Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions

: As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integrodifferential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our ﬁndings is provided.


Introduction
Of concern is the following general class of initial value problems modelled by: h t, s, H D α 2 t 0 u (s) ds , t > t 0 > 0, where H D α t 0 , H D α 1 t 0 and H D α 2 t 0 are the Hadamard fractional derivatives of orders α, α 1 and α 2 , respectively, 0 ≤ α 1 < α < 1 and 0 ≤ α 2 < α < 1. The operator H I ρ t 0 is the Hadamard fractional integral of order ρ ≥ 0. The definitions of these operators are given in Section 2.
We shall investigate the asymptotic behavior of solutions for Problem (1). Sufficient conditions on the nonlinear source term guaranteeing the convergence of solutions to logarithmic functions, for large values of time, are established. The importance of using analytical techniques to study the asymptotic behavior of solutions for Problem (1) arises from the lack of explicit solutions.
It is known that solutions for many kinds of (integer-order) ordinary differential equations may approach a certain function as time goes to infinity; in particular they may decay to zero, oscillate, or blow up in finite time. Many results in this regard exist in the literature. For example, we refer the reader to the papers [1][2][3][4][5][6], in which various classes of linear and nonlinear ordinary differential equations have been studied. Generalizing the existing results from integer orders to non-integer fractional orders is of great importance due to their numerous applications; see for instance, [7][8][9][10]. Unfortunately, imitating the techniques verbatim is not straightforward. Many difficulties arise when trying to do so. Some of these difficulties are due to the nature of the fractional derivatives themselves as they involve by definition all the past memory of solutions as well as nonregular kernels. In addition, many fundamental properties of integer-order derivatives are not valid for fractional-order derivatives. The chain rule is an example of such invalid properties. We will go around these difficulties by utilizing some adequate estimations, like desingularization methods, to deal with singular terms and by modifying and/or generalizing some versions of Bihari-LaSalle inequality.
The study of the asymptotic behavior of solutions for fractional differential equations, with Riemann-Liouville or Caputo fractional derivatives, has been investigated by many researchers, see e.g., [11][12][13][14][15][16][17][18][19][20]. The authors of [15] considered the fractional differential equation: under the condition is continuous in each argument and nondecreasing in the second one. They proved that the solutions can be represented asymptotically as a 1 + a 2 t α + O(t α−1 ), a 1 , a 2 ∈ R.
In [20], the case when the source function f depends on the solution and its sub-firstorder fractional derivative has been considered, namely, where C D α t + 0 is the Caputo fractional derivative of order α, 0 < α 1 < α < 1. The authors showed that any global solution of Problem (2) is asymptotic to ct α 1 for some real number c.
The present authors investigated the boundedness, power-type decay and asymptotic behavior of solutions for the initial value problems: Several different classes of source functions f such as and on the kernel h such as: or |h(t, s, u)| ≤ w(s)K t 1−α+α 2 |u| , for some functions k, l, P, Q, w and K have been treated, see [11,12,[21][22][23].
For fractional differential equations with Hadamard-type fractional derivatives, we found relatively few results in the literature tackling the long-time behavior of solutions of fractional initial value problems, see, [24][25][26][27][28]. The authors of [28] studied the stability and decay rate of the zero solution of the fractional differential problem: They considered first the linear case, f (u) = cu, c ∈ R, and established a criteria for the decay rate of solutions and the Lyapunov stability of the zero equilibrium. For the nonlinear case, they obtained the stability and decay rate of the hyperbolic zero equilibrium. They used a modified Laplace transform to express solutions by Mittag-Leffler functions and then used the asymptotic expansions of these functions to discuss the stability and logarithmic decay of the solutions. In [25], the authors discussed the stability of logarithmic type for the initial value problem: Under some sufficient growth conditions of f , it has been shown that the solutions decay to zero as the logarithmic function ln t Recently, the same authors considered Problem (4) in [26] with d dt H D α t 0 u (t) on the left hand side and the additional initial condition H D α t 0 u (t + 0 ) = u 1 ∈ R. They showed that solutions approach a logarithmic function as time goes to infinity.
To the best of our knowledge, the long-time behavior of solutions for the class of fractional integrodifferential equations with Hadamard fractional derivatives (1) has not been investigated so far. In this paper, we prove that solutions of (1) are asymptotic to the logarithmic function ln t t 0 α where α is the order of the involved fractional derivative. Under sufficient growth conditions, we show that there exist a real number r such that any solution for (1) in the space u ∈ C α+1 1−α,ln [t 0 , ∞), see (16), has the following property lim t→∞ u(t) The rest of this paper is organized as follows. In the next section, Section 2, we give some notations from fractional calculus and present some preliminary results. In Section 3, we introduce and prove our main results. Section 4 is devoted to an example that supports our results. A brief conclusion is presented at the end of the study in Section 5.

Preliminaries
This section is devoted to briefly introduce some basic definitions, notions, and properties from fractional calculus and fractional differential equations theory which will be used in further considerations.

Definition 3 ([7]
). The Hadamard left-sided fractional integral of order α > 0 is defined by: provided the right-hand side exists. We define H I 0 t 0 w = w. The function Γ is the Euler gamma function defined by

Definition 4 ([7]
). The Hadamard left-sided fractional derivative of order α > 0, is defined by: The next lemma shows that the Hadamard fractional derivative (or integral) of a logarithmic function results in a multiple of the same logarithmic function with the order of the fractional derivative (or integral) subtracted from (or added to) its power.
The composite of the Hadamard operators of fractional differentiation and integration with different orders is given next.
In particular, when α > β = m ∈ N, we obtain: The Hadamard differentiation operator is the left inverse to the associated Hadamard integration operator [7]. That is, for all t ∈ (t 0 , T]. In particular, for 0 ≤ α < 1, we have: For more about Hadamard fractional integral and derivative, we refer to the books [7,29,30]. The limit of the ratio of the Hadamard fractional integral H I α+1 t 0 s f (s, u(s), v(s)) (t) and the power function 1 Γ(α+1) ln t t 0 α as t → ∞ is treated in the lemma below. Proof. It is easy to see that: we get from the Dominated Convergence Theorem [31], This completes the proof.
The next lemma can be considered as a Hadamard fractional version of L'Hôpital's rule when applied to the solution of problem (1).

Lemma 5. Let u be a solution of problem
Apply H I α t 0 to (11) and use Lemmas 1 and 3, to find that: for all t > t 0 > 0. Reorder the double integral on the right-hand side and integrate by substitution to have: Dividing both sides of (13) by ln t t 0 α gives: for all t > t 0 > 0. Taking the limit of the resulting ratio at infinity and applying Lemma 4 leads to the desired result.
In the next lemma, we recall Bihari-LaSalle inequality which is a nonlinear generalization of the well-known Grönwall-Bellman inequality.
The following two types of functions are used repeatedly in the rest of this paper.

Definition 6.
A function ψ is said to be Ψ-type function if it is continuous, positive, nondecreasing on (0, ∞), The two classes of functions introduced above are not empty, see the example provided in Section 4.
Consider the following nonlinear inequality: for all t > t 0 , where a 1 , a 2 and a 3 are positive constants, g 1 , g 3 are Ω 1 -type functions, g 2 is Ω 0 -type function and ϕ j , j = 1, 2, 3 are Ψ-type functions with ϕ 1 ∝ ϕ 2 ∝ ϕ 3 . Let and where and Φ −1 j is the inverse functions of Φ j (x) = x ρ ds ϕ j (s) , j = 1, 2, 3, x > ρ > 0. A generalized version of Lemma 8 is introduced below. We shall provide an estimate for an integral term that arises later in our present problem. Although this estimate is not the best possible, it ensures that such an integral is bounded, which is exactly what we will need to prove our results below. Theorem 1. Suppose that w(t) is a continuous nonnegative function on (t 0 , ∞) satisfying Inequality (17) for all t > t 0 , a 1 , a 2 and a 3 are positive constants, g 1 , g 3 are Ω 1 -type functions, g 2 is Ω 0 -type function and ϕ j , j = 1, 2, 3, are Ψ-type functions with ϕ 1 ∝ ϕ 2 ∝ ϕ 3 . Then, and where K and K 1 are given in (18) and (19), respectively.
Proof. For t 0 < t < t 0 e, the inequality (17) becomes: and Lemma 8 is applicable. The relation (21) follows immediately. For the case t ≥ t 0 e, we have ln t t 0 ≥ 1 and (17) may be rewritten as: In virtue of the estimate (21) together with the continuity and monotonicity of the functions ϕ i , i = 1, 2, 3, we get: where K 2 is the constant given in (20).
From now on, we assume that the functions f and h on the right hand-side of (1), satisfy the conditions below: There are Ω 1 -type functions g 1 , g 3 , an Ω 0 -type function g 2 and Ψ-type functions ϕ i , i = 1, 2, 3 with ϕ 1 ∝ ϕ 2 ∝ ϕ 3 such that: Functions satisfying the above hypotheses are given in Section 4.
In the next result, we shall show that the solution for Problem (1) satisfies the useful nonlinear inequality below. and where and v j (t) = for all t > t 0 and j = 1, 2.
Proof. Firstly, we recall the relations (12) and (13) from the proof of Lemma 5, can be estimated in view of (47), Theorem 1 and the type of the functions ϕ i , i = 1, 2, 3 as follows: Obviously, Conditions (A 1 ), (A 2 ) and (A 3 ) are fulfilled with these functions and the hypotheses of Theorem 2 as well. Therefore, there exists a real constant r ∈ R such that solutions of Problem (57) satisfy lim t→∞ u(t) ln t t 0 α = r.

Conclusions
In this article, we considered a fractional integro-differential problem with Hadamard-Type fractional derivatives. The nonlinear source function depends on a lower-order Hadamard fractional derivative of the state and an integral involving another lower-order Hadamard fractional derivative of the state. We assumed the boundedness of the nonlinearities in question by some special kinds of functions in some appropriate spaces. Under these nonlinear growth conditions, we demonstrated that solutions of the initial value fractional problem under consideration are not only bounded by logarithmic functions but actually they converge asymptotically to logarithmic functions.