New gaps between zeros of fourth-order differential equations via Opial inequalities
© Saker et al.; licensee Springer 2012
Received: 30 January 2012
Accepted: 1 August 2012
Published: 30 August 2012
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.
where q is a real valued continuous function. This was strengthened in  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.
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 n th order differential equations -disconjugacy and disfocality are connected by the result of Nehari  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 .
obtain lower bounds for the spacing , where x is a solution of (1.1) satisfying for and ,
obtain lower bounds for the spacing , where x is a solution of (1.1) satisfying for and .
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  and . We note of particular interest in this paper is when q is oscillatory.
2 Main results
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 [, Theorem 3.9.1] that we will need in the proof of the main results.
Theorem 2.1 [, Theorem 3.9.1]
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 .
for any function satisfies .
for any function satisfies .
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 .
Theorem 2.3 
Using and instead of and in the proof of Theorem 2.2, we obtain the following result.
Substituting these last two inequalities into Theorem 2.4, we have the following result.
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.
and Γ is the Gamma function.
Using the inequalities (2.31) and (2.33) and proceeding as in the proof of Theorem 2.4, we obtain the following result.
Remark 3 In Theorem 2.6, if , then the term changes to .
where and are defined as in (2.34).
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.
where and are defined as in (2.34).
which are defined as in (2.8) and (2.44). The proof is complete. □
3 Discussions and examples
As a special case of Theorem 2.2, if , we have the following result.
As a special case of Theorem 2.5, if , then we have the following result.
As a special case of Theorem 2.7, if , we have the following result.
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.
The following examples illustrate the results.
which is satisfied for any and .
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 .
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