Lyapunov-Type Inequalities for Second-Order Boundary Value Problems with a Parameter

Up until now, integral inequalities have attracted the attention of many researchers, due to its wide applications in the research of qualitative and quantitative properties such as global existence, boundedness, and stability of differential and integral equations (see [1–26] and the references therein). Among these inequalities, one important kind is the Lyapunov-type inequality, which was originally presented by Lyapunov in [27] as follows. If u(t) is a solution of u′′ + q(t)u � 0, (1)


Introduction
Up until now, integral inequalities have attracted the attention of many researchers, due to its wide applications in the research of qualitative and quantitative properties such as global existence, boundedness, and stability of differential and integral equations (see  and the references therein). Among these inequalities, one important kind is the Lyapunov-type inequality, which was originally presented by Lyapunov in [27] as follows.
For example, in 2003, Yang [29] obtained the following result for the second-order half-linear equation: where q, r ∈ C( [a, b], R) such that r(t) > 0, for t ∈ [a, b], and p > 0.
In 2012, Tiryaki et al. [34] established an inequality for boundary value problem of the form where α > 1 and α * � (α/α − 1). eir result is as follows. Theorem 2 (see [34]). Assume boundary value problem (6) has a solution u(t); then, the following inequality holds: where In 2015, Agarwal et al. [36] established a Lyapunov-type inequality for the second-order forced boundary value problem of the form: in the subhalf-linear (0 < c < β) and the super-half-linear . eir result is as follows.
Theorem 3 (see [36]). Suppose that a, b, a < b, are consecutive zeros of a nontrivial solution of the first part of equation (8), then the inequality Agarwal andÖzbekler [36] also established a Lyapunovtype inequality for the second-order forced boundary value problem with mixed nonlinearities: where 0 < c < 1 < α < 2. e result is as follows.
Theorem 4 (see [36]). Suppose that a, b, a < b, are consecutive zeros of a nontrivial solution of the first part of equation (10), then the inequality We find that in [36], the authors studied the case 0 < c < 1 < α < 2 of equation (10). It will be interesting to prove Lyapunov-type inequalities for equation (10) or other equation when α and c have other relation. Motivated by [36], in this paper, we will establish a Lyapunov-type inequality for the nonlinear second-order boundary value problem of the form and λ ≥ 0 is a real parameter, and with the boundary condition (13)
2 Discrete Dynamics in Nature and Society (12) satisfying the boundary conditions (13). en,

Theorem 5. Assume u is a solution of equation
where h Using integration by parts to the first integral on the lefthand side of (20) and from (13) By using Hölder's inequality with indices On the contrary, from Lemma 1, we obtain en, from (23), (26), and (27), we obtain b a u ′ (t) dt Discrete Dynamics in Nature and Society For the first integral on the right-hand side of (28), inequality (18) in Lemma 2 with A � r(t), B � q(t), and x � |u(t)| ≥ 0 for t ∈ [a, b] implies that By (28) and (29), we obtain b a u ′ (t) dt From Lemma 1, we have In view of (30) and (31), we obtain that which also leads to (19). e proof is complete. If we take α � 2 and λ � 1 in inequality (19), we obtain the following result.

Theorem 6. Assume u is a solution of equation
where h Using integration by parts to the first and second integrals on the left-hand side of (37) and from (13) By using Hölder's inequality (24) From (26), (39), (41), and Lemma 1, we obtain Discrete Dynamics in Nature and Society b a u ′ (t) dt For the right-hand side of (42), inequality (18) in Lemma From (27) us, dividing both sides of (29) by ( b a |u ′ (t)|dt) α , we obtain which also leads to (36). e proof is complete.