Two generalized Lyapunov-type inequalities for a fractional p-Laplacian equation with fractional boundary conditions

In this paper, we investigate the existence of positive solutions for the boundary value problem of nonlinear fractional differential equation with mixed fractional derivatives and p-Laplacian operator. Then we establish two smart generalizations of Lyapunov-type inequalities. Some applications are given to demonstrate the effectiveness of the new results.


Introduction
Lyapunov's inequality [] has proved to be very useful in various problems where C D α a + is the Caputo fractional derivative of order  < α ≤ . For boundary conditions (.) and (.), two Lyapunov-type inequalities were established, respectively, as follows: Recently, we considered in [] the same equation (.) with the fractional boundary condition In [], Arifi et al. considered the following nonlinear fractional boundary value problem with p-Laplacian operator: a + are the Riemann-Liouville fractional derivative of orders α, β, p (s) = |s| p- s, p > , and χ : [a, b] → R is a continuous function. It was proved that if (.) has a nontrivial continuous solution, then More recently, Chidouh and Torres in [] considered the following boundary value problem: where D α a + is the Riemann-Liouville fractional derivative with  < α ≤ , and q : [a, b] → R + is a nontrivial Lebesgue integrable function. Under the assumption that the nonlinear term f ∈ C(R + , R + ) is a concave and decreasing function, it was proved that if (.) has a nontrivial solution, then Obviously, if we set f (y) = y in (.), one can obtain a Lyapunov inequality (.). Motivated by the above work, in this paper, we consider the fractional boundary value problem where  < α, β ≤ , and k : [a, b] → R is a continuous function. We write (.) as an equivalent integral equation and then, by using some properties of its Green function and the Guo-Krasnoselskii fixed point theorem, we can obtain our first result asserting existence of nontrivial positive solutions to problem (.). Then, under some assumptions on the nonlinear term f , we are able to get two corresponding Lyapunov-type inequalities. Finally in this paper, two corollaries and an example are given to demonstrate the effectiveness of the obtained results.

Preliminaries
In this section, we recall the definitions of the Riemann-Liouville fractional integral, fractional derivative, and the Caputo fractional derivative and give some lemmas which are useful in this article. For more details, we refer to [, ].
Definition . Let α ≥  and f be a real function defined on [a, b]. The Riemann-Liouville fractional integral of order α is defined by a I  f ≡ f and where n is the smallest integer greater or equal to α and denotes the Gamma function.
where n is the smallest integer greater or equal to α.

Lemma . (Guo-Krasnoselskii fixed point theorem []
) Let X be a Banach space and let P ⊂ X be a cone. Assume  and  are bounded open subsets of X with  ∈  ⊂¯  ⊂  , and let T : P ∩ (¯  \  ) → P be a completely continuous operator such that (i) Tu ≥ u for any u ∈ P ∩ ∂  and Tu ≤ u for any u ∈ P ∩ ∂  ; or (ii) Tu ≤ u for any u ∈ P ∩ ∂  and Tu ≥ u for any u ∈ P ∩ ∂  . Then T has a fixed point in P ∩ (¯  \  ).

Lemma . (Jensen's inequality [])
Let ν be a positive measure and let be a measurable set with ν( ) = . Let I be an interval and suppose that u is a real function in L(dν) with If f is concave on I, then the inequality (.) holds with ≤ substituted by ≥.

Main results
We begin to write problem (.) in its equivalent integral form. and Then BVP (.) can be turned into the following coupled boundary value problems: , we see that BVP (.) has a unique solution, which is given by where H(t, s) is as in (.). Moreover, by Lemma  of [], we see that BVP (.) has a unique solution, which is given by where G(t, s) is as in (.). Substitute (.) into (.), we see that BVP (.) has a unique solution which is given by (.).
Lemma . The Green's function H defined by (.) satisfies the following properties: The first three properties are proved in []. For convenience, we set where c β and A β are as in (.). It is easy to check that A β <   and c β < a+b  . On the other hand, since Remark . Since a+b  < b-a  implies a < b, we see that the conclusion of Lemma () in [] only holds for a < b  . (s, s) has a unique maximum given by Suppose that there exist two positive constants r  > r  >  such that the following assumptions:

Let E = C[a, b] be endowed with the norm x = max t∈[a,b] |x(t)|. Define the cone P ⊂ E by
Then FBVP (.) has at least one nontrivial positive solution u belonging to E such that Proof Let T : P → E be the operator defined by By using the Arzela-Ascoli theorem, we can prove that T : P → P is completely continuous. Let i = {u ∈ P : u ≤ r i }, i = , . From (H), and Lemmas . and ., we obtain Hence, Tu ≥ u for u ∈ P ∩ ∂  . On the other hand, from (H), Lemmas . and ., we have Thus, by Lemma ., we see that the operator T has a fixed point in u ∈ P ∩ (¯  \  ) with r  ≤ u ≤ r  , and clearly u is a positive solution for FBVP (.).
Next, we will give two Lyapunov inequalities for FBVP (.).
Proof Assume u ∈ P is a nontrivial solution for (.), then u = . From (.), and Lemmas . and ., ∀t ∈ [a, b], we have Theorem . Let k : [a, b] → R + be a real nontrivial Lebesgue function. Assume that f ∈ C(R + , R + ) is a concave and nondecreasing function. If (.) has a nontrivial solution u ∈ P, then b a k(τ ) dτ >  β- (β) p ( (α + )) p (η) Proof By (.), Lemmas . and ., we get Using Lemma ., and taking into account that f is concave and nondecreasing, we see that The proof is completed.

Applications
In the following, some applications of the obtained results are presented.
From Theorems . and ., we have the following.
Corollary . For fractional boundary value problem (.), let k : [a, b] → R + be a nontrivial Lebesgue integrable function, and f ∈ C(R + , R + ) be a concave and nondecreasing