A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces

: Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond- α integral operator M ca, δ to the norm of the centered fractional maximal diamond- α integral operator M ca on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

Dynamic equations and integral inequalities have many applications in different areas of science. Some areas are electrical engineering, fluid dynamics, quantum mechanics, physical problems, wave equations, heat transfer and economics [14][15][16][17][18][19][20][21][22][23][24]. Tisdell and Zaidi [15] demonstrated basic qualitative and quantitative results for solutions to non-linear dynamic equations on time scales with an application to economic modelling. Seadawy et al. [18] demonstrated non-linear wave solutions of the Kudryashov-Sinelshchikov dynamical equation in mixtures of liquid-gas bubbles under the consideration of heat transfer and viscosity. Akin [25] demonstrated fractional integral type inequalities on time scales. Higgins [26] demonstrated asymptotic behavior of second-order nonlinear dynamic equations on time scales. Ozturk and Higgins [27] demonstrated limit behaviors of non-oscillatory solutions of three-dimensional time scale systems.
In variable exponential spaces, problems such as the boundedness and compactness of integral operators, take an important place. The concept of norms has the most important place in solving these problems. If we can obtain the norms of integral operators by the method we apply, then we can see the boundedness and compactness of these operators. However, there are almost no studies on the problems of time scales. Our main purpose in this study is to examine the equivalence of the norms of fractional integral operators. Thanks to these results, we will be able to establish the constraint and compactness conditions of integral operators on time scales.
The organization of this article is as follows. In Section 2, we give necessary definitions, lemmas and theorems. In Section 3, we prove the equivalence of the norm variable exponent L p(.) of the restricted centered fractional maximal diamond-α integral M c a,δ to the norm variable exponent L p(.) of the centered fractional maximal diamond-α integral M c a for all 0 < δ < ∞ and 1 ≤ p(x) < ∞ on time scales. In Section 4, we give the conclusion.

Materials and Methods
In this section, we provide necessary concepts and statements related to time scale and variable exponent Lebesgue space. The reader can refer to the monographs  for details. for 1 < p(.) ≤ ∞, t ∈ R m and some real positive numbers δ, θ. We have seen here that the norm of the restricted operator is less than the norm of the unrestricted operator. The normed inequalities obtained above will help us prove our results.
To prove our main results, we first provide some definitions and lemmas which will be used as follows.
Let us give information about the time scales that will help us in our work. A time scale T is a nonempty closed subset of R (for details, see [19,20]).
If σ(t) > t, then t is right-scattered and if σ(t) = t, then t is called right-dense. If ρ(t) < t, then t is left-scattered and if ρ(t) = t, then t is called left-dense.
T k is defined as follows (for details, see [19,20,42]) by the same way Definition 5. [20] If H : T → R is defined as ∆-antiderivative of h : T → R , then H ∆ = h(t) holds for ∀t ∈ T and we define the ∆-integral of h by for s, t ∈ T.
We now give similar definitions for the nabla operator.
Definition 9. [12,25] If f ∈ C rd (T, R) and t ∈ T k , then Here, we can define the fractional maximal diamond-α integral Now, we can define the restricted centered fractional maximal diamond-α integral operator and the restricted uncentered fractional maximal diamond-α integral operator, for t ∈ R m and δ ∈ R + (for details, see [28]).
for some λ > 0. The norm in L p(x) is the generalization of the norm in L p (p is constant). The Luxemburg norm in L p(x) is defined as follows:  (6), and let δ > 0. Then Proof. From the definition of the operator M a,δ f in (6), we obtain where v m is the volume of the unit ball in R m and the dilation operator τ δ is defined as follows: (τ δ f )(x) = f (δx), for δ > 0 and x ∈ R m . It follows from (12) that Taking the supremum over all f ∈ L p(x) (R m ) with f L p(x) (R m ) = 0 for the two sides of the above equation, we have for all δ > 0 and 1 ≤ p(x) < ∞.
Next, we will prove that We will use Equation (13) for proof. If f ∈ L p(x) (R m ), then we have M f ∈ L p(x) (R m ). From Lemma 1, Lemma 4 and Equation (13), we obtain Now, taking advantage of inequality (14), we get the following inequality Hence, we obtain from (14) that .

Main Results
In this section we give statements and proofs of our results. holds for δ > 0 and 1 ≤ p(x) < ∞.
Proof. For 0 < δ < ∞, we first prove If we use the definition of the operator M c a,δ in (5), then we have for 0 < δ < ∞ and x ∈ R m , where v m is the volume of the unit ball in R m . Hence, for 0 < δ < ∞ and x ∈ R m . If we use (17), then we have If supremum is taken over all the f ∈ L p(x) (R m ) for the two sides of (18), we have Next, we will use Equation (19) to prove for ∀δ > 0 and 1 ≤ p(x) < ∞. We just need to prove If we use Lemma 5, for > 0, then there exists a function f ∈ C ∞ c (R m ), such that Since Now we set δ 0 = Z + + R (Z + is a positive integer and R is a real integer). Then it can be written from the definition of M c a,δ that holds for |x| < R. Hence, from (20)- (22), we obtain Obviously, (23) implies that Here, the inequality (24) yields From (19) and (25), we have for ∀δ > 0 and 1 ≤ p(x) < ∞. Thus, proof of Theorem 3 is complete.
Now, let us prove the weak (1,1) boundedness for the restricted centered diamond-α fractional maximal operator. holds for all δ > 0.
Proof. Let M c a,δ be defined by (5) for 0 < δ < ∞. First, we prove that From the identity (17), we get For any 0 < δ, we obtain from (26) that Thus (27) implies that If f L 1 (R m ) = 0, then it follows from (28) Now taking the supremum over all f ∈ L 1 (R m ) with f L 1 (R m ) = 0 for the two sides of (29), we obtain Next, we will use (30) to prove that holds for 0 < δ. Now, let us prove the correctness of the following equation.
holds for any f ∈ L 1 (R m ) with f L 1 (R m ) = 0. Clearly, the right side of (31) is not bigger than the left side, so it is enough to show opposite inequality. From Lemma 3, we have µd M c a f (µ). For 0 < , there must be a µ 0 ∈ R + such that We conclude that This is equivalent to A ≤ sup µ>0 µ lim δ→∞ d M c a,δ f (µ) . Herewith, (31) holds. If we use Equation (31), we obtain that Thus, we get the result we want to achieve.

Remark 2.
Let M a be the uncentered fractional maximal operator defined by (3). Define the iterated fractional maximal operator denoted by M i+1 a as follows: M i+1 a g(y) = M a M i a g (y) (33) for i = 1, 2, 3, . . . and y ∈ R m . Set M 1 a g (y) = (M a g)(y).

Lemma 6.
Assume that a sequence {d k } ∞ k=1 satisfies the following two conditions simultaneously: (a) d 1 = s ∈ (0.1), Then {d k } ∞ k=1 is strictly monotone increasing and we have Proof. By the mathematical induction and the two conditions (a) and (b), we can easily obtain 0 < d i < 1 for each i ∈ N. Furthermore, the condition (b) implies This shows that {d k } ∞ k=1 is strictly monotone increasing. Since {d k } ∞ k=1 is monotone increasing and has the upper bound, the limit of {d k } ∞ k=1 exists, and we can easily get By Lemma 6, we have the following corollary.

Conclusions
For more than a quarter century, the concept of time scales has taken an important place in the literature. Mathematicians and scientists working in other disciplines have demonstrated many applications of dynamic equations and integral inequalities; for example, transformations, inverse conversions, extensions, wave equations, heat transfer, optics, fluid dynamics, quantum calculus, economy, etc. The boundedness and compactness of the integral operators we know from harmonic analysis occupy an important place in the literature. Norms in variable exponential spaces are used to solve these problems. Previously, studies on the concept of the equivalence of norms in variable exponential spaces were conducted. In this way, we obtain the boundedness and compactness of integral operators that we do not have any information about. For more detailed information, we refer the reader to references.
In this study, we wanted to relate the norms of integral operators with time scales. In this article, we showed the equivalence of the norm variable exponent L p(x) of the restricted centered fractional maximal diamond-α integral M c a,δ with norm variable exponent L p(x) of centered fractional maximal diamond-α integral M c a for all 0 < δ < ∞ and 1 ≤ p(x) < ∞ on time scales. Hereby, we will be able to establish the boundedness and compactness conditions of fractional integral operators. In the future, we plan to carry these studies to variable exponent grand Lebesgue spaces, which is more general.  Acknowledgments: The author would like to thank the editor and the referees for his/her careful reading and valuable comments.

Conflicts of Interest:
The author declares that there is no conflict of interest.