Analysis of some Katugampola fractional di ﬀ erential equations with fractional boundary conditions

: In this work, some class of the fractional di ﬀ erential equations under fractional boundary conditions with the Katugampola derivative is considered. By proving the Lyapunov-type inequality, there are deduced the conditions for existence, and non-existence of the solutions to the considered boundary problem. Moreover, we present some examples to demonstrate the e ﬀ ectiveness and applications of the new results.


Introduction
The Lyapunov inequality, proved in 1907 by Russian mathematician Aleksandr Mikhailovich Lyapunov [1], is very useful in various problems related with oscillation theory, differential and difference equations and eigenvalue problems (see [2][3][4][5][6][7] and the references therein). The Lyapunov result states that, if a nontrivial solution to the following boundary value problem holds. This theorem formulates a necessary condition for the existence of solutions and allows to deduce sufficient conditions for non-existence of solutions to the considered boundary problem.
where 1 < α ≤ 2, 0 < β ≤ 1 and g : [a, b] → R is a continuous function. Thanks to the detailed analysis of the integral equation equivalent to (1.2) we are able to obtain a corresponding Lyapunov-type inequality. After that, we show some applications to present the effectiveness of the new Lyapunov-type inequality. We deduce some existence and non-existence results for the considered problem (1.2) which are very helpful for other researchers in this field. Furthermore, at the end of the article there will be graphs illustrating the applications of the proven theorems.

Preliminaries
In this section, we introduce the definitions and properties of the Katugampola fractional operators which are needed to prove the main results. For more details, we refer to [15][16][17].
for t ∈ (a, b) are called the left-sided and right-sided Katugampola integrals of fractional order α, respectively. The operators I α,ρ a+ f and I α, for t ∈ (a, b) are called the left-sided and right-sided Katugampola derivatives of fractional order α, respectively.
The Katugampola derivative generalizes two other fractional operators, by introducing a new parameter ρ > 0 in the definition. Indeed, if we take ρ → 1, we have the Riemann-Liouville fractional derivative, i.e., Moreover, if we take ρ → 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 [9,16,17].
It is worth to mention that the complex formula for the Katugampola operator is established in [18].

Main result
We start with writing problem (1.2) in its equivalent integral form.
where the Green function G is given by Proof. Integrating equation from (1.2) and using Lemma 2.6 we obtain that general solution is of the form where c 1 and c 2 are some real constants. Since u(a) = 0, we get c 2 = 0. Moreover, differentiating (3.3) in Katugampola sense with c 2 = 0, we have By Example 2.3 and Lemma 2.5 we obtain Therefore, which ends the proof.
The below theorem present the properties of the Green function G obtained in (3.2) , β ∈ (0, 1] , α > β + 1 and ρ > 0. The function G given by (3.2) satisfies the following estimates Proof. First we prove the positivity of function G. For t ≤ s it is obvious, but for s < t we can rewrite function G in the form Let us see that there is the following estimation Thus the function G is positive also for s < t.
(ii) Now, we prove that G(t, s) ≤ G(s, s). Firstly, we consider the interval a ≤ t ≤ s ≤ b.
Differentiating G with respect to t we have Therefore, because the function G with respect to t is increasing on the considered interval. Now, let we take the interval a ≤ s < t ≤ b. Taking the derivative of function G with respect to t, we obtain Therefore, because the function G is decreasing with respect to t on the considered interval. From (3.4) and (3.5) we get In order to find the maximum value of this function, we check the sign of the derivative on the interior (a, b). We have It follows that f (ŝ) = 0 if and only ifŝ It is easily seen that f (s) < 0 forŝ < s and f (s) > 0 forŝ > s.
It ends the proof.
We are ready to state and prove our main results in the Banach space C[a, b] with the maximum norm ||u|| = max t∈ [a,b] |u(t)|.
Proof. It follows from Theorem 3.1 that solution of the fractional boundary value problem (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 The proof is completed.
Due to the fact, that the Katugampola derivative has an additional parameter ρ (which by taking ρ → 0 + reduces to the Hadamard fractional derivative and for parameter ρ = 1 become the Riemann-Liouville fractional derivative) we get the Lyapunov-type inequality for both the Riemann-Liouville derivative D α a+ and the Hadamard derivative H D α a+ . In particular, if we take, in Corollary 3.4, β = 0 we obtain the main result of the work [10] proved by Ferreiro.
exists, where g is a real and continuous function, then b a |g(s)|ds ≥ Γ(α) 4 β max{a, b}

Applications and examples
In this section, we apply the results on the Lyapunov-type inequalities obtained previously to study the nonexistence of solutions for certain fractional boundary value problems.

(4.2)
If problem (4.2) admits a nontrivial solution u λ , we say that λ is an eigenvalue of problem (4.2). We have the following result which provides a lower bound of the eigenvalues of problem (4.2).