Superlinear solutions of sublinear fractional differential equations and regular variation

We consider a sublinear fractional equation of the order in the interval (1, 2). We give conditions guaranteeing that this equation possesses asymptotically superlinear solutions. We show that all of these solutions are regularly varying and establish precise asymptotic formulae for them. Further we prove non-improvability of the conditions. In addition to the asymptotically superlinear solutions we discuss also other classes of solutions, some of them having no ODE analogy. In the very special case, when the coefficient is asymptotically equivalent to a power function and the order of the equation is 2, we get known results in their full generality. We reveal substantial differences between the integer order and non-integer order case. Among other tools, we utilize the fractional Karamata integration theorem and the fractional generalized L’Hospital rule which are proved in the paper. Several examples illustrating our results but serving also in alternative proofs are given too. We provide also numerical simulations.


Introduction
We study asymptotic properties of solutions to the fractional differential equation of the form D α+1 y = p(t)Φ γ (y), (1.1) t ∈ [0, ∞), where Φ γ (u) = |u| γ sgnu, γ ∈ (0, 1) (the sublinearity condition), p is a continuous function on [0, ∞), positive on (0, ∞), and α ∈ (0, 1). Thus the order of the equation lies in the interval (1,2). One of our main results says that if p is regularly varying, then asymptotically superlinear solutions are regularly varying; all these concepts are recalled below in this section. For these solutions asymptotic formulae are established and some of their properties are discussed. In order to put the results into a broader context, also some other classes of solutions are treated. The conditions which are involved in the theorems are sharp and shows dependence on α. If we formally set α = 1, then we obtain the known results for the second order equation On the other hand, in some instances we encounter new phenomena occurring only in purely fractional case. Note that for historical reasons, (1.2) is sometimes called Emden-Fowler type or Lane-Emden type or Thomas-Fermi type equation, but it appears also in another applications. Our equation (1.1) can be understood as a fractional extension of these models. Besides, equation (1.1) can be considered also in the framework of space-fractional diffusion equations, which are studied very extensively in the last years, see, e.g., [7]. By D β , β > 0, we mean the Caputo fractional differential operator defined as D β = I β −β D β , where D n , n ∈ N, is the classical integer order differential operator and · denotes the upper integer part. The symbol I β , β > 0, denotes the Riemann-Liouville fractional integral operator and is defined by t ≥ 0, where Γ is the Gamma function. For β = 0, we set I 0 = id, the identity operator. As the main sources of basic information on fractional calculus, we use the monographs [6,21].
We are interested particularly in existence and asymptotic behavior of asymptotically superlinear solutions of (1.1), i.e., the solutions y satisfying lim t→∞ y (t) = ∞; in the ODE literature sometimes called strongly increasing solutions or fast growing solutions. The term superlinear is justified by the fact that lim t→∞ y(t)/t = lim t→∞ y (t) = ∞. We introduce the notation for the set of asymptotically superlinear solutions by ASL := y : y is a solution of (1.1), lim t→∞ y (t) = ∞ .
It is easy to see that y ∈ ASL is eventually positive and eventually increasing. Sometimes we will deal with ASL solutions which are positive and increasing on the entire interval (and possibly start with the zero value), thus with the solutions belonging to ASL + , where ASL + := y ∈ ASL : y(t) > 0 on [0, ∞) with y (0) ≥ 0 or y(t) > 0 on (0, ∞) with y (0) > 0 .
Because of the sign condition on p, there are only two possible types of eventually positive increasing solutions in the ODE case, namely ASL solutions and asymptotically linear solutions, where the latter ones satisfy lim t→∞ y(t)/t = y (t) = const ∈ (0, ∞); in both the cases y is eventually increasing. As a by product of our considerations, we will reveal that in the purely fractional case, there is much wider variety of the classes of eventually positive increasing solutions, see in particular Remark 2. For instance, one can have a positive increasing solution y with lim t→∞ y (t) = 0 or even with 0 = lim inf t→∞ y (t) < lim sup t→∞ y (t) = ∞; such a behavior is excluded in the ODE case under the given setting. Note also that in contrast to the integer order case, if y is a positive solution of (1.1), then y does not need to be increasing and even under quite reasonable setting, its behavior can be complex.
It could seem that asymptotically (super)linear behavior is not peculiar to fractional order equations. But this conclusion is not true. Indeed, equation (1.1) can be written as system (4.1) where β = 1, q(t) = 1, and λ = 1. In accordance with the ODE case, let us define a strongly increasing solution (x, u) of (4.1) as the solution satisfying lim t→∞ x(t) = lim t→∞ u(t) = ∞. If now y = u, then y = u = x, and so lim t→∞ y(t) = lim t→∞ y (t) = ∞, thus y ∈ ASL (note that the systems where one derivative is fractional and another one is of integer order commonly appear in applications, see, e.g., [26]). In fact, such a behavior belongs among the natural ones because we consider just the Caputo operator which has the property D α+1 = D α D 1 . Anywise, this problem can be considered in a broader context of systems and derivatives in the Caputo as well as the Riemann-Liouville sense. Introducing an extension of the concept of strongly increasing solutions, we can study in parallel ASL solutions along with some other types of strongly increasing solutions. These ideas are described in more details at the end of the next section where we present directions for a future research.
Asymptotic analysis of the ODE of the form (1.2) has a long history. Here we mention several of the works which stay at the origin of our results. Bellman in his monograph [2] provided asymptotic formulae for solutions under the condition p(t) = Ct δ , C ∈ (0, ∞). Kamo and Usami in [15,16] made an analysis under the relaxed condition p(t) ∼ Ct δ as t → ∞, C ∈ (0, ∞). In [14,24,28], another substantial generalization was made in the sense that p(t) ∼ Ct δ as t → ∞ was replaced by p(t) = t δ L p (t), where L p is a slowly varying function (defined below). Note that in contrast to the condition p(t) ∼ Ct δ as t → ∞, the presence of a slowly varying component L p enables us to include a wider variety of qualitatively different situations. As will be shown, this concerns, among others, somehow critical (borderline) settings. In fact, we will observe that the purely fractional case even underlines the possibilities offered by the theory regular variation, and thanks to this framework new phenomena are revealed.
As for asymptotic theory of fractional differential equations, quite often the situations are studied where coefficients and/or solutions are close to power functions, in the sense of = t ϑ or ∼ t ϑ or o(t ϑ ) or O(t ϑ ) for t → ∞. See [1,8,18,20] for the equations of order less than one and [5,11,12,17,19,25,27] for the equations of order greater than one; note that the mentioned papers [17,19] contain also a worthwhile survey of related results.
As will emerge from our observations, thanks to the concept of regular variation one can substantially generalize the power behavior setting and at the same time one can provide quite precise information about behavior of solutions. In contrast to the ODE theory (which is surveyed in the monographs [23] and [29]), regular variation seems to occur in studying the non-integer case only very rarely and more or less marginally, see [13,22,32,33], although with no doubts it can serve as a powerful tool for investigation of asymptotic properties to fractional differential equations.
Our results extend in particular the ones obtained for ODEs in [16] in two ways: By fractional setting instead of integer-order one, and by regularly varying setting instead of asymptotically power one. As far as we know these type of results for nonlinear equations of order between 1 and 2 have not occurred in the literature. Moreover, we believe that the methods and observations used in our paper have a potential to be applied in a broader context and in a systematic study related to asymptotic descriptions of fractional equations of various types.
Next we recall some elements of the theory of regular variation; for more information see the monographs [3,10,31].
we write f ∈ RV(ϑ). If the limit in (1.3) is equal to 1, we speak about slowly varying functions; we write f ∈ SV, thus SV = RV(0). If f ∈ RV(ϑ), then relation (1.3) holds uniformly on each compact λ-set in (0, ∞) (this is the so-called Uniform convergence theorem, see, e.g., [3]). It is easy to see that f ∈ RV(ϑ) if and only if there exists a function L ∈ SV such that f (t) = t ϑ L(t) for every t. The slowly varying component of f ∈ RV(ϑ) will be denoted by L f , i.e., unless stated otherwise. The so-called Representation theorem (see, e.g., [3]) says the following: f ∈ RV(ϑ) if and only if t ≥ a, for some a > 0, where ϕ, ψ are measurable with lim t→∞ ϕ(t) = C ∈ (0, ∞) and lim t→∞ ψ(t) = 0. Selected properties of RV functions are presented in Propositions 1, 3, 4, and 5. Among other tools which play an important role in our theory, we would like to underline the following two statements which are proved in the last section. The first one can be understood as a fractional extension of the Karamata integration theorem. Notice that in contrast to the classical Karamata theorem (see Proposition 4), the convergence of the integral is possible also in the case ϑ > −1. One could establish an analogue for the Riemann-Liouville type fractional integral operator involving improper (convergent) integral ∞ which extends applicability of this tool, but this case is not needed in our considerations. Then as t → ∞.
The case ϑ = −β, where 0 < β < 1 deserves a special attention. Indeed, it leads to the formula where L f is the slowly varying component of f . In the integer order case (β = 1) such a formula does not hold (in contrast to the choice ϑ = −β) since we find ourselves in the critical setting in the Karamata integration theorem (Proposition 4-(iii)). As we will show, this fact has interesting consequences in the framework of the analysis of equation (1.1). A variant of the fractional Karamata theorem is considered also in [32]. Note that the monograph [30] discusses asymptotic expansions of the fractional integral, but only in the case where the expansion involves the power, exponential, and logarithmic terms.
The second tool which is worthy of note is the fractional extension of the following generalized L'Hospital rule (where we assume, in particular, that lim t→∞ g(t) = ∞): . (1.6) It is clear from Proposition 2 that if the limit lim t→∞ D β f (t)/D β g(t) exists, then The paper is organized as follows. All the results are gathered and accompanied by comments in the next section and all the proofs are postponed into the last section. In Sect. 2 we present, in particular, the condition guaranteeing existence of ASL + solution which is regularly varying. We also discuss necessity of the condition and show its optimality. Further we establish a precise asymptotic formula for regularly varying solutions in ASL + . By means of the so-called Asymptotic equivalence theorem we prove that all asymptotically superlinear solutions are regularly varying (and we establish their asymptotics). It is worthy of note that the method used in the proof of this result would be essentially new even in the integer order case. As already indicated, the next section contains also a series of comments where a particular emphasize is put on revealing differences between the integer order and non integer order case. The most significant differences are summarized at the end of the section where we also indicate directions for a future research. Some examples are provided too.

Results, comments, examples
We start with showing that under certain conditions (1.1) possesses a (positive) regularly varying asymptotically superlinear solution.
In addition, the existence of such y ∈ ASL + ∩ RV(ϑ) is guaranteed that it satisfies any of the initial conditions

Remark 1
In [16], it is shown that under the condition p(t) ∼ t δ as t → ∞ (which is a very special case of p ∈ RV(δ) when L p (t) ∼ 1 as t → ∞) ODE (1.2) possesses an asymptotically superlinear solution if and only if 1+γ +δ ≥ 0. Looking at Theorem 1, we see that instead of the condition 1 + γ + δ ≥ 0, we employ (2.1), i.e., the condition α + γ + δ ≥ 0 which displays the dependence on the order α. Because of α < 1, condition (2.1) implies 1 + γ + δ ≥ 0 but not vice versa. Thus the natural question is whether (2.1) can be relaxed. As we will show later, the answer is no, since (2.1) (along with the assumption on L p ) is not only sufficient but also necessary for the existence of asymptotically superlinear solutions in the non-integer case. It is worthy of note that the borderline case α + γ + δ = 0 is somehow delicate and solutions can exhibit very interesting behavior when allowing more general assumption L p ∈ SV instead of L p (t) ∼ 1 as t → ∞. See, in particular, Remark 2.

Remark 2
As already indicated, in contrast to very special setting for p in ODE (1.2) studied in [16], we allow p to be merely regularly varying. This enables us to reach very interesting conclusions. The role of the slowly varying component L p becomes highly important especially in the critical case α + γ + δ = 0. However, first let us observe that even when L p (t) ∼ 1 as t → ∞, in the critical case we come to the discrepancy between the integer order and the non-integer case since then all positive solutions of (1.2) are asymptotically superlinear (see [16,Theorem 4.2]) while (1.1) has no ASL solution (see Theorem 6) and on the other hand, (1.1) possesses a solution y which is close to the asymptotically linear one, i.e., Indeed, we obtain such a solution practically by the same way as in the proof of Theorem 1, i.e., as a fixed point of the operator K 1 , K 2 being as in the proof. Having a solution y in Ω, from y = (T y) = I α py γ , utilizing the fractional Karamata integration theorem (Proposition 1), we obtain Let us examine the role of L p in more details and in a broader context. Note that as t → ∞, and thus y is increasing but not strongly, so it is not ASL; note that lim t→∞ y(t) = ∞ holds by Proposition 3 since y(t) t L 1 1−γ p (t) as t → ∞ and can therefore be estimated from below by a regularly varying function of index 1. This behavior is in accordance with the necessary condition (Theorem 6). It is well known in the second order ODE case (see, e.g., [16]) that the only positive unbounded solutions of (1.2) are either asymptotically linear (lim t→∞ y(t)/t = lim t→∞ y (t) = const ∈ (0, ∞)) or asymptotically superlinear (lim t→∞ y(t)/t = lim t→∞ y (t) = ∞). Here we encounter a substantial difference between the integer and non-integer case. Indeed, for fractional equations, in addition to the just mentioned classes, one can have another (nonempty) classes which do not occur in the ODE case under the given general conditions (i.e., γ < 1 and p > 0). For example, if lim t→∞ L p (t) = 0, then we obtain a solution y with lim t→∞ y(t) = ∞ and lim t→∞ y (t) = 0. Since slowly varying functions permit even the behavior like lim inf t→∞ L p (t) = 0, lim sup t→∞ L p (t) = ∞, there exist solutions monotonically tending to infinity with the derivative behaving just in this wild way. An example of the slowly varying component causing such behavior is L p (t) = exp{(ln t) 1/3 cos(ln t) 1/3 }.
Another phenomenon which occurs only in the purely fractional case is the coexistence of solutions in different asymptotic classes of increasing solutions. Let us still assume α +δ +γ = 0 and take L p ∈ SV such that lim t→∞ L p (t) = 0. In the previous paragraph we showed existence of the solution y such that lim t→∞ y(t) = ∞ and lim t→∞ y (t) = 0. Under the same setting consider now the operator T in the form as t → ∞. Thus we have obtained two solutions where one is asymptotically linear while the other one is unbounded with the derivative tending to zero (the so-called intermediate solution).
The next theorem says that all regularly varying ASL solutions satisfy certain asymptotic formula.
as t → ∞, which is exactly the formula obtained for ASL solutions of ODE (1.2) in this special setting in [16]. This somehow demonstrates the sharpness of our result. Note that we have obtained this formula for (1.1) also in the borderline case α+δ+γ = 0 when p ∈ RV(δ) with L p (t) → ∞ as t → ∞ which is possible thanks to the fractional Karamata integration theorem in the critical case (see the comment after Proposition 1) and has no integer order analogy.
In the following theorem we offer an asymptotic estimate for all ASL solutions under relaxed conditions.
If y ∈ ASL, then as t → ∞.
Specially (see the final part of the proof of Theorem 3), if p ∈ RV(δ) in the previous theorem, then as t → ∞, where Q is defined by (2.5). However, as we will see, the result in this case can be substantially improved. Indeed, thanks to the next statement (the so-called asymptotic equivalence theorem), if the coefficient p is regularly varying, then regular variation along with a precise asymptotic formula (rather than only onesided asymptotic estimate) is guaranteed for all elements in ASL + . Along with (1.1) consider the equation where q positive and continuous on [0, ∞). Let the set ASL + q for (2.9) be defined analogously as the set ASL + =: ASL + p for equation (2.9).
Theorem 4 (Asymptotic equivalence theorem) Let x ∈ ASL + q and y ∈ ASL + p . Then the following hold: The next theorem is one of the most important conclusions of our paper.
Remark 4 (i) If the assumption p ∈ RV(δ) in Theorem 5 is relaxed to p(t) q(t) as t → ∞, where q is some regularly varying function of index δ, then the asymptotic formula is modified to y(t) Q(t) as t → ∞.
(ii) Since we work with superlinear solutions in Theorem 5, in view of Theorem 6, conditions (2.1), (2.2) are not restrictive.
The next corollary easily follows from Theorem 5 and Remark 4. An alternative way of its proof can be based on Example 1. If we set α = 1, then the corollary yields the result for ODE (1.2) in [16], see also Remark 3. However our statement is an extension of that result not only in the sense of fractional order but also in the sense of the presence of a general SV component of p; for the analysis of sublinear ODEs in the framework of regular variation see also [14,24,28,29].

Remark 5
Note that the condition α + γ + δ > 0 in the previous corollary cannot be replaced by (2.1). Indeed, if we have the critical case α + γ + δ = 0 and p(t) ∼ Ct δ as t → ∞, C ∈ (0, ∞), then, in particular lim sup t→∞ L p (t) < ∞, and thus the condition which is necessary for existence of an ASL solution (see Theorem 6) is not fulfilled. Here we encounter another difference between the integer order and non-integer order case. Indeed, it is known (see [16]), that if p(t) ∼ t δ as t → ∞, where 1 + γ + δ = 0, then every positive solution y of (1.2) has the from y(t) ∼ (1 − γ ) 1/(1−γ ) t(ln t) 1/(1−γ ) as t → ∞ and, in particular is ASL. On the other hand, from Remark 2 it follows under the critical case α + γ + δ = 0, if p(t) t δ , then y(t) t and y (t) 1 as t → ∞, thus y is not ASL.
The next theorem provides a necessary condition for existence of ASL solutions. As a consequence we thus demonstrate non-improvability of the conditions posed on α + γ + δ in the above presented results.

Remark 6
A closer examination of the proof of Theorem 6 shows, that for a general p, i.e., not necessarily regularly varying, one can derive necessary condition in the integral form lim sup (2.10) Looking at Theorem 6, we see that if p ∈ RV(δ), then not only in the case α+γ +δ > 0 but also in the critical case α+γ +δ = 0 one can have a necessary condition in the nonintegral form. The latter case has no integer order analogy; the necessary condition for equation (1.2) takes the form ∞ p(t)/t dt = ∞ and cannot be simplified by "removing" the integral because of the critical setting in Proposition 4.
In the following example we find an exact solution of (1.1) in a certain special case which can be used in an alternative proof of Corollary 1.

Example 1
Let δ satisfy α + γ + δ > 0 and ϑ be defined by (2.3). Consider equation (2.9) where B(a, b) being the Beta function and B(t; a, b) being the incomplete Beta function; the function B(t; a, b) is considered here in the form B(a, b). We claim that is a solution of (2.9) with the q defined by (2.11). Indeed, Employing the substitution x = (t − s)/(s + 1) in the integral, we obtain where we used the identity K ϑ Moreover, x(t) ∼ (C K ) 1 1−γ t ϑ and q(t) ∼ Ct δ as t → ∞. Applying now Theorem 4 with our particular choice of q, we get the statement of Corollary 1.

Example 2 Consider equation (2.9) where
, (2.13) We claim that is a solution of (2.9). Indeed, first note that the substitution τ = (t − s)/(s + 1) in the integral yields Hence, We have lim t→∞ x(t) = ∞ and and thus x is ASL q (where ASL q denotes the set of asymptotically superliar solutions of (2.9)). It is easy to see that G(t) ∼ ln t as t → ∞. Consequently, as t → ∞, and so we are indeed in the borderline case α + γ + δ = 0. Since ln t ∈ SV, we have x ∈ ASL q ∩ RV(1). According to Theorem 2 (formula (2.4)) it should hold 1 1−γ as t → ∞. Moreover, combining these observations with Theorem 4 (the asymptotic equivalence theorem), we obtain that if p(t) ∼ Ct −α−γ ln 1−γ t as t → ∞ for some C > 0 and y ∈ ASL + , then as t → ∞. Thanks to our previous results this statement can be obtained in a different manner -without knowing an exact solution of the auxiliary equation. Indeed, having L p (t) ∼ C ln 1−γ t → ∞ as t → ∞, the assumptions of Theorem 5 are fulfilled, thus it can be then utilized to guarantee that each y ∈ ASL + satisfies y(t) ∼ K 1 1−γ Q(t) as t → ∞. Since Q(t) ∼ t ln t as t → ∞ and ϑ = 1, we get (2.15).

Numerical simulations
The simulations were performed using FDE12 Matlab code. This code was created for numerical solving fractional differential equations, for details see [9].

Conclusion
In this paper we derive, in particular, the conditions guaranteeing that the set of asymptotically superlinear solutions of (1.1) is nonempty, all its elements are regularly varying with known index, and they satisfy quite precise asymptotic formula. We also discus some other types of solutions and reveal several purely fractional phenomena; they are listed below for clarity. Moreover, we establish theoretical tools (such as the fractional Karamata integration theorem and the fractional L'Hospital rule) which are expected to find a wider use in qualitative analysis of fractional differential equations.
One of the crucial conditions in our results is the requirement on the sign of α + γ + δ; we emphasize that the expression is dependent on the order α. Moreover, this expression becomes 1 + γ + δ when passing to α = 1, thus we obtain exactly the condition known from the ODE case. On the other hand, we have revealed several differences between the integer order and the non-integer order case. Here is the list of the most significant ones: -In the fractional Karamata integration theorem (Proposition 1), the "critical" setting ϑ = −β is in fact not critical. At the very end we mention some directions for a future research. We also discuss how superlinear behavior can be natural for fractional differential equations. We expect that our approach would work also when taking some generalizations of equation (1.1) such as D α (r (t)Φ μ (y )) = p(t)Φ γ (y), where r , p are positive continuous functions and 0 < γ < μ. Possibly we could replace the power nonlinearities by regularly varying nonlinearities. Moreover, instead of scalar equations we could consider a more general system such as where p(t) > 0, q(t) > 0, 1 < α + β < 2, and μλ < 1 with μ > 0, λ > 0. The form of the system would enable us to examine a wider range situations concerning strongly increasing solutions which are introduced in the following extended mode. We say that (x, u) is a strongly increasing solution of (4.1) provided lim t→∞ x(t) = lim t→∞ u(t) = ∞. Such a setting then includes both type of behavior, asymptotically superlinear and also asymptotically super-β-power (i.e. "bigger than t β "). Indeed, let us work with strongly increasing solutions of (4.1) and set q(t) = 1 and λ = 1.
If β = 1, then lim t→∞ u(t) = ∞, lim t→∞ u (t) = lim t→∞ x(t) = ∞, thus u is asymptotically superlinear; note that here u is automatically eventually increasing. If β < 1, then lim t→∞ D β u(t) = lim t→∞ x(t) = ∞. By the fractional L'Hospital rule, thus u is asymptotically super-β-power. Here the monotonicity of u is not guaranteed. In fact, these considerations are valid no matter what type of the fractional derivative is involved, the Caputo one or the Riemann-Liouville one. On the other hand, passing to the associated scalar equations (of the order in the interval (1, 2)) we find that while the asymptotically superlinear behavior is somehow peculiar to the Caputo derivative, the asymptotically super-β-power behavior is peculiar to the Riemann-Liouville derivative. Indeed, the equation is obtained from (4.1) when q(t) = 1, λ = 1, and β = 1. Taking now the Riemann-Liouville derivative (denoted by R D β ) instead of the Caputo one in (4.1), then the system reduces to provided q(t) = 1, λ = 1, and α = 1. The property of being strongly increasing means lim t→∞ u(t) = lim t→∞ u (t) = ∞ in the former case while lim t→∞ u(t) = lim t→∞ R D β u(t) = ∞ in the latter case. Moreover, u is increasing but u does not need to be monotone and may have complex behavior in the former case, while u does not need to be monotone and may have complex behavior but R D β u is increasing in the latter case.
We have dealt just with increasing type solutions (mostly the strongly ones), so this suggests to make a similar analysis for other types of solutions in the solution space. We could see that -in contrast to ODEs -even in very special cases, the solution space of fractional differential equations may show a very complicated structure. However, we expect that tools such as the Karamata theory of regular variation adapted to fractional calculus could substantially improve our knowledge of this rich structure under a fairly general setting.
It is known from the ODE theory that passing from the sublinear case γ < 1 (or γ < μ in system (4.1)) to the superlinear one γ > 1 (or γ > μ) may substantially affect qualitative properties; the same is expected in the fractional setting. Very exceptional is then the case γ = μ (the so-called half-linear equations). The fact that we have successfully applied theory of regular variation makes us optimistic as for its wider use in asymptotic theory of fractional differential equations -not only in the sense of basic properties like Proposition 3 or the fractional Karamata integration theorem, but also in the sense of the tools such as Tauberian type theorems or the de Haan theory which can be understood as a refinement of the Karamata theory. The existing concept of discrete regular variation is expected to serve well in qualitative theory of fractional difference equations.

Proofs
We start with the auxiliary statements where selected properties of regularly varying functions are gathered.
The following statement (the so-called Karamata integration theorem) is of great importance in our theory. Its fractional version is presented in Proposition 1.  x −η f (x) dx be well defined for some given function f , and given numbers η > 0, T > 0. Then The previous proposition plays a role in the proof of the fractional Karamata integration theorem which is presented next.

Proof of Proposition 1 Since
as t → ∞, where the asymptotic relation in which L is put out of the integral follows from Proposition 5 (it is obvious that the desired positive η exists); B denotes the Beta function.

Proof of Proposition 2 First assume that
where t 0 can be taken such that D β g(t) > 0 and g(t) > 0, t ≥ t 0 , without loss of generality. From this estimation we get t ≥ t 0 . Consequently, in view of the equality lim t→∞ g(t) = ∞, we get lim inf t→∞ f (t)/g(t) ≥ m − ε. Since ε > 0 was arbitrary, we conclude that lim inf t→∞ f (t)/g(t) ≥ m. Similarly we obtain the inequality for lim inf if we assume that m / ∈ R. The inequality can be proved analogously.