Fractional Differentiation Inequalities

Frontmatter

1. Introduction

This monograph is about fractional differentiation inequalities. These inequalities have applications in fractional differential equations in establishing the uniqueness of the solution of initial value problems, giving upper bounds to their solutions, and these inequalities by themselves are of great interest and deserve to be studied thoroughly.

George A. Anastassiou

2. Opial–Type Inequalities for Functions and Their Ordinary and Canavati Fractional Derivatives

Several L _p -form Opial-type inequalities [315] are presented involving functions and their ordinary and generalized fractional derivatives. The above follow a generalization of Taylor’s formula for generalized fractional derivatives. The chapter ends with the application of the derived inequalities in proving the uniqueness of solution/upper bound to the solution of some known very general fractional differential equations. This treatment is based on [18].

George A. Anastassiou

3. Canavati Fractional Opial–Type Inequalities and Fractional Differential Equations

A series of very general Opial-type inequalities [315] is presented involving fractional derivatives of different orders. These are based on Taylor’s formula for fractional derivatives. The results are applied in proving uniqueness to the solutions of very general fractional initial value problems of fractional ordinary differential equations. This treatment is based on [61].

George A. Anastassiou

4. Riemann—Liouville Opial—type Inequalities for Fractional Derivatives

This chapter provides Opial-type inequalities for generalized Riemann–Liouville fractional derivatives. The inequalities are given for integrable functions with a minimal restriction on the order of the derivatives. This treatment relies on [64].

George A. Anastassiou

5. Opial–type L p –Inequalities for Riemann—Liouville Fractional Derivatives

This chapter presents a class of L ^p -type Opial inequalities for generalized Riemann—Liouville fractional derivatives of integrable functions. The novelty of this approach is the use of the index law for fractional derivatives instead of a Taylor’s formula, which enables us to relax restrictions on the orders of fractional derivatives. This treatment relies on [65].

George A. Anastassiou

6. Opial–Type Inequalities Involving Canavati Fractional Derivatives of Two Functions and Applications

TA wide variety of very general but basic L _p (1 ≤ p ≤ ∞)-form Opial-type inequalities [315] is established involving generalized Canavati fractional derivatives [17, 101] of two functions in different orders and powers.

George A. Anastassiou

7. Opial–Type Inequalities for Riemann—Liouville Fractional Derivatives of Two Functions with Applications

A wide variety of very general but basic L _p (1 ≤ p ≤ ∞)-form Opialtype inequalities [315] is established involving Riemann–Liouville fractional derivatives [17, 230, 295, 314] of two functions in different orders and powers.

George A. Anastassiou

8. Canavati Fractional Opial–Type Inequalities for Several Functions and Applications

A wide variety of very general L _p (1 ≤ p ≤ ∞)-form Opial-type inequalities [315] is presented involving generalized Canavati fractional derivatives [17, 101] of several functions in different orders and powers. The above are based on a generalization of Taylor–s formula for generalized Canavati fractional derivatives [17]. From the established results are derived several other particular results of special interest. Applications of some of these special inequalities are given in proving the uniqueness of solution and in giving upper bounds to solutions of initial value problems involving a very general system of several fractional differential equations. Upper bounds to various fractional derivatives of the solutions that are involved in the above systems are given too. This treatment is based on [27].

George A. Anastassiou

9. Riemann—Liouville Fractional–Opial Type Inequalities for Several Functions and Applications

A wide variety of very general L _p (1 ≤ p ≤ ∞)-form Opial-type inequalities [315] is presented involving Riemann—Liouville fractional derivatives [17, 230, 295, 314] of several functions in different orders and powers.

George A. Anastassiou

10. Converse Canavati Fractional Opial–Type Inequalities for Several Functions

A collection of very general L _p (0 < p < 1)-form converse Opial-type inequalities [315] is presented involving generalized Canavati fractional derivatives [17, 101] of several functions in different orders and powers. Other particular results of special interest are derived from the established results. This treatment is based on [51].

George A. Anastassiou

11. Converse Riemann—Liouville Fractional Opial–Type Inequalities for Several Functions

A collection of very general L _p (0 < p < 1)-form converse Opial-type inequalities [315] is presented involving Riemann—Liouville fractional derivatives [17, 230, 295, 314] of several functions in different orders and powers. Other particular results of special interest are derived from the established results. This treatment is based on [47].

George A. Anastassiou

12. Multivariate Canavati Fractional Taylor Formula

We present here is a multivariate fractional Taylor formula using the Canavati definition of fractional derivative. As related results we present that the order of fractional-ordinary partial differentiation is immaterial, we discuss fractional integration by parts, and we estimate the remainder of our multivariate fractional Taylor formula. This treatment is based on [40].

George A. Anastassiou

13. Multivariate Caputo Fractional Taylor Formula

This is a continuation of Chapter 12. We establish here a multivariate fractional Taylor formula via the Caputo fractional derivative. The fractional remainder is expressed as a composition of two Riemann—Liouville fractional integrals.

George A. Anastassiou

14. Canavati Fractional Multivariate Opial–Type Inequalities on Spherical Shells

Here we introduce the concept of multivariate Canavati fractional differentiation especially of the fractional radial differentiation, by extending the univariate definition of [101]. Then we present Opial-type inequalities over compact and convex subsets of ℝ ^N , N ≥ 2, mainly over spherical shells, studying the problem in all possibilities. Our results involve one, two, or more functions. This treatment is based on [44].

George A. Anastassiou

15. Riemann—Liouville Fractional Multivariate Opial–type inequalities over a spherical shell

Here we introduce the concept of the Riemann—Liouville fractional radial derivative for a function defined on a spherical shell. Using polar coordinates we are able to derive multivariate Opial-type inequalities over a spherical shell of ℝ ^N , N ≥ 2, by studying the topic in all possibilities. Our results involve one, two, or more functions. We also produce several generalized univariate fractional Opial-type inequalities, many of which are used to achieve the main goals. This treatment is based on [45].

George A. Anastassiou

16. Caputo Fractional Multivariate Opial–Type Inequalities over a Spherical Shell

Here is introduced the concept of the Caputo fractional radial derivative for a function defined on a spherical shell. Using polar coordinates we are able to derive multivariate Opial-type inequalities over a spherical shell of ℝ ^N , N ≥ 2, by studying the topic in all possibilities. Our results involve one, two, or more functions. We present many univariate Caputo fractional Opial-type inequalities, several of which are used to establish results on the shell. We give an application to prove the uniqueness of solution of a general partial differential equation on the shell. Also we apply our results for Riemann—Liouville fractional derivatives. This treatment relies on [58].

George A. Anastassiou

17. Poincaré–Type Fractional Inequalities

Here we present Poincaré-type fractional inequalities involving fractional derivatives of Canavati, Riemann—Liouville, and Caputo types. The results are general L _p inequalities forward and reverse, univariate and multivariate, on a spherical shell. We give applications to ODEs and PDEs. We present also mean Poincaré-type fractional inequalities. This treatment relies on [57].

George A. Anastassiou

18. Various Sobolev–Type Fractional Inequalities

Here we present various univariate Sobolev-type fractional inequalities involving fractional derivatives of Canavati, Riemann—Liouville, and Caputo types. The results are general L _p inequalities forward and converse on a closed interval. We give an application to a fractional ODE. We present also the mean Sobolev-type fractional inequalities. This treatment relies on [56].

George A. Anastassiou

19. General Hilbert—Pachpatte–Type Integral Inequalities

In this chapter we present very general weighted Hilbert—Pachpatte–type integral inequalities. These are with regard to ordinary derivatives and fractional derivatives of Riemann—Liouville and Canavati types, and also in regard to general derivatives of Widder-type and linear differential operators. These results apply to continuous functions and some to integrable functions. This treatment relies on [41].

George A. Anastassiou

20. General Multivariate Hilbert—Pachpatte–Type Integral Inequalities

In this chapter we present general weighted Hilbert—Pachpatte–type multivariate integral inequalities. These are with regard to ordinary partial derivatives and fractional partial derivatives of Canavati and Riemann—Liouville–types. These results are applications of a general theorem we present for integrable multivariate functions. This treatment relies on [42].

George A. Anastassiou

21. Other Hilbert—Pachpatte–Type Fractional Integral Inequalities

This is a continuation of Chapters 19 and 20. We present here very general weighted univariate and multivariate Hilbert—Pachpatte-type integral inequalities. These involve Caputo and Riemann—Liouville fractional derivatives and fractional partial derivatives of the mentioned types. This treatment is based on [54]. Of great motivation to write this chapter have been [191, Theorem 316], [320, Theorem 1], [322, Theorem 1], and [147, 187, 188].

George A. Anastassiou

22. Canavati Fractional and Other Approximation of Csiszar’s f–Divergence

Here are presented various sharp and nearly optimal probabilistic inequalities that give best or nearly best estimates for the Csiszar’s f -divergence. These involve Canavati fractional and ordinary derivatives of the directing function f. Also given are lower bounds for the Csiszar’s distance. The Csiszar’s discrimination is the most essential and general measure for the comparison between two probability measures. This treatment is based on [28].

George A. Anastassiou

23. Caputo and Riemann—Liouville Fractional Approximation of Csiszar’s f–Divergence

Here are presented various tight probabilistic inequalities that give nearly best estimates for the Csiszar’s f-divergence. These involve Riemann—Liouville and Caputo fractional derivatives of the directing function f. Also is given a lower bound for the Csiszar’s distance. The Csiszar’s discrimination is the most essential and general measure for the comparison between two probability measures. This is a continuation of Chapter 22 and is based on [55].

George A. Anastassiou

24. Canavati Fractional Ostrowski–Type Inequalities

In the fractional Ostrowski-type inequalities, under the same initial conditions–assumption, as in the integer ordinary derivative case–one can derive results for higher-order (fractional) derivatives, appearing in the R.H.S.s of the corresponding inequalities.

George A. Anastassiou

25. Multivariate Canavati Fractional Ostrowski–Type Inequalities

Optimal upper bounds are given to the deviation of a value of a multivariate function of a fractional space from its average, over convex and compact subsets of ℝ ^N , N ≥ 2. In particular we work over rectangles, balls, and spherical shells. These bounds involve the supremum and L∞ norms of related multivariate Canavati fractional derivatives of the involved function. The presented inequalities are sharp; namely they are attained. This chapter has been motivated by the works of Ostrowski [318], 1938, and Anasstasiou [24], 2003, and the chapter is based on [43].

George A. Anastassiou

26. Caputo Fractional Ostrowski–Type Inequalities

Optimal upper bounds are given to the deviation of a value of a univariate or multivariate function of a Caputo fractional derivative related space from its average, over convex and compact subsets of ℝ ^N , N ≥ 1. In particular we work over closed intervals, rectangles, balls, and spherical shells. These bounds involve the supremum and L∞ norms of related univariate or multivariate Caputo fractional derivatives of the involved functions. The derived inequalities are sharp; namely they are attained by simple functions. This chapter has been motivated by the works of Ostrowski [318], 1938, and of the author’s [24], 2003 and [43], 2007, and the chapter also relies on [52].

George A. Anastassiou

27. Appendix

Without loss of generality we work on [0, 1] and the anchor point is zero.

George A. Anastassiou

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