Abstract
Some new reproducing kernel functions on time scales are presented. Reproducing kernel functions have not been found on time scales till now. These functions are very important on time scales and they will be very useful for researchers. We need these functions to solve dynamic equations on time scales with the reproducing kernel method.
1. Introduction
A time scale (which is a special case of a measure chain) is an arbitrary nonempty closed subset of the real numbers. Thus, that is, the real numbers, the integers, the natural numbers, and the nonnegative integers are examples of time scales, as are while that is, the rational numbers, the irrational numbers, the complex numbers, and the open interval between and , are not time scales. The theory of time scales was presented by Stefan Hilger in order to unify continuous and discrete analysis [1]. For more details of time scales see [2].
A time scale is denoted by in this work. The delta derivative for a function defined on is given as [1] follows:(i) is the usual derivative if ;(ii) is the usual forward difference operator if .
Reproducing kernel functions on time scale have been found in this work. These new reproducing kernel functions correspond to old reproducing kernel functions when .
The paper is organized as follows. The history of the reproducing kernel method (RKM) is given in Section 2. Section 3 introduces several reproducing kernel spaces for differential equations. Section 4 is devoted to the reproducing kernel functions on time scales. Derivation of the reproducing kernel Hilbert space on time scales is presented in Section 5. It has been proved that the new reproducing kernel functions on time scales coincide with the old reproducing kernel functions when in Section 6. Examples are illustrated in Section 7. There are some conclusions in Section 8.
2. History of Reproducing Kernel Method
In this section, the history of RKM will be investigated. Reproducing kernel space is a special Hilbert space. In recent years, there are many papers on the solution of the nonlinear problems with RKM [3]. The concept of the reproducing kernel can be traced back to the paper of Zaremba [4] in 1908. It was proposed for discussing the boundary value problems of the harmonic functions. In the early development stage of the reproducing kernel theory, most of the works were applied by Bergman [5]. Bergman asserted the corresponding kernels of the harmonic functions with one or several variables and the corresponding kernel of the analytic function in squared metric and applied them in the research of the boundary value problem of the elliptic partial differential equation. This is the first stage in the development history of the reproducing kernel [3].
The second development stage of the reproducing kernel theory was started by Mercer [6]. Mercer discovered that the continuous kernel of the positive definite integral equation has the positive definite property [3] He named the kernel with this property as positive definite Hermite matrix. He also found out that the positive defined Hermite matrix corresponded to a function family, proposed a Hilbert space with inner product , and proved the reproducibility of the kernel in this space:
The third development stage of the reproducing kernel theory is related to Aronszajn [7]. In 1950, he reduced the works of the formers and studied a systematic reproducing kernel theory including the Bergman kernel function.
RKM, which accurately computes the series solution, is of great interest to applied sciences. Recently, a lot of research work has been devoted to the application of RKM to a wide class of stochastic and deterministic problems involving fractional differential equation, nonlinear oscillator with discontinuity, singular nonlinear two-point periodic boundary value problems, integral equations, and nonlinear partial differential equations [3]. The efficiency of RKM has been used by many authors to investigate several scientific applications. Geng and Cui [8] and Zhou et al. [9] applied RKM to handle second-order boundary value problems. Yao and Lin [10] and Wang et al. [11] investigated a class of singular boundary value problems by RKM. In [12], RKM was used to solve nonlinear infinite-delay-differential equations. Wang and Chao [13] and Zhou et al. [9] independently employed RKM to variable-coefficient partial differential equations. Geng and Cui [14] and Du and Cui [15] investigated the approximate solution of the forced Duffing equation with integral boundary conditions by combining the homotopy perturbation method and RKM. Lv and Cui [16] presented a new algorithm to solve linear fifth-order boundary value problems. In [17], the authors developed a new existence proof of solutions for nonlinear boundary value problems. Cui and Du [18] obtained the representation of the exact solution for nonlinear Volterra-Fredholm integral equations by using RKM. Wu and Li [19] applied an iterative reproducing kernel method to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. Recently, RKM was applied to fractional partial differential equations and multipoint boundary value problems [17]. For more details about RKM and the modified forms and its effectiveness, see [20] and the references therein.
3. Reproducing Kernel Functions for Differential Equations
In this section, we define some useful reproducing kernel functions for differential equations.
Definition 1 (reproducing kernel function). Let . A function is called a reproducing kernel function of the Hilbert space if and only if (a) for all ,(b) for all and all .
The last condition is called “the reproducing property” as the value of the function at the point is reproduced by the inner product of with .
Definition 2 (reproducing kernel Hilbert space). A Hilbert space which is defined on a nonempty set is called a reproducing kernel Hilbert space if there exists a reproducing kernel function .
Definition 3. We define the space by where denotes the space of absolutely continuous functions. The inner product and the norm in are defined by
Theorem 4. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. By Definition 3, we have Integrating this equation by parts one time, we get Note that property of the reproducing kernel is If then (9) gives When , we have Therefore, Since we get The unknown coefficients and can be obtained by (12)–(18). Thus, is acquired as
Definition 5. We define the space by The inner product and the norm in are defined by
Theorem 6. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. By Definition 5, we have Integrating (23) by parts two times, we get Note that property of the reproducing kernel is If then (23) gives When , we get Thus, Since we have The unknown coefficients and can be obtained by (26)–(32). Thus, is achieved as
Definition 7. We define the space by The inner product and the norm in are defined by
Theorem 8. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. By Definition 7, we get
Integrating (37) by parts three times, we obtain
Note that property of the reproducing kernel is
If
then, (37) gives
When , we know
Consequently, we attain
Since
we have
The unknown coefficients and can be obtained by (40)–(46). Thus, is gained as
4. Reproducing Kernel Functions on Time Scales
In this section, we define some useful reproducing kernel functions on time scale . All functions are new in the literature.
Definition 9 (see [1, page 22]). A function is called regulated provided its right-sided limits exist (finite) at all right-dense points in and its left-sided limits exist (finite) at all left-dense points in .
Definition 10 (see [1, page 22]). A function is called rd-continuous provided it is continuous at right-dense points in and it is left-sided limits exist (finite) at left-dense points in . The set of rd-continuous functions will be denoted by .
Lemma 11 (see [1, page 28]). If , and , then (i);(ii);(iii);(iv);(v);(vi);(vii);(viii)if on , then (ix)if for all , then .
Definition 12. We define the space as the set of all functions defined on . The inner product and the norm in are defined by
Theorem 13. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. Define by (50) and note that Let and let . Then, by Definition 12, we have This completes the proof.
Definition 14. The inner product and the norm in are defined by
Theorem 15. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. Define by (54) and note that By Definition 14 and integration by parts, we have This completes the proof.
Definition 16. The inner product and the norm in are defined by
Theorem 17. The space is a reproducing kernel space, and its reproducing kernel function is given by
Proof. Define by (58) and note that
By Definition 16, we have
This completes the proof.
5. Derivation of Reproducing Kernel Hilbert Space on Time Scales
Derivation of the new reproducing kernel functions are presented in this section. By Definition 9, we have We get by integration by parts. We have We acquire Table 1 by (63). We get by Table 1, and can be found in a similar way.
Theorem 18. The space is a reproducing kernel Hilbert space.
Proof. Assume that is a Cauchy sequence in . Thus, we have Therefore, we acquire which shows that for any , the sequence is a Cauchy sequence in and is a Cauchy sequence in space . Consequently, there exist a unique number and a unique function , satisfying Now, suppose that is right scattered. It can be seen that the sequence is a Cauchy sequence and converges to . Hence, Therefore, is a Hilbert space. This space is a reproducing kernel space by Definition 2. Consequently, is a reproducing kernel Hilbert space.
Theorem 19. and are reproducing kernel Hilbert spaces.
Proof. The proof of this theorem is similar to the proof of the Theorem 18. Therefore, the proof is omitted.
Remark 20. Old reproducing kernel functions have been in the literature and many authors have used them to solve first-order and second-order differential equations. However, our new reproducing kernel functions are new in the literature and no one has used them before. These new reproducing kernel functions will be used to solve dynamic equations on time scales by RKM in my forthcoming papers.
6. Comparing Reproducing Kernel Functions When
(i)When , it is obvious that (ii)When , we have
Therefore, If we substitute (71) into the , we get This shows that when .(iii)When , we know that If we put (73) into the , we acquire
This proves that when .
7. Examples
In this section, common reproducing kernel functions will be obtained. These functions have been used to solve first and second order differential equations with RKM by many researchers. The function space is defined as follows: The inner product and the norm in the function space are defined as follows: Function space is a reproducing kernel Hilbert space and its reproducing kernel function is given as [3]
Example 1 (see [3, page 17]). When we take , then we get We acquire at .
Example 2 (see [3, page 11]). When we choose , then we acquire We obtain at .
Example 3 (see [3, page 17]). If we take , then we obtain We get at .
8. Conclusion
In this work, reproducing kernel functions on time scales were found. It was proved that reproducing kernel functions on time scale coincide with the old reproducing kernel functions when . These reproducing kernel functions are new on time scales. Therefore, they will be very useful to solve dynamic equations on time scales.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author would like to express his thanks to unknown referees for their careful reading and helpful comments. This paper is supported by Dicle University.