Abstract
We prove Hartman-type and Lyapunov-type inequalities for a class of Riemann–Liouville fractional boundary value problems with fractional boundary conditions. Some applications including a lower bound for the corresponding eigenvalue problem are obtained.
1. Introduction
In [1], Lyapunov established the following striking inequality:
Theorem 1. Let . Assume that the problemhas a solution such that for . Then,and constant 4 is the best possible largest number.
It has been shown that this result serves as a good tool in the study of several properties of solutions of differential equations (such as eigenvalue problems and eigenvalue inequalities) (see, for example, [2–5] and the references therein). Many authors have worked on generalizations of classical inequalities (see, for instance, [4–16] and the references therein).
In [11], the authors use the Hahn integral operator to prove a description of new generalization of Minkowski’s inequality.
In [5], the authors improve inequality in (2) by proving the following Hartman–Winter inequality:where is the nonnegative part of .
Inequality (3) is also known as the best Lyapunov inequality.
In [17], Ferreira considered the following fractional differential problem:where and denotes the Riemann–Liouville fractional derivative of order (see Definition 2 in the following).
The author established the following Lyapunov-type inequality for problem (4).
Theorem 2. (see [17]). Assume that problem (4) has a solution such that for . Then,
Remark 1. Note that if we let in (5), one obtains Lyapunov’s classical inequality (2).
For the convenience of the reader, we recall the concept of fractional integral and derivative of order .
Definition 1. (see [18, 19]). The Riemann–Liouville fractional integral of order for a real-valued function is defined by andwhere is the Euler gamma function.
Definition 2. (see [18, 19]). The Riemann–Liouville fractional derivative of order for function is defined by andwhere with the integer part of .
The new development in fractional calculus has attracted the attention of researchers of various disciplines. Different mathematical procedures have been considered by several authors through different research-oriented aspects of fractional differential equations (see, for instance, [20–22] and the references therein).
Our goal in this paper is to establish Hartman-type and Lyapunov-type inequalities for the following problem:where and . Some applications are given to illustrate our result.
The organization of the paper is as follows. In Section 2, we derive the explicit expression of the Green function corresponding to problem (8) and we establish some properties on it. This allows us to prove Hartman-type and Lyapunov-type inequalities for problem (8). In Section 3, we present some applications including a lower bound for the corresponding eigenvalue problem.
2. Main Results
2.1. Green’s Function
First, we recall the following well-known properties (see, for example, [18, 19]).
Lemma 1. Let and . Then,(i)For, and (ii) if and only if , where , for (iii)Assume that ; then,where , for
Lemma 2. Let be a solution of problem (8). Then,where is Green’s function of problem (8) given by
Proof. Let be such solution. By Lemma 1, we haveUsing the fact that , we obtain and .
Therefore,This ends the proof.
To get a quick perspective, in Figure 1, we have the representation of Green’s function with the contours and some projections.
One can see from Figure 1 that Green’s function and it is nondecreasing with respect to the first variable. This important observation will be proved for with .
(a)
(b)
(c)
Definition 3. Let with . We say thatif there exists such that for all .
Remark 2. Let and . Then,Next, we establish some properties on Green’s function given by (11).
Proposition 1. (i)On ,(ii)On ,(iii)The function satisfies the following property:
Proof. (i)From Lemma 2, for , we have where . Now, since , for , then by using Remark 2, with and , we obtain Hence, inequalities in (16) follow by observing that(ii)We have Similar to case , by using the fact that and applying Remark 2 with and , we obtain the required result.(iii)Let . Since the function is nondecreasing on , we deduce thatThis completes the proof.
2.2. Statements and Proofs of Main Results
Theorem 3. (Hartman–Winter-type inequality). Let . Assume that problem (8) has a solution such that for . Then,where .
Proof. From Lemma 2, we know thatWithout loss of generality, we may assume that for .
Using (26), Proposition 1 (iii) and the fact that , we deduce thatHence,or equivalently,Therefore,from which inequality (25) follows.
Remark 3. Let . Under the same conditions as in Theorem 3, we haveBy applying the previous theorem with , we obtain the following:
Corollary 1. Let . Assume that the problemsadmit a solution such that for . Then,In particular,
Corollary 2. ()
Under the same conditions as in Theorem 3, we have
Proof. By Theorem 3, we havewhere .
For , we haveNote thatFurthermore, on and on .
Hence,So Lyapunov-type inequality (35) follows from (36) and (39).
Corollary 3. Let . Assume that the problemsadmit a solution such that for . Then,
Proof. Inequality (41) follows from (39) with .
3. Applications
3.1. Lower Bound for the Eigenvalues
Consider the following eigenvalue problem:
Theorem 4. Assume that eigenvalue problem (42) has a solution such that for . Then,
Proof. By Remark 3 (with and ), we havefrom which inequality (43) follows by observing that
3.2. Nonexistence Results
Consider the following problem:where . Then, we have the following result.
Theorem 5. Assume thatThen, problem (46) has no nontrivial solution.
Proof. The assertion follows from Corollary 2.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding research group no. RG-1435-043.