Let Fx,y=asxys+as-1xys-1+⋯+a0x be a real-valued polynomial function in which the degree s of y in Fx,y is greater than or equal to 1. For any polynomial yx, we assume that T:Rx→Rx is a nonlinear operator with Tyx=Fx,yx. In this paper, we will find an eigenfunction yx∈Rx to satisfy the following equation: Fx,yx=ayx for some eigenvalue a∈R and we call the problem Fx,yx=ayx a fixed point like problem. If the number of all eigenfunctions in Fx,yx=ayx is infinitely many, we prove that (i) any coefficients of Fx,y,asx,as-1x,…,a0x, are all constants in R and (ii) yx is an eigenfunction in Fx,yx=ayx if and only if yx∈R.

1. Introduction and Preliminaries

Lenstra [1] investigated that
(1)Fx,yx=0
in which F(x,y) is a polynomial function over the algebraic rational number field Q(α) (where α is an algebraic number). He found a polynomial y=y(x)∈Q(α)[x] satisfying the polynomial equation
(2)Fx,yx=x.
Further, Tung [2] extended (2) to solve polynomial solutions (near solutions) y(x)∈K[x] (K is a field) for the following equation:
(3)Fx,yx=axm,
where a∈K is a constant depending on the polynomial solution y(x) and m∈N a given nonnegative integer.

Moreover, Lai and Chen [3–5] extended (3) to solve y(x)∈R[x] satisfying the polynomial equation as the form
(4)Fx,yx=apmx,x∈R,
where a∈R, m∈N, p(·) is an irreducible polynomial in x∈R, and the polynomial function F(x,y):R×R→R is written by
(5)F(x,y)=∑i=0saixyiwiths≥1,
where s denotes the degree degyF of y in F(x,y).

Recently, Chen and Lai [6, 7] research a quasicoincidence problem in which an arbitrary nonzero polynomial function f(x)∈R[x] is given as follows:
(6)Fx,yx=afx,
where a is a constant.

Definition 1 (Chen and Lai, [<xref ref-type="bibr" rid="B10">6</xref>]).

A polynomial function y=y(x) satisfying (6) is called a quasicoincidence solution corresponding to some real number a. This number a is called a quasicoincidence value corresponding to the polynomial solutions y=y(x).

In this paper, we define a fixed point like problem in which the y(x)∈R[x] is replaced by the arbitrary polynomial f(x)∈R[x] throughout this paper. Then we restate (6) as the following equation:
(7)Fx,yx=ay(x).
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 a, for convenience, we use the following notations to represent different situations:

Efunction, the set of all eigenfunctions “y(x)” satisfying (7);

Evalue, the set of all eigenvalues “a” satisfying (7);

Efunction(a), the set of all eigenfunctions “y(x)” corresponding to a fixed eigenvalue “a”.

For each a∈R, the cardinal number of Efunction(a), denoted by Efunction(a), satisfies the following condition:
(8)Efunction(a)≤degyF(x,y).

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

Throughout the paper, we denote the polynomial function by
(9)Fx,y=asxys+as-1xys-1+⋯+a1xy+a0x=∑i=0sai(x)yi
with s≥1. Moreover, we may assume that a0(x) is nonzero. Since a0(x)=0, problem (7) may become
(10)asxysx+as-1xys-1x+⋯+a1xyx=ayx.
Moreover, if problem (7) has infinitely many eigenfunctions, dividing y(x) by both sides of the above equation, then there may exist infinitely many nonzero eigenfunctions y(x) satisfying
(11)asxys-1x+as-1xys-2x+⋯+a1x=a
for some a∈R. 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 y(x)∈Efunction. Then
(12)yx=dpxforsomed∈R,
and this p(x) is divisible a0(x) and is denoted by p(x)∣a0(x).

Proof.

Since y(x)∈Efunction, we have F(x,y(x))=ay(x) for some a∈R. This means
(13)as(x)ys(x)+as-1(x)ys-1(x)+⋯+a1(x)y(x)+a0(x)=ay(x)
for some a∈R. It leads to
(14)y(x)asxys-1x+as-1xys-2x+⋯+a1x-a=-a0x;
then y(x) is a factor of a0(x).

In Lemma 2, any eigenfunction is a factor p(x) of a0(x). Thus we define a class of this factor as follows.

Notation 1.

Let p(x)∈R[x], and we denote Φ(p(x))={αp(x):α∈R}.

According to Notation 1, it is obvious that for any p(x) in R[x], we have the cardinal number
(15)Φpx=∞.

For convenience, we explain the relations of Efunction and Φ(p(x)) in the following lemma.

Lemma 3.

Consider
(16)Efunction=⋃px∣a0xΦpx∩Efunction.

Proof.

For any y(x)∈Efunction, by Lemma 2, we have y(x)∣a0(x). That is,
(17)yx∈Φpx
for some factor p(x) of a0(x). It follows that
(18)Efunction⊆⋃px∣a0xΦpx
and we obtain
(19)Efunction=⋃p(x)∣a0(x)Φpx∩Efunction.

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 Efunction=∞; there exists a factor p(x) of a0(x) such that the cardinal number
(20)Φpx∩Efunction=∞.

Proof.

By Lemma 3, we obtain
(21)∞=Efunction=⋃p(x)∣a0(x)Φpx∩Efunction≤∑p(x)∣a0(x)Φpx∩Efunction.
Since the number of all factor p(x) of a0(x) is at most 2dega0(x), by pigeonhole’s principle, it yields
(22)Φpx∩Efunction=∞
for some factor p(x) of a0(x).

In order to solve the problem (7), [6, Lemma 3 and Theorem 11] are needed as follows.

Lemma 5 (see [<xref ref-type="bibr" rid="B10">6</xref>, Lemma 3]).

Assume that the number of all quasicoincidence solutions (defined in Definition 1) is infinitely many; then, for any two quasicoincidence solutions y1(x) and y2(x), we have
(23)y1x-y2x=λgx
for some constant λ∈R and some factor g(x) of f(x).

Theorem 6 (see [<xref ref-type="bibr" rid="B10">6</xref>, Theorem 11]).

Assume that the number of all quasicoincidence solutions (defined in Definition 1) is infinitely many; then
(24)Fx,y=∑i=0scifxgixy-y1xi
for some y1(x)∈R[x], g(x) is a factor of f(x), and ci∈R for i=0,1,…,s.

3. Main Theorems

In this section, we consider F(x,y)=ay(x) for the polynomial function F(x,y) defined in (9).

We investigate the fixed point like problem of simple polynomial functions F(x,y) with s=1 at first. Theorems 7 and 8 describe the necessary and sufficient results of these simple functions.

Theorem 7.

Let F(x,y) be a polynomial function with degyF=1 as the form F(x,y)=a1(x)y+a0(x) for some a1(x), a0(x)∈R[x]. If the cardinal number Efunction=∞, then

a1(x)∈R;

any eigenfunction y(x) of (7) is of the form
(25)y(x)=λa0(x)

for some λ∈R.

Proof.

Since Efunction=∞, by Lemma 4, there exists a factor p(x) of a0(x) such that
(26)Φpx∩Efunction=∞.
There exist two different eigenfunctions y1(x),y2(x)∈Φ(p(x))∩Efunction with
(27)y1x=α1px,y2x=α2px,
for different constants α1,α2∈R. Since y1(x),y2(x)∈Efunction, we have
(28)Fx,y1x=ay1x,Fx,y2x=by2x,
where a, b∈Evalue. It follows that
(29)Fx,y1x=a1xy1x+a0x=ay1x,Fx,y2x=a1xy2x+a0x=by2x.
By (27) and (29), we have
(30)a1xα1px+a0x=aα1px,a1xα2px+a0x=bα2px.
By (30), we get
(31)a1xα1px-α2px=aα1px-bα2px.
Since α1≠α2 and p(x) is nonzero, it follows that
(32)a1(x)=aα1-bα2α1-α2∈R.
For any y(x)∈Efunction, we have
(33)Fx,yx=a1xyx+a0x=ayx.
By (32), we let a1(x)=a1~∈R, (33) becomes
(34)a1~yx+a0x=ayx,
and it follows that
(35)a0x=a-a1~yx.
Owing to a0(x)≠0, then we obtain
(36)yx=λ~a0x,
where λ~=1/a-a1~.

The following theorem is the sufficient conditions of Theorem 7.

Theorem 8.

Let F(x,y) be a polynomial function with degyF=1 as the form F(x,y)=a1(x)y+a0(x) for some a1(x), a0(x)∈R[x]. If

a1(x)∈R,

any eigenfunction y(x) of (7) is of the form
(37)yx=λa0x,

for some λ∈R,

then Efunction=∞.Proof.

By (i), we let a1(x)=a1∈R, then F(x,y(x))=ay(x) for some a∈R. This implies
(38)a1yx+a0x=ayx
and then y(x)=1/a-a1a0(x) is an eigenfunction of (7) for any constant a≠a1. It follows that
(39)∞=1a-a1a0(x):a∈R-a1≤Efunction;
then Efunction=∞.

In Theorems 7 and 8, problem (7) with degyF=1 is introduced. In the following theorems, we deal with (7) with degyF≥2 when the number of all eigenfunctions is infinitely many.

Theorem 9.

Suppose that the cardinal number Efunction=∞ and degyF(x,y)≥2. Then the polynomial F(x,y) can be represented as
(40)Fx,y=∑i=0sβiyi
for some constants βi∈R.

Proof.

Since Efunction=∞, by Lemma 4, there exists a factor p(x) of a0(x) satisfying
(41)Φpx∩Efunction=∞.
Let y1(x) be an eigenfunction in Φ(p(x))∩Efunction such that
(42)Fx,y1x=a1y1(x)
for some eigenvalue a1∈R. By Remainder Theorem, we get
(43)Fx,y=y-y1xF1x,y+a1y1x,
where F1(x,y) is the quotient and a1y1(x) is the remainder.

From the above identity and considering any eigenfunction y(x) in Φ(p(x))∩Efunction/{y1(x)} with F(x,y(x))=ay(x), we substitute (43) by taking y=y(x) above and it becomes
(44)ayx=Fx,yx=yx-y1xF1x,yx+a1y1x.
Since y1(x), y(x)∈Φ(p(x)), we have
(45)y1x=λ1px,(46)y(x)=λp(x)
for some different constants λ1 and λ. Substituting (45) and (46) into (44), it becomes
(47)aλpx=λpx-λ1pxF1x,yx+a1λ1px
and it leads to
(48)F1x,yx=aλ-a1λ1λ-λ1∈R
for any eigenfunction y(x)∈Φ(p(x))∩Efunction/{y1(x)}.

By (48), there exist infinitely many quasicoincidence solutions in Φ(p(x))∩Efunction/{y1(x)} to satisfy
(49)F1x,y=afx
with f(x)=1. This problem is a quasicoincidence problem; then by Theorem 6, we have
(50)F1x,y=∑i=0s-1cifxgixy-y1xi.
Moreover, since f(x)=1 and f(x)/gi(x)∈R[x] for any i=0,1,2,…,s-1, it implies that g(x)∈R and by Lemma 5, any y2x,y3(x)∈Φ(p(x))∩Efunction/{y1(x)}, we have
(51)y2(x)-y3(x)=dg(x)=d′∈R.
By definitions of Φ(p(x)), y2(x), and y3(x) can also be represented as
(52)y2x=λ2px,y3(x)=λ3p(x)
for some λ2,λ3∈R. By (51), it follows that
(53)y2x-y3x=λ2-λ3px∈R.
Moreover, by (53), this implies that p(x)∈R and by (45), y1(x)∈R, say, y1(x)=b1 and (50) implies that
(54)F1x,y=∑i=0s-1ciy-b1i=∑i=0s-1diyi
for some di∈R, i=0,1,2,…,s-1.

By (54), (43) implies that
(55)Fx,y=y-y1xF1x,y+a1y1x=y-b1∑i=0s-1diyi+a1b1=∑i=0sβiyi.

Conversely, if F(x,y) can be expressed as in Theorem 9, then the cardinal number Efunction=∞; this problem becomes the sufficient conditions of Theorem 9.

Theorem 10.

Assume that
(56)Fx,y=∑i=0sβiyi
for some βi∈R for i=0,1,…,s; then
(57)|Efunction|=∞.

Proof.

For any y(x)=c∈R,
(58)Fx,yx=∑i=0sciyix=∑i=0scici(thisisaconstant)=ac,
for some a=∑i=0scici/c∈R. Then R⊆Efunction and then |Efunction|=∞.

In fact, if Efunction=∞, then Efunction=R and we prove it as follows.

Theorem 11.

If |Efunction|=∞, we have
(59)Efunction=R.

Proof.

Since Efunction=∞, by the proof of Theorem 10, we have
(60)R⊆Efunction.

Conversely, considering any y(x)∈Efunction, we have
(61)Fx,yx=∑i=0sβiyix=ayx
for some eigenvalue a∈R. By Lemma 2, we have y(x)∣β0; this implies y(x)∈R and then Efunction⊆R. This proof is completed.

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

Corollary 12.

Let F(x,y)=∑i=0sai(x)yi∈R[x,y], s≥2, with aj(x)∉R for some j, then |Efunction|<∞.

Proof.

This result can be immediately obtained from Theorem 9.

Corollary 13.

If there exists an eigenfunction y(x)∈Efunction with y(x)∉R, then |Efunction|<∞.

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 a∈R, 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 F(x,y), we can decide whether the number of all eigenfunctions in (7) is infinitely many or not.

If (x,y)=xy+1, there do not exist infinitely many eigenfunctions (Theorem 7).

If F(x,y)=xy+x, there do not exist infinitely many eigenfunctions (Theorem 7).

If F(x,y)=y+x, there exist infinitely many eigenfunctions and
(62)Efunction={λx:λ∈R}(Theorem8).

If F(x,y)=-y2+7y+1, there exist infinitely many eigenfunctions and
(63)Efunction=R(Theorem10).

If F(x,y)=xy2+xy+1, there do not exist infinitely many eigenfunctions (Corollary 12).

If F(x,y)=∑i=0sciyi+x, s≥2, for any constants ci∈R, 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 F:R×R→R, if |Efunction|<∞, can we find a co-NP complete algorithm to solve all eigenfunctions y=y(x) 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.

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