Asymptotically Linear Solutions for Some Linear Fractional Differential Equations

Abstract

We establish here that under some simple restrictions on the functional coefficient 𝑎(𝑡) the fractional differential equation ₀𝐷^𝛼_𝑡[𝑡𝑥^′−𝑥+𝑥(0)]+𝑎(𝑡)𝑥=0,𝑡>0, has a solution expressible as 𝑐𝑡+𝑑+𝑜(1) for 𝑡→+∞, where ₀𝐷^𝛼_𝑡 designates the Riemann-Liouville derivative of order 𝛼∈(0,1) and 𝑐,𝑑∈ℝ.

1. Introduction

Consider the ordinary differential equation𝑥^′′+𝑓(𝑡,𝑥)=0,𝑡≥1, (1.1) where the function 𝑓∶[1,+∞)×ℝ→ℝ is continuous such that 𝑓((𝑡)|𝑥|𝑡,𝑥)≤ℎ⋅𝑔𝑡,𝑡≥1,𝑥∈ℝ. (1.2) Here, the functions ℎ∶[1,+∞)→[0,+∞) and 𝑔∶[0,+∞)→[0,+∞) are continuous, and there exists 𝜀∈[0,1] with ₁^+∞𝑡^𝜀ℎ(𝑡)𝑑𝑡<+∞. (1.3)

Then, given 𝑐, 𝑑∈ℝ, (1.1) has a solution 𝑥(𝑡), defined in a neighborhood of +∞, which is expressible as 𝑐𝑡+𝑜(𝑡) for 𝜀=0, as 𝑐𝑡+𝑜(𝑡^1−𝜀) for 𝜀∈(0,1) and, finally, as 𝑐𝑡+𝑑+𝑜(1) for 𝜀=1 when 𝑡→+∞. Such a solution is called asymptotically linear in the literature. In particular, these developments apply to the homogeneous linear differential equation 𝑥^′′+𝑎(𝑡)𝑥=0.

A unifying technique of proof for such estimates can be read in [1] and is based on the next reformulation of the differential equation (1.1) 𝑦(𝑡)=𝑡^−𝜀𝑑−_𝑡^+∞𝜏^𝜀𝑓((𝜏,𝑥(𝑡)=𝜏,𝑥))𝑑𝜏𝑐−𝑑sgn𝜀−1𝑡+𝜀𝑡_𝑡^+∞𝑦(𝜏)𝜏𝑑𝜏−(1−𝜀)^𝑡_𝑡0𝑦(𝜏)𝑑𝜏 (1.4) for some 𝑡₀≥1 large enough. For a different approach, the so-called Riccatian method, in the case of intermediate asymptotic (𝜀∈(0,1), 𝑐=0), see the technique from [2, 3].

The study of asymptotically linear solutions to linear and nonlinear ordinary differential equations is of importance in fluid mechanics, differential geometry (Jacobi fields, e.g., [4, page 239]), bidimensional gravity (the geodesics of the Euclidean planar spray 𝑥^′′=0 being the asymptotically linear solutions 𝑥(𝑡)=𝑐𝑡+𝑑), and others.

In this note, we are interested in the existence of a fractional variant for the problem of asymptotically linear solutions which can be formulated as follows: are there any nontrivial fractional differential equations which have only asymptotically linear solutions and also their solution sets contain solutions (asymptotically linear) for all the prescribed values of numbers 𝑐, 𝑑, and 𝜀? To the best of our knowledge, this is an open problem in the theory of fractional differential equations.

Fractional differential equations have been of great interest during the last few years. This follows from the intensive development of the theory of fractional calculus [5, 6] followed by the applications of its methods in various sciences and engineering [7]. We can mention that the fractional differential equations are playing an important role in fluid dynamics, traffic model with fractional derivative, measurement of viscoelastic material properties, modeling of viscoplasticity, control theory, economy, nuclear magnetic resonance, mechanics, optics, signal processing, and so on. Basically, the fractional differential equations are used to investigate the dynamics of the complex systems; the models based on these derivatives have given superior results as those based on the classical derivatives, see [8, page 305], [9–11].

To introduce a fractional differential operator of order 1+𝛼, there are three options. The first two consist of a mixed ordinary differential-Caputo fractional differential operator, namely, (^𝐶₀𝐷^𝛼_𝑡𝑥)^′(𝑡)=₀𝐷^𝛼_𝑡(𝑥^′)(𝑡), and, respectively, a Riemann-Liouville fractional differential operator (₀𝐷^𝛼_𝑡𝑥)^′(𝑡)=(₀𝐷_𝑡^1+𝛼𝑥)(𝑡).

We recall that (₀𝐷^𝛼_𝑡𝑓)(𝑡)=(1/Γ(1−𝛼))⋅(𝑑/𝑑𝑡)[∫^𝑡₀(𝑓(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠] represents the Riemann-Liouville derivative of order a of some function 𝑓, cf. [8, page 68], and Γ stands for Euler's function Gamma. Remark as well that, in general, ₀𝐷_𝑡^𝛼+𝛽𝑥≠₀𝐷^𝛼_𝑡(₀𝐷^𝛽_𝑡𝑥) for a, 𝛽∈(0,1), see [8, page 74]. To deal with iterations, Miller and Ross [12] coined the term sequential fractional differential operator of order 𝛼+𝛽 for the quantity ₀𝐷^𝛼_𝑡(₀𝐷^𝛽_𝑡𝑥), cf. [8, pages 108, 122].

Also, the quantity (^𝐶₀𝐷^𝛼_𝑡𝑓)(𝑡)=(1/Γ(1−𝛼))∫^𝑡₀(𝑓^′(𝑠)/(𝑡−𝑠)^𝛼)𝑑𝑠 has been called the Caputo derivative in physics, see [8, page 79], and it is often preferred due to its sound explanation of what the initial data signify.

The first variant of differential operator was used in [13] to study the existence of solutions 𝑥(𝑡) of nonlinear fractional differential equations that obey the restrictions 𝑥(𝑡)⟶1when𝑡⟶+∞,𝑥^′∈𝐿¹∩𝐿^∞((0,+∞),ℝ). (1.5)

The second variant of differential operator, see [14], was employed to prove that, for any real numbers 𝑥₀, 𝑥₁, the linear fractional differential equation ₀𝐷_𝑡^1+𝛼𝑥+𝑎(𝑡)𝑥=0,𝑡>0, (1.6) possesses a solution 𝑥(𝑡) with the asymptotic development 𝑥(𝑡)=𝑥₀+𝑂(1)𝑡^𝛼−1+𝑥₁𝑡^𝛼when𝑡⟶+∞. (1.7)

A recent application of the Caputo derivative can be found in [15].

All of these fractional differential operators are based upon the natural splitting of the second-order operator 𝑑²/𝑑𝑡², namely, 𝑥^′′=(𝑥^′)^′. Here, we shall introduce a different fractionalizing of 𝑥^′′ which is based on the identities 𝑡𝑥^′′=𝑡𝑥^′−𝑥^′=𝑡𝑥^′−𝑥+𝑥(0)^′,𝑡>0, (1.8) stemming from the integration technique in the Lie algebra 𝐿₂, cf. [16, page 23].

In the following section, we give a positive (partial) answer to the preceding open question. In fact, we produce some simple conditions regarding the continuous function 𝑎∶[0,+∞)→ℝ such that, given 𝑐∈ℝ−{0}, the fractional differential equation (FDE) below₀𝐷^𝛼_𝑡𝑡𝑥^′−𝑥+𝑥(0)+𝑎(𝑡)𝑥=0,𝑡>0, (1.9) possesses a solution with the asymptotic development 𝑥(𝑡)=𝑐𝑡+𝑥(0)+𝑜(1) when 𝑡→+∞.

2. Asymptotically Linear Solutions

Let us start with a result regarding the case of intermediate asymptotic.

Proposition 2.1. Set the numbers 𝜀∈(0,1), 𝑐≠0, and 𝑐₁∈(0,1), 𝐴>0, such that 𝑐|,1max|()1−𝜀⋅Γ1−𝛼𝐴≤𝑐₁. (2.1) Assume also that 𝑎∈𝐶([0,+∞),ℝ) is confined to 1+𝑡^1−𝜀𝑎(𝑡≤𝐴|)|𝑡^𝛼,𝑡>0. (2.2) Then, the FDE ₀𝐷^𝛼_𝑡𝑡𝑥^′−𝑥+𝑎(𝑡)𝑥=0,𝑡>0, (2.3) has a solution 𝑥∈𝐶([0,+∞),ℝ)∩𝐶¹((0,+∞),ℝ), with lim_𝑡↘0[𝑡^2−𝛼𝑥^′(𝑡)]=0, which verifies the asymptotic formula 𝑥(𝑡)=𝑐𝑡+𝑂(𝑡^𝜀) when 𝑡→+∞.

Proof. Introduce the complete metric space ℳ=(𝐷,𝛿), where 𝐷={𝑦∈𝐶((0,+∞),ℝ)∶sup_𝑡>0[𝑡^−𝜀|𝑦(𝑡)|]≤𝑐₁,𝑡>0} and the metric 𝛿 is given by the usual formula 𝛿𝑦₁,𝑦₂=sup_𝑡>0𝑦₁(𝑡)−𝑦₂(𝑡)𝑡^𝜀,𝑦₁,𝑦₂∈𝐷. (2.4) In particular, lim_𝑡↘0𝑦(𝑡)=0 for all 𝑦∈𝐷.
Introduce the function 𝑥∶(0,+∞)→ℝ via the formulas 𝑦=𝑡𝑥^′−𝑥,𝑥(𝑡)=𝑐𝑡−𝑡_𝑡^+∞𝑦(𝑠)𝑠²𝑑𝑠,𝑡>0. (2.5) Since lim_𝑡↘0𝑥(𝑡)=0, we deduce that 𝑥 can be continued backward to 0; so, its extension 𝑥 belongs to 𝐶([0,+∞),ℝ)∩𝐶¹((0,+∞),ℝ). Also, lim_𝑡↘0[𝑡^1−𝛼𝑦(𝑡)]=lim_𝑡↘0[𝑡^2−𝛼𝑥^′(𝑡)]=0.
Define further the integral operator 𝑇∶ℳ→ℳ by the formula (𝑇)𝑦𝑡)1(=−Γ(𝛼)^𝑡₀𝑎(𝑠)(𝑡−𝑠)^1−𝛼𝑐𝑠−𝑠_𝑠^+∞𝑦(𝜏)𝜏²𝑑𝜏𝑑𝑠,𝑡>0. (2.6) The estimate 𝑇)𝑦𝑡≤1(()Γ(𝛼)^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑐|𝑐|𝑠+¹𝑠1−𝜀^𝜀𝑑𝑠≤𝑡^𝜀1Γ(𝛼)^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑐₁+|𝑐|𝑠1−𝜀^1−𝜀𝑑𝑠≤𝑡^𝜀1Γ(𝛼)^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼1+𝑠^1−𝜀𝑐|,𝑐𝑑𝑠⋅max|¹1−𝜀≤𝑡^𝜀1Γ(𝛼)^𝑡₀𝑑𝑠(𝑡−𝑠)^1−𝛼𝑠^𝛼𝑐|,𝑐⋅𝐴max|¹()𝑐|,11−𝜀≤Γ1−𝛼⋅𝐴max|1−𝜀⋅𝑡^𝜀≤𝑐₁𝑡^𝜀,𝑡>0, (2.7) shows that 𝑇 is well defined by taking into account (2.1), (2.2).
Now, given 𝑦₁, 𝑦₂∈𝐷, we have (𝑇)𝑦₁(𝑡)−(𝑇)𝑦₂(𝑡≤1)Γ(𝛼)^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼⋅𝑠_𝑠^+∞𝑑𝜏𝜏^2−𝜀𝑑𝑠⋅𝛿𝑦₁,𝑦₂≤1Γ(𝛼)(1−𝜀)^𝑡₀𝑠^𝜀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑑𝑠⋅𝛿𝑦₁,𝑦₂≤1Γ(𝛼)𝑐|,1⋅max|⋅1−𝜀^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑑𝑠⋅𝑡^𝜀⋅𝛿𝑦₁,𝑦₂≤1Γ(𝛼)𝑐|,1max|⋅1−𝜀^𝑡₀1+𝑠^1−𝜀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑑𝑠⋅𝑡^𝜀𝛿𝑦₁,𝑦₂()𝑐|,1≤Γ1−𝛼⋅𝐴max|1−𝜀⋅𝑡^𝜀𝛿𝑦₁,𝑦₂≤𝑡^𝜀⋅𝑐₁𝛿𝑦₁,𝑦₂,𝑡>0, (2.8) and so 𝛿(𝑇(𝑦₁),𝑇(𝑦₂))≤𝑐₁𝛿(𝑦₁,𝑦₂).
The operator 𝑇 being a contraction, it has a unique fixed point 𝑦₀∈𝐷. Since 𝑡∫_𝑡^+∞(𝑦₀(𝑠)/𝑠²)𝑑𝑠=𝑂(𝑡^𝜀) when 𝑡→+∞, the proof is complete.

Theorem 2.2. Assume that (2.1) holds true and 𝑎∈𝐶([0,+∞),ℝ) verifies the sharper restriction 1+𝑡^1−𝜀|𝑎(𝑡1)|≤𝐴min𝑡^𝛼,1𝑡^𝛽,𝑡>0, (2.9) where 1>𝛽>𝛼+𝜀. Then, the solution 𝑥 of FDE (2.3) from Proposition 2.1 has the asymptotic development 𝑥(𝑡)=𝑐𝑡+𝑜(1) when 𝑡→+∞.

Proof. Notice that ^𝑡₀𝑑𝑠(𝑡−𝑠)^1−𝛼𝑠^𝛽=𝑡^𝛼−𝛽1₀𝑑𝑢(1−𝑢)^1−𝛼𝑢^𝛽=𝑡^𝛼−𝛽𝐵𝛼,1−𝛽, (2.10) where 𝐵 is the Beta function, cf. [8, page 6].
Via (2.9), we have the estimate 𝑦₀(𝑡)=𝑇𝑦₀(𝑡)≤𝑡^𝜀1Γ(𝛼)^𝑡₀1+𝑠^1−𝜀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑐|,𝑐𝑑𝑠⋅max|¹1−𝜀≤𝑡^𝜀1Γ(𝛼)^𝑡₀𝑑𝑠(𝑡−𝑠)^1−𝛼𝑠^𝛽𝑐|,𝑐⋅𝐴max|¹1−𝜀≤𝑡^{𝜀+𝛼−𝛽}⋅Γ1−𝛽Γ𝑐|,1𝛼+1−𝛽𝐴max|(1)1−𝜀=𝑜when𝑡⟶+∞. (2.11)
By means of L’Hôpital’s rule, we conclude that (recall (2.5)) lim_𝑡→+∞−𝑡_𝑡^+∞𝑦₀(𝑠)𝑠²𝑑𝑠=lim_𝑡→+∞𝑦₀(𝑡)=0. (2.12)
The proof is complete.

Our main contribution is given next.

Theorem 2.3. Set the numbers 𝜀∈(0,1−𝛼), 𝛽∈(𝛼+𝜀,1), 𝑐, 𝑑 with 𝑐²+𝑑²>0, and 𝑐₁∈(0,1), 𝐴>0, such that 𝑐|,|𝑑|,1max|()1−𝜀⋅Γ1−𝛼𝐴≤𝑐₁. (2.13) Assume also that 𝑎∈𝐶([0,+∞),ℝ) satisfies the inequality 1𝑡^𝜀+1+𝑡^1−𝜀|𝑎(𝑡1)|≤𝐴min𝑡^𝛼,1𝑡^𝛽,𝑡>0. (2.14) Then the FDE (1.9) has a solution 𝑥∈𝐶([0,+∞),ℝ)∩𝐶¹((0,+∞),ℝ), with 𝑥(0)=𝑑 and lim_𝑡↘0[𝑡^2−𝛼𝑥^′(𝑡)]=0, which has the asymptotic development 𝑥(𝑡)=𝑐𝑡+𝑑+𝑜(1)when𝑡⟶+∞. (2.15)

Proof. Keeping the notations from Proposition 2.1, introduce the change of variables 𝑦=𝑡𝑥^′−𝑥+𝑑,𝑥(𝑡)=𝑐𝑡+𝑑−𝑡_𝑡^+∞𝑦(𝑠)𝑠²𝑑𝑠,𝑡>0,𝑦∈𝐷, (2.16) and the integral operator 𝑇∶ℳ→ℳ with the formula (𝑇)𝑦𝑡)1(=−Γ(𝛼)^𝑡₀𝑎(𝑠)(𝑡−𝑠)^1−𝛼𝑐𝑠+𝑑−𝑠_𝑠^+∞𝑦(𝜏)𝜏²𝑑𝜏𝑑𝑠,𝑡>0. (2.17)
As before, we have the estimates 𝑇𝑦(𝑡)≤𝑡^𝜀1Γ(𝛼)^𝑡₀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼1𝑠^𝜀+1+𝑠^1−𝜀𝑐|,|𝑑|,𝑐𝑑𝑠⋅max|¹1−𝜀≤𝑐₁𝑡^𝜀𝑇)𝑦,𝑡>0,(₁(𝑡)−(𝑇)𝑦₂(𝑡≤𝑡)^𝜀Γ(𝛼)𝑐|,|𝑑|,1max|1−𝜀^𝑡₀1/𝑠^𝜀+1+𝑠^1−𝜀|𝑎(𝑠)|(𝑡−𝑠)^1−𝛼𝑑𝑠⋅𝛿𝑦₁,𝑦₂≤𝑡^𝜀⋅𝑐₁𝛿𝑦₁,𝑦₂,𝑡>0, (2.18) for all 𝑦, 𝑦₁, 𝑦₂∈𝐷.
Finally, for the fixed point 𝑦₀ of the operator 𝑇, we have that 𝑦₀(𝑡)=𝑇𝑦₀(𝑡)≤𝑡^{𝜀+𝛼−𝛽}⋅Γ1−𝛽Γ𝑐|,|𝑑|,1𝛼+1−𝛽𝐴max|(1)1−𝜀=𝑜when𝑡⟶+∞. (2.19)
The proof is complete.

3. Conclusion

A particular case of Theorem 2.3 is when 𝑐=0, 𝑑=1, that is, when the solution of (1.9) reads as 𝑥(𝑡)=1+𝑜(1) for 𝑡→+∞. Notice from (2.14) that the behavior of the functional coefficient 𝑎(𝑡) is confined to lim_𝑡→+∞𝑎(𝑡)=0. However, there is no restriction with respect to the (eventual) zeros of 𝑎(𝑡). On the other hand, in the recent contribution [13, Section??3], we were forced to request that the functional coefficient of the FDE has a unique zero in (0,+∞). In conclusion, the fractional differential operators proposed in (1.9), (2.3) allow more freedom for the functional coefficient.

Acknowledgment

We are indebted to a referee for several insightful suggestions.