Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives

Miroslav Bartušek; Zuzana Došlá

doi:10.1515/dema-2025-0138

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Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives

Miroslav Bartušek and Zuzana Došlá

Published/Copyright: July 16, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

Monotonicity, oscillation, and asymptotic behavior of solutions of nonlinear fractional differential equations are investigated. Fractional differential equations are classified according to their oscillation properties, and a comparison between the classical differential operator and the Riemann-Liouville operator is made.

Keywords: fractional differential equation; Riemann-Liouville derivative; oscillation; monotonicity; asymptotic behavior

MSC 2010: 34A08; 34C10; 34E10

1 Introduction

Differential equations of non-integer orders represent an important tool for modeling many problems in the natural and technical sciences. The global nature of a non-integer-order differential operator not only allows a more realistic description of the given problems, but also brings new challenges for theoretical and computational analysis.

In this article, we study the oscillatory behavior of solutions of higher-order differential equations with the Riemann-Liouville differential operator. It is known that the choice of the fractional differential operator can affect the behavior of solutions, as well as the methods for their investigation. This is documented, e.g., in [1–4] or [5,6], where the asymptotic properties of solutions of differential equations with the Caputo derivative or with the Weyl derivative of the order α ∈ ( 1 , 2 ) have been treated. To our best knowledge, the oscillation and asymptotic behavior of solutions of differential equations with the Riemann-Liouville differential operator of the order α > 1 have not been systematically studied, in contrast with the ordinary differential equations.

Consider the differential equations

(1) D 0 α x ( t ) + q ( t ) f ( x ( t ) ) = 0 , t ∈ ( 0 , ∞ ) ,

and

(2) D 0 α x ( t ) − q ( t ) f ( x ( t ) ) = 0 , t ∈ ( 0 , ∞ ) ,

where α ∈ ( n − 1 , n ] , and n ∈ { 2 , 3 , … } and where D 0 α denotes the Riemann-Liouville fractional differential operator of the order α , i.e., for α ∈ ( n − 1 , n ) , we have

D 0 α x ( t ) = 1 Γ ( n − α ) d n d t n ∫ 0 t ( t − s ) n − α − 1 x ( s ) d s ,

and D 0 n x ( t ) = d n d t n x ( t ) = x ( n ) ( t ) . As usual, Γ denotes the Gamma function.

Throughout this article, we assume the following:

( i 1 ) the function q is positive and continuous on the open interval ( 0 , ∞ ) such that

(3) ∫ 0 1 t α − 1 − λ 0 ( n − α ) q ( t ) d t < ∞ , for some λ 0 > 0 ;

( i 2 ) the function f is continuous on R satisfying f ( u ) u > 0 for u ≠ 0 such that

(4) ∣ f ( u ) ∣ ≥ ∣ u ∣ λ , for all u ∈ R and for some λ ∈ ( 0 , 1 ] with λ ≤ λ 0 ,

(5) limsup ∣ u ∣ → ∞ f ( u ) ∣ u ∣ λ 0 sgn u < ∞ .

By a solution of (1), resp. (2), we mean a real-valued function x in the space C ( 0 , ∞ ) of continuous functions such that

(6) t n − α x ( t ) ∈ C [ 0 , ∞ ) ,

the fractional-order derivative D 0 α x exists on the interval ( 0 , ∞ ) , and x satisfies equation (1), resp. (2), for all t ∈ ( 0 , ∞ ) . A solution x is said to be oscillatory, if it has arbitrary large zeros; otherwise, it is called nonoscillatory. Let x be a solution of (1), resp. (2). For α ∈ ( n − 1 , n ) we define the function

(7) x 1 ( t ) ≔ J 0 n − α x ( t ) = 1 Γ ( n − α ) ∫ 0 t ( t − s ) n − α − 1 x ( s ) d s , t ∈ ( 0 , ∞ ) ,

where J 0 n − α is the Riemann-Liouville fractional integral operator. Moreover, for α = n , we set x 1 ( t ) ≔ x ( t ) . We will show later (Lemma 1) that for every solution x of (1), the associated function x 1 belongs to the space C [ 0 , ∞ ) and satisfies x 1 ( n ) = D n J 0 n − α x = D 0 α x . Thus, from (1), it follows that

(8) x 1 ( n ) ( t ) = − q ( t ) f ( x ( t ) ) , t ∈ ( 0 , ∞ ) .

And similarly, every solution x of (2) satisfies x 1 ( n ) ( t ) = q ( t ) f ( x ( t ) ) for t ∈ ( 0 , ∞ ) .

We note that under some stronger assumptions, special cases of equations (1) and (2) have been studied in [7,8]. In [8], these equations have been investigated for the order α ∈ ( 2 , 3 ] , the function f ( u ) = u , and q being positive and continuous on ( 0 , ∞ ) with q ∈ L 1 ( 0 , 1 ) and ∫ 0 ∞ q ( t ) d t = ∞ . Here, the numerical observations show the change of the behavior of x 1 and its derivatives (see [8, Example 3]). Under the same conditions on q as in [8], equation (1) was considered in [7] for α > 3 and the function f satisfying (5) with λ 0 = 1 . It is worth mentioning that the existence and multiplicity of positive solutions for (1) satisfying certain boundary conditions at zero and at infinity have been recently investigated in [9]. Also observe that equation (1) with λ = 1 is a special case of the equations treated in [10, Chapter 5], where the long-time behavior of time-fractional partial differential equations has been studied.

The plan of this article is follows. In accordance with the integer-order case, we introduce in Section 2 a classification of the equations according to their oscillation properties, as equations having Property A or Property B . Section 3 is devoted to the basic properties of solutions of equations (1) and (2). Our main results are presented in Section 4. Section 5 contains examples and comments.

2 Classification of equations with respect to their oscillation properties

Recall that in case α = n , D 0 α coincides with the ordinary differential operator, it means that D 0 α x ( t ) = x ( n ) ( t ) , and x 1 ( t ) = x ( t ) .

It is natural to consider the oscillation and asymptotic problems for equations (1) and (2) as an extension of these problems for ordinary differential equations

(9) x ( n ) ( t ) + p ( t ) f ( x ( t ) ) = 0 , t ∈ [ a , ∞ ) ,

and

(10) x ( n ) ( t ) − p ( t ) f ( x ( t ) ) = 0 , t ∈ [ a , ∞ ) ,

where the function p is continuous and positive on ( a , ∞ ) , p ∈ L loc 1 [ a , ∞ ) and f satisfies the assumption ( i 2 ) with λ 0 = 1 . Oscillation properties for these equations have been deeply studied in the literature (see, e.g., monographs [11,12]), and they are sometimes described using the so-called Property A and Property B, introduced by Kiguradze and Chanturia (see [12, Sections 1.1, 10.2]). The prototypes of this classification are equations x ( n ) + k x = 0 , k ∈ R .

Definition 1

Equation (9) is said to have Property A if for n even each of its solution is oscillatory, and for n odd every solution is either oscillatory or strongly decreasing in the sense that for large t ,

( − 1 ) i x ( t ) x ( i ) ( t ) > 0 ( i = 1 , … , n − 1 ) , lim t → ∞ x ( i ) ( t ) = 0 ( i = 0 , 1 , … , n − 1 ) .

Equation (10) is said to have Property B if for n odd each solution is either oscillatory or strongly increasing, i.e., for large t ,

x ( t ) x ( i ) ( t ) > 0 ( i = 1 , … , n − 1 ) , lim t → ∞ ∣ x ( i ) ( t ) ∣ = ∞ ( i = 0 , 1 , … , n − 1 ) ,

and for n even every solution is either oscillatory or strongly decreasing or strongly increasing.

This traditional definition has been modified to nonlinear dynamical systems and to equations with the disconjugate differential operator [11, Section 8.5]. Property A and Property B are sometimes called strong oscillation [11] or almost oscillation [8,13].

The following holds for the integer-order case.

Theorem A

[12, Theorem 10.4] Let λ < 1 . Equation (9) has Property A and equation (10) has Property B if and only if

(11) ∫ a ∞ t λ ( n − 1 ) p ( t ) d t = ∞ .

Theorem B

[12, Corollary 10.4]. Let λ = 1 and

(12) limsup t → ∞ t ∫ t ∞ s n − 2 p ( s ) d s = ∞ .

Then, equation (9) has Property A and equation (10) has Property B.

In accordance with the integer-order case, we introduce the following classification.

Definition 2

Equation (1) is said to have weak Property A if for n odd every its nonoscillatory solution x satisfies for large t ,

(13) ( − 1 ) i x ( t ) x 1 ( i ) ( t ) > 0 ( i = 0 , … , n − 1 ) ,

and for n even every its nonoscillatory solution x satisfies either

(14) ( − 1 ) i + 1 x ( t ) x 1 ( i ) ( t ) > 0 ( i = 0 , … , n − 1 )

(15) x ( t ) x 1 ( t ) > 0 , ( − 1 ) i + 1 x ( t ) x 1 ( i ) ( t ) > 0 ( i = 1 , … , n − 1 ) .

Moreover, solutions of type (13)–(15) satisfy

(16) lim t → ∞ x ( t ) = 0 , lim t → ∞ x 1 ( i ) ( t ) = 0 , ( i = 2 , … , n − 1 ) ,

and solutions of type (15) satisfy lim t → ∞ x 1 ( t ) = c ≠ 0 .

Equation (1) is said to have Property A if it has weak Property A, there exist no solutions of type (15), and solutions of type (13) and (14) satisfy lim t → ∞ x 1 ( t ) = 0 .

Definition 3

Equation (2) is said to have weak Property B if for n even, every its nonoscillatory solution x satisfies for large t either (13) or

(17) x ( t ) x 1 ( i ) ( t ) > 0 ( i = 0 , … , n − 1 ) ,

and for n odd either (14) or (15) or (17). Moreover, solutions of type (13) and (14) satisfy (16), solutions of type (15) satisfy lim t → ∞ x 1 ( t ) = c ≠ 0 , and solution of type (17) satisfies

(18) lim t → ∞ ∣ x ( t ) ∣ = lim t → ∞ ∣ x 1 ( i ) ( t ) ∣ = ∞ ( i = 0 , 1 , … , n − 1 ) .

Equation (2) is said to have Property B if it has weak Property B, there exist no solutions of type (15) and solutions of type (13), and (14) satisfy lim t → ∞ x 1 ( t ) = 0 .

Remark 1

Let α = n . Then, x 1 ( t ) = x ( t ) for t ∈ ( 0 , ∞ ) , and solutions satisfying (14) as well as solutions satisfying (15) together with (16) cannot exist. Thus, weak Property A and weak Property B coincide with Property A and Property B from Definition 1.

3 Basic properties of solutions

In the sequel, we prove auxiliary results for eventually positive solutions of (1) and (2); the analogous results hold for eventually negative solutions. We assume that x is a solution of (1) or (2) and the corresponding function x 1 is defined by (7). We refer [14,15] for the basic results of fractional calculus.

Throughout this section, we assume

n − 1 < α < n , λ > 0 .

Lemma 1

If x is a solution of (1) or (2), then there exists a proper limit

lim t → 0 + x 1 ( t ) = c Γ ( α − n + 1 ) ,

where c is defined by (6).

Proof

Let ε > 0 . Then, by (6), there exists t 0 > 0 such that

c − ε ≤ t n − α x ( t ) ≤ c + ε , t ∈ ( 0 , t 0 ) ,

and by (7),

x 1 ( t ) = 1 Γ ( n − α ) ∫ 0 t s n − α x ( s ) ( t − s ) α + 1 − n s n − α d s .

Using the relation (see, e.g., [14, Theorem D.6])

(19) ∫ 0 t s a − 1 ( t − s ) b − 1 d s = Γ ( a ) Γ ( b ) Γ ( a + b ) t a + b − 1 , a , b ∈ ( 0 , ∞ ) ,

we obtain

( c − ε ) Γ ( α − n + 1 ) ≤ x 1 ( t ) ≤ ( c + ε ) Γ ( α − n + 1 ) , t ∈ ( 0 , t 0 ) ,

from where the conclusion follows for ε → 0 + .□

The following expression for the solution x using x 1 plays the crucial role in our later consideration.

Lemma 2

Let x be a solution of (1) or (2). Then, for T > 0 fixed, there exists M > 0 such that

(20) ∣ x 1 ′ ( t ) ∣ ≤ M t α − n + 1 , for t ∈ ( 0 , T ] ,

and

(21) x ( t ) = 1 Γ ( α − n + 1 ) x 1 ( 0 ) t n − α + ∫ 0 t x 1 ′ ( s ) ( t − s ) n − α d s , t ∈ ( 0 , ∞ ) .

Proof

Let x be a solution of (1) or (2). First, we prove estimation (20) for t ∈ ( 0 , 1 ] . By Taylor’s theorem, we have

(22) x 1 ′ ( t ) = x 1 ′ ( 1 ) + x 1 ″ ( 1 ) 1 ! ( t − 1 ) + … + x 1 ( n − 1 ) ( 1 ) ( n − 2 ) ! ( t − 1 ) n − 2 + ∫ 1 t ( t − s ) n − 2 ( n − 2 ) ! x 1 ( n ) ( s ) d s .

Put

K 1 = max ∣ u ∣ ≤ 1 ∣ f ( u ) ∣ , K 2 = max t ∈ [ 0 , 1 ] t n − α ∣ x ( t ) ∣ , M 1 = ∑ i = 1 n − 1 ∣ x ( i ) ( 1 ) ∣ ( i − 1 ) ! .

Using (5), there exists K > 0 such that

∣ f ( u ) ∣ ≤ K 1 + K ∣ u ∣ λ 0 , for u ∈ R .

From here, (3), (8), and (22), we have

∣ x 1 ′ ( t ) ∣ ≤ M 1 + 1 ( n − 2 ) ! ∫ t 1 s n − 2 ∣ q ( s ) ∣ ( K 1 + K ∣ x ( s ) ∣ λ 0 ) d s ≤ M 1 + K 1 ( n − 2 ) ! ∫ t 1 s α − 1 − λ 0 ( n − α ) q ( s ) s n − α − 1 d s + K ( n − 2 ) ! ∫ t 1 s n − 2 − λ 0 ( n − α ) q ( s ) ( s n − α ∣ x ( s ) ∣ ) λ 0 d s ≤ M 1 + K 1 + K K 2 λ 0 ( n − 2 ) ! 1 t α − n + 1 ∫ 0 1 s α − 1 − λ 0 ( n − α ) q ( s ) < ∞ ,

from where (20) follows for T ≤ 1 . Since x 1 ′ ∈ C ( 0 , ∞ ) , (20) also holds for T > 1 .

Let t > 0 be fixed, ε ∈ ( 0 , t ) . Denote I = ( 0 , ε ) . Since x is continuous on I , we have D 0 n − α J 0 n − α x ( t ) = x ( t ) on I (see [14, Theorems 2.14]). Thus, applying [15, Corollary 2.2] to x 1 , we obtain

(23) x ( t ) = D 0 n − α x 1 ( t ) = lim ε → 0 + D ε n − α x 1 ( t ) = lim ε → 0 + 1 Γ ( α − n + 1 ) x 1 ( ε ) ( t − ε ) n − α + ∫ ε t x 1 ′ ( s ) ( t − s ) n − α d s .

By Lemma 1, we can put x 1 ( 0 ) = lim t → 0 + x 1 ( t ) . Moreover, there exists

lim ε → 0 + ∫ ε t x 1 ′ ( s ) ( t − s ) n − α d s .

Indeed, using (19) and (20) for T = t , we have

(24) ∫ 0 t ∣ x 1 ′ ( s ) ∣ ( t − s ) n − α d s ≤ M ∫ 0 t d s ( t − s ) n − α s α − n + 1 = M Γ ( n − α ) Γ ( α − n + 1 ) .

From here and (23), we obtain (21).□

Remark 2

In [7], we have proved relation (21) under the stronger assumption q ∈ L 1 ( 0 , 1 ] and λ 0 = 1 .

The next lemmas extend the well-known Kiguradze lemma ([12, Lemma 1.1]).

Lemma 3

If x is an eventually positive solution of (1), then there exists k ∈ { 0 , 1 , … , n } such that for large t, either n − k is odd and

(25) x 1 ( i ) ( t ) > 0 ( i = 0 , 1 , … , k ) , ( − 1 ) n + i + 1 x 1 ( i ) ( t ) > 0 ( i = k + 1 , … , n ) ,

or n − k is even and

(26) x 1 ( i ) ( t ) < 0 ( i = 0 , 1 , … , k ) , ( − 1 ) n + i + 1 x 1 ( i ) ( t ) > 0 ( i = k + 1 , … , n ) .

Proof

Let x be an eventually positive solution of (1) and the corresponding function x 1 be defined by (7). Denote x i ( t ) = x 1 ( i − 1 ) ( t ) for i = 2 , … , n . Then, x 1 ( n ) = D n J 0 n − α x = D 0 α x , D 0 n − α x 1 ( t ) = x ( t ) , and the vector function ( x 1 , … , x n ) is a solution of the differential system

(27) x i ′ = x i + 1 ( t ) , i = 1 , … , n − 1 , x n ′ ( t ) = − q ( t ) f ( x ( t ) ) .

Since x is eventually positive solution, x n is decreasing, the functions x i , i = 1 , … , n − 1 are monotone for large t , and there exist lim t → ∞ x i ( t ) for i = 2 , … , n . In addition, if x k ( t ) x k + 1 ( t ) > 0 , then x k − 1 ( t ) x k ( t ) > 0 and the statement follows from (8).□

By the same way as in the proof of Lemma 3, we obtain the result for (2).

Lemma 4

If x is an eventually positive solution of (2), then there exists k ∈ { 0 , 1 , … , n } such that for large t, either n − k is even and

(28) x 1 ( i ) ( t ) > 0 ( i = 0 , 1 , … , k ) , ( − 1 ) n + i x 1 ( i ) ( t ) > 0 ( i = k + 1 , … , n ) ,

or n − k is odd and

(29) x 1 ( i ) ( t ) < 0 ( i = 0 , 1 , … , k ) , ( − 1 ) n + i x 1 ( i ) ( t ) > 0 ( i = k + 1 , … , n ) .

Lemma 5

If x is an eventually positive solution of (1) or (2) such that x 1 ( t ) < 0 for large t, then lim t → ∞ x 1 ( t ) = 0 , and x 1 ′ ( t ) > 0 for large t.

Proof

The proof is similar as that one of [7, Lemma 3] for (1).□

The aforementioned sign properties of x 1 indicates a natural classification of differential equations according to their order.

Corollary 1

If x is an eventually positive solution of (1) satisfying (26), then n is even and
(30) ( − 1 ) i + 1 x 1 ( i ) ( t ) > 0 , i = 0 , 1 , … , n .
If x is an eventually positive solution of (2) satisfying (29), then n is odd and
(31) ( − 1 ) i x 1 ( i ) ( t ) > 0 , i = 0 , 1 , … , n .

Proof

Let x be a solution of (1) satisfying (26). By Lemma 5, k = 0 and the conclusion follows from Lemma 3. Similarly, Claim (b) follows from Lemmas 5 and 4.□

Lemma 6

Let k ∈ { 1 , … , n − 1 } and x be an eventually positive solution of (1) satisfying (25) or solution of (2) satisfying (28) for t ≥ t 0 > 0 . For k = 1 , assume that lim t → ∞ x 1 ( t ) = ∞ .

Then, there exists T ≥ t 0 such that for t ≥ T , the following hold:

(32) x ( t ) ≥ c 0 2 t α − n x 1 ( t ) ,

where c 0 = 1 ⁄ Γ ( α − n + 1 ) ;

(33) x 1 ( k − 1 ) ( t ) ≥ x 1 ( k − 1 ) ( t 0 ) + ( − 1 ) n − k 1 2 ( n − k ) ! ∫ T t s n − k x 1 ( n ) ( s ) d s ,

and there exists c > 0 such that

(34) I x ≔ ∫ T ∞ s n − k ∣ x 1 ( n ) ( s ) ∣ d s ≥ c ∫ T ∞ s n − k + λ ( k − 1 + α − n ) q ( s ) d s .

In addition, if I x = ∞ , then for t ≥ T ,

(35) x 1 ( t ) ≥ t k − 1 k ! x 1 ( k − 1 ) ( t ) ,

and

(36) x 1 ( k − 1 ) ( t ) ≥ t ( n − k ) ! ∫ t ∞ s n − k − 1 ∣ x 1 ( n ) ( s ) ∣ d s .

Proof

Since k ≥ 1 , we have x 1 ′ ( t ) > 0 for large t . Without loss of generality, let x 1 ′ ( t ) > 0 for t ≥ t 0 . Using (21), we have for t > t 0 ,

x ( t ) = c 0 x 1 ( 0 ) t n − α + ∫ 0 t x 1 ′ ( s ) d s ( t − s ) n − α = c 0 x 1 ( 0 ) t n − α + ∫ 0 t 0 x 1 ′ ( s ) d s ( t − s ) n − α + ∫ t 0 t x 1 ′ ( s ) d s ( t − s ) n − α ≥ c 0 x 1 ( 0 ) t n − α + c 1 ( t − t 0 ) n − α + 1 ( t − t 0 ) n − α ( x 1 ( t ) − x 1 ( t 0 ) ) ,

where c 1 = − ∫ 0 t 0 ∣ x 1 ′ ( s ) ∣ d s . From here, we obtain (32). Since for t ≥ t 0

x 1 ( t ) = x 1 ( t 0 ) + x 1 ′ ( t 0 ) 1 ! ( t − t 0 ) + … + x 1 ( k − 1 ) ( t 0 ) ( k − 1 ) ! ( t − t 0 ) k − 1 + ∫ t 0 t ( t − s ) k − 1 ( k − 1 ) ! x 1 ( k ) ( s ) d s ≥ x 1 ( k − 1 ) ( t 0 ) ( k − 1 ) ! ( t − t 0 ) k − 1 ,

there exists t 2 ≥ t 1 such that

x 1 ( t ) ≥ 1 2 x 1 ( k − 1 ) ( t 0 ) ( k − 1 ) ! t k − 1 , for t ≥ t 2 .

From here and (32), we obtain

(37) x ( t ) ≥ c 2 t k − 1 + α − n , t ≥ t 2 ,

where c 2 = c 0 x 1 ( k − 1 ) ( t 0 ) ⁄ ( 4 ( k − 1 ) ! ) .

Using Taylor’s theorem with the center t , we have

x 1 ( k − 1 ) ( t 2 ) = x 1 ( k − 1 ) ( t ) + x 1 ( k ) ( t ) 1 ! ( t 2 − t ) + … + x 1 ( n − 1 ) ( t ) ( n − k ) ! ( t 2 − t ) n − k + ∫ t t 2 ( t 2 − s ) n − k ( n − k ) ! x 1 ( n ) ( s ) d s .

Taking into account x 1 ( j ) ( t ) ( t 2 − t ) j − k + 1 < 0 , j = k , … , n for t ≥ t 2 , we obtain

x 1 ( k − 1 ) ( t ) ≥ x 1 ( k − 1 ) ( t 2 ) + ( − 1 ) n − k ∫ t 2 t ( s − t 2 ) n − k ( n − k ) ! x 1 ( n ) ( s ) d s ,

and there exists t 3 ≥ t 2 such that (33) holds with T = t 3 . Furthermore, ∣ x 1 ( n ) ( s ) ∣ = ( − 1 ) n − k x 1 ( n ) ( s ) and by (5), (8), and (37), we have

I x = ∫ t 3 ∞ s n − k ∣ x 1 ( n ) ( s ) ∣ d s ≥ c 2 λ ∫ t 3 ∞ s n − k + λ ( k − 1 + α − n ) q ( s ) d s .

If I x = ∞ , then from here and [12, Lemma 1.3], we obtain (36) and there exists t 4 ≥ t 3 such that for t ≥ t 4 ,

i x 1 ( k − i ) ( t ) ≥ t x 1 ( k − i + 1 ) ( t ) , i = 1 , … , k .

By induction, we obtain (35) for t ≥ T = t 4 .□

Lemma 7

Let x be an eventually positive solution of (1) or (2) satisfying (13) or (14) or (15) with bounded x 1 . Then, there exists c > 0 such that

x ( t ) ≤ c t α − n , for l a r g e t .

Thus, lim t → ∞ x ( t ) = 0 .

Proof

For the proof of the statement for (1), see [7, Lemma 7], taking into account Lemma 2. The proof for (2) is similar.□

Consider the differential equation

(38) u ′ ( t ) = − φ ( t , u ) ,

where φ ∈ C ( [ a , ∞ ) × R ) . Solution u of (38) is said to be regular if it is defined on [ a , ∞ ) and it is nontrivial in any neighborhood of infinity.

Lemma 8

[12, Lemma 10.4] Let φ ∈ C ( [ a , ∞ ) × R ) be such that

(39) φ ( t , u ) ≥ φ ( t , v ) , for t ≥ a , u ≥ v ≥ 0 ,

and the differential equation (38) has no positive regular solution.

Let t 0 ∈ [ a , ∞ ) and h : [ t 0 , ∞ ) → ( 0 , ∞ ) be continuous nondecreasing function. Then, there exists no continuous function y : [ t 0 , ∞ ) → ( 0 , ∞ ) satisfying the inequality

(40) y ( t ) ≥ ∫ t ∞ h ( s ) φ s , y ( s ) h ( t ) d s , t ≥ t 0 .

Remark 3

Lemmas 1–7 and Corollary 1 hold for any λ > 0 .

4 Main results

Our main results are the following.

Theorem 1

Let n − 1 < α < n , λ < 1 , and

(41) ∫ 1 ∞ t λ ( α − 1 ) q ( t ) d t = ∞ .

Then, equation (1) has weak Property A and equation (2) has weak Property B.

Proof

Let x be a positive solution of (1), and let k ∈ { 1 , … , n − 1 } . By Lemma 3, it is either of type (25) or (26). When k = 1 , let lim t → ∞ x 1 ( t ) = ∞ .

Assume by contradiction that x is of type (25) for k ≥ 1 and t ≥ t 0 > 0 . By Lemma 6 x satisfies (32) and (35), i.e., there exists T such that for t ≥ T ,

(42) x ( t ) ≥ c 0 2 t α − n x 1 ( t ) ≥ c 0 2 ( k ! ) t k − 1 + α − n x 1 ( k − 1 ) ( t ) ,

where c 0 = 1 ⁄ Γ ( α − n + 1 ) as (34) and (41) imply

(43) I x = ∫ T ∞ s n − k ∣ x 1 ( n ) ( s ) ∣ d s = ∞ .

Using (36) and (42), we have for t ≥ t 1 ,

x 1 ( k − 1 ) ( t ) ≥ c 0 2 ( k ! ) λ t ( n − k ) ! ∫ t ∞ s n − k − 1 + λ ( k − 1 + α − n ) q ( s ) ( x 1 ( k − 1 ) ( s ) ) λ d s .

Putting

y ( t ) = x 1 ( k − 1 ) ( t ) t , φ ( t , u ) = t λ ( α − 1 ) q ( t ) ∣ u ∣ λ , h ( t ) = c 1 t n − 1 − k ,

where

c 1 1 − λ = c 0 2 k ! λ 1 ( n − k ) ! ,

we have for t ≥ t 1 ,

(44) y ( t ) ≥ ∫ t ∞ h ( s ) s λ ( α − 1 ) q ( s ) y ( s ) h ( s ) λ d s = ∫ t ∞ h ( s ) φ s , y ( s ) h ( s ) d s .

Consider equation

(45) u ′ ( t ) = − t λ ( α − 1 ) q ( t ) ∣ u ∣ λ , t ≥ 1 ,

where λ < 1 . If there exists a solution of (45) such that u ( t ) > 0 for t ≥ t 0 ≥ 1 , then

∫ t 0 t s λ ( α − 1 ) q ( s ) d s = − ∫ t 0 t u ′ ( s ) u λ ( s ) d s = ∫ u ( t ) u ( t 0 ) 1 τ λ d τ ≤ ∫ 0 u ( t 0 ) 1 τ λ d τ < ∞ ,

which is a contradiction with (41). Thus, (45) has no positive regular solution. Applying Lemma 8 to (38), we obtain a contradiction with the fact that y is a solution of (44). Hence, solution of type (25) for k ∈ { 1 , … , n − 1 } does not exist. The case that x is a solution of type (26) can be eliminated by the similar way. Thus, k = 0 and the conclusion follows from Lemmas 3 and 7.

Now, consider equation (2). By the analogous arguments, the possible types of nonoscillatory solutions follow from Lemma 4.

It remains to prove that solution of type (17) satisfies (18). By contradiction, let k = n and lim t → ∞ x 1 ( n − 1 ) ( t ) = d , d < ∞ . Using (4), (8), (37), and (41), we obtain

x 1 ( n − 1 ) ( t ) ≥ ∫ t 0 t x 1 ( n ) ( s ) d s ≥ c 2 λ ∫ t 0 t q ( s ) s λ ( α − 1 ) d s = ∞ ,

which is a contradiction with the boundedness of x 1 ( n − 1 ) . Therefore, we have lim t → ∞ x 1 ( n − 1 ) ( t ) = ∞ , and from (37), lim t → ∞ x ( t ) = ∞ . Thus, (18) holds.□

Theorem 2

Let n − 1 < α < n , λ = 1 , and

(46) limsup t → ∞ t ∫ t ∞ s α − 2 q ( s ) d s = ∞ .

Then, equation (1) has weak Property A and equation (2) has weak Property B.

Proof

Let x be a positive solution of (1), and let k ∈ { 1 , … , n − 1 } . By Lemma 3, x is either of type (25) or (26). When k = 1 , let lim t → ∞ x 1 ( t ) = ∞ .

Assume by contradiction that x is of type (25) for t ≥ t 0 > 0 . Obviously, (4) holds for λ = 1 . Thus, (34) with λ = 1 and (46) imply (43). Applying Lemma 6, x satisfies (32) and (35), which gives (42). Using this, (8), and (36), we have

x 1 ( k − 1 ) ( t ) ≥ c t x 1 ( k − 1 ) ( t ) ∫ t ∞ s α − 2 q ( s ) d s ,

where c = [ 2 Γ ( α − n + 1 ) ( n − k ) ! ] − 1 . Thus,

1 c ≥ t ∫ t ∞ s α − 2 q ( s ) d s ,

which gives a contradiction with (46).

Using (4), we obtain by the similar way that possible types of nonoscillatory solutions of (2) are (13) or (17) when n is even, and (14) or (15) or (17) when n is odd. The proof of the limit properties (18) is similar as in the proof of Theorem 1.□

From Theorems 1 and 2, we obtain the following result.

Theorem 3

Let n − 1 < α < n and λ ≤ 1 . In case λ < 1 , assume that (41) hold. If there exist d > 0 and σ ∈ [ 0 , α − n + 1 ) such that

(47) t n − 1 + ( 1 − λ ) ( n − α ) + σ q ( t ) ≥ d , for l a r g e t ,

then equation (1) has Property A and equation (2) has Property B.

Proof

Observe that condition (47) implies the validity of (46). Thus, by Theorems 1 and 2, equation (1) has weak Property A and equation (2) has weak Property B. It remains to prove that solutions of type (15) do not exist and solutions x of type (13) and (14) satisfy

(48) lim t → ∞ x 1 ( t ) = 0 .

For equation (1), the statement follows from the proof of [7, Theorem 2]. Consider equation (2). If x is its solution of type (14), then (48) follows from Lemma 5. If x is a solution of type (13) or (15), then proceeding by the similar way as in the proof of [7, Theorem 2], we obtain that x 1 satisfies (48), too. If x is solution of type (15), this is a contradiction with the fact that x 1 is positive increasing. Thus, solutions of type (15) do not exist and the proof is complete.□

Remark 4

Theorem 2 substantially improves [8, Theorems 11,14] stated for the linear equations (1) and (2), where n = 3 . Theorem 3 extends [7, Theorem 2].

We complete our results with the case α = n . In this case, weak Property A and weak Property B coincide with Property A and Property B from Definition 1, and Theorems 1 and 2 can be viewed as an extension of Theorems A and B.

The following result slightly improves Theorems A and B with a = 0 in the sense that the assumption q ∈ L loc 1 ( 0 , ∞ ) is replaced by (3).

Theorem 4

Let α = n . In case λ < 1 , assume (41), and in case λ = 1 , assume (46). Then (1) has Property A and (2) has Property B.

Proof

Assume by contradiction that Property A (Property B) fails and x is a solution of (1), which disrupts properties determined in Definition 2 (Definition 3). By definition of solution, x is defined on ( 0 , ∞ ) . Obviously, x causes that these properties are not satisfied on [ 1 , ∞ ) , which is a contradiction with Theorem A (Theorem B) with a = 1 .□

5 Examples and concluding remarks

The following examples illustrate our results.

Example 1

Consider equation

(49) D 0 α x ( t ) + q ( t ) ∣ x ( t ) ∣ λ sgn x = 0 , t ∈ ( 0 , ∞ ) ,

where n − 1 < α < n , n ≥ 3 , λ ∈ ( 0 , 1 ] , q ( t ) ≡ 1 for t ∈ [ 0 , 1 ] and q ( t ) = t μ for t > 1 , μ ∈ R .

We show that (49) has Property A in the following cases:

λ = 1 , μ > − α , and λ < 1 , μ ≥ − 1 − λ ( α − 1 ) .

Obviously, α > 2 . Taking λ 0 = λ , conditions (3)–(5) are satisfied. Indeed, condition (3) reads as

∫ 0 1 t α − 1 − λ 0 ( n − α ) q ( t ) d t = ∫ 0 1 t α − 1 − λ ( n − α ) + μ d t ≤ ∫ 0 1 t α − 2 d t < ∞ .

Now, we verify conditions of Theorem 3. If λ < 1 , then (41) is satisfied because

∫ 1 ∞ t λ ( α − 1 ) + μ d t ≥ ∫ 1 ∞ t λ ( α − 1 ) − 1 − λ ( α − 1 ) d t = ∫ 1 ∞ t − 1 d t = ∞ .

It remains to check the validity of (47). Let λ = 1 . Then, μ + α > 0 and σ ∈ [ 0 , α − n + 1 ) exists such that σ + μ + α ≥ α − n + 1 , i.e., σ + μ ≥ − n + 1 . Hence,

t n − 1 + ( 1 − λ ) ( n − α ) + σ + μ ≥ 1 , for t ≥ 1 ,

and (47) holds with d = 1 . Similarly, if λ < 1 , then taking σ = max { 0 , α − n + 1 − ( 1 − λ ) ( n − 1 ) } , we have

t n − 1 + ( 1 − λ ) ( n − α ) + σ + μ ≥ t n − 1 + ( 1 − λ ) ( n − α ) + σ − 1 − λ ( α − 1 ) = t ( n − 1 ) ( 1 − λ ) + n − α − 1 + σ ≥ 1 ,

for t ≥ 1 . By Theorem 3, (49) has Property A.

The next example illustrates that solutions of (2) satisfying (15) and lim t → ∞ x 1 ( t ) = ∞ can exist. Moreover, it shows that conditions in Theorem 1 are optimal in a certain sense.

Example 2

Consider equation

(50) D 0 α x ( t ) − c 1 t μ ∣ x ( t ) ∣ λ sgn x = 0 , t ∈ ( 0 , ∞ ) ,

where 2 < α < 3 , λ ∈ ( 0 , 1 ) , λ < 2 α − 5 , and

μ = ( 5 ⁄ 2 − α ) λ − 5 ⁄ 2 , c 1 = 3 8 2 Γ ( α − 3 ⁄ 2 ) Γ ( 1 ⁄ 2 ) λ .

Putting λ 0 = λ , conditions (3)–(5) are satisfied. Indeed, condition (3) reads as

∫ 0 1 t α − 1 − λ ( 3 − α ) + μ d t = ∫ 0 1 t α − 1 − λ ( 3 − α ) + ( 5 ⁄ 2 − α ) λ − 5 ⁄ 2 = ∫ 0 1 t α − λ ⁄ 2 − 7 ⁄ 2 d t < ∞ .

We prove that the function

x ( t ) = Γ ( 1 ⁄ 2 ) 2 Γ ( α − 3 ⁄ 2 ) t α − 5 ⁄ 2

is a solution of (50). According to (7) and (19) with a = α − 3 ⁄ 2 and b = 3 − α , we have

x 1 ( t ) = 1 Γ ( 3 − α ) ∫ 0 t x ( s ) ( t − s ) α − 2 d s = Γ ( 1 ⁄ 2 ) 2 Γ ( 3 − α ) Γ ( α − 3 ⁄ 2 ) ∫ 0 t s α − 5 ⁄ 2 ( t − s ) α − 2 d s = Γ ( 1 ⁄ 2 ) 2 Γ ( 3 − α ) Γ ( α − 3 ⁄ 2 ) Γ ( α − 3 ⁄ 2 ) Γ ( 3 − α ) Γ ( 3 ⁄ 2 ) t = t .

Thus,

D 0 α x ( t ) = x 1 ‴ ( t ) = 3 8 t − 5 ⁄ 2

and (50) holds. The solution x satisfies (15); however, lim t → ∞ x 1 ( t ) = ∞ . Therefore, (50) does not have weak Property B, and in view of Theorem 1, condition (41) cannot be valid.

Let ε ∈ ( 0 , 3 ⁄ 2 ) . Consider equation (50) with

λ = 1 − 2 ε ⁄ 3 and α = 3 − ε ⁄ 4 .

Note that the condition λ < 2 α − 5 is valid. We claim that

(51) ∫ 1 ∞ t λ ( α − 1 ) + ε q ( t ) d t = ∞ .

Indeed,

∫ 1 ∞ t λ ( α − 1 ) + ε q ( t ) d t = c 1 ∫ 1 ∞ t 3 λ ⁄ 2 + ε − 5 ⁄ 2 d t = c 1 ∫ 1 ∞ t − 1 d t = ∞ .

Since (50) does not have weak Property B, condition (41) in Theorem 1 cannot be replaced by the weaker condition (51).

6 Concluding remarks

(1) In comparison, the integer and fractional case, we have showed that the nonlocal character of the operator D α , n − 1 < α < n , i.e., D n − 1 + β , where β ∈ ( 0 , 1 ) , can influence properties of solutions of equations (1) and (2) for large t . This fact arises in the behavior of the integral operator J n − α = J β , i.e.,

x 1 ( t ) = J 0 β x ( t ) = 1 Γ ( β ) ∫ 0 t ( t − s ) β − 1 x ( s ) d s , β ∈ ( 0 , 1 ) .

It would be interesting to study the asymptotic behavior of solutions for equations (1) and (2) with Weyl fractional operators.

(2) In the integer-order case n ≥ 3 , it holds that the linear equation x ( n ) + p ( t ) x = 0 , t ∈ [ t 0 , ∞ ) , has Property A if and only if the linear equation x ( n ) − p ( t ) x = 0 , t ∈ [ t 0 , ∞ ) , has Property B (see [12, Theorems 1.3, 1.3’] or [11, Theorem 8.24]). It is an open problem whether this property remains to hold for the linear equations (1) and (2).

Funding information: Authors state no funding involved.
Author contributions: Both authors have equal contributions. Both authors read and approved the final manuscript.
Conflict of interest: No potential conflict of interest was reported by the authors.

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Received: 2024-02-19

Revised: 2024-11-04

Accepted: 2025-04-29

Published Online: 2025-07-16

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

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0138

Keywords for this article

fractional differential equation; Riemann-Liouville derivative; oscillation; monotonicity; asymptotic behavior

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