Abstract
This paper is devoted to establishing some criteria for the existence of non-trivial solutions for a class of fractional q-difference equations involving the p-Laplace operator, which is nowadays known as Lyapunov's inequality. The method employed for it is based on a construction of a Green's function and its maximum value. Parallel to this result, it is worth mentioning that the Hartman-Wintner inequality for the q-fractional p-Laplace boundary value problem is also provided. It covers all previous results known in the literature on the fractional case as well as that on the classical ordinary case. The non-existence of non-trivial solutions to the q-difference fractional p-Laplace equation subject to the Riemann-Liouville mixed boundary conditions will obey such integral inequalities. The tools mainly rely on an integral form of the solution construction of a Green function corresponding to the considered problem and its properties as well as its maximum value in consideration where the kernel is the Green's function. The example that we consider here for applying this result is an eigenvalue fractional problem. To be more specific, we provide an interval where an appropriate Mittag-Leffler function to the given eigenvalue fractional boundary problem has no real zeros.
2010 Mathematics Subject Classification: 34A08, 34A40, 26D10, 33E12
1. Introduction
The field of fractional calculus and its applications to the class of partial differential equations, as well as ordinary equations, gained a rapid development. Interesting fractional results turn, in general, on the existence and non-existence of solutions. Such kinds of fractional equations come from different disciplines in sciences, covering medical and engineering matters. Techniques used in this kind of work recourse mainly to the use of Green's function and its corresponding maximum values, which is not always an easy approach. The fractional differential equations with the p-Laplacian operator involves this mathematical tool. However, to overcome this kind of difficulty, another approach can be taken, namely the use of the Cauchy-Schwarz inequality and related inequalities as holders. Different aspects have been considered by many different researchers in this respect. They have treated the existence of either a single or multiple solutions for linearity, but there have been few cases for non-linearity. In addition to this, the manner to extend these results to a general case with more a general operator seems non-evident and requires a thorough analysis of the maximum value of Green's functions. This paper is devoted to tackling this problem with the p-Laplace operator using the Green's function method for the non-linearity case.
Some results focusing on the existence of positive solutions of boundary value problems for a class of fractional differential equations with the p-Laplacian operator have been raised in previous papers (see [1–22] and the references therein). Ren and Chen [15] and Su et al. [17] established the existence of positive solutions to four-point boundary value problems for non-linear fractional differential equations with the p-Laplacian operator. However, for papers on this line concerning the q-difference type of fractional problems, we refer the reader to references [1–4, 8–12, 15, 18, 19, 19–33].
It is worthy of notice that the q-fractional calculus was introduced by Jackson [30, 31], as the reader may observe in consulting the article of Ernst [28], where he attributed the work to Jackson.
Accordingly, we mention the recent developments related to this subject (see [5, 12, 13, 32, 34–46]) and the references therein.
For multiple solutions for the non-linear case, we refer to the work done by El-Shahed and Al-Askar [47], whereas Graef et al. [48] deal with positive solutions by applying different methods.
The first result came from Liapunov [6], in the second ordinary differential equation. It was shown that if u is a non-trivial solution of
where a < b, a and b are two real constants, and the function q ∈ C([a, b]; ℝ), then the function q must satisfy the following integral inequality:
After this result, several extensions are derived from this one, and consequently, analogous inequalities are obtained for a class of fractional differential equations subject to different kind of boundary conditions (see [5, 12–14, 29, 32, 34–36, 39–41, 44, 45, 49, 50]). However, concerning the fractional q-difference boundary value problem, it was shown in Jleli and Samet [42] that a non-trivial solution of
where denotes the fractional q-derivative of Riemann-Liouville type [43, 51], and Q:[a, b] → ℝ is a continuous function, exists if the following integral inequality
is satisfied.
In the opinion of the authors, there are no articles dealing with these types of inequalities for the study of non-trivial solutions for the p-Laplacian operator involving the q-fractional case. We therefore fill the gap in the literature with this paper.
Our result generalizes that one investigated in Jleli and Samet [42].
In this work, we aim to investigate the following q-fractional boundary value problem with the p Laplace operator
where are the fractional q-derivative of the Riemann-Liouville type with 1 < α, β < 2, 0 ≤ A, B ≤ 1, 0 < ξ, δ < 1, p > 1, and Q:[a, b] → ℝ is a continuous function on [a, b].
We prove that the necessary condition of the existence of non-trivial solutions of (4) is the following:
where and are defined respectively by
where
and
Besides, we show that from this inequality derive several existing previous results in the literature as well as the standard Lyapunov inequality (1): those of Hartman and Wintner [52], Ferreira [39], and so on.
2. Definitions and Lemmas
In this section, we adopt the main tools that will be needed in the subsequent sections; these belong to q fractional calculus. Notations, definitions and lemmas are recalled in order to cover the goal of this paper, whereas, for consistency, we conserve the same notations for q fractional material as adopted in Jleli and Samet [42].
Let q ∈ (0, 1), N0 = {0, 1, 2, …}, and define
The similar q formula to the power (a − b)n with n ∈ N0 is
More generally, if α ∈ ℝ, then
For the particular case when b = 0, we note a(α) = aα. Also, the similar q formula to the power function
(x − y)n, with n ∈ N0 is
For the general case, when γ ∈ ℝ, then
It has the following properties
.
When derivatives are involved, it holds:
(t − a)α ≥ (t − b)α, for a ≤ b ≤ t, and α > 0.
We define the q-Gamma function by
In particular one has
Here and further, we recall some properties of the q-fractional derivative of a function f defined on [a, b], a < b, to ℝ.
The q-fractional derivative of a function f: [a, b] → ℝ, is defined by
and
Remark:
By using the following changes:
it is easy to conclude that if (aDqf)(t) ≤ 0 (respectively, (aDqf)(t) ≥ 0) then f is decreasing (respectively, f is increasing).
Remark:
If f is differentiable in (a, b) then
The q-fractional derivative of a function f: [a, b] → ℝ of higher order is defined by
The q-derivative of a product and a quotient of functions f and g defined on [a, b] follows as
and
Lemma 2.1. [44] For t, s ∈ [a, b], the following formulas hold:
and
wherei(aDq) denotes the q-derivative with respect to the variable i.
Remark:
If γ > 0, a ≤ b ≤ t, then
Next, we recall the q-integral of a function f defined on [a, b], a < b, to ℝ and its properties.
The q-integral of a function f: [a, b] → R is defined by
One may see that the above series is convergent if f is continuous.
If a < c < b, then the following integral equality is satisfied
The following two relations are also satisfied
An essential and important theorem that is known for the classical ordinary case is also valid for the fractional one; it is the fundamental theorem of calculus. Once applied to the fractional operator, we get
if the continuity of the function f is provided. When the continuity of f is avoided, we obtain
Another crucial integration that is very useful in dealing with non-existence of solutions for a class of fractional boundary value problems is the integration by parts. It follows as
The rule of q-integration by parts is also expressed by (see [24])
If and g are q-regular at zero, then the limit on the right-hand-side of (9) can be replaced by (For more details, see [24]).
In what follows, we define the q-fractional Riemann-Liouville integral of a function f defined on [a, b] as follows
Let us assume that f and g are two functions defined on [a, b] such that f ≤ g, then the following properties are satisfied
and
As auxiliary results, we need to use the following two lemmas. The reader may consult [23, 30, 31] for more details.
Lemma 2.2. [30, 50] Let f :[a, b] → ℝ be a continuous function. Then
Lemma 2.3. [30, 50] Let α > p − 1 and p be a positive integer. The following then holds:
2.1. Results and Consequences
The method that we would like to apply here consists of getting an equivalent integral representation of the non-trivial solution of the considered fractional boundary value problem. It therefore necessitates an appropriate construction of the Green function, which plays a crucial role in getting Lyapunov's inequalities.
In order to reduce the q fractional boundary value problem (4) to an equivalent integral equation, an auxiliary result is needed. It is formulated in the following Lemma.
Lemma 2.4. Let u ∈ AC([a, b]). The unique non-trivial solution of the q fractional boundary value problem
where 1 < α < 2, a < ϵ < b, and 0 ≤ A ≤ 1, is then given by
where
Proof. We apply Lemma 2.3 in order to reduce the fractional boundary value problem (4) to an equivalent integral one
where c1, c2 are real constants.
From u(a) = 0 and (4), we get c2 = 0. Therefore, the general solution of (4) is given by
From (14), we deduce that
and
Now, the boundary condition u(b) = Au(ϵ) yields
where
Thus, the non-trivial solution of (4) is uniquely given by
where the Green function G is defined in (11) and (12), and the proof is finished.
Lemma 2.5. Let u ∈ AC[a, b]. The q fractional boundary value problem
1 < α, β < 2, a < ϵ < b, and 0 ≤ A, B ≤ 1,
then admits a non-trivial unique solution defined by
where G(t, s) is defined in (11), (12) and
where
Proof. We use Lemma 2.4 in order to reduce the fractional differential Equation (4) to an equivalent integral one
In view of the boundary condition aDqu(a) = 0 and (23), we obtain c4 = 0. Hence the non-trivial solution of the fractional boundary value (4) is given by
Now in light of (24), we get
By the boundary condition yields
where
One may observe that, in a similar way to Lemma 2.4, we get
Thus, the given fractional boundary value problem (4) may be re-written equivalently as
Again by Lemma 2.4, the non-trivial solution of (4) is uniquely given by
The proof of the desired result is achieved.
Next we shall focus on finding the properties of the Green functions as well as their maximum principle. In order to do so, we express this fact in the following lemma.
Lemma 2.6. Let u ∈ C[a, b]. The Green functions G and H defined respectively in (11), (12) and (21), (22) are then continuous and satisfy
1 < α, β < 2, a < ϵ < b, and 0 ≤ A, B ≤ 1.
Proof. Before starting the proof of Lemma 2.6, let us mention that γ and are positive, since a < ϵ, δ < b, and 0 ≤ A, B ≤ 1.
We consider
Let us differentiate g(t, s) defined in (12) with respect to t, for s ≤ t, by
which is non-positive, since a < s < t < b.
Therefore, the function g is decreasing in its argument t, and the following inequality is satisfied
To this end, one may conclude that the right-hand-side of (35) may be expressed as
Thus, G(t, ((qs + (1 − q)a))a) is non-negative and satisfies
For t ≤ s, G is defined by
where
Similarly to above, we make differentiation with respect to t, and then we get
which is non-negative, and consequently the function g is non-decreasing in its argument t. We have
where .
Now, to prove the inequality involving H, we consider H(t, s) defined in (21) − (22) by
For t ≤ s, we have
For t ≥ s, we consider
We claim that h(t, (qs + (1 − q)a)a) is non-negative too. It is sufficient to replace α − 1 by β − 1 in all the steps of the proof of Lemma 2.6 (b), and we get the same result. So the proof is omitted, since it is similar to that of Lemma 2.6. Therefore
Likely, one may conclude that the right-hand-side h(s, (qs + (1 − q)a)) appearing in the previous inequality may be expressed as
Thus, H(t, (qs + (1 − q)a)) is non-negative and satisfies
The main result of this paper, which is a Lyapunov's inequality for a q- fractional difference p-Laplacian boundary value problem (4), will be formulated in the next theorem. We state and prove it in light of the previous lemmas.
Theorem 2.1. Assume that u is a non-trivial solution of the q fractional boundary value problem
whereare the fractional q-derivative of the Riemann-Liouville type with 1 < α, β < 2, 0 ≤ A, B ≤ 1, a < ϵ, δ < b, p > 1, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
whereandare defined in (37) and (49), respectively.
Proof. Let us define the norm of u, where u is a non-trivial solution of the q-fractional difference boundary value problem (4) by
Then, in view of Lemma 2.5, the non-trivial solution u ∈ AC([a, b], ℝ) may be re-written for all t ∈ [a, b] as follows
We then deduce
Based on the non-triviality of the solution u and the fact that p and r are conjugates, one may observe that
Due to Lemma 2.5 and Lemma 2.6, it holds that
To this end, it is worth noticing that, by letting q to 1−, we retrieve the following integral inequality due to Hartman and Wintner (see [52])
Due to this fact, the obtained integral inequality (49) may be viewed as the q-fractional integral Hartman and Wintner inequality for the p-Laplacian case. However, it is easy for the reader to get an analogous result to this fundamental inequality by considering p = 2, α = 2, and q → 1−.
Several types of Lyapunov's inequality were derived from Theorem 2.1. Hereafter, we formulate and express all of them in the following corollaries. We shall focus on covering both cases, ordinary differential equations and fractional differential equations. In addition, we illustrate this Theorem by giving an example. It consists of getting the interval of non-zeros of an appropriate eigenvalue fractional boundary value problem.
Let us start with the first result derived from Theorem 2.1.
Corollary 2.1. Suppose that u is a non-trivial solution of the q fractional boundary value problem
whereare the fractional q-derivative of Riemann-Liouville type with 1 < α, β < 2, 0 ≤ A, B ≤ 1, a < ϵ, δ < b, p > 1, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
Proof. It is sufficient to let q → 0+ and consider two cases: t ≤ s and s ≤ t. In the first case, and defined in (37) and (47) tend to zero. In the second case, they take the following form
and we get the result of this corollary.
Corollary 2.2. Suppose that u is a non-trivial solution of the q fractional boundary value problem
whereare the fractional q-derivative of the Riemann-Liouville type with 1 < α, β < 2, 0 ≤ A, B ≤ 1, a < ϵ, δ < b, p > 1, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
Proof. The result is achieved by letting q → 1− in (49).
Remarks:
The result of this corollary (Corollary 2.2) represents a Hartman-Wintner inequality for the q-fractional difference p-Laplacian boundary value problem (54). For the particular case when A = B = 0 and α = β, we obtain
When α = β = 2, we retrieve the necessary condition of existence of non-trivial solutions investigated by Lyapunov for the second ordinary differential equation subject to the Dirichlet boundary conditions, and therefore one may conclude that if the non-trivial solution corresponding to this problem exists, then the non-trivial solution of (54) exists too, and vice-versa.
Indeed, in that case, for α = β = 2, we find:
Now we focus on a second mixed-order differential inequality by taking α = β = 2. For the next derived result from Theorem 2.1, we provide an important inequality that is very useful. This is the arithmetic-geometric-harmonic inequality. It says that:
Corollary 2.3. Suppose that u is a non-trivial solution of the q fractional boundary value problem
whereare the fractional q-derivative of the Riemann-Liouville type with 1 < α, β < 2, 0 ≤ A, B ≤ 1, a < ϵ, δ < b, p > 1, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
Proof. We use the result of Corollary 2.2 by considering the arithmetic-geometric-harmonic inequality, and we get the desired result. Now we focus on a second mixed-order differential inequality by taking α = β = 2, p = 2 and therefore r = 2, since p and r are conjugates.
Corollary 2.4. Suppose that u is a non-trivial solution of the fractional q-difference boundary value problem
whereare the fractional derivative of the Riemann-Liouville type of order 2, 0 ≤ A, B ≤ 1, a < ϵ, δ < b, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
where g and h are defined in (12) and (22), respectively (with α = β = 2), and γ andare defined by
Proof. We set α = β = 2, p = r = 2, and we let q → 1− in Corollary 2.2, and the desired result is therefore established.
Remark:
The result obtained in Corollary 2.4 is more general than the Hartman Wintner inequality. For the particular case when A = B = 0, we get the classical Hartman-Wintner inequality.
Corollary 2.5. Suppose that u is a non-trivial solution of the fractional boundary value problem
whereare the fractional derivative of the Riemann-Liouville type of order 2, and Q:[a, b] → ℝ is a continuous function on [a, b].
The following integral inequality is then satisfied
Proof. It is sufficient to use the arithmetic-geometric-harmonic inequality to the conclusion of Corollary 2.4 and set A = B = 0, and the result will follow.
Corollary 2.6. Suppose that u is a non-trivial solution of the fractional boundary value problem
whereare the fractional derivative of the Riemann-Liouville type of order 2, and Q:[a, b] → R is a continuous function on [a, b].
The following integral inequality is then satisfied
Proof. Similarly to the above, we apply two times the arithmetic-geometric-harmonic inequality to the result of Corollary 2.4, and the desired result is achieved.
3. On an Interval of Real Zeros of the Mittag-Leffler Function
In this section, we are interested in getting the interval of real zeros of the following Mittag-Leffler function [43]:
where C denotes the set of complex numbers, and R(α) is the real part of α. The key tool in proving this result consists of an appropriate integral inequality of the following fractional boundary value problem.
Theorem 3.1. Let u be a non-trivial solution of
then |λ| ≥ (Γ(α)4α − 1)2.
Proof. We apply Corollary 2.3 with A = B = 0, α = β, p = 2, r = 2, and a = 0, b = 1. We obtain
and the proof is completed.
Statements
Data availability statement
All datasets generated for this study are included in the article/supplementary material.
Author contributions
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
Funding
FTc and FTa extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through Research Group no. RG-1440-010.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
Lyapunov's inequality, q-fractional integral, Green's function, p-Laplacian, mixed boundary conditions
Citation
Ragoub L, Tchier F and Tawfiq F (2020) Criteria of Existence for a q Fractional p-Laplacian Boundary Value Problem. Front. Appl. Math. Stat. 6:7. doi: 10.3389/fams.2020.00007
Received
30 November 2019
Accepted
24 February 2020
Published
22 April 2020
Volume
6 - 2020
Edited by
Mustafa Inc, Firat University, Turkey
Reviewed by
HongGuang Sun, Hohai University, China; Marin I. Marin, Transilvania University of Braşov, Romania
Updates
Copyright
© 2020 Ragoub, Tchier and Tawfiq.
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*Correspondence: Lakhdar Ragoub radhkla@hotmail.com
This article was submitted to Mathematical Physics, a section of the journal Frontiers in Applied Mathematics and Statistics
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