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The θ-derivative as unifying framework of a class of derivatives

  • Muneerah AL Nuwairan EMAIL logo
Published/Copyright: November 11, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we develop a unified framework for studying some derivatives defined as limits. This framework, the θ -derivative, is used to investigate the relationships between these derivatives and their relation to the ordinary derivative. It is shown that the existence of any of these derivatives is equivalent to the existence of the ordinary derivative. By using these results, we show that two derivatives that appear in the literature under different names are actually identical, and an infinite family of derivatives actually consists of only one member. We also give a unified form for the integral corresponding to these derivatives, generalize the standard analysis theorems to this setting, and relate our results to those of other researchers. Finally, we address the question of whether these derivatives should be considered fractional derivatives.

Keywords: conformable derivative; new conformable derivative; M-fractional derivative; beta derivative; Katugampola derivative
MSC 2010: 26A33

1 Introduction

Fractional calculus is almost as old as ordinary calculus. Nowadays, it is a thriving field with many applications [1–10]. Of the two branches of fractional calculus, integration is the more settled with the Riemann-Liouville being the main integral. There has even been an axiomatic characterization of this integral [11]. Differentiation, on the other hand, remains highly splintered. There has been many proposed definitions including Riemann-Liouville [12,13], Grünwald-Letnikov [14,15], Caputo [12,13,16] Caputo-Fabrizio [17,18], Atangana-Baleanu [19,20], and many others. For a survey, see the study by de Oliveira and Machado [21]. Among the proposed definitions of fractional derivatives were several that define the derivative as a limit. These include the conformable [22–25], Katugampola [26,27], new conformable [28], Beta derivative [29,30], and M-fractional derivative [31–33]. A cursory survey of the literature reveals that these derivatives are still being used by many researchers. The main objective of this study is to treat these derivatives, and similar derivatives that arise in the literature [34], in a uniform way and explore the relationship between them. The main contributions of this work can be summarized as follows:

  1. Proving that all the truncated Mittag-Leffler derivatives coincide (Corollary 4.3).

  2. Showing that the conformable and the Kaugampola derivatives coincide (Theorem 3.2).

  3. A method was given for simplifying the derivatives studied (Corollary 4.4).

  4. A unified expression for the integrals corresponding to these derivatives was given (Definition 6.1 and Theorem 6.1).

  5. Obtaining the conformable derivative as the simplification of Kaugampola derivative (Example 4.2).

  6. A unified discussion of some failings of these derivatives is given in Section 5.

  7. A unified treatment of the generalizations of some classical analysis theorems to these derivatives is given (Lemmas 7.2 (2), 7.3, and 7.4).

  8. A unified critique of these derivatives is given in Section 9.

This article is organized as follows. In Section 2, the notion of θ -derivative was defined, and examples are provided. In Section 3, the relationship of θ -derivatives with ordinary derivatives, the conformable, and Katugampola derivatives is explained. We also show that the conformable and Katugampola derivatives coincide. In Section 4, we deduce differentiabillity from θ -differentiability in a more general setting, and show that all truncated M-fractional derivatives coincide. We also describe a method for replacing the function θ by a simpler one that gives identical θ -derivative. In Section 5, we show that θ -derivatives do not satisfy some properties desirable for fractional derivatives including the semigroup property. In Section 6, we derive the θ -integrals corresponding to θ -derivatives, and show that these also do not satisfy the semigroup property. In Section 7, we give the θ -derivative version of some classical theorems in analysis including Rolle’s theorem, the mean value theorem, and Darboux’s theorem. In Section 8, we state some of Tarasov’s results and discuss their relationship with the current work. In Section 9, we discuss the view of some researchers that these derivatives should not be classified as fractional derivatives. Finally, Section 10 is devoted to the conclusions.

2 Definitions and some properties of the derivative D θ

In this section, we state the main definition, express several derivatives as examples of θ -derivative, and fix the notation used in the rest of the article.

Definition 2.1

Let O ⊂ R 2 be an open set, and U = π 1 ( O ) be the projection of O onto the first coordinate. Let θ : O ⟶ R be a function satisfying θ ( z , 0 ) = z for all z ∈ U . If I ⊆ U is open, and f : I ⟶ R is a function such that

lim h → 0 f ( θ ( z , h ) ) − f ( z ) h

exists, then f is said to be θ -differentiable at z with θ -derivative

(1) D θ f ( z ) = lim h → 0 f ( θ ( z , h ) ) − f ( z ) h .

Note that

(2) D θ f ( z ) = lim h → 0 f ( θ ( z , h ) ) − f ( z ) h = lim h → 0 f ( θ ( z , h ) ) − f ( θ ( z , 0 ) ) h = ∂ ( f ∘ θ ) ∂ h ( z , 0 )

is a partial derivative of the function f ∘ θ ( z , h ) .

Example 2.1

The following is a list of some well-known derivatives, shown as special cases of the θ -derivative defined in Definition 2.1.

  1. If θ = θ 0 ( z , h ) = z + h , with O = R 2 , U = R , we obtain the ordinary derivative

    D θ f ( z ) = f ′ ( z ) = lim h → 0 f ( z + h ) − f ( z ) h .

  2. For 0 < α ≤ 1 , and θ = θ C , α ( z , h ) = z + h z 1 − α , with O = { ( z , h ) ∈ R 2 , z > 0 } , U = ( 0 , ∞ ) , we obtain the α -conformable derivative

    D C , α θ f ( z ) = lim h → 0 f ( z + h z 1 − α ) − f ( z ) h .

  3. For 0 < α ≤ 1 , and θ = θ K , α ( z , h ) = z e h z − α , with O = { ( z , h ) ∈ R 2 , z > 0 } , U = ( 0 , ∞ ) , we obtain the α -Katugampola derivative

    D K , α θ f ( z ) = lim h → 0 f ( z e h z − α ) − f ( z ) h .

  4. For 0 < α ≤ 1 , θ = θ N , α = z + h e ( α − 1 ) z , O = R 2 , and U = R , we obtain the new conformable derivative

    D N , α θ f ( z ) = lim h → 0 f ( z + h e ( α − 1 ) z ) − f ( z ) h .

  5. For 0 < α ≤ 1 , β > 0 , 1 ≤ j ∈ N , and θ = θ β , j , α ( z , h ) = z E β , j ( h z − α ) , where

    E β , j ( y ) = ∑ k = 0 j y k Γ ( β k + 1 )

    is the truncated Mittag-Leffler function, and we obtain the truncated M-fractional derivative, with O = { ( z , h ) ∈ R 2 , z > 0 } , U = ( 0 , ∞ ) , and

    D β , j , α θ f ( z ) = lim h → 0 f ( z E β , j ( h z − α ) ) − f ( z ) h .

  6. For 0 < β ≤ 1 , and θ = θ B , β ( z , h ) = z + h z + 1 Γ ( β ) 1 − β with O = { ( z , h ) ∈ R 2 , z > − 1 Γ ( β ) } , U = − 1 Γ ( β ) , ∞ , we obtain the β -derivative

    D B , β θ f ( z ) = lim h → 0 f z + h z + 1 Γ ( β ) 1 − β − f ( z ) h .

Remark 2.1

We use ε θ ( z ) to denote the θ -derivative of the identity function ε θ ( z ) = D θ f ( z ) whenever the right hand side exists, and it is undefined otherwise, where f ( z ) = z is the identity function. Note that by the definition of θ -derivative, it follows that

(3) ε θ ( z ) = lim h → 0 θ ( z , h ) − z h = ∂ θ ∂ h ( z , 0 ) .

We shall see below that the function ε θ determines the derivative D θ .

In the following, we will not explicitly specify the set O , with the understanding that it is the set of points where θ is defined. The following lemma lists some elementary properties of the θ -derivative.

Lemma 2.1

Let a , b , c ∈ R . If θ : O ⟶ R , I ⊂ U = π 1 ( O ) and f , g : I ⟶ R , then

  1. If f ≡ c is a constant, then D θ f ≡ 0 .

  2. If D θ f ( z ) and D θ g ( z ) exist, then D θ ( a f + b g ) ( z ) = a D θ f ( z ) + b D θ g ( z ) .

Without further restrictions on θ , we cannot conclude that θ -differentiability implies continuity, as shown by the following example.

Example 2.2

Let

θ ( z , h ) = z + ∣ h ∣ , and f ( z ) = 1 z ≥ 0 0 z < 0 .

By direct computation, one obtains that D θ f ≡ 0 , but f is not continuous at 0.

Definition 2.2

The function θ : O ⟶ R is called z -crossing at z ∈ U = π 1 ( O ) if for each δ > 0 , there exist h 1 , h 2 ∈ ( − δ , δ ) such that θ ( z , h 1 ) < z < θ ( z , h 2 ) . The function θ is called z -crossing on I if it is z -crossing at every z ∈ I .

Remark 2.2

The z-crossing property is fulfilled in most cases:

  1. If ε θ = ∂ θ ∂ h ( z , 0 ) ≠ 0 , then since θ ( z , h ) ≃ θ ( z , 0 ) + h ε θ ( z ) = z + h ε θ ( z ) , θ is z -crossing at z .

  2. For the functions in Example 2.1, except possibly θ β , j , α , one can verify by inspection that θ ( z , h ) < z for h < 0 and θ ( z , h ) > z for h > 0 , which implies that θ is z -crossing, and θ β , j , α is z -crossing since ε β , j , α ( z ) ≠ 0 for z > 0 .

Lemma 2.2

Let θ be z-crossing at z, and θ ( z , h ) be continuous for h in an interval around 0. If D θ f ( z ) exists, then f is continuous at z.

Proof

Let ε > 0 be given. Let A = D θ f ( z ) = lim h → 0 f ( θ ( z , h ) ) − f ( z ) h , then there exists δ > 0 such that h ⟶ θ ( z , h ) is continuous on ( − δ , δ ) , and A − f ( θ ( z , h ) ) − f ( z ) h < 1 for 0 < ∣ h ∣ < δ .

Thus, we have

(4) ∣ f ( θ ( z , h ) ) − f ( z ) ∣ < ( ∣ A ∣ + 1 ) ∣ h ∣ for 0 < ∣ h ∣ < δ .

Since θ ( z , 0 ) = z , the inequality holds for ∣ h ∣ < δ . Let δ 1 = min ( δ , ε ∣ A ∣ + 1 ) , then for ∣ h ∣ < δ 1 , we have ∣ f ( θ ( z , h ) ) − f ( z ) ∣ < ε . Pick h 1 , h 2 ∈ ( − δ 1 , δ 1 ) such that θ ( z , h 1 ) < z < θ ( z , h 2 ) . Since the image of ( − δ 1 , δ 1 ) under the map h ⟶ θ ( z , h ) is connected, it contains [ θ ( z , h 1 ) , θ ( z , h 2 ) ] . Let δ 2 = min { z − θ ( z , h 1 ) , θ ( z , h 2 ) − z } . If ∣ y − z ∣ < δ 2 , then y ∈ [ θ ( z , h 1 ) , θ ( z , h 2 ) ] ⊆ { θ ( z , h ) : ∣ h ∣ < δ 1 } , so, ∣ f ( y ) − f ( z ) ∣ < ε .□

We postpone developing further property of θ -differentiable functions till later sections, and will devote the following two sections to examining the notion of θ -differentiability.

3 The relation between θ -derivatives and ordinary derivatives

In this section, we express θ -derivatives in terms of ordinary derivatives. We also express ordinary derivatives using θ -derivatives in certain cases and show that the conformable and Katugampola derivatives coincide. Theorem 3.1 plays an important role in this section, so we have taken extra care in proving it. This is partly warranted, since one can fall in such elementary errors as assuming that lim z → a g ( z ) = b and lim z → b f ( z ) = c implies lim z → a f ∘ g ( z ) = c . This is easily seen to be false by taking f ( z ) = χ { 0 } ( z ) , the characteristic function of the set { 0 } , and g ( z ) = z sin 1 z . These functions satisfy the hypotheses with a = b = c = 0 , but with f ∘ g ( z ) having no limit as z → 0 . This cannot occur if we add the requirement that there exist δ > 0 such that 0 < ∣ z − a ∣ < δ implies g ( z ) ≠ b . The following lemma is an elementary result in Calculus, and we provide its proof for completeness.

Lemma 3.1

The following statements hold.

  1. If lim z → b f ( z ) = c , lim z → a g ( z ) = b , and there is δ > 0 such that 0 < ∣ z − a ∣ < δ implies that g ( z ) ≠ b , then lim z → a f ∘ g ( z ) = c .

  2. If f is differentiable at a and f ′ ( a ) ≠ 0 , then there is δ > 0 such that f ( z ) ≠ f ( a ) for 0 < ∣ z − a ∣ < δ .

Proof

We show the results as follows:

  1. Let ε > 0 be given. Since lim z → b f ( z ) = c , there exist δ 1 > 0 such that 0 < ∣ z − b ∣ < δ 1 implies that ∣ f ( z ) − c ∣ < ε . Since lim z → a g ( z ) = b , there is δ 2 > 0 such that 0 < ∣ z − a ∣ < δ 2 implies ∣ g ( z ) − b ∣ < δ 1 . By hypothesis, there is δ 3 > 0 such that 0 < ∣ z − a ∣ < δ 3 gives that g ( z ) ≠ b . Take δ = min { δ 2 , δ 3 } > 0 , then 0 < ∣ z − a ∣ < δ implies that 0 < ∣ g ( z ) − b ∣ < δ 1 , which in turn gives that ∣ f ( g ( z ) ) − a ∣ < ε .

  2. Let K = f ′ ( a ) ≠ 0 . By the definition of f ′ ( a ) , there exists δ > 0 such that 0 < ∣ z − a ∣ < δ implies that ∣ f ( z ) − f ( a ) z − a − K ∣ < ∣ K ∣ 2 , which implies that ∣ f ( z ) − f ( a ) z − a ∣ > ∣ K ∣ 2 > 0 , establishing the desired result.□

Theorem 3.1

Suppose θ 2 ( z , h ) = θ 1 ( z , H ( z , h ) ) with H ( z , 0 ) = 0 and ∂ H ∂ h ( z , 0 ) ≠ 0 . If k is a function such that D θ 1 k ( z ) exists, then D θ 2 k ( z ) exists and

(5) D θ 2 k ( z ) = ∂ H ∂ h ( z , 0 ) D θ 1 k ( z ) .

Proof

Fix z , and let g ( h ) = H ( z , h ) . The map g is differentiable, thus continuous at 0 with lim h → 0 g ( h ) = g ( 0 ) = H ( z , 0 ) = 0 and g ′ ( 0 ) = ∂ H ∂ h ( z , 0 ) ≠ 0 . By Lemma 3.1 (2), since g ′ ( 0 ) ≠ 0 , there is a δ > 0 such that g ( h ) ≠ g ( 0 ) = 0 for 0 < ∣ h − 0 ∣ < δ . In Lemma 3.1 (1), take f ( u ) = k ( θ 1 ( z , u ) ) − k ( z ) u . By the assumption on the θ 1 -differentiability of k at z , we have lim u → 0 f ( u ) = D θ 1 k ( z ) . By applying Lemma 3.1 (1), we obtain

(6) lim h → 0 f ∘ g ( h ) = lim u → 0 f ( u ) = D θ 1 k ( z ) .

Since

lim h → 0 H ( z , h ) h = lim h → 0 H ( z , h ) − H ( z , 0 ) h = ∂ H ∂ h ( z , 0 ) ,

we obtain

□ lim h → 0 k ( θ 2 ( z , h ) ) − k ( z ) h = lim h → 0 k ( θ 1 ( z , H ( z , h ) ) ) − k ( z ) h = lim h → 0 H ( z , h ) h k ( θ 1 ( z , H ( z , h ) ) ) − k ( z ) H ( z , h ) = ∂ H ∂ h ( z , 0 ) lim h → 0 k ( θ 1 ( z , H ( z , h ) ) ) − k ( z ) H ( z , h ) = ∂ H ∂ h ( z , 0 ) lim h → 0 f ∘ g ( h ) = ∂ H ∂ h ( z , 0 ) D θ 1 k ( z ) .

Theorem 3.1 is fairly straightforward, but it is useful. If we take θ 1 ( z , h ) = θ 0 ( z , h ) = z + h (ordinary derivative), then for any θ ( z , h ) , we have θ ( z , h ) = z + ( θ ( z , h ) − z ) . So, θ ( z , h ) = θ 0 ( z , H ( z , h ) ) with H ( z , h ) = θ ( z , h ) − z . Thus, we have

∂ H ∂ h ( z , 0 ) = ∂ θ ∂ h ( z , 0 ) = ε θ ( z ) .

Note that for any θ , H ( z , 0 ) = θ ( z , 0 ) − z = z − z = 0 . From this and Theorem 3.1, we obtain the following corollary.

Corollary 3.1

If ε θ ( z ) ≠ 0 and f is differentiable at z, then it is θ -differentiable and D θ f ( z ) = ε θ ( z ) f ′ ( z ) .

Note that for the derivative in Example 2.1, we have

  1. For the conformable derivative: θ C , α ( z , h ) = z + h z 1 − α and ε θ c , α ( z ) = z 1 − α .

  2. For the Katugampola derivative: θ K , α ( z , h ) = z e h z − α and ε θ K , α ( z , 0 ) = z 1 − α .

  3. For the new conformable derivative: θ N , α ( z , h ) = z + h e ( α − 1 ) z and ε θ N , α ( z ) = e ( α − 1 ) z .

  4. For the M-fractional derivative: θ β , j , α ( z , h ) = z E β , j ( h z − α ) . Since E β , j ( h z − α ) = ∑ i = 0 j h i z − α i Γ ( β i + 1 ) , we have

    ∂ E β , j ( h z − α ) ∂ h = ∑ i = 1 j i h i − 1 z − α i Γ ( β i + 1 ) ,

    ∂ E β , j ( h z − α ) ∂ h h = 0 = z − α Γ ( β + 1 ) ,

    and

    ε θ β , j , α ( z ) = ∂ θ β , j , α ∂ h ( z , 0 ) = z ∂ E β , j ( h z − α ) ∂ h h = 0 = z − α + 1 Γ ( β + 1 ) .

  5. For the β -derivative: θ B , β = z + h ( z + 1 Γ ( β ) ) 1 − β and ε θ B , β ( z ) = ( z + 1 Γ ( β ) ) 1 − β .

By using the aforementioned observation, and Corollary 3.1, we have the following corollary.

Corollary 3.2

If f is differentiable at z and 0 < α ≤ 1 , then

  1. If z > 0 , then D C , α θ f ( z ) = z 1 − α f ′ ( z ) .

  2. If z > 0 , then D K , α θ f ( z ) = z 1 − α f ′ ( z ) .

  3. For any z , D N , α θ f ( z ) = e ( α − 1 ) z f ′ ( z ) .

  4. If z > 0 , then D β , j , α θ f ( z ) = z 1 − α Γ ( β + 1 ) f ′ ( z ) .

  5. If z > − 1 Γ ( β ) , then D B , β θ f ( z ) = z + 1 Γ ( β ) 1 − β f ′ ( z ) .

Some of the results in Corollary 3.2 have appeared in the literature [22,27] and are not hard to deduce directly. A more interesting application of Theorem 3.1 is to obtain ordinary differentiability from fractional θ -differentiability. So, we take θ 2 ( z , h ) = θ 0 ( z , h ) = z + h .

  1. For the conformable derivative θ 1 ( z , h ) = θ C , α ( z , h ) = z + h z 1 − α , H ( z , h ) must satisfy z + h = θ 2 ( z , h ) = θ 1 ( z , H ( z , h ) ) = z + H z 1 − α . Thus, H z 1 − α = h or H ( z , h ) = h z α − 1 . Note that ∂ H ∂ h ( z , h ) = z α − 1 ≠ 0 for z > 0 .

  2. For the Katugampola derivative θ 1 ( z , h ) = θ K , α ( z , h ) = z e h z − α , H ( z , h ) must satisfy z + h = z e H z − α . That is, 1 + h z = e H z − α , or equivalently, H ( z , h ) = z α ln ( 1 + h z ) , which is well defined for z > 0 and h > − z with partial derivative ∂ H ∂ h ( z , h ) = z α 1 + h z 1 z = z α z + h and ∂ H ∂ h ( z , 0 ) = z α − 1 .

  3. For the new conformable derivative θ 1 ( z , h ) = θ N , α ( z ) = z + h e ( α − 1 ) z . We need H such that z + h = θ 2 ( z , h ) = θ 1 ( z , H ) = z + H e ( α − 1 ) z . This is equivalent to h = H e ( α − 1 ) z , or H = h e ( 1 − α ) z , so ∂ H ∂ h ( z , h ) = e ( 1 − α ) z .

  4. For the beta derivative, we have θ B , β ( z , h ) = z + h z + 1 Γ ( β ) 1 − β . We need H ( z , h ) such that z + H z + 1 Γ ( β ) 1 − β = z + h . That is, H z + 1 Γ ( β ) 1 − β = h , or H = h z + 1 Γ ( β ) β − 1 , which is well defined for z > − 1 Γ ( β ) with ∂ H ∂ h ( z , 0 ) = z + 1 Γ ( β ) β − 1 .

Note that H ( z , 0 ) = 0 , ∂ H ∂ h ( z , 0 ) is well defined and nonzero for z > 0 for cases 1 and 2, for all z in case 3, and for z > − 1 Γ ( β ) in case 4. Thus, by using Theorem 3.1, we obtain the following corollary.

Corollary 3.3

Let f be any function.

  1. If 0 < α ≤ 1 , z > 0 , and D C , α θ f ( z ) exists, then f is differentiable at z and f ′ ( z ) = z α − 1 D C , α θ f ( z ) .

  2. If 0 < α ≤ 1 , z > 0 , and D K , α θ f ( z ) exists, then f is differentiable at z and f ′ ( z ) = z α − 1 D θ K , α f ( z ) .

  3. If 0 < α ≤ 1 , z ∈ R , and D N , α θ f ( z ) exists, then f is differentiable at z and f ′ ( z ) = e ( 1 − α ) z D N , α θ f ( z ) .

  4. If 0 < β ≤ 1 , z > − 1 Γ ( β ) , and D B , β θ f ( z ) exists, then f is differentiable at z and f ′ ( z ) = z + 1 Γ ( β ) β − 1 D B , β θ f ( z ) .

Remark 3.1

The fact that conformable differentiability is equivalent to ordinary differentiability was noted in [35, 36]. From corollaries 3.2 and 3.3, it follows that the conformable and the Katugampola derivatives are identical on z > 0 . Indeed, for z > 0 , both are equal to z 1 − α f ′ ( z ) . The equality of these two derivatives is sufficiently surprising to warrant listing it as a theorem and giving another proof.

Theorem 3.2

Let z > 0 and f be any function. For 0 < α ≤ 1 , D C , α θ f ( z ) exists if and only if D K , α θ f ( z ) , in which case, D C , α θ f ( z ) = D K , α θ f ( z ) .

Proof

We can establish the theorem by directly using Theorem 3.1. Let θ 1 = θ C , α and θ 2 = θ K , α . We have θ 1 ( z , h ) = θ 2 ( z , H ) is equivalent to z + h z 1 − α = z e H z − α . That is, H = z α ln 1 + h z α . Thus,

∂ H ∂ h ( z , h ) = z α h + z α , and ∂ H ∂ h ( z , 0 ) = 1 .

From Theorem 3.1, it follows that if D C , α θ f ( z ) exists, then so does D K , α θ f ( z ) , and they are equal. Likewise, taking θ 1 = θ K , α , and θ 2 = θ C , α , we have θ 1 ( z , h ) = θ 2 ( z , H ) is equivalent to z e h z − α = z + H z 1 − α , which is equivalent to H = z α ( e h z − α − 1 ) . Thus,

∂ H ∂ h ( z , h ) = e h z − α and ∂ H ∂ h ( z , 0 ) = 1 .

By Theorem 3.1, it follows that if D K , α θ f ( z ) exists, then so does D C , α θ f ( z ) , and they are equal.□

Another application of Theorem 3.1 is to show that the existence of the derivatives in Example 2.1 is independent of the order of the derivative. Thus, we have

Corollary 3.4

If z > 0 and 0 < α 1 , α 2 ≤ 1 , then the existence of any of D C , α 1 θ f ( z ) , D C , α 2 θ f ( z ) , D K , α 1 θ f ( z ) , or D K , α 2 θ f ( z ) implies the existence of the others. In which case, we have

D C , α 1 θ f ( z ) = D K , α 1 θ f ( z ) = z α 2 − α 1 D C , α 2 θ f ( z ) = z α 2 − α 1 D K , α 2 θ f ( z ) .

Proof

By Corollary 3.3, if any of these derivatives exist, then f ′ ( z ) exists and by Corollary 3.2, the existence of the ordinary derivative implies the existence of each of the derivatives listed with

D C , α i θ f ( z ) = D K , α i θ f ( z ) = z 1 − α i f ′ ( z ) from which the corollary follows.□

Similarly, we can show that the existence of the β -derivatives is independent of the order of the derivative.

Example 3.1

Since D C , α θ f ( z ) = z 1 − α f ′ ( z ) , a conformable differential equation is just an ordinary differential equation. Thus, the differential equation in Example 4.1 of [22], namely, y ( 1 ⁄ 2 ) + y = z 2 + 2 z 3 ⁄ 2 , where y ( 1 ⁄ 2 ) denotes D C , 1 ⁄ 2 θ y , is just the equation z 1 ⁄ 2 y ′ + y = z 2 + 2 z 3 ⁄ 2 , which is easily solved by using the integrating factor e 2 z to obtain y ( z ) = z 2 + C e − 2 z .

4 Simplifying θ -derivatives

In Section 3, we avoided addressing the M-fractional derivative. This was done to avoid the difficulties in solving z + h = z E β , j ( H z − α ) for H . In this section, we develop tools that enable us to bypass the need to compute H . We use them to deduce differentiabillity from θ -differentiability in a more general setting, and use these results to show that all truncated M-fractional derivatives coincide. We also describe a method for replacing the function θ by a simpler one that gives identical θ -derivative, and we illustrate this simplification with some examples.

Theorem 4.1

Let z be fixed with θ ( z , h ) , ∂ θ ∂ h ( z , h ) defined and continuous for h in a neighborhood of 0. Let ε θ ( z ) = ∂ θ ∂ h ( z , 0 ) ≠ 0 . If D θ k ( z ) exists, then k is differentiable at z with k ′ ( z ) = 1 ε θ ( z ) D θ k ( z ) .

Proof

Fix z as in the theorem. Let ψ ( h ) = θ ( z , h ) − z . We have ψ ( 0 ) = 0 and ψ ′ ( 0 ) = ∂ θ ∂ h ( z , 0 ) = ε θ ( z ) ≠ 0 . By the inverse function theorem, there are U 1 and U 2 neighborhoods of 0, so that ψ : U 1 → U 2 is a diffeomorphism. Let ζ = ψ − 1 : U 2 → U 1 and note that ζ ′ ( h ) = 1 ψ ′ ( ζ ( h ) ) . For each h ∈ U 2 , we have h = ψ ( ζ ( h ) ) = θ ( z , ζ ( h ) ) − z . Thus, θ 0 ( z , h ) = z + h = θ ( z , ζ ( h ) ) . The rest of the proof mimics that of Theorem 3.1. Since g ( h ) = ζ ( h ) satisfies g ′ ( 0 ) ≠ 0 , by Lemma 3.1(2), there exists δ > 0 such that g ( h ) ≠ g ( 0 ) = 0 for 0 < ∣ h ∣ < δ . Since g ( h ) and f ( u ) = k ( θ ( z , u ) ) − k ( z ) u satisfy the conditions of Lemma 3.1(1) with lim u → 0 f ( u ) = D θ k ( z ) and lim h → 0 g ( h ) = 0 , we have lim h → 0 f ∘ g ( h ) = D θ k ( z ) . Now

k ( z + h ) − k ( z ) h = k ( θ ( z , ζ ( h ) ) ) − k ( z ) h = k ( θ ( z , ζ ( h ) ) ) − k ( z ) ζ ( h ) ζ ( h ) h = f ∘ g ( h ) g ( h ) h .

Since

lim h → 0 g ( h ) h = ζ ′ ( 0 ) = 1 ψ ′ ( ζ ( 0 ) ) = 1 ψ ( 0 ) = 1 ε θ ( z ) ,

it follows that

□ k ′ ( z ) = D θ k ( z ) 1 ε θ ( z ) .

By using Theorems 3.1 and 4.1, we have

Corollary 4.1

Let z be fixed with θ ( z , h ) , ∂ θ ∂ h ( z , h ) defined and continuous for h in a neighborhood of 0 with ∂ θ ∂ h ( z , 0 ) ≠ 0 . Then D θ f ( z ) exists if and only if f ′ ( z ) does. In which case, D θ f ( z ) = ε θ ( z ) f ′ ( z ) .

Example 4.1

In [34], the limit

D f ( z ) = lim h → 0 f z + h Γ ( β ) z 1 − α Γ ( β − α + 1 ) − f ( z ) h

is proposed as the definition of a new fractional derivative. This is a θ -derivative with θ ( z , h ) = z + h Γ ( β ) z 1 − α Γ ( β − α + 1 ) .

Clearly, ε θ ( z ) = Γ ( β ) z 1 − α Γ ( β − α + 1 ) ≠ 0 for z > 0 . Thus, by Corollary 4.1, for z > 0 , this derivative exists if and only if the ordinary derivative does, in which case, it is Γ ( β ) z 1 − α Γ ( β − α + 1 ) f ′ ( z ) .

Applying the above corollary to the two functions θ 1 and θ 2 , we obtain:

Corollary 4.2

Let θ 1 ( z , h ) , θ 2 ( z , h ) , ∂ θ 1 ∂ h ( z , h ) and ∂ θ 2 ∂ h ( z , h ) be defined and continuous for h in a neighborhood of 0. If ∂ θ 1 ∂ h ( z , 0 ) ≠ 0 and ∂ θ 2 ∂ h ( z , 0 ) ≠ 0 , then D θ 1 f ( z ) exits if and only if D θ 2 f ( z ) exists. In that case, D θ 1 f ( z ) = ε θ 1 ( z ) ε θ 2 ( z ) D θ 2 f ( z ) . In particular, if ε θ 1 ( z ) = ε θ 2 ( z ) ≠ 0 , then D θ 1 f ( z ) = D θ 2 f ( z ) .

By using Corollary 4.2, we have that the M-fractional derivatives are independent of the number of terms retained in the sum, and their common value is a multiple of the conformable derivative.

Corollary 4.3

For z > 0 and j 1 , j 2 > 0 , we have

D β , j 1 , α θ f ( z ) = D β , j 2 , α θ f ( z ) = 1 Γ ( β + 1 ) D C , α θ f ( z ) .

We use Corollary 4.2, to simplify the function θ used in computing derivatives, as given in the following corollary.

Corollary 4.4

Let θ ( z , h ) and ∂ θ ∂ h ( z , h ) be defined and continuous for h in a neighborhood of 0. If ∂ θ ∂ h ( z , 0 ) ≠ 0 , then D θ f ( z ) = D θ 1 f ( z ) , where θ 1 ( z , h ) = z + h ε θ ( z ) .

Example 4.2

By using Corollary 4.4, we obtain the following simplifications:

  • For θ = θ K , α , the simplified version is θ 1 = θ C , α the conformable derivative.

  • For θ = θ β , j , α , the simplified version is θ 1 ( z , h ) = z + h z 1 − α Γ ( β + 1 ) = θ β , 1 , α ( z , h ) .

5 Some desirable properties of fractional derivative not satisfied by θ -derivatives

There is no general agreement on what constitute the desirable properties for a fractional derivative [37–40]. We consider three properties and show that θ -derivatives fail to satisfy two of them.

  1. The most import property is the semigroup property, i.e., the requirement that D α D β f ( z ) = D α + β f ( z ) . This is never satisfied by θ -derivative. In general, applying two θ -derivatives to a function does not result in a θ -derivative of that function. Indeed, for D θ 1 D θ 2 f ( z ) = D θ 3 f ( z ) to hold for twice differentiable function f , where ε θ 1 ( z ) ≠ 0 and ε θ 2 ( z ) ≠ 0 , we would need to have

    ε θ 1 ( z ) ε θ 2 ( z ) f ″ ( z ) + ε θ 1 ( z ) ε θ 2 ′ ( z ) f ′ ( z ) = ε θ 1 ( z ) ( ε θ 2 ( z ) f ′ ( z ) ) ′ = ε θ 3 ( z ) f ′ ( z ) .

    Taking f to be a polynomial with f ″ ( z ) = 1 and f ′ ( z ) = 0 , we obtain that ε θ 1 ( z ) ε θ 2 ( z ) = 0 , which is impossible.

  2. Another desirable property for derivatives that is not satisfied by θ -derivatives is the requirement that lim α → 0 D α f ( z ) = f ( z ) . Indeed, all θ -derivatives are multiples of f ′ , so will not converge to f .

  3. A third desirable property is lim α → 1 D α f ( z ) = f ′ ( z ) . Whether this property holds depends on the family of θ ’s used. For conformable and Katugampola derivatives, we have that D α f ( z ) = z 1 − α f ′ ( z ) , which clearly converge to f ′ ( z ) . For the M-fractional derivative, we have seen that D α f ( z ) = z 1 − α Γ ( β + 1 ) f ′ ( z ) , which will converge to f ′ ( z ) Γ ( β + 1 ) . Thus, the property will be satisfied only when Γ ( β + 1 ) = 1 . For the beta derivative, we have D β f ( z ) = z + 1 Γ ( β ) 1 − β f ′ ( z ) , which will converge to f ′ ( z ) when β approaches 1.

6 Integrals corresponding to θ -derivatives

We can easily deduce the integral corresponding to D θ . We seek an operator I θ that is a right inverse of D θ . That is, D θ I θ f ( z ) = f ( z ) . Let g ( z ) = I θ f ( z ) . The requirement is that f ( z ) = D θ g ( z ) = ε θ ( z ) g ′ ( z ) , so g ′ ( z ) = 1 ε θ ( z ) f ( z ) . This leads to the following definition.

Definition 6.1

Suppose ε θ ( z ) is nonzero and continuous on [ a , b ] . The θ -integral is defined by

I a θ f ( z ) = ∫ a z f ( t ) ε θ ( t ) d t .

Theorem 6.1

Let θ ( z , t ) and ∂ θ ∂ h ( z , t ) be defined and continuous for h in a neighborhood of 0. If ε θ ( z ) nonzero on [ a , b ] and f is continuous at z, then D θ I a θ f ( z ) = f ( z ) .

Proof

Since f ε θ is continuous at z , the fundamental theorem of calculus implies d d z I a θ f ( z ) = f ( z ) ε θ ( z ) . Thus, D θ I a θ f ( z ) = ε θ ( z ) f ( z ) ε θ ( z ) = f ( z ) , establishing the desired result.□

Note that if θ ( z , h ) and ∂ θ ∂ h ( z , h ) are continuous and ε θ is nonzero on [ a , b ] , then

I a θ D θ f ( z ) = ∫ a z 1 ε θ ( t ) D θ f ( t ) d t = ∫ a z 1 ε θ ( t ) ε θ ( t ) f ′ ( t ) d t = f ( z ) − f ( a ) .

Example 6.1

By applying Theorem 6.1 to the derivatives given in Example 2.1, we obtain

  1. The integral corresponding to the conformable and Katugampola derivative is given by

    I θ f ( z ) = ∫ a z f ( s ) ε θ ( s ) d s = ∫ a z f ( s ) s 1 − α d s .

  2. The integral corresponding to the new conformable derivative is given by

    I N , α θ f ( z ) = ∫ z a f ( s ) e ( 1 − α ) s d s .

  3. The integral corresponding to the β -derivative is

    I θ f ( z ) = ∫ a z f ( s ) ε θ ( s ) d s = ∫ a z f ( s ) s + 1 Γ ( β ) 1 − β d s .

  4. For the M-fractional derivative, we obtain the integral

    I θ f ( z ) = ∫ a z f ( s ) ε θ ( s ) d s = Γ ( β + 1 ) ∫ a z f ( s ) s 1 − α d s .

In Section 5, we showed that the semigroup property does not hold for θ -derivative. It should also be noted that the corresponding property for fractional integrals I a α I a β = I a α + β does not hold for θ -integrals. Indeed, suppose that I a θ 2 I a θ 1 f ( z ) = I a θ 3 f ( z ) . The left hand side is

∫ a z 1 ε θ 2 ( t ) ∫ a t f ( s ) ε θ 1 ( s ) d s d t = ∫ a z f ( s ) ε θ 1 ( s ) ∫ s z 1 ε θ 2 ( t ) d t d s = ∫ a z f ( s ) ( C ( z ) − C ( s ) ) ε θ 1 ( s ) d s = C ( z ) ∫ a z f ( s ) ε θ 1 ( s ) d s − ∫ a z f ( s ) C ( s ) ε θ 1 ( s ) d s ,

where C ′ ( z ) = 1 ε θ 2 ( z ) .

To obtain the desired equality, we need to have

C ( z ) ∫ a z f ( s ) ε θ 1 ( s ) d s − ∫ a z f ( s ) C ( s ) ε θ 1 ( s ) d s = ∫ a z f ( s ) ε θ 3 ( s ) d s .

Differentiating both sides with respect to z , we obtain 1 ε θ 2 ( z ) ∫ a z f ( s ) ε θ 1 ( s ) d s = f ( z ) ε θ 3 ( z ) . If we choose f to be smooth and zero on [ a , z − δ ] for δ small and with f ( z ) = 1 , we can make the integral, and hence, the whole left hand side arbitrarily small, while the right hand side has the constant value 1 ε θ 3 ( z ) . Thus, this value must be zero, which is impossible.

7 Adaption of classical theorems to θ -derivatives

In this section, we derive the θ -derivatives versions of some results from classical analysis.

Lemma 7.1

If θ : O → R is continuous and z-crossing, I ⊆ U = π 1 ( O ) . If f , g : I → R are θ -differentiable, then

  1. (Leibniz rule) D θ f g ( z ) = f ( z ) D θ g ( z ) + g ( z ) D θ f ( z ) .

  2. If g ( z ) ≠ 0 , then D θ f g ( z ) = D α f ( z ) g ( z ) − f ( z ) D α f ( z ) g ( z ) 2 .

Proof

Since θ is z -crossing, θ -differentiability at z implies continuity at z , and we have

D θ f g ( z ) = lim h → 0 f ( θ ( z , h ) ) g ( θ ( z , h ) ) − f ( z ) g ( z ) h = lim h → 0 f ( θ ( z , h ) ) ( g ( θ ( z , h ) ) − g ( z ) ) h + lim h → 0 g ( z ) ( f ( θ ( z , h ) ) − f ( z ) ) h = f ( z ) D θ g ( z ) + g ( z ) D θ f ( z ) ,

and

□ D θ f g ( z ) = lim h → 0 f g ( θ ( z , h ) ) − f g ( z ) h = lim h → 0 f ( θ ( z , h ) ) g ( z ) − f ( z ) g ( θ ( z , h ) ) h g ( z ) g ( θ ( z , h ) ) = lim h → 0 ( f ( θ ( z , h ) ) − f ( z ) ) g ( z ) − f ( z ) ( g ( θ ( z , h ) ) − g ( z ) ) h g ( θ ( z , h ) ) g ( z ) = g ( z ) D θ f ( z ) − f ( z ) D θ g ( z ) g ( z ) 2 .

Lemma 7.2

Let f : [ a , b ] → R be a function.

  1. If c ∈ ( a , b ) and f is θ -differentiable at c and has a minimum or maximum at c, then D θ f ( c ) = 0 .

  2. (Rolle’s theorem) Let θ be continuous and z-crossing. If f is θ -differentiable on [ a , b ] with f ( a ) = f ( b ) , then there exists c ∈ ( a , b ) such that D θ f ( c ) = 0 .

Proof

We proceed as follows:

  1. If c is a minimum, then f ( θ ( c , h ) ) − f ( c ) ≥ 0 . Thus, lim h → 0 + f ( θ ( c , h ) ) − f ( c ) h ≥ 0 , lim h → 0 − f ( θ ( c , h ) ) − f ( c ) h ≤ 0 , and D θ f ( c ) = lim h → 0 + f ( θ ( c , h ) ) − f ( c ) h = lim h → 0 − f ( θ ( c , h ) ) − f ( c ) h = 0 . The proof for maximum is similar.

  2. By Lemma 2.2, f is continuous on [ a , b ] , so has a minimum and maximum. If f is constant, then D θ f ( c ) = 0 for each c ∈ ( a , b ) . If f is not constant, then its minimum or maximum must differ from f ( a ) = f ( b ) , so must be achieved on an interior point c . By the first part of the Lemma, D θ f ( c ) = 0 .□

Lemma 7.3

(Mean value theorem) Assume that θ and ∂ θ ∂ h are continuous with ε θ nonzero on [ a , b ] . Let B ( z ) = ∫ a z 1 ε θ ( t ) d t . If f is θ -differentiable on [ a , b ] , then there exists a c ∈ ( a , b ) such that

D θ f ( c ) = f ( b ) − f ( a ) B ( b ) − B ( a ) .

Proof

Since ε θ ≠ 0 on [ a , b ] and f is θ -differentiable, it is differentiable, hence continuous on [ a , b ] . The fundamental theorem of calculus and our assumptions that ε θ ≠ 0 gives that B ′ ( z ) = 1 ε θ ( z ) .

Thus, by Theorem 3.1, B is θ -differentiable with D θ B ( z ) = 1 . Let g ( z ) = f ( z ) − f ( a ) − ( f ( b ) − f ( a ) ) B ( z ) − B ( a ) B ( b ) − B ( a ) , then g is continuous and θ -differentiable. By Lemma 7.2, since g ( a ) = g ( b ) = 0 , there exists c ∈ ( a , b ) such that 0 = D θ g ( c ) = D θ f ( c ) − f ( b ) − f ( a ) B ( b ) − B ( a ) .□

Lemma 7.4

(Darboux’s theorem) Let θ and ∂ θ ∂ h be continuous with ε θ positive on [ a , b ] . If f is θ -differentiable on [ a , b ] , then

  1. If D θ f ( a ) < 0 < D θ f ( b ) , then there exists c ∈ ( a , b ) such that D θ f ( c ) = 0 .

  2. If D θ f ( a ) < λ < D θ f ( b ) , then there exists c ∈ ( a , b ) such that D θ f ( c ) = λ .

Proof

Under the assumption of Lemma 7.4, we have

  1. By Theorem 4.1, f is differentiable, hence continuous on [ a , b ] ; thus, it has a minimum. Since f ′ ( a ) = D θ f ( a ) ε θ ( a ) < 0 and f ′ ( b ) = D θ f ( b ) ε θ ( b ) > 0 this minimum is not at an endpoint. Thus, the minimum is attained at some c ∈ ( a , b ) . By Lemma 7.2(1), we have D θ f ( c ) = 0 .

  2. By Theorem 4.1, f is differentiable. Let B ( z ) = ∫ a z 1 ε θ ( t ) d t , then by Theorem 3.1 and the fundamental theorem of calculus, D θ B ( z ) = ε θ ( z ) B ′ ( z ) = ε θ ( z ) 1 ε θ ( z ) = 1 . Let g ( z ) = f ( z ) − λ B ( z ) then D θ g ( z ) = D θ f ( z ) − λ . Thus, D θ g ( a ) < 0 and D θ g ( b ) > 0 . By the first part of the lemma, there is a c ∈ ( a , b ) so that 0 = D θ g ( c ) = D θ f ( c ) − λ .□

Note that by replacing f by − f , we obtain: If D θ f ( a ) > λ > D θ f ( b ) , then there exits a c ∈ ( a , b ) such that D θ f ( c ) = λ .

8 Tarasov’s theorem and its relation to the obtained results

In this section, the author investigate the relation of the obtained results with Tarasov’s theorem. In [41], Tarsov proved the following theorem

Theorem 8.1

If the operator D α can be applied to functions from C 2 ( U ) , where U is a neighborhood of z 0 , and if D α is linear and satisfies Leibniz rule D α f g ( z ) = f ( z ) D α g ( z ) + g ( z ) D α f ( z ) , then

(7) D α f ( z ) = ( D α ( z ) ) f ′ ( z ) .

Remark 8.1

In [41], the claim was that D α f ( z ) = a ( z ) f ′ ( z ) for some function a ( z ) , but from the proof it is clear that a ( z ) = D α z . This coincides with the multiplier we found for θ -derivatives.

It might be worth noting that the proof in [41] does not require that D α be applied to twice continuously differentiable functions. The proof is valid if we assume that C 1 -functions are D α -differentiable.

Tarasov’s theorem states that for a derivative satisfying Leibniz Rule, the derivative of a smooth function is the product of the derivative of the identity and the ordinary derivative of the function, i.e., D α f ( z ) = D α ( z ) f ′ ( z ) . This expression is identical to the one we derived. However, the assumptions and proof were very different:

  1. In Theorem 4.1, we deduced the differentiability of a function from the existence of its θ -derivative. Clearly, this result would be meaningless, if we can only apply θ -derivatives to smooth functions.

  2. In terms of its assumptions, Tarasov’s theorem is closer to Theorem 3.1, where we assumed differentiability, but not smoothness.

  3. Tarasov’s theorem makes no assumptions about the way the fractional derivative is computed, so can be used with a wider class of fractional derivatives. Thus, in [42], this theorem is applied to, among others, the Kolwankar-Gangal derivative [43] under the assumption of smoothness.

9 Are θ -derivatives fractional derivatives?

As we have noted in Sections 5 and 6, θ -derivatives and the corresponding integrals do not satisfy important properties desirable for fractional derivatives and integrals such as the semigroup property. In addition, since these derivatives are multiples of ordinary derivatives, they do not expand the class of differentiable functions and as a consequence do not give more flexibility in modelling physical phenomena. Thus, the author feels, that the critics insisting that these derivatives should not be classified as fractional derivatives might be justified. However, this does not mean that these derivatives are not worthy of study as mathematical constructs.

10 Conclusions

In this article, we developed a framework for examining a class of derivatives. We showed that the existence of these derivatives is equivalent to ordinary differentiability. We obtained an expression for these derivative in terms of the ordinary derivative. We showed that some of these derivatives that were assumed to be different are actually identical. We also derived a form for the corresponding integrals and showed that these derivatives and integrals fail to satisfy desirable properties for fractional derivatives and integrals. We adapted some of the theorem of classical analysis to these derivatives. Finally, we discussed the issue of whether these should be classified as fractional and ventured the opinion that they probably should not be thus classified.

Acknowledgments

The author acknowledge the Deanship of Scientific Research at King Faisal University for the financial support.

  1. Funding information: This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. GRANT3222).

  2. Conflict of interest: The author states no conflict of interest.

  3. Data availability statement: All relevant data are within the manuscript.

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Received: 2023-08-15
Revised: 2023-10-04
Accepted: 2023-10-16
Published Online: 2023-11-11

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2023-0143
Keywords for this article
conformable derivative; new conformable derivative; M-fractional derivative; beta derivative; Katugampola derivative
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Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
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  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
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  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
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  60. On semigroups of transformations that preserve a double direction equivalence
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  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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