) with the property that given any 1 . 0; there is a neighborhood U of t such that j f ðrðtÞÞ 2 f ðsÞ 2 f 7 ðtÞ½rðtÞ 2 sŠj # 1jrðtÞ 2 sj; The following theorems can be found in Refs. [3,5].

for all s [ U:

THEOREMS ON TIME SCALES

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Theorem 2.1

335

For f: T ! C and t [ Tk the following hold:

i) If f is D differentiable at t, the f is continuous at t. ii) If f is continuous at t and t is right scattered, then f is D differentiable at t and f D ðtÞ ¼

f ðsðtÞÞ 2 f ðtÞ : sðtÞ 2 t

iii) If t is right dense, then f is D differentiable at t if and only if the limit lim s!t

f ðtÞ 2 f ðsÞ t2s

exists as a finite number. In this case f D ðtÞ is equal to this limit. iv) If f is D differentiable at t, then f ðsðtÞÞ ¼ f ðtÞ þ ðsðtÞ 2 tÞf D ðtÞ:

Theorem 2.2

For f: T ! C and t [ Tk the following hold:

i) If f is 7 differentiable at t, then f is continuous at t. ii) If f is continuous at t and t is left scattered, then f is 7 differentiable at t and f ðrðtÞÞ 2 f ðtÞ : f 7 ðtÞ ¼ rðtÞ 2 t iii) If t is left dense, then f is 7 differentiable at t if and only if the limit lim s!t

f ðtÞ 2 f ðsÞ : t2s

exist as a finite number. In this case f 7 ðtÞ is equal to this limit. iv) If f is 7 differentiable at t, then f ðrðtÞÞ ¼ f ðtÞ þ ðrðtÞ 2 tÞ f 7 ðtÞ: Throughout this paper, we make the assumption that a and b are points in T with a # b: We then define interval [a,b ] in T by ½a; bŠ U {t [ T : a # t # b}: Open intervals and half open intervals etc. are defined accordingly.

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Our understanding of monotone increasing (decreasing) function on T is as on R. For the next three propositions, we assume the existence of D and 7 derivatives of the function f on T. Suppose f is monotone increasing on [a,b ].

Proposition 2.1 Then

ðiÞ f D ðtÞ $ 0; ðiiÞ f 7 ðtÞ $ 0;

for all t [ ½a; bÞ: for all t [ ða; bŠ:

Proof (i) Let t0 be in ½a; bÞ: Here we have two cases to consider. Case 1: If t0 is right scattered, then f D ðt0 Þ ¼

f ðsðt0 ÞÞ 2 f ðt0 Þ $ 0; sðt0 Þ 2 t0

since f ðsðt0 ÞÞ $ f ðt0 Þ and sðt0 Þ . t0 :

Case 2: If t0 is right dense, then lim f D ðt0 Þ ¼ s!t 0

f ðt0 Þ 2 f ðsÞ $ 0; t0 2 s

since f ðsÞ $ f ðt0 Þ when s . t0 :

This completes the proof of part (i). The proof of part (ii) is similar. A Proposition 2.2 Then,

Suppose f is monotone decreasing on [a,b]. f D ðtÞ # 0; f 7 ðtÞ # 0;

Proof

for all t [ ½a; bÞ: for all t [ ða; bŠ:

It can be proven similarly as in the proof of Proposition 2.1.

Proposition 2.3 Then,

Suppose f has local extremum at t0 [ T: f D ðt0 Þf 7 ðt0 Þ # 0:

Proof Suppose t0 is local maximum point of the function f(t ). Here we have two observations: i) If t0 is right dense, then f D ðt0 Þ # 0: In fact, f D ðt0 Þ ¼ limþ s!t0

f ðt0 Þ 2 f ðsÞ # 0; t0 2 s

since f ðt0 Þ $ f ðsÞ and t0 , s:

A

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ii) If t0 is left dense, then f 7 ðt0 Þ $ 0: In fact, f 7 ðt0 Þ ¼ lim2 s!t0

f ðt0 Þ 2 f ðsÞ $ 0; t0 2 s

since f ðt0 Þ $ f ðsÞ when t0 . s:

To complete the proof, there are four cases to consider. Case 1: If t0 is right dense and left scattered, then f D ðt0 Þf 7 ðt0 Þ # 0; since f D ðt0 Þ # 0 from observation (i) and f 7 ðt0 Þ ¼

f ðrðt0 ÞÞ 2 f ðt0 Þ $ 0: rðt0 Þ 2 t0

Case 2: If t0 is right scattered and left dense, then f D ðt0 Þf 7 ðt0 Þ # 0; since f D ðt0 Þ ¼

f ðsðt0 ÞÞ 2 f ðt0 Þ #0 sðt0 Þ 2 t0

and f 7 ðt0 Þ $ 0 from observation (ii). Case 3: If t0 is right scattered and left scattered, then the result f D ðt0 Þf 7 ðt0 Þ # 0 follows from the definitions of f D ðt0 Þ and f 7 ðt0 Þ: Case 4: If t0 is left dense and right dense, then the result f D ðt0 Þf 7 ðt0 Þ # 0 follows from the above observations (i) and (ii).

A

HILGER’S MEAN VALUE THEOREM Pre D Differentiability [7]. The mapping f : T ! R is called pre D differentiable on D # T; provided that for the pair ( f, D ) the following are satisfied: i) f is continuous on T and at each t [ D; it is D differentiable,

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ii) for arbitrary t1 ; t2 [ T; t1 , t2 ; the set ½t1 ; t2 Š=D consists of countably many right dense points. Theorem 3.1 [7] Let the mappings f : T ! R N ; g : T ! R be pre D differentiable on D and for t1 ; t2 [ T; t1 , t2 let 0 # j f D ðtÞj # g D ðtÞ;

k

t [ D > jt1 ; t2 j

be valid. Then it follows that 0 # j f ðt2 Þ 2 f ðt1 Þj # gðt2 Þ 2 gðt1 Þ: Now we state the Hilger’s Mean Value Theorem with 7 derivative. For this purpose, we define pre 7 derivative. Definition 3.1 The mapping f : T ! R is called pre 7 differentiable on D # T; provided that for the pair ( f, D ) the following are satisfied: i) f is continuous on T and at each t [ D; it is 7 differentiable, ii) for arbitrary t1 ; t2 [ T; t1 , t2 ; the set ½t1 ; t2 Š=D consists of countably many left dense points. Theorem 3.2 Let the mappings f : T ! R N ; g : T ! R be pre 7 differentiable on D and for t1 ; t2 [ T; t1 , t2 let 0 # j f 7 ðtÞj # g 7 ðtÞ;

t [ D > ½t1 ; t2 Šk

be valid. Then it follows that 0 # j f ðt2 Þ 2 f ðt1 Þj # gðt2 Þ 2 gðt1 Þ: We sketch the proof of the theorem. Proof Let 1 . 0 be given. Mathematical induction principle in [t1,t2] for the statement " # X 22n AðtÞ : j f ðt2 Þ 2 f ðtÞj # gðt2 Þ 2 gðtÞ þ 1 t2 2 t þ t,sn

works backward.

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The validity of

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" Aðt1 Þ : j f ðt2 Þ 2 f ðt1 Þj # gðt2 Þ 2 gðt1 Þ þ 1 t2 2 t1 þ

X

# 2

2n

t1 ,sn

for arbitrary 1 . 0; follows from the statement of the theorem. One can see that the following four conditions of mathematical induction method in [t1,t2] are satisfied. (I): The statement A(t2) is trivially satisfied. (II): Let t [ D be left scattered and A(t ) is true, then A(r(t )) is also true. (III): Let t be left dense and not minimal. Then rðtÞ ¼ t ¼ t1 Case 1: t [ D > ðt1 ; t2 Š Case 2: t [ ðt1 ; t2 Š=D There exists a neighborhood U such that whenever A(t ) is true, A(r ) is also true for all r [ U; r # t: (IV): Let t be right dense point. A(r ) is true for all r . t implies A(t ) is true. A

Corollary 3.1 [7] If f : T ! R N is D differentiable on D and f D ðtÞ # M for all t [ D; then for t1 ; t2 [ T; j f ðt2 2 f ðt1 Þj # Mjt2 2 t1 j: Corollary 3.2 [7] If f : T ! R N is D differentiable on D and f D ðtÞ0 for all t [ D; then f(t ) is constant. Corollary 3.3 [7] If f : T ! R N is D differentiable on D and f D ðtÞ $ 0 for all t [ D; then f(t ) is monotone increasing. Similar results can be drawn for pre 7 differentiable functions. Corollary 3.4 If f : T ! R N is 7 differentiable on D and j f 7 ðtÞj # M for all t [ D; then for t1 ; t2 [ T; j f ðt2 Þ 2 f ðt1 Þj # Mjt2 2 t1 j: Corollary 3.5 If f : T ! R N is 7 differentiable on D and f 7 ðtÞ ¼ 0 for all t [ D; then f(t ) is constant.

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Corollary 3.6 If f : T ! R N is 7 differentiable on D and f 7 ðtÞ $ 0 for all t [ D; then f(t ) is monotone increasing.

MAIN RESULTS The Mean Value Theorem, which relates the values of a function to values of its derivative, is one of the most useful results in Real Analysis. One version of this theorem on time scales has been studied by Hilger [11]. Discrete Mean Value result is given in [2]. The other related result can be found in [4]. Here we state and prove Rolle’s Theorem on time scale T. Theorem 4.1 Suppose f is continuous on [a,b] and it has D and 7 derivatives on ½a; bŠ > T* : If f ðaÞ ¼ f ðbÞ; then there is a t0 in ½a; bŠ > T* such that f D ðt0 Þf 7 ðt0 Þ # 0: Proof Since f ðaÞ ¼ f ðbÞ; we say that either f(t ) is constant or there is a t0 in ½a; bŠ > T* such that f has local extremum at this point. Case 1: If f(t ) is constant, then f D ðt0 Þ ¼ 0 and f 7 ðt0 Þ ¼ 0: Hence we have f D ðt0 Þf 7 ðt0 Þ ¼ 0: Case 2: If t0 is extremum point of f(t ), because of the Proposition 2.3, we have f D ðt0 Þf 7 ðt0 Þ # 0: Theorem 4.2 (Generalized Mean Value Theorem) Assume f(t ) and g(t ) are continuous on [a,b ], they have D and 7 derivatives on ½a; bŠ > T* ; and g D ðtÞg 7 ðtÞ . 0 for all ½a; bŠ > T* : Then there is a t0 [ ½a; bŠ > T* such that f D ðt0 Þ f ðbÞ 2 f ðaÞ f 7 ðt0 Þ # # g D ðt0 Þ gðbÞ 2 gðaÞ g 7 ðt0 Þ or f D ðt0 Þ f ðbÞ 2 f ðaÞ f 7 ðt0 Þ $ $ : g D ðt0 Þ gðbÞ 2 gðaÞ g 7 ðt0 Þ

THEOREMS ON TIME SCALES

Proof

341

Let h(t ) be a function which is defined on [a,b ] by

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hðtÞ ¼ f ðtÞ½gðbÞ 2 gðaފ 2 gðtÞ½f ðbÞ 2 f ðaފ: So hðaÞ ¼ hðbÞ ¼ f ðaÞgðbÞ 2 gðaÞf ðbÞ: By Rolle’s Theorem, there is a t0 [ ½a; bŠ > T* such that D h (t0)h 7(t0) # 0. Hence we say either h D ðt0 Þ # 0 and

h 7 ðt0 Þ $ 0

h D ðt0 Þ $ 0 and

h 7 ðt0 Þ # 0

or are satisfied. W.L.O.G. we assume that g D ðtÞ . 0; g 7 ðtÞ . 0; for t [ ½a; bŠ > T* : When the case h D ðt0 Þ # 0 and h 7 ðt0 Þ $ 0 happens, we have h D ðt0 Þ ¼ f D ðt0 Þ½gðbÞ 2 gðaފ 2 g D ðt0 Þ½f ðbÞ 2 f ðaފ # 0 and h 7 ðt0 Þ ¼ f 7 ðt0 Þ½gðbÞ 2 gðaފ 2 g 7 ðt0 Þ½f ðbÞ 2 f ðaފ $ 0: It follows that f D ðt0 Þ f ðbÞ 2 f ðaÞ f 7 ðt0 Þ # : # 7 D g ðt0 Þ gðbÞ 2 gðaÞ g ðt0 Þ The proof of the other case is similar. A Our result works for the class of functions where D and 7 derivatives are not the same on closed interval of T. Here, we give one example for several time scales. pffi Example 4.1 Let T be any time scale. The functions f ðtÞ ¼ tðt . 0Þ and gðtÞ ¼ t 2 are D and 7 differentiable. It can be shown that 1 f D ðtÞ ¼ pffi pffiffiffiffiffiffiffiffi ; t þ sðtÞ

g D ðtÞ ¼ sðtÞ þ t;

1 f 7 ðtÞ ¼ pffiffiffiffiffiffiffi pffi ; rðtÞ þ t

g 7 ðtÞ ¼ rðtÞ þ t

for every t [ Tk : for every t [ Tk :

Then we have Table I: If we take gðtÞ ¼ t in the Generalized Mean Value Theorem, we have the following result.

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TABLE I

T [a,b ] t0

R

Z

hZ

ð1:1ÞN < {0}

2N < {0}

N20

0 < {1/n}n[N < j3/2,2j

[0,2] pffiffi 2=3 3

[0,2] 1

[0,2h ] h

[0,1.1] 1=ð1:1Þ9

[0,2] 1

[0,22] 1

[0,2] 1

Corollary 4.1 Assume f(t ) is continuous on [a,b] and has D and 7 derivatives on ½a; bŠ > T* : Then there is a t0 [ ½a; bŠ > T* such that f D ðt0 Þ #

f ðbÞ 2 f ðaÞ # f 7 ðt0 Þ b2a

or f 7 ðt0 Þ #

f ðbÞ 2 f ðaÞ # f D ðt0 Þ: b2a

Here we state two theorems in [3] to derive next two corollaries. Theorem 4.3 [3] If f : T ! R is D differentiable on T k and if f D is continuous on T k, then f is 7 differentiable on Tk and f 7 ðtÞ ¼ f D ðrðtÞÞ

for all t [ Tk :

Theorem 4.4 [3] If f : T ! R is 7 differentiable on Tk and if f 7 is continuous on Tk, then f is D differentiable on T k and f D ðtÞ ¼ f 7 ðsðtÞÞ

for all t [ Tk :

Corollary 4.2 If f : T ! C is D differentiable on ½a; bŠ > Tk and f D ðtÞ ¼ 0 for all t [ ½a; bŠ > Tk ; then f is constant. Proof By theorem 4.3, we have that f is 7 differentiable on ½a; bŠ > Tk and f 7 ðtÞ ¼ 0: Let t1 [ ða; bŠ: Then, by Corollary 4.1, there exists t0 [ ½a; t1 Š > T* such that 0 ¼ f D ðt0 Þ #

f ðt1 Þ 2 f ðaÞ # f 7 ðt0 Þ ¼ 0: t1 2 a

Consequently, f ðt1 Þ ¼ f ðaÞ: Since t1 is arbitrary point in ða; bŠ; f is constant. A Corollary 4.3 If f : T ! R N is 7 differentiable on D and f 7 ðtÞ ¼ 0 for all t [ ½a; bŠ > Tk ; then f is constant.

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References [1] Agarwal, R. and Bohner, M. (1999) “Basic calculus on time scales and some of its applications”, Results Math. 35(1–2), 3–22. [2] Agarwal, R.P. (1992) Difference Equations and Inequalities (Marcel Dekker, New York). [3] Atici, F.M. and Guseinov, G.Sh. (2001) “On Green’s functions and positive solutions for boundary value problems on time scales”, J. Comput. Appl. Math., Special issue on “Dynamic Equations on Time Scales”, edited by Agarwal, R.P., Bohner, M. and O’Regan, D, 2002. [4] Atici, F.M., Guseinov, G.Sh. and Kaymakc¸alan, B. (2000) “On Lyapunov inequality in stability theory for Hill’s equation on time scales”, J. Inequal. Appl. 5, 603–620. [5] Aulbach, B. (1990) “Analysis auf Zeitmengen”, Lecture Notes (Universitat Augsburg, Augsburg). [6] Bohner, M. and Peterson, A. (2001) Dynamic Equations on Time Scales: An Introduction with Applications (Birkhauser, Boston). [7] Erbe, L. and Hilger, S. (1993) “Sturmian theory on measure chains”, Differ. Eqs Dynam. Syst. 1(3), 223–244. [8] Erbe, L. and Peterson, A. (1999) “Green’s functions and comparison theorems for differential equations on measure chains”, Dynam. Contin. Discrete Impuls. Syst. 6, 121–137. [9] Erbe, L. and Peterson, A. (2000) “Eigenvalue conditions and positive solutions”, J. Differ. Eqs Appl. 6, 165 –191. [10] Erbe, L. and Peterson, A. (2000) “Positive solutions for a nonlinear differential equation on a measure chain”, Math. Comput. Model. 32, 571–585. [11] Hilger, S. (1990) “Analysis on measure chains, a unified approach to continuous and discrete calculus”, Res. Math. 18, 18 –56. [12] Kaymakc¸alan, B., Laksmikantham, V. and Sivasundaram, S. (1996) Dynamical Systems on Measure Chains (Kluwer Academic Publishers, Boston).