Analytic Solutions of a Second-Order Functional Differential Equation with a State Derivative Dependent Delay
We investigate an analytic solution of the second-order differential equation with a state derivative dependent delay of the form . Considering a convergent power series of an auxiliary equation with the relation , we obtain an analytic solution . Furthermore, we characterize a polynomial solution when is a polynomial.
The functional differential equation,where all , , provides a mathematical model for a physical or biological system in which the rate of change of system is determined not only by its present state, but also by its history (see [1, 2]). In recent years, many authors studied the existence and the uniqueness of an analytic solution of a variety of these equations. In 1984, Eder  classified solutions of the functional differential equation by using the Banach fixed point theorem. Let and be nonzero complex constants and a complex function. The first-order functional differential equation , , and has been studied by Si and Cheng , Qiu and Liu , and Zhang , respectively.
In 2001 , Si and Wang investigated the existence of analytic solution of the second-order functional differential equation:In 2009, Liu and Li  studied the equationObserve that (3) can be reduced to (2) by setting .
Next, the equationhas been studied by Si and Wang .
In order to obtain analytic solutions of (4), they constructed a corresponding auxiliary equation with parameter . The existence of solutions of an auxiliary equation depends on the condition of a parameter , such as is in the unit circle and is a root of unity which satisfies the Diophantine condition.
In this paper, we study the existence of analytic solutions of the second-order differential equation with a state derivative dependent delay of the formIf , then (5) reduces to (4).
Note that in this paper, we will study three cases of parameter in a corresponding auxiliary equation. One of them is the case that is a root of unity satisfying the Brjuno condition.
To construct an auxiliary equation, we setThenwhere is a complex constant. In particular, we haveApplying relations (6) and (8) to (5), we obtainWe construct the corresponding equation by differentiating both sides of (9) with respect to . This yields
2. Analytic Solutions of (10)
Consider the auxiliary equationwith initial value conditions and , where are complex numbers. Observe that if is an analytic solution of (11), then (10) has an analytic solution of the from . Equation (11) can be reduced equivalently to the integro-differential equationwhere and To construct analytic solutions of (12), we separate our study on the conditions of the parameter as follows:(H1);(H2), where is a Brjuno number; that is, , where denotes the sequence of partial fraction of the continued fraction expansion of ;(H3) for some integers with and , and for all and .
From now on, we let be an analytic function in a neighborhood of the origin. Then we represent by a power series .
Theorem 1. Let satisfy condition (H1). Then (11) has an analytic solutionin a neighborhood of the origin such that , , where is a nonzero complex number.
Proof. Since is analytic in a neighborhood of the origin, there exists a constant such that for . Substituting (13) into (12) and comparing coefficients of , we getand in general for The first expression allows us to choose and the second expression implies Consequently, the sequence is successively determined by the last expression in a unique manner. This implies that (11) has a formal power series solution.
Next, we show that the power series converges in a neighborhood of the origin. Since and for , there exists a positive constant such thatLet us define a power series , where a positive sequence is determined by , and for It follows that for That is, is a majorant series of We show that is analytic in a neighborhood of the origin. Note that if we let , thenConsider the equationSince is continuous in a neighborhood of the origin, , and , the implicit function theorem implies that there exists a unique function which is analytic in a neighborhood of the origin with a positive radius. Because is a majorant series of , is also analytic in a neighborhood of the origin with a positive radius. This completes the proof.
Now, we consider an analytic solution of the auxiliary equation (11) in the case of satisfies condition In order to study the existence of under the Brjuno condition, we will recall the definition of Brjuno number and some basic facts. As stated in , for a real number , we let be an integer part of and let be a fractional part of . Observe that if is an irrational number, then it has a unique expression of Gauss’s continued fractiondenoted simply by , where ’s and ’s are calculated by the following algorithm:(a), ;(b), for all .Define the sequences and by the following recursive relation:Note that . For each , we consider an arithmetical function When satisfies condition , we call it a Brjuno number. Consider in which for each , , where is a positive constant. We can show that is a Brjuno number but is not a Diophantine number. Therefore, Brjuno condition is weaker than the Diophantine condition. So, condition contains both Diophantine condition and a part of near resonance.
Let and be the sequence of partial denominators of Gauss’s continued fraction for As in , letLet be the set of integers such that either or for some and in , with , one has and divides . For any nonnegative integer , we definewhere . Let be a function defined byLet , and let be defined by condition . Note that is nondecreasing.
Lemma 2 (Davie’s lemma ). LetThen(a)there is a universal constant (independent of and ) such that(b) for all and ,(c).
Theorem 3. Assume that satisfies condition (H2). Then there exists an analytic solutionof (11) in a neighborhood of the origin such that , , where is a nonzero complex number.
Proof. We now imitate the proof of Theorem 1 with approximate new bound. The sequence is defined similar to the proof of Theorem 1. Note that and . Since and is analytic near the origin, there exists a positive constant so that for To construct a governing series of , we let be a nonnegative sequence determined by , and for all From this construction, we can demonstrate that a power series satisfies the implicit functional equationwith and . This yields the power series converges in a neighborhood of the origin. Hence, there exists a positive constant such that for .
Let be a function defined in Lemma 2. By mathematical induction, we can show that for Lemma 2 yields . This implies that has a convergence radius at least . The proof is completed.
Finally, we consider the case that satisfies condition . In this case, is not only on the unit circle, but also a root of unity. Let be a sequence defined by , with , , andwhere is a positive constant defined as in the proof of Theorem 3.
Theorem 4. Assume that satisfies condition (H3). Let be a power series determined by , , andwhereIf for , then (11) has an analytic solution in a neighborhood of the origin such that , , and , where is an arbitrary constant satisfying , where is defined as in (32).
Otherwise, if for some , then (11) has no analytic solution in a neighborhood of the origin.
Proof. Observe that if then is a trivial analytic solution of (11). So, we consider only the case .
If for some positive number , then . But implies , which is a contradiction. This concludes that (11) has no analytic solution in a neighborhood of the origin.
Assume that for . Then . So, there are infinitely many choices of . Choose so that , where is defined in (32).
Note that for , where We can see thatfor .
Likewise, the remaining proof is similar to one of Theorem 1. Consider the implicit functional equationSince ,