New gaps between zeros of fourth-order differential equations via Opial inequalities

In this paper, for a fourth-order differential equation, we will establish some lower bounds for the distance between zeros of a nontrivial solution and also lower bounds for the distance between zeros of a solution and/or its derivatives. We also give some new results related to some boundary value problems in the theory of bending of beams. The main results will be proved by making use of some generalizations of Opial and Wirtinger-type inequalities. Some examples are considered to illustrate the main results.

MSC: 34K11, 34C10.

In this paper, we are concerned with the lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives for the fourth-order differential equation

where are continuous measurable functions and is a nontrivial interval of reals. By a solution of (1.1) on the interval , we mean a nontrivial real-valued function , which has the property that and satisfies equation (1.1) on J. We assume that (1.1) possesses such a nontrivial solution on . An existence result in the literature, which is related to our results in this paper, was proved by Lyapunov [1] for second-order differential equations. This result states that if is a solution of the differential equation

with () and , then

where q is a real valued continuous function. This was strengthened in [2] with replaced by where . Since the appearance of this inequality various proofs and generalizations or improvements have appeared in the literature for different types of differential equations. For contributions, we refer the reader to the papers [3-22] and the references cited therein.

The nontrivial solution of (1.1) is said to oscillate or to be oscillatory, if it has arbitrarily large zeros. Equation (1.1) is oscillatory if one of its nontrivial solutions is oscillatory. Equation (1.1) is said to be -disconjugate if i and j are positive integers such that and no solution of (1.1) has an -distribution of zeros, i.e., no nontrivial solution has a pair of zeros of multiplicities i and j, respectively. In general, an nth order differential equation

is said to be -disconjugate on an interval I in case no nontrivial solution has a zero of order k followed by a zero of order . This means that, for every pair of points , , there does not exist a nontrivial solution of (1.1) which satisfies

The least value of b such that there exists a nontrivial solution which satisfies (1.5) is called the -conjugate point of a. The differential equation (1.4) is said to disconjugate on an interval I if one of its nontrivial solutions has at most zeros. For our case, if no nontrivial solution of (1.1) has more than three zeros, the equation is termed disconjugate. Together with -disconjugacy, we consider the related concept which is -disfocality. The differential equation (1.4) is said to be disfocal on an interval I if, for every nontrivial solution x at least one of the functions does not vanish on I. If the equation is not disfocal on I, then there exists an integer k (), a pair of points , and a nontrivial solution x such that k of the functions vanishes at a and the remaining functions at b, i.e.,

Equation (1.1) is said to be -disconjugate on if there is no nontrivial solution and , such that . Equation (1.1) is said to be -disfocal on an interval I for some in case there does not exist a solution x with a zero of order k followed by a zero of of order , where for and . For nth order differential equations -disconjugacy and disfocality are connected by the result of Nehari [23] which states that if (1.4) is -disfocal on it is disconjuguate on . For more details about disconjugacy and disfocality and the relation between them, we refer the reader to the paper [24].

In [4] and [25], the authors established some new Lyapunov type inequalities for higher order differential equations. Next, we present some special cases of their results for fourth-order differential equations. In [25], it is proved that if is a solution of the fourth-order differential equation

which satisfies , then

and if satisfies , then

In [4], the author proved that if is a solution of (1.7), which satisfies , then

In this paper, we are concerned with the following problems for the general equation (1.1):

(i) obtain lower bounds for the spacing , where x is a solution of (1.1) satisfying for and ,

(ii) obtain lower bounds for the spacing , where x is a solution of (1.1) satisfying for and .

We will also establish some new results related to some boundary value problems in the theory of bending of beams. In particular, we consider the boundary conditions

which correspond to a beam clamped at each end. Second, we consider the boundary conditions

which correspond to a beam clamped at and free at . The study of the boundary conditions which correspond to a beam clamped at and free at , and the boundary conditions , which correspond to a beam hinged or supported at both ends are similar to the proof of the boundary conditions (1.8)-(1.9) and will be left to the interested reader. For more discussions of boundary conditions of the bending of beams, we refer to [26,27].

The paper is organized as follows: In Section 2, we prove several results related to the problems (i)-(ii) and also prove some results related to the boundary value problems of the bending of beams with the boundary conditions (1.8) and (1.9). The main results will be proved by employing some Opial and Wirtinger type inequalities. The results yield conditions for disfocality and disconjugacy. In Section 3, we will discuss some special cases of the results to derive some new results for equation (1.7) and give some illustrative examples. To the best of the authors knowledge, this technique has not been employed before on equation (1.1), and the ideas are different from the techniques employed in [4] and [25]. We note of particular interest in this paper is when q is oscillatory.

In this section, we will prove the main results by employing some Opial and Wirtinger type inequalities. In the following, we present a generalization of Opial’s inequality due to Agarwal and Pang [[28], Theorem 3.9.1] that we will need in the proof of the main results.

Theorem 2.1 [[28], Theorem 3.9.1]

Assume that the functionsϑandϕare non-negative and measurable on the interval, m, nare real numbers such that, and () fixed. Letbe such thatis absolutely continuous on. Iffor () then

where

If we replaceby, (), then (2.1) holds whereis replaced bywhich is given by

where

In the following, we present a special case of the Wirtinger type inequality due to Agarwal et al. [29] that we will need in the proof of the main results. For and a positive function with either or on , we have

for any with .

Remark 1 It is clear that the inequality (2.4) is satisfied for any function y satisfying the imposed assumptions. If we put with or , or and , then we have the following inequality which gives a relation between and on the interval .

Lemma 2.1For, then we have

for anywith, wheresatisfies the equation

for any functionsatisfies.

Remark 2 Note that equation (2.6) holds if one chooses , where in this case

Also, the inequality (2.5) holds if . In this case, the function satisfies the differential equation

for any function satisfies .

Now, we are ready to state and prove the main results when . For simplicity, we introduce the following notations:

and

Theorem 2.2Suppose thatxis a nontrivial solution of (1.1). If, forand, then

where. If, forand, then

where.

Proof We prove (2.10). Multiplying (1.1) by and integrating by parts, we have

Using the assumptions , and , we have

Integrating by parts the right-hand side, we see that

Using the assumption , we see that

Substituting (2.13) into (2.12), we have

Applying the inequality (2.1) on the integral

with , , , , , and , we get (note that , for ) that

where is defined as in (2.8). Applying the inequality (2.1) again on the integral

with , , , , and , we see that

where is defined as in (2.9). Applying the Wirtinger inequality (2.5) on the integral

where , we see that

where satisfies equation (2.7) for any positive function . Substituting (2.17) into (2.16), we have

Substituting (2.15) and (2.18) into (2.14) and canceling the term , we have

which is the desired inequality (2.10). The proof of (2.11) is similar by using integration by parts and the constants and are replaced by and which are defined as in (2.8). The proof is complete. □

Next, we recall the following inequality in Agarwal and Pang [30].

Theorem 2.3[30]

Let, be nonnegative measurable functions onand () fixed. Ifsuch that, , is absolutely continuous on, then

where

Ifsuch that, , is absolutely continuous onthen (2.20) holds withis replaced bywhere

Suppose that the solution of (1.1) satisfies . Applying the inequality (2.20) with , and on the term , leads to

where

If instead , then (2.21) holds where is replaced by

Using and instead of and in the proof of Theorem 2.2, we obtain the following result.

Theorem 2.4Suppose thatxis a nontrivial solution of (1.1). If, for, and, then

where. If, forand, then

where.

If the function is nonincreasing on , then we see that

and

Substituting these last two inequalities into Theorem 2.4, we have the following result.

Theorem 2.5Assume thatis a nonincreasing function andxis a nontrivial solution of (1.1). If, forand, then

If, forand, then

We mention here that if we use the maximum value of , we see that

It is worth mentioning here that the inequality due to Fink [31] can be applied on the term . This in fact will give new results when is a nonincreasing function. We now state this inequality. Let () but fixed and let such that , , is absolutely continuous on and . Then

where

The details of the application of (2.23) will be left to the interested reader. One can note that the inequality (2.23) has been proved without weighted functions, so it will be interesting to extend this inequality and prove an inequality similar to the inequality (2.23) with weighted functions.

In the following, we apply an inequality due to Boyd [32] and the Schwarz inequality to obtain results similar to Theorem 2.5. The Boyd inequality states that if with (or ), then

where , , ,

and

Note that the inequality (2.24) has immediate application to the case where . Choose and apply (2.24) to and , and then add to obtain

where is defined as in (2.25). The inequality (2.24) has an immediate application when , to the case where (or ). In this case, equation (2.24) becomes

where

and Γ is the Gamma function.

Applying the Schwarz inequality,

on the term

we see that

Applying the inequality (2.27) on the integral

with and (note that ), we see that

where we assumed that is a nonincreasing function. Substituting (2.31) into (2.30), we have

Applying the Wirtinger inequality (2.5) on the integral

where , we see that

where satisfies equation (2.7) for any positive function . This implies that

where . If we replace by , then (2.32) becomes

where .

Using the inequalities (2.31) and (2.33) and proceeding as in the proof of Theorem 2.4, we obtain the following result.

Theorem 2.6Assume thatis a nonincreasing function andxis a nontrivial solution of (1.1). If, forand, then

If, forand, then

Remark 3 In Theorem 2.6, if , then the term changes to .

In the following, we will prove some results related to the boundary value problems in the theory of bending of beams. We begin with the boundary conditions , which correspond to a beam clamped at each end. Let

Theorem 2.7Suppose thatxis a nontrivial solution of (1.1) and. If, then

whereandare defined as in (2.34).

Proof Multiplying (1.1) by and integrating by parts, we have

Using the assumptions that and , we get that

Integrating by parts the right-hand side, we see that

Using the assumption , we see that

Integrating by parts the left-hand side of (2.37), we see that

Using the assumption , we have

Substituting (2.38) and (2.40) into (2.37), we have

Applying the inequality (2.1) on the integral

with , , , , and , we see that

where and are defined as in (2.34). Applying the inequality (2.20) on the integral

with , and , we see that

where . Substituting (2.42) and (2.43) into (2.41) and canceling the term , we have

which is the desired inequality (2.35). The proof is complete. □

Remark 4 One can use the condition instead of in the proof of Theorem 2.8. In this case the term is replaced by and also the term is replaced by .

In the following, we consider the boundary conditions , which correspond to a beam hinged or supported at both ends. The proof will be as in the proof of Theorem 2.7, by using these boundary conditions to get that and . This gives us the following result.

Theorem 2.8Suppose thatxis a nontrivial solution of (1.1) and. If, then

whereandare defined as in (2.34).

Next, in the following, we establish some results, which allow us to consider the case when . Let

where

Theorem 2.9Suppose thatxis a nontrivial solution of (1.1). If, forand, then

where. If, forand, then

where.

Proof We prove (2.45). Multiplying (1.1) by and integrating by parts, we have

Using the assumptions and , we have

Integrating by parts the last term in the right-hand side, we see that

Using the assumption , we see that

Substituting (2.13) into (2.12), we have

Applying the inequality (2.1) on the integral

with , , , , , , and , we get (note that , for ) that

where is defined as in (2.8) and Q is replaced by . Applying the inequality (2.1) again on the integral

with , , , , and , we see that

where is defined as in (2.9) and Q is replaced by . Applying the Wirtinger inequality (2.5) on the integral

where , we see that

where satisfies equation (2.7) for any positive function . Substituting (2.52) into (2.51), we have

Applying the inequality (2.1) on the integral

with , , , , , , and , we get (note that , for ) that

where is defined as in (2.44). Substituting (2.50), (2.53), and (2.54) into (2.49) and canceling the term , we have

which is the desired inequality (2.45). The proof of (2.46) is similar to (2.45) by using the integration by parts and the constants

are replaced by

which are defined as in (2.8) and (2.44). The proof is complete. □

In this section, we present some special cases of the results obtained in Section 2 and also give some illustrative examples. We begin with Theorem 2.2 and consider the case when . In this case, equation (1.1) becomes the fourth-order differential equation

Using the definitions of the functions and , and putting , we see after simplifications that

which lead to

and

As a special case of Theorem 2.2, if , we have the following result.

Theorem 3.1Suppose thatxis a nontrivial solution of (3.1). If, forand, then

where. If, forand, then

where.

As a special case of Theorem 2.5, if , then we have the following result.

Theorem 3.2Suppose thatxis a nontrivial solution of (3.1). If, forand, then

If, forand, then

As a special case of Theorem 2.7, if , we have the following result.

Theorem 3.3Suppose thatxis a nontrivial solution of (3.1). If, then

Remark 5 Note that the violation of the conditions in Theorem 3.3 yield sufficient conditions for disconjugacy of equation (3.1).

As a special case of Theorem 2.8, if , we have the following result.

Theorem 3.4Suppose thatxis a nontrivial solution of (3.1). If, then

The following examples illustrate the results.

Example 1 Consider the equation

where λ and α are positive constants. If be a solution of (3.3) with , then

Then if , the condition (3.2) of Theorem 3.3 is given by

which is satisfied for any and .

Example 2 Consider the equation

where λ and α are positive constants. By Theorem 3.3, we see that the equation (3.5) is 2-2 disconjugate on if

That is,

From this, we conclude that the interval of disconjugacy is bounded below by a constant times the cubic root of the frequency for , i.e., if is the interval of disconjugacy, then . In fact, this is compatible with the special case of the results that has been proved in [33].

The authors declare that they have no competing interests.

All the authors contributed equaly and significantly in writting this paper.

1. Lyapunov, AM: Probleme General de la Stabilitie du Movement, Princeton University Press, Princeton (1949)

2. Wintner, A: On the nonexistence of conjugate points. Amer. J. Math.. 73, 368–380 (1951). Publisher Full Text

3. Almenar, P, Jódar, L: A Lyapunov inequality for a second order nonlinear differential equations. Appl. Math. Lett.. 24, 524–527 (2001)

4. Cakmak, D: Lyapunov-type integral inequalities for certain higher order differential equations. Appl. Math. Comput.. 216, 368–373 (2010). Publisher Full Text

5. Chen, LS: A Lyapunov inequality and forced oscillations in general nonlinear nth order differential-difference equations. Glasg. Math. J.. 18, 161–166 (1977). Publisher Full Text

6. Eliason, SB: Lyapunov type inequalities for certain second-order functional differential equations. SIAM J. Appl. Math.. 27, 180–199 (1974). Publisher Full Text

7. Parhi, N, Panigrahi, S: On Liapunov-type inequality for third order differential equations. J. Math. Anal. Appl.. 233, 445–460 (1999). Publisher Full Text

8. Parhi, N, Panigrahi, S: Disfocality and Liapunov-type inequalities for third-order differential equations. Comput. Math. Appl.. 16, 227–233 (2003)

9. Pachpatte, BG: A note on Lyapunov type inequalities. Indian J. Pure Appl. Math.. 21, 45–49 (1990)

10. Pachpatte, BG: On the zeros of solutions of certain differential equations. Demonstr. Math.. 25, 825–833 (1992)

11. Pachpatte, BG: A Lyapunov type inequality for a certain second-order differential equation. Proc. Natl. Acad. Sci., India. 64, 69–73 (1994)

12. Pachpatte, BG: An inequality suggested by Lyapunov’s inequality. Centre Rech. Math. Pures, Neuchatel Chambery, Fac. 26, Ser. I, 1-4 (1995)

13. Pachpatte, BG: Lyapunov type integral inequalities for certain differential equations. Georgian Math. J.. 4, 139–148 (1997). Publisher Full Text

14. Pachpatte, BG: Inequalities related to the zeros of solutions of certain second order differential equations. Facta Univ. Ser. Math. Inform.. 16, 35–44 (2001)

15. Patula, WT: On the distance between zeros. Proc. Am. Math. Soc.. 52, 247–251 (1975). Publisher Full Text

16. Panigrahi, S: Lyapunov-type integral inequalities for certain higher-order differential equations. Electron. J. Differ. Equ.. 2009, 1–14 (2009)

17. Reid, WT: A generalized Lyapunov inequality. Electron. J. Differ. Equ.. 13, 182–196 (1973). Publisher Full Text

18. Saker, SH: Lyapunov inequalities for half-linear dynamic equations on time scales and disconjugacy. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms. 18, 149–161 (2011)

19. Saker, SH: Applications of Wirtinger inequalities on the distribution of zeros of the Riemann Zeta-function. J. Inequal. Appl.. 2010, Article ID 215416 (2010)

20. Saker, SH: Applications of Opial and Wirtinger inequalities on zeros of third order differential equations. Dyn. Syst. Appl.. 20, 479–494 (2011)

21. Saker, SH: Lyapunov type inequalities for a second order differential equations with a damping term. Ann. Pol. Math.. 103, 37–57 (2012). Publisher Full Text

22. Saker, SH: Opial’s type inequalities on time scales and some applications. Ann. Pol. Math.. 104, 234–260 (2012)

23. Nehari, Z: Green’s functions and disconjugacy. Arch. Ration. Mech. Anal.. 62, 53–76 (1976)

24. Elias, U: Necessary conditions and sufficent conditions for disfocality and disconjugacy of a differential equation. Pac. J. Math.. 81, 379–397 (1979). Publisher Full Text

25. Yang, X: On a Liapunov-type inequality for a certain higher-order differential equations. Appl. Math. Comput.. 134, 307–317 (2003). Publisher Full Text

26. Courant, R, Hilbert, D: Methods of Mathematical Physics, Wiley, New York (1989)

27. Zill, DG, Wright, WS, Cullen, MR: Advanced Engineering Mathematics, Jones and Bartlett Publishers, Canada (2011)

28. Agarwal, RP, Pang, PYH: Opial Inequalities with Applications in Differential and Difference Equations, Kluwer, Dordrechet (1995)

29. Agarwal, RP, Bohner, M, O’Regan, D, Saker, SH: Some dynamic Wirtinger-type inequalities and their applications. Pac. J. Math.. 252, 1–18 (2011). Publisher Full Text

30. Agarwal, RP, Pang, PYH: Sharp Opial-type inequalities involving higher order derivatives of two functions. Math. Nachr.. 174, 5–20 (1995). Publisher Full Text

31. Fink, AM: On Opial’s inequality for . Proc. Am. Math. Soc.. 115, 177–181 (1992)

32. Boyd, D: Best constants in class of integral inequalities. Pac. J. Math.. 30, 367–383 (1969). Publisher Full Text

33. Brown, R, Hinton, D: Lyapunov inequalities and their applications. In: Rassias T (ed.) Survey on Classical Inequalities, pp. 1–25. Kluwer, Drodrecht (2000)