1 Introduction
The Lyapunov inequality and its generalizations have been indispensable in the investigation of various topics of differential equations including oscillation theory, stability theory, intervals of disconjugacy, and eigenvalue problems [
1‐
5]. The problem was initiated by Lyapunov himself who established a necessary condition for the existence of solutions for the boundary value problem (BVP)
$$\begin{aligned} \textstyle\begin{cases} x''(t)+r(t)x(t)=0,&t \in (a,b), \\ x(a)=x(b)=0. \end{cases}\displaystyle \end{aligned}$$
(1.1)
Indeed, he proved in [
6] that if BVP (
1.1) has a non-trivial solution, then the inequality
$$ \int_{a}^{b} \bigl\vert r(t) \bigr\vert \,\mathrm{d}t> \frac{4}{b-a} $$
(1.2)
holds, where
r is a real-valued integrable function. Since then (
1.2) is referred to as the Lyapunov inequality. In [
7], Wintner was ahead and replaced
\(\vert r(t) \vert \) by the function
\(r^{+}(t)\) and obtained the following slightly different version of the Lyapunov inequality:
$$ \int_{a}^{b} r^{+}(t)\,\mathrm{d}t> \frac{4}{b-a}, $$
(1.3)
where
\(r^{+}(t)=\max \{r(t),0\}\). This inequality is considered to be the best reachable one in the sense that the constant 4 in (
1.3) cannot be replaced by any other larger constant (see [
6] and [
8, Theorem 5.1]). In his remarkable book, Hartman [
8] was beyond this estimate and obtained a generalized version as follows:
$$ \int_{a}^{b}(b-t) (t-a)r^{+}(t)\,\mathrm{d}t>b-a. $$
(1.4)
The study of this problem on various types of differential and difference equations over the last years has resulted in different versions of Lyapunov-type inequalities; the reader may consult the papers [
9‐
13] and the first chapter in [
14] for a complete view. In parallel to the intensive investigation tendency amongst researchers, Agarwal et al. in [
15] has recently considered the mixed non-linear BVP of the form
$$\begin{aligned} \textstyle\begin{cases} x''(t)+r_{1}(t)\vert x(t) \vert ^{\eta -1}x(t)+r_{2}(t)\vert x(t) \vert ^{\delta -1}x(t)=0, \\ x(a)=x(b)=0, \end{cases}\displaystyle \end{aligned}$$
(1.5)
where the non-linearities satisfy
$$\begin{aligned} 0< \eta < 1< \delta < 2 \end{aligned}$$
(1.6)
and no sign restrictions are imposed on the real-valued integrable potential functions
\(r_{1}\),
\(r_{2}\), and obtained the following Hartman-type and Lyapunov-type inequalities.
At some earlier time in [
16] and under the same conditions, the same authors considered the mixed non-linear forced BVP of the form
$$\begin{aligned} \textstyle\begin{cases} x''(t)+r_{1}(t)\vert x(t) \vert ^{\eta -1}x(t)+r_{2}(t)\vert x(t) \vert ^{\delta -1}x(t)=g(t), \\ x(a)=x(b)=0, \end{cases}\displaystyle \end{aligned}$$
(1.10)
where
g is a real-valued integrable function and obtained the following Hartman-type and Lyapunov-type inequalities.
It is to be noted that the forcing function
g in (
1.10) requires no sign restriction as well. Furthermore, inequality (
1.11) implies (
1.12) because, upon applying the arithmetic–geometric mean, we get
\((b-t)(t-a)\leq (b-a)^{2}/4\) for all
\(t\in [a,b]\). The reader can also figure out that both inequalities (
1.11) and (
1.12) reduce to the classical Hartman-type and Lyapunov-type inequalities as
\(\eta \to 1^{-}\) and
\(\delta \to 1^{+}\), respectively.
Fractional differential equations have proved direct evolvement into multidisciplinary subjects such as viscoelasticity, ground water flows, boundary layer theory, granular flows, dynamics of cold atoms in optical lattices, plasma turbulence, and dynamics of polymeric materials; see, for instance, [
17,
18]. The development of these equations in the last years has recently led to a tremendous number of papers which have studied different qualitative topics. Amongst them is the investigation of Lyapunov inequality which was initiated by Ferreria in [
19] and continued by other scholars [
20‐
35]. On the other hand, the newly defined conformable fractional calculus was initiated in [
36] and studied later on in the papers [
37,
38] where many properties of conformable operators were introduced. Apart from its simple application, nevertheless, it has been realized that this topic proves to be essential and profitable in generating new types of fractional operators [
39]. However, the progress in this direction is still at its earliest stage [
40‐
43].
The objective of this paper is to state and prove new generalized Lyapunov-type and Hartman-type inequalities for a conformable boundary value problem of order
\(\alpha \in (1,2]\) with mixed non-linearities of the form
$$ \bigl(\mathbf{T}_{\alpha }^{a} x\bigr) (t)+r_{1}(t) \bigl\vert x(t) \bigr\vert ^{\eta -1}x(t)+r_{2}(t)\bigl\vert x(t) \bigr\vert ^{ \delta -1}x(t)=g(t), \quad t\in (a,b), $$
satisfying the Dirichlet boundary conditions
\(x(a)=x(b)=0\), where
\(r_{1}\),
\(r_{2}\), and
g are real-valued integrable functions, and the non-linearities satisfy the conditions
\(0<\eta <1<\delta <2\). Moreover, Lyapunov-type and Hartman-type inequalities are obtained when the conformable derivative
\(\mathbf{T}_{\alpha }^{a}\) is replaced by a sequential conformable derivative
\(\mathbf{T}_{\alpha }^{a} \circ \mathbf{T}_{\alpha }^{a}\),
\(\alpha \in (1/2,1]\). The potential functions
\(r_{1}\),
\(r_{2}\) as well as the forcing term
g require no sign restrictions. The obtained inequalities generalize and compliment some existing results in the literature.
This section is devoted to stating some preliminaries on higher-order fractional conformable derivatives. We borrow the notations and terminology from the recent papers [
36,
37].
If
g is differentiable, then one should note the following essential identity:
$$ \bigl(T_{\alpha }^{a} g\bigr) (t)=(t-a)^{1-\alpha } g^{\prime }(t). $$
(2.2)
Moreover, the conformable fractional integral of order
\(0<\alpha \leq 1\) starting at
\(a\geq 0\) is defined by
$$ \bigl(I_{\alpha }^{a} g\bigr) (t)= \int_{a}^{t}g(x) (x-a)^{\alpha -1}\,\mathrm{d}x $$
(2.3)
or following the notation in [
36] as
$$ \bigl(I_{\alpha }^{a} g\bigr) (t)= \int_{a}^{t}g(x)x^{\alpha -1}\,\mathrm{d}x. $$
Throughout this article, we shall apply the conformable integral in (
2.3). In case of higher order, the following definition is adopted.
Note that if
\(\alpha =n+1\), then
\(\gamma =1\) and the fractional derivative of
g becomes
\(g^{(n+1)}(t)\). Also, when
\(n=0\) (or
\(\alpha \in (0,1)\)), then
\(\gamma =\alpha \) and the definition coincides with that in Definition
2.1. From (
2.4), it is an immediate consequence that if
\(n< \alpha \leq n+1\), then
\(\gamma =\alpha -n\) and if, moreover, the
\((n+1)\)st derivative (or the derivative of
\(g^{(n)}\)) exists, then we have
$$ \bigl( \mathbf{T}_{\alpha }^{a} g\bigr) (t)= \bigl(T_{a}^{\gamma }g^{(n)}\bigr) (t)=(t-a)^{1- \gamma } g^{(n+1)}(t)=(t-a)^{1-\alpha +n}g^{(n+1)}(t). $$
(2.5)
In case of higher order, the following definition is valid.
Notice that if
\(\alpha =n+1\) then
\(\gamma =1\) and hence
$$ \bigl(I_{\alpha }^{a} g\bigr) (t)=\bigl(\textbf{I}_{n+1}^{a} g\bigr) (t)=\frac{1}{n!} \int_{a} ^{t}(t-x)^{n} g(x)\,\mathrm{d}x, $$
which is the iterative integral of
g,
\(n+1\) times over
\((a,t]\).
Recalling that the left Riemann–Liouville fractional integral of order
\(\alpha >0\) starting from
a is defined by
$$ \bigl(_{a}\textbf{I}^{\alpha }g\bigr) (t)= \frac{1}{\Gamma (\alpha )} \int_{a} ^{t}(t-s)^{\alpha -1} g(s)\,\mathrm{d}s, $$
(2.7)
we see that
\((I_{\alpha }^{a} g)(t)= ({}_{a}\textbf{I}^{\alpha }g)(t)\) for
\(\alpha = n+1\),
\(n=0,1,2,\ldots \) .
The following is a generalization of Lemma
2.3.
4 Conclusion
Conformable derivatives are naturally local fractional derivatives which allow deriving with respect to arbitrary order. Recently, it has been realized that conformable derivatives are essential in generating new types of fractional operators; see, for instance, the results reported in [
39]. In this article, we have accommodated the concept and the properties of conformable derivatives to establish new generalized Lyapunov-type and Hartman-type inequalities for a boundary value problem with mixed non-linearities,
\(\alpha \in (1,2]\). In addition, the main results are carried out for the sequential conformable derivatives of the form
\(\mathbf{T}_{\alpha }^{a} \circ \mathbf{T}_{\alpha }^{a}\),
\(\alpha \in (1/2,1]\). The corresponding classical Lyapunov-type and Hartman-type inequalities are obtained in the limiting cases
\(\alpha \rightarrow 2^{-}\) and
\(\alpha \rightarrow 1^{-}\), respectively. The obtained inequalities generalize and compliment some existing results in the literature.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.