Infinitely Many Eigenfunctions for Polynomial Problems: Exact Results

Abstract

Let be a real-valued polynomial function in which the degree of in is greater than or equal to 1. For any polynomial , we assume that is a nonlinear operator with . In this paper, we will find an eigenfunction to satisfy the following equation: for some eigenvalue and we call the problem a fixed point like problem. If the number of all eigenfunctions in is infinitely many, we prove that (i) any coefficients of , are all constants in and (ii) is an eigenfunction in if and only if .

1. Introduction and Preliminaries

Lenstra [1] investigated that in which is a polynomial function over the algebraic rational number field (where is an algebraic number). He found a polynomial satisfying the polynomial equation Further, Tung [2] extended (2) to solve polynomial solutions (near solutions) ( is a field) for the following equation: where is a constant depending on the polynomial solution and a given nonnegative integer.

Moreover, Lai and Chen [3–5] extended (3) to solve satisfying the polynomial equation as the form where , , is an irreducible polynomial in , and the polynomial function is written by where denotes the degree of in .

Recently, Chen and Lai [6, 7] research a quasicoincidence problem in which an arbitrary nonzero polynomial function is given as follows: where is a constant.

Definition 1 (Chen and Lai, [6]). A polynomial function satisfying (6) is called a quasicoincidence solution corresponding to some real number . This number is called a quasicoincidence value corresponding to the polynomial solutions .

In this paper, we define a fixed point like problem in which the is replaced by the arbitrary polynomial throughout this paper. Then we restate (6) as the following equation: It is a new developed fixed point like problem. We call the polynomial (7) as a fixed point like problem. The number of all eigenfunctions in (7) may be infinitely many, or finitely many, or not solvable.

Since there may exist many eigenfunctions corresponding to the eigenvalue , for convenience, we use the following notations to represent different situations:(1), the set of all eigenfunctions “” satisfying (7);(2), the set of all eigenvalues “” satisfying (7);(3), the set of all eigenfunctions “” corresponding to a fixed eigenvalue “”. For each , the cardinal number of , denoted by , satisfies the following condition:

In Section 2, we derive some properties of eigenfunctions of . If (7) has infinitely many eigenfunctions, the concerned properties are described in Section 3.

Throughout the paper, we denote the polynomial function by with . Moreover, we may assume that is nonzero. Since , problem (7) may become Moreover, if problem (7) has infinitely many eigenfunctions, dividing by both sides of the above equation, then there may exist infinitely many nonzero eigenfunctions satisfying for some . Therefore, this problem becomes a special case of (3).

2. Some Lemmas and a Former Theorem

Throughout this paper, we consider (7) for the polynomial function (9).

Lemma 2. Let . Then and this is divisible and is denoted by .

Proof. Since , we have for some . This means for some . It leads to then is a factor of .

In Lemma 2, any eigenfunction is a factor of . Thus we define a class of this factor as follows.

Notation 1. Let , and we denote .

According to Notation 1, it is obvious that for any in , we have the cardinal number

For convenience, we explain the relations of and in the following lemma.

Lemma 3. Consider

Proof. For any , by Lemma 2, we have . That is, for some factor of . It follows that and we obtain

We will use the definitions of “the pigeonhole principle,” which concert with Grimaldi [8] and the above relation can be explained as the following lemma.

Lemma 4. Suppose that the cardinal number ; there exists a factor of such that the cardinal number

Proof. By Lemma 3, we obtain Since the number of all factor of is at most , by pigeonhole’s principle, it yields for some factor of .

In order to solve the problem (7), [6, Lemma and Theorem ] are needed as follows.

Lemma 5 (see [6, Lemma 3]). Assume that the number of all quasicoincidence solutions (defined in Definition 1) is infinitely many; then, for any two quasicoincidence solutions and , we have for some constant and some factor of .

Theorem 6 (see [6, Theorem 11]). Assume that the number of all quasicoincidence solutions (defined in Definition 1) is infinitely many; then for some , is a factor of , and for .

3. Main Theorems

In this section, we consider for the polynomial function defined in (9).

We investigate the fixed point like problem of simple polynomial functions with at first. Theorems 7 and 8 describe the necessary and sufficient results of these simple functions.

Theorem 7. Let be a polynomial function with as the form for some , . If the cardinal number , then (i);(ii)any eigenfunction of (7) is of the form  for some .

Proof. Since , by Lemma 4, there exists a factor of such that There exist two different eigenfunctions with for different constants . Since , we have where , . It follows that By (27) and (29), we have By (30), we get Since and is nonzero, it follows that For any , we have By (32), we let , (33) becomes and it follows that Owing to , then we obtain where .

The following theorem is the sufficient conditions of Theorem 7.

Theorem 8. Let be a polynomial function with as the form for some , . If (i) ,(ii)any eigenfunction of (7) is of the form  for some , then .

Proof. By (i), we let , then for some . This implies and then is an eigenfunction of (7) for any constant . It follows that then .

In Theorems 7 and 8, problem (7) with is introduced. In the following theorems, we deal with (7) with when the number of all eigenfunctions is infinitely many.

Theorem 9. Suppose that the cardinal number and . Then the polynomial can be represented as for some constants .

Proof. Since , by Lemma 4, there exists a factor of satisfying Let be an eigenfunction in such that for some eigenvalue . By Remainder Theorem, we get where is the quotient and is the remainder.
From the above identity and considering any eigenfunction in with , we substitute (43) by taking above and it becomes Since , , we have for some different constants and . Substituting (45) and (46) into (44), it becomes and it leads to for any eigenfunction .
By (48), there exist infinitely many quasicoincidence solutions in to satisfy with . This problem is a quasicoincidence problem; then by Theorem 6, we have Moreover, since and for any , it implies that and by Lemma 5, any , we have By definitions of , , and can also be represented as for some . By (51), it follows that Moreover, by (53), this implies that and by (45), , say, and (50) implies that for some , .
By (54), (43) implies that

Conversely, if can be expressed as in Theorem 9, then the cardinal number ; this problem becomes the sufficient conditions of Theorem 9.

Theorem 10. Assume that for some for ; then

Proof. For any , for some . Then and then .

In fact, if , then and we prove it as follows.

Theorem 11. If , we have

Proof. Since , by the proof of Theorem 10, we have
Conversely, considering any , we have for some eigenvalue . By Lemma 2, we have ; this implies and then . This proof is completed.

From Theorems 9 and 11, we can easily obtain the following two corollaries.

Corollary 12. Let , , with for some , then .

Proof. This result can be immediately obtained from Theorem 9.

Corollary 13. If there exists an eigenfunction with , then .

Proof. This result can be immediately obtained from Theorem 11.

From Theorems 7, 8, and 10 and Corollary 12, we provide some examples of fixed point like problem (7) for some , which have infinitely many eigenfunctions and do not have infinitely many eigenfunctions as follows.

Example 14. In the following examples, we by the form of , we can decide whether the number of all eigenfunctions in (7) is infinitely many or not.(1)If , there do not exist infinitely many eigenfunctions (Theorem 7).(2)If , there do not exist infinitely many eigenfunctions (Theorem 7).(3)If , there exist infinitely many eigenfunctions and (4)If , there exist infinitely many eigenfunctions and (5)If , there do not exist infinitely many eigenfunctions (Corollary 12).(6)If , , for any constants , there do not exist infinitely many eigenfunctions (Corollary 12).

We would like to provide one open problem as follows.

Further Development. For a real-valued polynomial function , if , can we find a co-NP complete algorithm to solve all eigenfunctions satisfying (7)?

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was partly supported by NSC 103-2115-M-539-001, Taiwan.