Asymptotic behavior of fractional super-linear differential equations

Miroslav Bartušek; Zuzana Došla

doi:10.1515/ms-2025-0024

Article Open Access

Asymptotic behavior of fractional super-linear differential equations

Miroslav Bartušek and Zuzana Došla

Published/Copyright: May 7, 2025

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From the journal Mathematica Slovaca Volume 75 Issue 2

Abstract

This work is devoted to the nonlinear fractional differential equations of arbitrary order, considering the Riemann-Liouville derivative. We extend well-known oscillation criteria for super-linear integer-order differential equations. Finally, we show some discrepancies between the asymptotic behavior of solutions for fractional and integer order case.

Keywords: Ffractional differential equations; Riemann-Liouville derivative; super-linearity; asymptotic behavior; oscillation

MSC 2010: Primary 34C10; 34E11; 65L17

1 Introduction

Consider the fractional differential equations

(1.1) D0αx(t)+q(t)f(x(t))=0,t>0,

and

(1.2) D0αx(t)−q(t)f(x(t))=0,t>0,

where n – 1 < α ≤ n and n ∈ {3, 4, . . . }, and D0α denotes the Riemann-Liouville fractional differential operator of the order ff, that is, for α < n,

D0αx(t)=1Γ(n−α)dndtn∫0t(t−s)n−α−1x(s)ds,

and D0nx(t)=dndtnx(t)=x(n)(t). As usually, Γ denotes the Gamma function.

Throughout the paper, we assume.

(i₁) the function q is a real-valued positive continuous defined on (0,∞) such that for some λ₀ > 1

(1.3) ∫01tα−1−λ0(n−α)q(t)dt<∞;

(i₂) the function f ∈ C⁰(ℝ), f(u)u > 0 for u ≠ 0 satisfies for some λ such that 1 < λ ≤ λ₀

(1.4) |f(u)|≥|u|λfor u∈R,

and

(1.5) lim sup|u|→∞f(u)|u|λ0sgnu<∞.

In last few years, the study of fractional differential equations and their applications have been developed in various directions. The current research in the asymptotic and oscillation theory is devoted to the differential and difference equations with various types of fractional operators, see, e.g., [1, 4, 5, 7, 8, 9, 12, 13, 15-17] and references therein.

Our aim here is to study asymptotic and oscillatory behavior of solutions to equations (1.1) and (1.2) when the nonlinearity term f is super-linear. In particular, we extend well-known oscillation criteria for super-linear integer-order differential equations [14. Corollary 10.1] and we point out some similarities and discrepancies between fractional and integer-order case.

If α = n, then (1.1) and (1.2) are integer-order differential equations, which can be considered on [t₀,∞) for some t₀ > 0. The asymptotic and oscillatory behavior of integer-order differential equations have been deeply studied in the literature, see, e.g., monographs [6, 11, 14] and recent contributions [2, 3]. The classification of integer-order differential equations as having Property A or Property B has been introduced by I. T. Kiguradze, see [14. Sections 1.1,10.2] and the prototypes of such classification are equations x⁽ⁿ⁾ + kx = 0, k ∈ ℝ. Roughly speaking, Property A and Property B state that every solution which may oscillate, does oscillate ([11. Section 8.5]). The natural question which arises whether it is possible to extend these properties for nonlinear fractional differential equations (1.1) and (1.2).

A partial answer was given in [4, 5, 15]. In [15], the fractional linear equations of the order α ∈ (2, 3] were investigated assuming q is positive continuous function on (0,∞) such that q ∈ L¹(0,1) and ∫0∞q(t)dt=∞. Under the same conditions on q, oscillatory properties of solutions of (1.1) were treated in [4] for any α > 2 and f satisfying (1.5) for λ₀ = 1. In [5], asymptotic and oscillatory behavior for (1.1) and (1.2) was investigated under the conditions (1.3)-(1.5) for 0 < λ ≤ 1, λ ≤ λ₀. This paper extend and completes [4, 5] for the super-linear case λ > 1.

By a solution x of (1.1) or (1.2), we mean a real-valued function in C(0,∞) such that

(1.6) tn−αx(t)∈C[0,∞),

D0αx exists on (0,∞) and x satisfies (1.1) or (1.2) for t > 0. A solution x is said to be oscillatory if it has arbitrary large zeros, otherwise is called nonoscillatory.

Let x be a solution of equation (1.1) or (1.2). Denote for α ∈ (n – 1, n)

(1.7) x1(t)=J0n−αx(t)=1Γ(n−α)∫0t(t−s)n−α−1x(s)ds,t>0,

where J0n−α is the Riemann-Liouville fractional integral operator. For α = n, we set x₁(t) = x(t). Any solution x of (1.1) satisfies x₁ ∈ C[0,∞), and x1(n)=DnJ0n−αx=D0αx, see [5. Lemma 1 and Remark 1]. Thus from (1.1)

(1.8) x1(n)(t)=−q(t)f(x(t)),t>0,

and similarly, any solution x of (1.2) satisfies x1(n)(t)=q(t)f(x(t)) for t > 0.

To describe oscillatory properties of solutions of (1.1), (1.2), the following definitions were introduced in [5] as an extension of Property A and Property B for integer-order differential equations.

Definition 1

Equation (1.1) is said to have weak Property A if for n odd, every its nonoscillatory solution x satisfies for large t

(1.9) (−1)ix(t)x1(i)(t)>0(i=0,…,n−1),

and for n even, every its nonoscillatory solution x satisfies either

(1.10) (−1)i+1x(t)x1(i)(t)>0(i=0,…,n−1),

(1.11) x(t)x1(t)>0,(−1)i+1x(t)x1(i)(t)>0(i=1,…,n−1).

Moreover, solutions of type (1.9), (1.10) and (1.11) satisfy

(1.12) limt→∞x(t)=0,limt→∞x1(i)(t)=0,(i=2,…,n−1),

and solutions of type (1.11) satisfy limt→∞x1(t)=c≠0.

Equation (1.1) is said to have Property A if it has weak Property A, there exist no solutions of type (1.11), and solutions of type (1.9) and (1.10) satisfy limt→∞x1(t)=0.

Definition 2

Equation (1.2) is said to have weak Property B if for n even, every its nonoscillatory solution x satisfies for large t either (1.9) or

(1.13) x(t)x1(i)(t)>0(i=0,…,n−1),

and for n odd, either (1.10) or (1.11) or (1.13). Moreover, solutions of type (1.9) and (1.10) satisfy (1.12), solutions of type (1.11) satisfy limt→∞x1(t)=c≠0 and solutions of type (1.13) satisfy

(1.14) limt→∞|x(t)|=limt→∞|x1(i)(t)|=∞(i=0,1,…,n−1).

Equation (1.2) is said to have Property B if it has weak Property B, there exist no solutions of type (1.11), and solutions of type (1.9) and (1.10) satisfy limt→∞x1(t)=0.

Remark 1

If α = n, then x₁(t) = x(t) for t > 0 and solutions satisfying (1.10) as well as solutions satisfying (1.11) and (1.12) cannot exist. Thus weak Property A and weak Property B coincide with Property A and Property B for integer-order differential equations, see [14. Sections 1.1,10.2]. Moreover, solutions satisfying (1.9) are sometimes called Kneser solutions and are typical for (1.1) [(1.2)] when n is odd [even], while solutions satisfying (1.13) are called strongly increasing solutions and are typical for (1.2).

2 Preliminaries

Define ℝ⁺ = [0,∞). For β > 0, the operator J0β is defined on L¹(0,b] by

J0β[y](t)=J0βy(t):=1Γ(β)∫0t(t−s)β−1y(s)ds

for 0 < t ≤ b, and is called the Riemann-Liouville fractional integral operator of order fi. For β = 0, we set J00:=I, the identity operator.

By D, we denote the operator that maps differentiable functions onto its derivative, i.e.,

Dy(t)=ddty(t)=y′(t).

For β > 0, the operator D0β is the Riemann-Liouville fractional differential operator of order β

D0β[y](t)=D0βy(t):=DnJ0n−βy(t),

where n = Гβ˥ is the ceiling function and Dⁿ denotes the n-fold iterates of D. For β = n ∈ ℕ, D0β coincides with the classical differential operator, it means that D0βy(t)=y(n)(t). It is worth to note that y⁽ⁿ⁾ is local, while D0αy is non-local, depending on (0, b]. We refer [10] for the basic results of fractional calculus.

In the sequel, x is a solution of (1.1) or (1.2) and x₁ is defined by (1.7). We prove some properties for eventually positive solutions of (1.1); the analogous results hold for eventually negative solutions.

The next two lemmas can be viewed as an extension of the well-known Kiguradze lemma ([14. Lemma 1.1]) for integer-order differential equations.

Lemma 2.1

Let x be an eventually positive solution of (1.1). Then there exists k ∈ {0, 1, … , n–1} such that for large t either n – k is odd and

(2.1) x1(i)(t)>0 (i=0,1,…,k),(−1)n+i+1x1(i)(t)>0(i=k+1,…,n),

or n is even and

(2.2) (−1)i+1x1(i)(t)>0(i=0,1,…,n),limt→∞x1(t)=0.

Proof. It follows from Lemmas 3 and 5, Corollary 1(a) and Remark 3 in [5]. □

Lemma 2.2

If x is an eventually positive solution of (1.2), then there exists k ∈ {0, 1, . . . , n} such that for large t either n – k is even and

(2.3) x1(i)(t)>0(i=0,1,…,k),(−1)n+ix1(i)(t)>0(i=k+1,…,n),

or n is odd and (2.2) holds.

Proof. It follows from Lemmas 4 and 5, Corollary 1(b), Remark 3 in [5]. □

Lemma 2.3

Let k ∈ {1, . . . , n – 1} be fixed and

(2.4) ∫1∞tn−k−λ(n−α−k+1)q(t)dt=∞.

Assume that x is an eventually positive solution of (1.1) satisfying (2.1) or solution of (1.2) satisfying (2.3) for t ≥ t₀ > 0. In addition, for k = 1 assume that limt→∞x1(t)=∞. Then there exists T ≥ t₀ such that for t ≥ T,

(2.5) x(t)≥c02tα−nx1(t)≥c02k!tα−n+k−1x1(k−1)(t)≥c02k!tα−n+kx1(k)(t),

where c₀ = 1/Γ(α – n + 1), and

(2.6) x1(k−1)(t)≥x1(k−1)(T)+12(n−k)!∫Ttsn−k|x1(n)(s)|ds.

Proof. Using [5. Lemma 6 and Remark 3], we get the first two inequalities in (2.5). Moreover, by Lemmas 2.1 and 2.2, we have (−1)n−kx1(n)≥0 and (2.6) holds. By (2.4), we have

∫T∞tn−k|x1(n)(t)|dt=∞.

From here the last inequality in (2.5) follows, taking into account [14. Lemma 1.3], choosing i = 1 and l = k in (1.20). □

Lemma 2.4

Let x be an eventually positive solution of (1.1) [(1.2)] satisfying either (1.9) or (1.11) with limt→∞x1(t)<∞. Then

(2.7) ∫1∞tn−1q(t)f(x(t))dt<∞.

Proof. It is similar as the proof of [4. Lemma 9] for (1.1). Assume that x is an eventually positive solution of (1.1) [(1.2)] satisfying (1.11) such that for t ≥ t₀,

x1(t)>0,(−1)i+1x1(i)(t)>0,i=1,…,n−1,

and limt→∞x1(t)=d<∞. The Taylor series at t, where t ≥ t₀, gives

−d≤x1(t0)−x1(t)=x1′(t)1!(t0−t)+x1′′(t)2!(t0−t)2+⋯+x1(n−1)(t)(n−1)!(t0−t)n−1+∫tt0(t0−s)n−1(n−1)!x1(n)(s)ds.

If x is a solution of (1.1) [(1.2)], then by Lemma 2.1 [Lemma 2.2] with k = 1, we have n is even and x1(n)(t)<0 [n is odd and x1(n)(t)>0] for t ≥ t₀. Thus all members on the right-hand side are negative, and we get (2.7).

Similarly, if x is an eventually positive solution of (1.1) [(1.2)] satisfying (1.9), then x₁ is positive decreasing. Thus, there exists limt→∞x1(t)=d<∞ and the rest of the proof is the same as above. □

Lemma 2.5

Let x be an eventually positive solution of (1.1) or (1.2) satisfying (1.9) or (1.10) or (1.11) with bounded x₁. Then there exists c > 0 such that

(2.8) 0< x(t)≤ctn−αforlarget.

Proof. By [5. Lemma 2 and Remark 3], for T > 0 fixed, there exists M > 0 such that

|x1′(t)|≤Mtn−α−1 for t∈(0,T],

and

x(t)=1Γ(α−n+1)(x1(0)tn−α+∫0tx1′(s)(t−s)n−αds) for t>0.

Under these conditions, the estimation (2.8) has been proved in [4. Lemma 7]. □

We close this section by the following inequality.

Lemma 2.6

([14. Lemma 10.3]). Let φ ∈ C([a,∞) × ℝ) be such that

φ(t,x)≥φ(t,y)fort≥a, x≥y≥0

and the differential equation

(2.9) x′(t)=φ(t,x)

has no positive solution. Let c₀ ∈ (0,∞), t₀ ∈ [a,∞) and h. [t₀,∞) → (0,∞) be continuous nondecreasing function. Then there exists no continuous function y: [t₀,∞) → (0,∞) satisfying the inequality

(2.10) y(t)≥c0+∫t0t1h(s)φ(s,h(s)y(s))ds,t≥t0.

3 Property A and Property B

Our main results are the following.

Theorem 3.1

Let n – 1 < α ≤ n, λ > 1 and

(3.1) ∫1∞tn−1−λ(n−α)q(t)dt=∞.

Then equation (1.1) has weak Property A and equation (1.2) has weak Property B.

Proof. Let n – 1 < α < n, k ∈ {1, . . . , n – 1} and x be an eventually positive solution of (1.1). When k = 1, let limt→∞x1(t)=∞. By Lemma 2.1, x is either of type (2.1) or (2.2). First, we prove that solutions satisfying (2.1) do not exist.

Assume that x satisfies (2.1) for t ≥ t₀ > 0. By Lemma 2.1, n – k is odd. The condition (3.1) implies the validity of (2.4), and Lemma 2.3 can be applied. From (1.4), (1.8), (2.5) and (2.6), we have that there exists T ≥ t₀ such that

(3.2) x1(k−1)(t)≥x1(k−1)(T)+c1∫Ttsn−k+λ(k−1+α−n)q(s)x1(k−1)(s)λds,

where c1=12(n−k)!2k!Γ(α−n+1)−λ. Put for t ≥ T

y(t)=x1(k−1)(t),φ(t,x)=tn−1−λ(n−α)q(t)|x|λ,h(t)=c11/(λ−1)tk−1.

Since h is nondecreasing, from (3.2) we have for t ≥ T

y(t)≥y(T)+∫Tt1h(s)sn−1−λ(n−α)q(s)h(s)y(s)λds,

thus

(3.3) y(t)≥y(T)+∫Tt1h(s)φ(s,h(s)y(s))ds.

Consider equation (2.9) where a = T. If there exists a solution y of (2.9) such that y(t) > 0 for t ≥ ≥ T, then from (2.9) and the fact that λ > 1, we get

∫σtsn−1−λ(n−α)q(s)ds=∫σty′(s)yλ(s)ds=∫y(σ)y(t)1τλdτ≤∫y(σ)∞1τλdτ<∞,

which is a contradiction with (3.1). Hence, (2.9) has no positive solution. Applying Lemma 2.6 with t₀ = T, there exists no function y satisfying (2.10), which is a contradiction with (3.3). Thus every nonoscillatory solution of (1.1) is one of types (1.9)-(1.11) and their limit properties (1.12) follow from [4. Lemma 2.10].

By the similar way, we get that equation (1.2) does not have solution of type (2.3) for k ∈ {1, . . . , n – 1} with limt→∞x1(t)=∞ in case k = 1. Therefore, every eventually positive solution is of type either (1.9) or (1.10) or (1.11) or (1.13).

It remains to prove that solution of (1.2) given by (1.13) satisfies (1.14). Let k = n – 1. From (2.5) and the fact that x1(n−1) is increasing, we have

(3.4) x(t)≥c2tα−1,t≥T,

where c2=c0x1(n−1)(T)/(2(n−1)!) and T, c₀ are given by Lemma 2.3. From (3.1) and (3.4), we have for t ≥ T,

x1(n−1)(t)≥∫Ttx1(n)(s)ds≥c2λ∫Ttq(s)sλ(α−1)ds≥c2λ∫Ttq(s)sn−1+λ(α−n)ds →∞as t→∞.

Thus, limt→∞x1(n−1)(t)=∞, and from (3.4), we have limt→∞x(t)=∞, so (1.14) holds.

If α = n, then the result follows from [14. Corollary 10.1]. □

As claimed above, for α = n, weak Property A for (1.1) coincides with Property A and weak Property B for (1.2) coincides with Property B introduced in [14]. In the fractional case, the following holds.

Theorem 3.2

Let n – 1 < α < n, λ > 1 and (3.1) hold. If there exists ε ∈ (0, 1) such that

(3.5) lim inft→∞t−ε(α−n+1)(∫1ts−n−1λ−1q(s)−1λ−1)λ−1λds=0,

then equation (1.1) has Property A and equation (1.2) has Property B.

Proof. First observe that the solution of type (1.10) satisfies limt→∞x1(t)=0, see Lemmas 2.1, 2.2. We have to prove that there exists no solution of type (1.11) and no solution of type (1.9) such that both solutions satisfy

(3.6) limt→∞x1(t)=c,c∈R, c≠0.

Assume by contradiction that there exists an eventually positive solution x either of type (1.11) or (1.9) satisfying (3.6) for c > 0. We proceed following the idea as in the proof of in [4. Theorem 1.2]. Let t₀ ≥ 1 such that x(t) > 0 for t ≥ t₀ and ε be such that 0<ε<1. By Lemma 2.5, there exists c₀ > 0 such that

(3.7) 0< x(t)≤1,x(t)≤c0tn−α for t≥t0,

and for t – t^ε ≥ t₀,

x1(t)=1Γ(n−α)(∫0t0x(s)(t−s)α−n+1ds+∫t0t−tεx(s)(t−s)α−n+1ds+∫t−tεtx(s)(t−s)α−n+1ds).

By (1.6), there exists c=max0≤s≤t0∣sn−αx(s)∣. Thus,

|∫0t0x(s)(t−s)α−n+1ds|≤c(t−t0)α−n+1∫0t0dssn−α→0

as t → ∞. By Lemma 2.4 and (1.4), we can set

L=∫t0∞sn−1q(s)xλ(s)ds<∞.

Using Hölder inequality, we have

∫t0t−tεx(s)(t−s)α−n+1ds≤1tε(α−n+1)∫t0t(sn−1q(s))1/λx(s)(sn−1q(s))1/λds≤1tε(α−n+1)(∫t0tsn−1q(s)xλ(s)ds)1/λ(∫t0ts−(n−1)/(λ−1)q−1/(λ−1)(s)ds)(λ−1)/λ≤L1/λt−ε(α−n+1)(∫t0ts−n−1λ−1q−1λ−1ds)(λ−1)/λ.

Thus, from (3.5), we have

lim inft→∞∫t0t−tεx(s)(t−s)α−n+1ds=0,

and using (3.7)

∫t−tεtx(s)(t−s)α−n+1ds≤∫t−tεtc0(t−s)α−n+1sn−αds≤c0(t−tε)n−α1n−αtε(n−α) →0

as t → ∞. Therefore, lim inft→∞x1(t)=0, so limt→∞x1(t)=0, which is a contradiction with (3.6). □

4 Examples

The following example illustrates our results.

Example 1

Consider equation

(4.1) D0αx(t)+q(t)∣x(t)∣λsgnx=0,t>0,

where n – 1 < α < n, n ≥ 3, λ > 1 and

q(t)=tλ(n−α)for t∈[0,1],tμfor t>1, μ∈R.

Obviously, (1.3) is satisfied. Moreover, (3.1) holds for

(4.2) μ≥−n+λ(n−α).

Applying Theorem 3.1, equation (4.1) has weak Property A for μ satisfying (4.2). Similarly, one can check that (3.5) is satisfied if the strict inequality in (4.2) holds. Thus, by Theorem 3.2, equation (4.1) has Property A for μ > –n + λ(n – α).

Observe that this result is in accordance with the oscillation criterion for the integer-order equation

x(n)(t)+tμ∣x(t)∣λsgnx=0,t≥t0>0,

where λ > 1. Using [14. Corollary 10.1], this equation has Property A for μ ≥ –n, i.e., for n even all solutions are oscillatory and for n odd any solution is either oscillatory or Kneser solution tending to zero as t → ∞.

Next example illustrates that (1.1) can have “purely fractional” solution satisfying (1.11) and limt→∞x1(t)=∞. Moreover, it shows that conditions in Theorem 3.1 are optimal in a certain sense.

Example 2

Consider equation

(4.3) D0αx(t)−q(t)∣x(t)∣λsgnx(t)=0,t>0,

where 5=2 < α < 3, λ > 1,

q(t)=3t−5/2c1tα−3+c2tα−5/2−λ

and c₁ = 1/Γ(α – 2), c2=8Γ(32)/Γ(α−32). The function

x(t)=c1tα−3+c2tα−5/2

satisfies (1.6), i.e., t^3–βx(t) ∈ C[0,∞) and, using (1.7)

x1(t)=1+8t,D0αx(t)=x1′′′(t)=3t−5/2,

and x is a solution of (4.3). Hence (4.3) does not have weak Property B. Observe that condition (1.3) is satisfied for λ = λ₀. Indeed, we have

q(t)≤c3t(3−α)λ−5/2 for t∈(0,1],

where c₃ = 3(c₁ + c₂)^–λ, and for λ > 1,

∫01tα−1−λ0(3−α)q(t)dt≤c3∫01tα−1−λ(3−α)t(3−α)λ−5/2dt=∫01tα−7/2dt<∞.

On the other hand, if ε > 0 and 1 < λ ≤ 1 + 2ε, then

∫1∞t2−λ(3−α)q(t)dt<∞,∫1∞t2−λ(3−α)+εq(t)dt=∞.

Thus, the condition (3.1) in Theorem 3.1 is not satisfied and cannot be replaced by the weaker condition

∫1∞tn−1−λ(n−α)+εq(t)dt=∞,ε>0.

5 Concluding remarks

We close the paper with some suggestions for possible further progress.

(1) If α = n and f(u) = |u|^≥ sgn u, λ > 1, then the condition (3.1) is necessary and sufficient in order to have (1.1) [(1.2)] Property A [Property B], see [14. Corollary 10.1 and Theorem 15.1].

If n–1 < α < n and f(u) = |u|^≥ sgn u, λ > 1, it is open problem whether the condition (3.1) is also necessary in order to have (1.1) [(1.2)] weak Property A [weak Property B].

(2) As claimed above, Property A for even-order equations means that all solutions are oscillatory. If n – 1 < α < n and n is even, it is an open problem whether solutions of (1.1) satisfying (1.10) exist, i.e., whether property A means oscillation of all solutions of (1.1).

(3) The typical phenomena for the super-linear integer-order equation

(5.1) x(n)−q(t)|x|λsgnx=0,t≥t0,

where λ > 1, n ≥ 2, is the existence of noncontinuable to infinity nonoscillatory solutions (blowup solutions). Such solutions are defined on [t₀,τ), τ < ∞, and different from zero in a left neighbourhood of τ such that limt→τ−|x(t)|=∞. Their existence can be proved by considering the Cauchy problem with the initial conditions x⁽ⁱ⁾(t₀) > 0, i = 0, 1, . . . , n – 1. If (5.1) has Property B and no solutions satisfying (1.13) exist, then solutions of the Cauchy problem must be noncontinuable to infinity, see [14. Theorem 11.4].

Hence, it is natural to study the existence of blow-up solutions for (1.2). In view of Theorem 2, this problem leads to the problem of the nonexistence of solutions satisfying (1.13).

((Communicated by Michal Fečkan)

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Received: 2024-06-28

Accepted: 2024-10-05

Published Online: 2025-05-07

Published in Print: 2025-04-28

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

Articles in the same Issue

https://doi.org/10.1515/ms-2025-0024

Keywords for this article

Ffractional differential equations; Riemann-Liouville derivative; super-linearity; asymptotic behavior; oscillation

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