LYAPUNOV TYPE INEQUALITIES FOR n TH ORDER FORCED DIFFERENTIAL EQUATIONS WITH MIXED NONLINEARITIES

. In the case of oscillatory potentials, we present Lyapunov type inequalities for n th order forced diﬀerential equations of the form satisfying the boundary conditions a 1 < a 2 < No sign restriction is imposed on the forcing term and the nonlinearities satisfy The obtained inequalities generalize and compliment the existing results in the literature.


(Communicated by Rafael Ortega)
Abstract. In the case of oscillatory potentials, we present Lyapunov type inequalities for nth order forced differential equations of the form satisfying the boundary conditions where a 1 < a 2 < · · · < ar, 0 ≤ k i and r ∑ j=1 k j + r = n; r ≥ 2.
The obtained inequalities generalize and compliment the existing results in the literature.

Introduction.
Consider the Hill's equation where q(t) ∈ L 1 [a, b] is a real-valued function. If there exists a nontrivial solution x(t) of Eq. (1) satisfying the Dirichlet boundary conditions where a, b ∈ R with a < b and x(t) ̸ = 0 for t ∈ (a, b), then the inequality holds. This striking inequality was first proved by Lyapunov [37] and it is known as "Lyapunov inequality". Later Wintner [57] and thereafter some authors achieved 2282 R. P. AGARWAL AND A.ÖZBEKLER to replace the function |q(t)| in Ineq. (3) by the function q + (t) i.e. they obtained the following inequality: where q + (t) = max{q(t), 0}. The constant "4" in the right hand side of inequalities (3) and (4) is the best possible largest number (see [37] and [29,Thm. 5.1]).
Thereafter, using analogous techniques as in the proof of Thm.  (8) where a k ∈ R, k = 1, 2, . . . n, with a 1 < a 2 < · · · < a n−1 < a n are consecutive zeros. Then the inequality holds, where a = a 1 and b = a n .
It appears that an explicit form of Green's function g n (t, s) for the n-point boundary value Prb. (14) -(8) was first given by Das and Vatsala [19] in 1973. One of the aims of this paper is to extend and improve the results given in Thm.'s 1.1 and 1.2 to the forced differential equations with mixed nonlinearities under the general boundary conditions than those in (8). Our proofs are based on Green's function G n (t, s) of Eq. (14) satisfying the boundary conditions where a 1 < a 2 < · · · < a r and r ∑ j=1 k j + r = n; k i ≥ 0 for all i = 1, 2, . . . r.
In 1962, Beesack [11] proved that Green's function G n (t, s) of Prb. (14)-(15) satisfies the inequality for a 1 < s < a r and −∞ < t < ∞. He also showed that for a 1 < t, s < a r by using the inequality for a 1 < t < a r .
Motivated by the above works for linear equations, in this paper we will find analogs of well-known Lyapunov type inequalities for more general equations of the form satisfying the r-point boundary conditions (15), where n, m ∈ N, the potentials q i (t), i = 1, . . . , m, the forcing term f (t) are real-valued functions, and no sign restrictions are imposed on them. Further, the exponents in Eq. (21) satisfy 0 < α 1 < · · · < α j < 1 < α j+1 < · · · < α m < 2.
It is clear that the two special cases of Eq. (21) are the nth order forced sub-linear equation and the nth order forced super-linear equation Further, we note that letting α i → 1 − , i = 1, . . . , j and α i → 1 + , i = j + 1, . . . , m in (21) results in where Since when r = n the boundary conditions (15)  We further remark that the Lyapunov type inequalities have been studied by many authors, see for instance the survey paper [52] and the references therein, but to the best of our knowledge there are no results in the literature for Eq. (21), and in particular for Eq.'s (22), (23) and (24).

Main results. Throughout this paper we shall assume that
In what follows we will need the following lemma, see [8, 10, Lemma 2.1] (we include its proof here for completeness).
for any µ ∈ (0, 2) with equality holding if and only if B = z = 0.
Thus, (26) holds. Note that if B > 0, then Ineq. (26) is strict. Now we state and prove our first result.
Remark 1. It is of interest to find analogues of Cor.'s 2 and 3 for Eq.'s (22) and (23) dropping the forcing term f (t), i.e., for the nth order sub-linear and the nth order super-linear equation respectively. We state these results in the following: Proposition 1 (Lyapunov type inequality). Let x 1 (t) and x 2 (t) be nontrivial solutions of Eq.'s (38) and (39) satisfying the n-point boundary conditions (15), respectively. If hold, where σ = min{k 1 , k r }, and the constants γ 0 and β 0 are defined in (36) and (37).
The following result is analogous to that of our first main result, i.e., Thm. 2.2.
3. (k, n − k) Conjugate boundary value problem. In this section, we further consider the nth order forced mixed nonlinear differential equation of the form satisfying the (k, n − k) conjugate boundary conditions (43).

Theorem 3.2 (Lyapunov type inequality). Let Eq. (51) has a nontrivial solution
However the function ψ 1 (s) given in (49) is not correct, we will give the modified form of ψ 1 (s). For this, we need the following lemma. (46)-(47). Then

Lemma 3.3. Let the function G n (t, s) be defined as in
where .

Last two inequalities imply that
In a similar way, it can be shown that We note that and for Employing the handy inequalities (57) and (58), we can improve and extend the results of Yang [55] for (k, n − k) conjugate boundary value Prb. (45)- (43).

Theorem 3.4 (Hartman type inequality). Let x(t) be a nontrivial solution of Eq.
holds, where the functions Q m (t) and Q m (t) are defined in (29) and for k = 1, . . . , n − 1.
We will need to prove the following lemma for the next result.
(62), we obtain the quadratic inequality where But Ineq. (73) is possible if and only if S 1 S 2 > 1/4. The proof of Thm. 3.6 is completed.
On account of the limit process (35), as a direct consequence of Thm.'s 3.4 and 3.6, we have following result for the equation where ν(t) is defined in (25).  (a, b), then the following hold: where Q m (t), Φ(t) and Φ nk are defined in (29), (60) and (68) respectively.
Remark 5. When n = 2 or (and) f (t) = 0 the results given in Cor.'s 8 and 9 are still new.
3.1. Further estimations for n = 2ℓ. In this section, we consider Prb. (45)- (43) with n = 2k = 2ℓ, i.e., the equation satisfying the 2-point boundary conditions where a, b ∈ R with a < b are consecutive zeros. It appears that the first generalization of Hartman and Lyapunov results for the linear equation  = 0 in (a, b), then the following hold: The proof of Thm. 3.7 is based on Green's function G ℓ (t, s) of the 2-point bound- satisfying (87), obtained in the same paper [20] as follows:   (a, b), then the following hold: (i) Hartman type inequality; Here the constant ρ i is defined in (30).
The results those given in Sec. 3 can be extended to even-order forced differential equations of the form (86) satisfying 2-point boundary conditions (87) and now we present some new Hartman and Lyapunov type inequalities for Prb. (86)-(87).  = 0 in (a, b), then the following hold: (i) Hartman type inequality; (ii) Lyapunov type inequality; where the functions Q m (t) and Q m (t) are defined in (29) and for ℓ = 1, . . . , n.
It will be of interest to find similar results for the nth order mixed nonlinear equations of the form Eq. (21) and Eq. (45) for some α k ≥ 2, or super-linear Eq. (23) and Eq. (81) for β ∈ [2, ∞) even though f (t) = 0. In fact, the case when n = 2 (Emden-Fowler super-linear), is of immense interest. Finally, we note that the inequalities obtained in this paper for nonlinear equations may not be the best ones. Therefore, it will be of immense interest to develop new Lyapunov and Hartman type inequalities better than the inequalities presented in this paper. For this, we suggest to the readers to prove an new inequality better than inequality (26) given in Lemma 2.1.