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Function spaces and their dual spaces on time scales.

Abstract

In this paper we introduce some function spaces on time scales along with their dual spaces and obtain some relations between them. As an application, the infinite matrix transformation on times scales is studied.

AMS subject classification: 40C05, 46A20, 46E15.

Keywords: Time scales, function spaces, duals, matrix transformations.

1. Introduction

Calculus on time scales has been introduced by Bernd Aulbach and Stefan Hilger [1, 9] to unify discrete and continuous analysis. A time scale T is a nonempty closed subset of the real numbers, so that it is a complete metric space with the metric d(t, s) = |t - s|. The books by Bohner and Peterson [4, 5] are excellent references for calculus on time scales.

In the paper by Kizmaz [10], the following sequence spaces are defined:

[l.sub.[infinity]]([DELTA]) = {x = ([x.sub.k]) : [DELTA]x [member of] [l.sub.[infinity]]}, c([DELTA]) = {x = ([x.sub.k]) : [DELTA]x [member of] c}, [c.sub.0]([DELTA]) = {x = ([x.sub.k]) : [DELTA]x [member of] [c.sub.0]},

where [DELTA][x.sub.k] = [x.sub.k+1] - [x.sub.k], k [member of] [Z.sup.+]. These sequence spaces are Banach spaces with norm

[parallel]x[parallel] = [absolute value of [x.sub.1]] + [parallel][DELTA]x[[parallel].sub.[infinity]],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also some properties of these spaces and of their [alpha], [beta], [gamma] dual spaces have been given. Finally, matrix classes related to these sequence spaces were studied. Later Et [6], Et and Colak [7] generalized the results of [10] considering the n-th power of the operator [DELTA].

In [2], the results on dual spaces in which methods of [10] were adapted, had been devoted to the function space

[L.sub.[infinity]](D) = {f | f : [0,[infinity]) [right arrow] R, Df = f' [member of] [L.sub.[infinity]]}.

This function space is a Banach space with norm

[parallel]f[parallel] = [absolute value of f (0)] + [parallel]f'[[parallel].sub.[infinity]],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper our aim is to unify and generalize the above mentioned results by defining the function spaces and their dual spaces on time scales.

The paper is organized as follows. In Section 2, we give some basic concepts of the time scale calculus and also introduce some function spaces and define [alpha], [beta], [gamma] -dual spaces. In Section 3, we obtain a relation between the [alpha]-dual of some function spaces and construct the [alpha][alpha]-dual of [L.sub.[infinity]]([DELTA]). In Section 4, we study infinite matrix transformations on special time scales as an application of the results in the earlier sections.

2. Preliminaries

First, we shall briefly mention some basic definitions of time scale calculus for the reader's convenience. For t [member of] T we define the forward jump operator [sigma] : T [right arrow] T by

[sigma](t) = inf {s [member of] T : s > t}.

If [sigma](t) > t, we say that t is right-scattered, and if [sigma](t) = t, then t is called right-dense. The graininess [mu] : T [right arrow] [0,[infinity]) is defined by

[mu](t) := [sigma](t) - t.

For a, b [member of] T with a [less than or equal to] b we define the closed interval [a, b] in T by [a, b] = {t [member of] T : a [less than or equal to] t [less than or equal to] b}. The set [T.sup.[kappa]] is defined to be T\{[t.sub.0]} if T has a left-scattered maximum [t.sub.0], otherwise T = [T.sub.[kappa]].

Now, let f be a function defined on T and let t [member of] [T.sup.[kappa]]. Then we define [f.sup.[DELTA}](t) to be the number (provided it exists) with the property that given any [epsilon] > 0, there is a neighborhood U of t (i.e., U = (t - [delta], t + [delta]) [intersection] T for some [delta] > 0) such that

[absolute value of [f ([sigma](t)) - f (s)] - [f.sup.[DELTA]](t)[[sigma](t) - s]] [less than or equal to] [epsilon][absolute value of [sigma](t) - s]

for all s [member of] U. We call [f.sup.[DELTA]] (t) the delta derivative of f at t. Moreover, we say that f is delta differentiable on [T.sup.[kappa]] provided [f.sup.[DELTA]](t) exists for all t [member of] [T.sup.[kappa]].

Note that in the case T = R we have [f.sup.[DELTA]] (t) = f'(t) and in the case T = Z we have [f.sup.[DELTA]](t) = f (t + 1) - f (t).

Here, F is called an antiderivative of a function f defined on T if [F.sup.[DELTA]] = f holds on [T.sup.[kappa]]. In this case we define a Cauchy integral by

[[integral].sup.t.sub.s] f ([tau])[DELTA][tau] = F(t) - F(s),

where s, t [member of] T.

Throughout this paper, we assume that T is unbounded above. Now let us suppose that a real-valued function f is defined on [a,[infinity]) = {t [member of] T : t [greater than or equal to] a} and is integrable from a to any point A [member of] T with A [greater than or equal to] a. If the integral

F(A) = [[integral].sup.A.sub.a] f (t)[DELTA]t

approaches a finite limit as A [right arrow] [infinity], then we call that limit the improper integral of first kind of f from a to[infinity]and write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In the papers by Guseinov [8] and by Bohner and Guseinov [3], many properties of the [DELTA]-integral on time scales are given.

Let T be a time scale such that T [subset] [0,[infinity]) and there exists a subset {[t.sub.k] : k [member of] [N.sub.0]} [subset] T with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We define the spaces of continuous functions

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

One can easily see that these are normed linear spaces and [C.sub.0] [subset] C [subset] [L.sub.[infinity]]. Next we define

[L.sub.[infinity]]([DELTA]) = {f|f [member of] K, [f.sup.[DELTA]] [member of] [L.sub.[infinity]]}, C([DELTA]) = {f|f [member of] K, [f.sup.[DELTA]] [member of] C}, [C.sub.0]([DELTA]) = {f|f [member of] K, [f.sup.[DELTA]] [member of] [C.sub.0]},

where K = {f|f : T [right arrow] R and f is [DELTA]-differentiable on [T.sup.[kappa]]}. It is also easy to show that these function spaces are Banach spaces with the norm

[parallel]f[[parallel].sub.[DELTA]] = [absolute value of f (0)] + [parallel][f.sup.[DELTA]][[parallel].sub.[infinity]]

and [C.sub.0]([DELTA]) [subset] C([DELTA]) [subset] [L.sub.[infinity]]([DELTA]).

Next we define the operator

[phi] : [L.sub.[infinity]]([DELTA]) [right arrow] [L.sub.[infinity]]([DELTA])

with [phi](f (t)) = f(t) - f(0). It is clear that [phi] is a bounded linear operator on [L.sub.[infinity]]([DELTA]). Also [phi][[L.sub.[infinity]]([DELTA])] = {g [member of] [L.sub.[infinity]]([DELTA]) : g(0) = 0} is a subspace of [L.sub.[infinity]]([DELTA]) and is a space with norm [parallel]f[[parallel].sub.[DELTA]] = [parallel][f.sup.[DELTA]] [[parallel].sub.[infinity]].

On the other hand we define the operator

D : [phi][L.sub.[infinity]]([DELTA]) [right arrow] [L.sub.[infinity]]

with D(f) = [f.sup.[DELTA]]. D is a linear isometry so that the spaces [phi][L.sub.[infinity]]([DELTA]) and [L.sub.[infinity]] are equivalent normed spaces.

Definition 2.1. Let X be a function space and a [member of] T. We define the dual spaces of F [subset] X in the following way:

(i) [F.sup.[alpha]] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] [absolute value of f (t)g(t)][DELTA]t < [infinity] for all g [member of] F},

(ii) [F.sup.[beta]] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] f(t)g(t)[DELTA]t is convergent for all g [member of] F},

(iii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[F.sub.[alpha]], [F.sub.[beta]] and [F.sub.[gamma]] are called [alpha], [beta] and [gamma]-dual spaces of F, respectively.

The proof of the following theorem easily follows from the above definition.

Theorem 2.2. Let F and G be function spaces. Then

(i) [F.sup.[alpha]] [subset or equal to] [F.sup.[beta]] [subset or equal to] [F.sup.[gamma]],

(ii) F [subset or equal to] G implies [G.sup.*] [subset or equal to] [F.sup.*], * = [alpha], [beta], [gamma].

3. Main Results

Lemma 3.1. If f [member of] [phi][L.sub.[infinity]]([DELTA]), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. If f [member of] [phi] [L.sub.[infinity]]([DELTA]), then [absolute value of [f.sup.[DELTA]](t)] [less than or equal to] M for all t [member of] T \ {0}, where M is a positive constant. Let t [member of] T \ {0}. Then we use the properties of the [DELTA]-integral on time scales to reach the desired result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This completes the proof. []

Theorem 3.2. We have

[[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)] [DELTA]t < [infinity], [[integral].sub.T]t [absolute value of f (t)][DELTA]t < [infinity]}.

Proof. Define

[D.sub.1] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] t [absolute value of f(t)][DELTA]t < [infinity]}.

One can easily see that [[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]] [subset] [D.sub.1]. If f [member of] [D.sub.1], then for an [t.sub.0] [member of] T \ {0},

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all g [member of] [phi][L.sub.[infinity]]([DELTA]). This implies that f [member of] [[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]]. []

Remark 3.3. It is easy to see that the set [D.sub.1] is nonempty since one can consider f as zero function. The Dirichlet-Abel test is a main tool for presenting a nontrivial example.

Theorem 3.4. (Dirichlet-Abel Test [3]) Let the following conditions be satisfied.

(i) f is integrable from a to any point A [member of] T with A [greater than or equal to] a, and the integral F(A) = [[integral].sup.A.sub.a] f(t)[DELTA]t is bounded for all A [greater than or equal to] a.

(ii) g is monotone on [a,[infinity]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the improper integral of first kind of the form [[infinity].sup.[infinity].sub.a] f(t)g(t)[DELTA]t is convergent.

Example 3.5. Let f (t) = t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2]([t.sup.2] + 1), t [member of] T. We now use the Dirichlet-Abel test to see the convergence of the integral

[[integral].sup.a.sub.0]|t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2]([t.sup.2] + 1)|[DELTA]t.

If we take [g.sub.1](t) = t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2], [h.sub.1](t) = 1/[t.sup.2] + 1, t [member of] T, then we have that

[[integral].sup.a.sub.0] [absolute value of [g.sub.1](t)][DELTA]t = -1/[(A + 1).sup.2] + 1

for all A [member of] T (A [greater than or equal to] 0) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that [h.sub.1] is monotone decreasing on [T.sup.[kappa]]. Hence

[[integral].sup.a.sub.0] [g.sub.1](t)[h.sub.1](t)[DELTA]t = [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity]

for all a [member of] T. To verify that the second condition is valid, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [s.sub.0] > 0, [s.sub.0] [member of] T. We take

[g.sub.2](t) = t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2], [h.sup.2](t) = t/[t.sup.2] + 1, t [member of] [[s.sub.0],[infinity]).

We have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is bounded for all A [member of] T (A [greater than or equal to] [s.sub.0]) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that [h.sub.2] is decreasing on [[s.sub.0],[infinity]). This implies

[[integral].sub.T] t [absolute value of f (t)][DELTA]t = [[integral].sub.T] t |t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2]([t.sup.2] + 1)|[DELTA]t < [infinity].

Hence we obtain f [member of] [D.sub.1].

Lemma 3.6. If f [member of] [L.sub.[infinity]]([DELTA]), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. If f [member of] [L.sub.[infinity]]([DELTA]), then [absolute value of [f.sup.[DELTA]](t)] [less than or equal to] N for all t [member of] T, where N is a positive constant. Let t [member of] T. Then we use the properties of the [DELTA]-integral on time scales to reach the desired result

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then it follows that

[absolute value of f(t)] [less than or equal to] Nt + [absolute value of f(0)] < A(t + 1), A = max {N, [absolute value of f(0)]}.

This completes the proof. []

Theorem 3.7. We have

[[[phi]C([DELTA])].sup.[alpha]] = [[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]].

Proof. Let f [member of] [[[phi]C([DELTA])].sup.[alpha]]. Then [[integral].sub.T] [absolute value of f(t)g(t)][DELTA]t is convergent for all g [member of] [phi]C([DELTA]). We can take g(t) = t, t [member of] T. Therefore f [member of] [[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]] by Theorem 3.2. It can easily seen that [[[phi][L.sup.[infinity]]([DELTA])].sup.[alpha]] [subset or equal to] [[[phi]C([DELTA])].sup.[alpha]] by Theorem 2.2. []

Theorem 3.8. We have

[[C([DELTA])].sup.[alpha]] = [[[L.sub.[infinity]]([DELTA])].sup.[alpha]].

Proof. Let f be an element of the space [[C([DELTA])].sup.[alpha]]. Then f [member of] [[[phi][L.sub.[infinity]]([DELTA])].sup.[alpha]] by Theorem 3.2. If we use Lemma 3.6, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all g [member of] [L.sub.[infinity]]([DELTA]). This implies that f [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha]]. The other side of the inclusion follows from Theorem 2.2. []

Corollary 3.9. Let F stand for [L.sub.[infinity]] or C. Then

[[F([DELTA])].sup.[alpha]] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] t [absolute value f(t)][DELTA]t < [infinity]}.

Theorem 3.10. Let F stand for [L.sub.[infinity]] or C. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. From Definition 2.1, we can write that

[[[L.sub.[infinity]]([DELTA])].sup.[alpha][alpha]] = {f : [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] [absolute value of f(t)g(t)][DELTA]t < [infinity] for all g [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha]]}.

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If f [member of] [D.sub.2], then for all g [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha]],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that f [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha][alpha]].

Now suppose that f [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha][alpha]] and f [not member of] [D.sub.2]. Then we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. So there is a strictly increasing sequence <[t.sub.n]> such that [t.sub.n] [member of] T, with 0 < [t.sub.1] < [t.sub.2] < [t.sub.3] < ... and

[t.sup.-1.sub.k] [absolute value of f ([t.sub.k])]/[t.sub.k+1] - [t.sub.k] > [([t.sub.k] + 1).sup.-1][absolute value of f ([t.sub.k])]/[t.sub.k+1] - [t.sub.k] > [k.sup.2].

Without loss of generality, f (t) [not equal to] 0, t [member of] <[t.sub.n]>. We define the function g by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to see [[integral].sup.a.sub.0] [absolute value of g(t)][DELTA]t < [infinity]. Hence g [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha]] and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is a contradiction to our assumption that f [member of] [[[L.sub.[infinity]]([DELTA])].sup.[alpha][alpha]]. Hence f [member of] [D.sub.2]. []

Remark 3.11. It is easy to see [D.sub.2] is nonempty. Let f (t) = t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2]. Then we have

[[integral].sup.a.sub.0] [absolute value of f(t)]{DELTA]t = [[integral].sup.a.sub.0] t + [sigma](t) + 2/[(t + 1).sup.2][([sigma] (t) + 1).sup.2] [DELTA]t = -1/[(a + 1).sup.2] + 1

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence f [member of] [D.sub.2].

4. Matrix Transformations

In this section, as an application, we shall study infinite matrix transformations on special time scales of the form T = {[t.sub.k] : [t.sub.1] = 0, [t.sub.k] < [t.sub.k+1], k [member of] N}. Let X and Y be function spaces defined on T. We denote the set of all infinite matrices from space X to space Y by (X, Y).

Let A = ([g.sub.n]([t.sub.k])[mu]([t.sub.k])) be an infinite matrix of real valued functions [g.sub.i](i [member of] N) which are continuous functions on T and [mu]([t.sub.k]) = [t.sub.k+1] - [t.sub.k]. Define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We write formally, provided the series converges for each n and [A.sub.n](f) [member of] Y whenever f [member of] X,

[A.sub.n](f) = [[infinity].summation over (k=1)] [g.sub.n]([t.sub.k])f([t.sub.k])[mu]([t.sub.k]), n [member of] N

such that

[DELTA][A.sub.n](f) = [[infinity].summation over (k=1)] [DELTA][g.sub.n]([t.sub.k])f([t.sub.k])[mu]([t.sub.k]), n [member of] N.

Lemma 4.1. The following is valid:

[L.sup.[beta].sub.[infinity]] = [L.sub.1] = {f |f : T [right arrow] R, [[integral].sub.T] [absolute value of f(t)][DELTA]t < [infinity]}.

Proof. Let

f [member of] [L.sup.[beta].sub.[infinity]] = {f | [[integral].sup.a.sub.0] [absolute value of f(t)][DELTA]t < [infinity], [[integral].sub.T] f(t)g(t)[DELTA]t is convergent for all g [member of] L[infinity]}.

Then we let g be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence [[integral].sub.T] [absolute value of f(t)][DELTA]t < [infinity] and f [member of] [L.sub.1]. If f is an element of [L.sub.1], then for all g [member of] [L.sub.[infinity]]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Hence f [member of] [L.sup.[beta].sub.[infinity]]. []

Theorem 4.2. A [member of] ([L.sub.[infinity]],C([DELTA])) if and only if

(i) [[infinity].summation over (k=1)] [absolute value of [g.sub.n]([t.sub.k])] [mu]([t.sub.k]) < [infinity] for each n [member of] N,

(ii) B [member of] ([L.sub.[infinity]],C), where B = ([h.sub.n]([t.sub.k])[mu]([t.sub.k])) = (([g.sub.n+1]([t.sub.k]) - [g.sub.n]([t.sub.k]))[mu]([t.sub.k])).

Proof. ([??]) Let A [member of] ([L.sub.[infinity]],C([DELTA])). Then for all n [member of] N, [[infinity].summation over (k=1)] [g.sub.n]([t.sub.k])f([t.sub.k])[mu]([t.sub.k]) is convergent and [A.sub.n](f) [member of] C([DELTA]) for all f [member of] [L.sub.[infinity]].

(i) For all f [member of] [L.sub.[infinity]],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to see that [integral].sup.a.sub.0] [absolute value of [g.sub.n]([t.sub.k])] [DELTA][t.sub.k] < [infinity]. Hence [g.sub.n] [member of] [L.sup.[beta].sub.[infinity]] and [g.sub.n] [member of] [L.sub.1] by Lemma 4.1. This implies that

[[integral].sub.T] [absolute value of [g.sub.n](t)] [DELTA]t = [[infinity].summation over (k=1)] [absolute value of [g.sub.n]([t.sub.k])] ([t.sub.k+1] - [t.sub.k]) = [[infinity].summation over (k=1)] [absolute value of [g.sub.n]([t.sub.k])] [mu]([t.sub.k]) < [infinity].

(ii) We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for all f [member of] [L.sub.[infinity]]. Therefore [B.sub.n](f) [member of] C. Hence B [member of] ([L.sub.[infinity]],C).

([??]) Let (i) and (ii) be true. Then for all n [member of] N and f [member of] [L.sub.[infinity]],

|[g.sub.n]([t.sub.k])f ([t.sub.k])[mu]([t.sub.k])[absolute value of [less than or equal to] M][absolute value of [g.sub.n]([t.sub.k])] [mu]([t.sub.k]),

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with M [member of] R. We get that [[infinity].summation over (k=1)][g.sub.n]([t.sub.k])f([t.sub.k])[mu]([t.sub.k]) is convergent for all n [member of] N. Also we have [DELTA][A.sub.n](f) = [B.sub.n](f) and [B.sub.n](f) [member of] C. Hence [A.sub.n] (f) [member of] C([DELTA]). This completes the proof. []

Acknowledgments

I would like to thank Professor Ferhan Atici for her advise, valuable experience and help.

Received January 31, 2007; Accepted February 20, 2007

References

[1] Bernd Aulbach and Stefan Hilger, Linear dynamic processes with inhomogeneous time scale, In Nonlinear dynamics and quantum dynamical systems (Gaussig, 1990), volume 59 of Math. Res., pages 9-20. Akademie-Verlag, Berlin, 1990.

[2] Ozlem Batit, Some function spaces and their dual spaces, Master's thesis, Ege University, Izmir, Turkey, 2001.

[3] Martin Bohner and Gusein Sh. Guseinov, Improper integrals on time scales, Dynam. Systems Appl., 12(1-2):45-65, 2003. Special issue: dynamic equations on time scales.

[4] Martin Bohner and Allan Peterson, Dynamic equations on time scales, Birkhauser Boston Inc., Boston, MA, 2001. An introduction with applications.

[5] Martin Bohner and Allan Peterson, Advances in dynamic equations on time scales, Birkhauser Boston Inc., Boston, MA, 2003.

[6] Mikail Et, On some difference sequence spaces, Doga Mat., 17(1):18-24, 1993.

[7] Mikail Et and Rifat Colak, On some generalized difference sequence spaces, Soochow J. Math., 21(4):377-386, 1995.

[8] Gusein Sh. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285(1):107-127, 2003.

[9] Stefan Hilger, Analysis on measure chains--a unified approach to continuous and discrete calculus, Results Math., 18(1-2):18-56, 1990.

[10] H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24(2):169-176, 1981.

[11] Erwin Kreyszig, Introductory functional analysis with applications, John Wiley & Sons, New York-London-Sydney, 1978.

[12] I.J. Maddox, Elements of functional analysis, Cambridge University Press, Cambridge, second edition, 1988.

Ozlem Batit

Department of Mathematics, Ege University, 35100 Bornova, Izmir, Turkey

E-mail: ozlem.batit@ege.edu.tr
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