Abstract
In this article, we prove a Lyapunov-type inequality for a fractional differential equation under mixed boundary conditions. As applications, we deduce nonexistence results for some fractional boundary value problems. Moreover, we obtain numerical approximations of a lower bound for the eigenvalues of the corresponding equations.
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1 Introduction
The well-known Lyapunov result states that if a nontrivial solution to the boundary value problem
exists, where \(g:[a,b]\rightarrow {\mathbb {R}}\) is a continuous function, then
By now, this result and its extensions have been found to be useful, e.g., in oscillation theory, disconjugacy, eigenvalue problems, and many other theories based on differential and difference equations (see [1,2,3,4,5,6,7] and the references therein).
Together with the rising popularity of fractional operators, which are interesting because of their non-local character allowing us to model non-local or time dependent processes, many modifications of the Lyapunov inequality appeared. The first work in this direction is due to Ferreira [8], who derived a Lyapunov-type inequality for differential equations depending on the Riemann–Liouville fractional derivative. For other, similar works we refer the reader to [9,10,11,12,13,14].
Motivated by the above works as well as useful applications of Lyapunov’s inequalities, we focus here on the Katugampola fractional differential equation with mixed boundary conditions. More precisely, we consider the fractional boundary value problem
where \(D_{a+}^{\alpha , \rho }\) denotes the Katugampola fractional derivative of order \(\alpha \) and \(g:[a,b]\rightarrow {\mathbb {R}}\) is a continuous function. The Katugampola fractional derivative was chosen because it generalizes two other fractional operators: the Riemann–Liouville and the Hadamard fractional derivatives. The main goal of this work is to obtain a Lyapunov-type inequality for the above fractional boundary value problem and to present applications to demonstrate the effectiveness of this inequality.
2 Preliminaries
Before presenting the main results, let us start by recalling the concept of Katugampola fractional operators which was introduced in 2014 by Udita Katugampola. For more details, we refer to [15, 16].
Definition 2.1
Let \(\alpha>0,\ \rho >0,\ -\infty<a<b<\infty \). The operators
for \(t\in (a,b)\), are called the left-sided and right-sided Katugampola integrals of fractional order \(\alpha \), respectively. The operators \(I^{\alpha ,\rho }_{a+}\) and \(I^{\alpha ,\rho }_{b-}\) are well defined in \(L^p(a,b),\ p\ge 1.\)
Definition 2.2
Let \(\alpha>0,\ \rho >0,\ n=[\alpha ]+1,\ 0<a<t<b\le \infty \). The operators
for \(t\in (a,b)\), are called the left-sided and right-sided Katugampola derivatives of fractional order \(\alpha \), respectively.
The Katugampola derivative generalizes two other fractional operators by introducing a new parameter \(\rho > 0\) in the definition. Indeed, if we take \(\rho \rightarrow 1\), we have the Riemann–Liouville fractional derivative i.e.,
Moreover, if we take \(\rho \rightarrow 0^+\), we get the Hadamard fractional derivative, i.e.,
The higher order Katugampola fractional operators satisfy the following properties, which were precisely discussed and proven in [17].
-
1.
If \(\alpha >0\), \(\rho >0\) and \(\lambda >-1\) then
$$\begin{aligned} I^{\alpha ,\rho }_{a+}\left( \frac{t^\rho -a^\rho }{\rho }\right) ^{\lambda }= & {} \frac{\Gamma (\lambda +1)}{\Gamma (\lambda +\alpha +1)}\left( \frac{t^\rho -a^\rho }{\rho }\right) ^{\alpha +\lambda }. \end{aligned}$$ -
2.
For \(\rho>0,\ \alpha>0,\ \lambda >\alpha -1\), we have
$$\begin{aligned} D^{\alpha ,\rho }_{a+}\left( \frac{t^\rho -a^\rho }{\rho }\right) ^\lambda= & {} \frac{\Gamma (\lambda +1)}{\Gamma (\lambda +1-\alpha )}\left( \frac{t^\rho -a^\rho }{\rho }\right) ^{\lambda -\alpha }. \end{aligned}$$ -
3.
If \(n-1<\alpha <n\), \(n\in {\mathbb {N}}\), \(\rho >0\) then
$$\begin{aligned} I^{\alpha ,\rho }_{a+}D^{\alpha ,\rho }_{a+}f(t)= & {} f(t)+\sum _{i=0}^{n-1}{{\tilde{c}}_i\left( \frac{t^\rho -a^\rho }{\rho }\right) ^{i-n+\alpha }}, \end{aligned}$$where \({\tilde{c}}_i\) are real constants.
-
4.
If \(\alpha>0,\ \rho >0\) and \(f\in L^p(a,b)\) then
$$\begin{aligned} D^{\alpha ,\rho }_{a+}I^{\alpha ,\rho }_{a+}f(t)=f(t). \end{aligned}$$
3 Main results
In this section we prove a necessary condition for the existence of a solution to the boundary value problem with Katugampola derivative of order \(1 <\alpha \le 2\).
3.1 Integral representation of the solution
We start by writing (1.1)–(1.2) in its equivalent integral form.
Theorem 3.1
The function \(u \in C[a, b]\) is a solution to the boundary value problem (1.1)–(1.2) if and only if u is a solution to the integral equation
where the Green function G is given by
Proof
The general solution to (1.1) is
Taking the derivative of u, we obtain
Using the boundary conditions (1.2), we get \(c_2=0\) and
Therefore,
which concludes the proof. \(\square \)
3.2 Green function estimates
Theorem 3.2
The function G defined by (3.2) satisfies the following properties:
-
1.
\(G(t,s) \ge 0 \) for \(t \in [a,b]\), \(s \in [a,b]\);
-
2.
\(\max \limits _{t \in [a,b]}G(t,s) = G(s,s) \le \displaystyle \frac{\rho ^{1-\alpha }(b^{\rho } - a^{\rho })}{\Gamma (\alpha )}s^{\rho - 1}(b^{\rho } - s^{\rho })^{\alpha -2}.\)
Proof
First we prove the positivity of G. For \(t\le s\) it is obvious, but for \(s < t\) we can rewrite G in the form
Let us note that
because the following inequalities hold:
and
Therefore \(G(t,s)\ge 0\) also for \(s<t\).
Now, we show that \(G(t,s) \le G(s,s)\). First, let us take the interval \(a \le t \le s \le b\). Differentiating G with respect to t we get
This means that the function G with respect to t is increasing on the given interval. We conclude
Now, we turn our attention to the function G on the interval \(a \le s < t \le b\). We start by fixing an arbitrary \(s\in [a,b]\). Differentiating G with respect to t we have
Since the inequalities (3.3) hold, it follows that \(\frac{\partial G}{\partial t} <0\). Therefore, the function G with respect to t is decreasing on the given interval. We get
for all \(t,s\in [a,b]\).
The proof is completed. \(\square \)
3.3 A Lyapunov-type inequality
We are ready to state and prove our main results in the Banach space C[a, b] with the maximum norm \(\Vert u\Vert = \max \limits _{t \in [a,b]}\vert u(t)\vert \).
Theorem 3.3
If a nontrivial continuous solution of the fractional boundary value problem (1.1)–(1.2) exists, where g is a real and continuous function, then
Proof
It follows from Theorem 3.1 that the solution of the fractional boundary value problem (1.1)–(1.2) satisfies the integral equation (3.1). Thus,
Using the estimation of the function G which was obtained in Theorem 3.2 we get
Thus, we have
The proof is completed. \(\square \)
Note that taking \(\rho =1\) in Theorem 3.3 we have Lyapunov’s inequality with the Riemman–Liouville fractional derivative \(D^\alpha _{a+}\).
Corollary 3.4
If a nontrivial continuous solution of the fractional boundary value problem
exists, where g is a real and continuous function, then
Moreover, taking \(\rho \rightarrow 0^+\) in Theorem 3.3 we have Lyapunov’s inequality with the Hadamard fractional derivative \(^{H}{D}^\alpha _{a+}\).
Corollary 3.5
If a nontrivial continuous solution of the fractional boundary value problem
exists, where g is a real and continuous function, then
4 Applications
In this section, we apply the results on the Lyapunov-type inequality obtained previously to study the nonexistence of solutions for certain fractional boundary value problems.
Theorem 4.1
If
then (1.1)–(1.2) has no nontrivial solution.
Proof
Assume the contrary, i.e., (1.1)–(1.2) has a nontrivial solution u. By Theorem 3.3, we obtain
which contradicts assumption (4.1). \(\square \)
The other application is that we derive an estimation related to the eigenvalue of the corresponding equation by using our Lyapunov-type inequality (3.6). For given \(\lambda \in {\mathbb {R}}\), we consider the following boundary value problem:
If (4.2) admits a nontrivial solution \(u_\lambda \), we say that \(\lambda \) is an eigenvalue of (4.2).
Corollary 4.2
If \(\lambda \) is an eigenvalue of (4.2), then
Proof
Since \(\lambda \) is an eigenvalue of (4.2), it has a nontrivial solution \(u_\lambda \). According to Theorem 3.3, we get
Therefore, calculating the integral by using the substitution \(b^\rho -s^\rho =t\), we obtain
which is the desired result. \(\square \)
Example 4.3
Let us consider the following problem:
By Corollary 4.2, if a continuous solution to (4.3) exists, then necessarily
Note that the existence of solutions depends on the parameters \(\alpha ,\ \lambda \) and \(\rho \). Indeed, for \(\rho =1\) inequality (4.4) is satisfied if and only if
and for \(\rho \rightarrow 0^+\) we have
The Fig. 1 shows the behavior of \(C_{\alpha ,\rho }\ (\rho =1,2)\) with respect to \(\alpha \in (1, 2)\) (Fig. 1).
Moreover, let \(\lambda =0.5\). If we take \(\alpha =1.25\), then inequality (4.4) is satisfied if and only if \(\rho >0\). Hence, in this case, a solution to (4.3) may exist for all values of \(\rho \) for which the Katugampola derivative is defined. Now, let us choose \(\alpha =1.5\). In this case inequality (4.4) is satisfied if and only if \(\rho >0.7893\). Therefore, for \(\rho = 1\), a solution to (4.3) may exist but for \(\rho \rightarrow 0^+\) it does not. Finally, for \(\alpha =1.95\), inequality (4.4) is satisfied if and only if \(\rho > 1.79\). In particular, for \(\rho =1\) and \(\rho \rightarrow 0^+\) a solution to (4.3) does not exist.
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Łupińska, B. Existence and nonexistence results for fractional mixed boundary value problems via a Lyapunov-type inequality. Period Math Hung 88, 118–126 (2024). https://doi.org/10.1007/s10998-023-00542-5
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DOI: https://doi.org/10.1007/s10998-023-00542-5