On Oscillations of Solutions of Third-Order Dynamic Equation
We are proving the new oscillation theorems for the solutions of third-order linear nonautonomous differential equation with complex coefficients. In the case of real coefficients we derive the oscillation criterion that is invariant with respect to the adjoint transformation. Our main tool is a new version of Levinson's asymptotic theorem.
Consider an ordinary nonautonomous differential equation of the third order with complex valued variable coefficients , , and .
A solution of (1.1) is said to be oscillatory if it has an infinite sequence of zeros in , and nonoscillatory, otherwise. Equation (1.1) is said to be non-oscillatory if all solutions are non-oscillatory and is said to be oscillatory if there exists at least one oscillatory solution.
Oscillation theorems for ordinary differential equation of the third order in the case of real variable coefficients have been studied in [1–7]. To the best of the author's knowledge, the oscillations of the solutions of nonautonomous third order equations with complex coefficients have not been studied yet.
Let be the set of times differentiable functions on . By substitution equation (1.1) with turns to the following equation: where the functions and are given by
Define characteristic (Weierstrass) function of (1.2) depending on a phase function By direct calculations To consider the case of complex coefficients, we are using asymptotic solutions of (1.2) in Euler form with phase functions , , that are approximate solutions of the characteristic equation Char.
The adjoint transformation , or transforms (1.2) to its adjoint equation . Note that condition (1.9) is not invariant with respect to the adjoint transformation . In the case of real coefficients under some restrictions, we will give the criterion of oscillations of solutions of (1.2) that is invariant with respect to the adjoint transformation (see Theorem 2.9 below).
2. Main Theorems
Let be the Wronskian of two differentiable functions and . The following asymptotic theorem is proved by using Levinson's asymptotic theorem .
Theorem 2.1. Assume that there exists complex-valued phase functions , , such that expressions , do not change a sign, that is,
Then solutions of (1.2) may be represented in the form where , are defined in (1.5), (1.6), and
Theorem 2.2. The solution of (1.2) corresponding to the asymptotic solution with the phase is oscillatory if and only if
The following theorem we deduce from Theorem 2.2 by choosing
Theorem 2.3. Assume that there exists a complex-valued function such that , and Then (1.2) with complex coefficients has one nonoscillatory solution and two linearly independent oscillatory solutions if and only if
By taking from Theorem 2.3, we get the following corollary.
Corollary 2.4. Assume that for some complex number Then (1.2) with complex coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
Corollary 2.5. Assume that conditions are satisfied. Then (1.2) with complex coefficients is non-oscillatory.
By taking from Theorem 2.3 we get another corollary.
Corollary 2.6. Assume that conditions are satisfied. Then (1.2) with complex coefficients is non-oscillatory.
In the case by taking , from Theorem 2.3, we deduce the following theorem.
Theorem 2.8. Assume that , , and Then (1.2) with the real coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
Another result may be proved by the different choice of the phase functions as follows: where is defined in (2.4), and
Define 3 auxiliary regions on the real plane
Theorem 2.9. Assume that , is simply connected region for some , and conditions (2.2) and (2.3), are satisfied. Then (1.2) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if for at least one of cubic roots.
Note that condition (2.25) is invariant with respect to the adjoint transformation .
For the case of the real constant coefficients , from Theorem 2.9, one can deduce the obvious result that (1.2) is oscillatory if and only if . Indeed in this case condition (2.25) turns to , and conditions (2.2), (2.3), and (2.24) could be dropped.
Theorem 2.11. Assume that is real, it does not change the sign, and conditions are satisfied. Then equation has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
In the case (self-adjoint equation (1.2)), condition (2.25) turns to (2.20) (see Theorem 2.8 above), which is Leighton's (see ) necessary condition of oscillations for solutions of the second-order equation .
Example 2.12. Equation where is a real number and is oscillatory by Corollary 2.4 since conditions (2.13) and (2.14) are satisfied with . Note that for this example Theorem 1.2 is not applicable since both conditions (1.8) and (1.9) fail even when .
Our main tool is Levinson's asymptotic theorem.
Theorem 3.1 (see ). Let be an diagonal matrix function which satisfies dichotomy condition.
For each pair of integers and in exist constants such that for all x and t, Let the matrix satisfy or by which we mean that each entry in has an absolutely convergent infinite integral. Then the system has a vector solution with the asymptotic form where is the identity matrix, is the error matrix-function, and is a constant column vector.
Proof of Theorem 2.1. Rewrite (1.2) as a system
where matrix function is defined via phase functions as follows:
we get the following:
Choosing specific auxiliary functions we have Here and further we suppress the time variable for the simplicity.
From Liouville's formula applied to (3.7) with the assumption that are solutions of , we get the following The Liouville's formula may be written in the form We always are choosing the phase functions such that (3.13) is satisfied (see (2.4)). From (3.13), we get that To apply Theorem 3.1 to system (3.8) note that from (2.1) it follows dichotomy condition (3.1) of Theorem 3.1: Condition (3.2) of Theorem 3.1 turns to , and it is followed from
One can drop condition since from (3.15) we have Assuming that condition may be dropped as well since So condition (3.2) of Theorem 3.1 turns to or (2.2) and (2.3). From Theorem 3.1 applied to system (3.8) and we get that or representation (2.5).
Proof of Theorem 2.2. Theorem 2.2 is followed from Theorem 2.1 since in representation (2.5) one may choose asymptotic solutions as follows: which are oscillating if and only if condition (2.7) is satisfied.
Proof of Theorem 2.3. Theorem 2.3 is deduced from Theorem 2.2 by choosing phase functions as in (2.8). From
condition (2.1) turns to condition (2.9). From (2.4), we get that
Since in conditions (2.2)-(2.3) the function appears in denominator we should assume that , or . By direct calculations, we get that
In view of
conditions (2.2) and (2.3) of Theorem 2.2 turn to (2.10) and (2.11).
Further the asymptotic solution is oscillating if and only if (2.12) is satisfied. Indeed, the solution corresponding to the asymptotic solution is non-oscillatory ( does not have zeros; otherwise is undefined at some points).
Proof of Corollaries 2.4 and 2.6. We deduce Corollaries 2.4 and 2.6 from Theorem 2.3 by the special choice of function as follows:
From (2.10) and (2.11) we get that
or in the case
which is equivalent to (2.13).
Further from (2.12) we get condition (2.14) in the case : The proof of Corollary 2.6 is followed from (3.30) to (3.27) by choosing .
Case 2 (). In this case is complex valued, are real. In view of we get that Condition (2.25) fails in this case: and (1.2) is non-oscillatory. Further If then , , , and we get the following: If then , , , and we get the following:
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