Given a vector space X, we investigate the solutions f:R→X of the linear functional equation of third order fx=pfx-1+qfx-2+rf(x-3), which is strongly associated with a well-known identity for the Fibonacci numbers. Moreover, we prove the Hyers-Ulam stability of that equation.

1. Introduction

The problem of stability of functional equations was motivated by a question of Ulam [1] and a solution to it by Hyers [2]. Since then, numerous papers have been published on that subject and we refer to [3–6] for more details, some discussions, and further references; for examples of very recent results, see, for example, [7].

In this paper, as usual, C, R, Z, and N stand for the sets of complex numbers, real numbers, integers, and positive integers, respectively. For a nonempty subset S of a vector space, let ξ:S→S be a function. Moreover, ξ0(x)=x, ξn+1(x)=ξ(ξn(x)), and (only for bijective ξ) ξ-n-1(x)=ξ-1(ξ-n(x)) for x∈S and n∈N0:=N∪{0}.

Jung has proved in [3] (see also [8]) some results on solutions and Hyers-Ulam stability of the functional equation
(1)f(x)=pf(ξ(x))-qf(ξ2(x)),
in the case where S=R and ξ(x)=x-1 for x∈R.

If S:=N0 and p,q∈Z, then solutions x:N0→Z of the difference equation f(x)=pf(x-1)-qf(x-2) are called the Lucas sequences (see, e.g., [9]). In some special cases they are called with specific names, for example, the Fibonacci numbers (p=1, q=-1, x(0)=0, and x(1)=1), the Lucas numbers (p=1, q=-1, x(0)=2, and x(1)=1), the Pell numbers (p=2, q=-1, x(0)=0, and x(1)=1), the Pell-Lucas (or companion Lucas) numbers (p=2, q=-1, x(0)=2, and x(1)=2), and the Jacobsthal numbers (p=1, q=-2, x(0)=0, and x(1)=1).

For some information and further references concerning the functional equations in a single variable, we refer to [10–12]. Let us mention yet that the problem of Hyers-Ulam stability of functional equations is connected to the notions of controlled chaos and shadowing (see [13]).

We remark that if ξ:S→S is bijective, then (1) can be written in the following equivalent form:
(2)f(η2(x))=pf(η(x))-qf(x),
where η:=ξ-1.

In view of the last remark, the following Hyers-Ulam stability result concerning (1) can be derived from [14, Theorem 2] (see also [15]).

Theorem 1.

Let p,q∈R be given with q≠0 and let S be a nonempty subset of a vector space. Assume that a1, a2 are the complex roots of the quadratic equation x2-px+q=0 with |ai|≠1 for i∈{1,2}. Moreover, assume that X is either a real vector space if p2-4q>0 or a complex vector space if p2-4q<0. Let ξ:S→S be bijective. If a function f:S→X satisfies the inequality
(3)∥f(x)-pf(ξ(x))+qf(ξ2(x))∥≤ɛ
for all x∈S and for some ɛ≥0, then there exists a unique solution F:S→X of (1) with
(4)∥f(x)-F(x)∥≤ɛ|(|a1|-1)(|a2|-1)|
for all x∈S.

In [16, Theorem 1.4], the method presented in [3] was modified so as to prove a theorem which is a complement of Theorem 1. Note that, for bijective ξ, the following theorem improves the estimation (4) in some cases (e.g., a1=3/2, a2=-3/2, or a1=1/2, a2=-1/2). However, in some other situations (e.g., a1=3, a2=-3), the estimation (4) is better than (5). The following theorem also complements Theorem 1, because ξ can be quite arbitrary in the case of (α).

Theorem 2.

Given p,q∈R with q≠0, assume that the distinct complex roots a1, a2 of the quadratic equation x2-px+q=0 satisfy one of the following two conditions:

|ai|<1 for i∈{1,2};

|ai|≠1 for i∈{1,2} and ξ:S→S is bijective.

Moreover, assume that X is either a real vector space if p2-4q>0 or a complex vector space if p2-4q<0. If a function f:S→X satisfies inequality (3), then there exists a solution F:S→X of (1) such that
(5)∥f(x)-F(x)∥≤ɛ|a1-a2|(|a1|||a1|-1|+|a2|||a2|-1|)
for all x∈S. Moreover, if the condition (β) is true, then the F is the unique solution of (1) satisfying (5).

In this paper, we investigate the solutions of the functional equation
(6)f(x)=pf(x-1)+qf(x-2)+rf(x-3),
where p, q, r are real constants. Moreover, we also prove the Hyers-Ulam stability of that equation. Equation (6) is a kind of linear functional equations of third order because it is of the form
(7)f(x)=a1(x)f(ξ(x))+a2(x)f(ξ2(x))+a3(x)f(ξ3(x))
for the case of a1(x)=p, a2(x)=q, a3(x)=r, and ξ(x)=x-1.

2. General Solution

In the following theorem, we apply [16, Theorem 1.1] for the investigation of general solutions of the functional equation (6).

Theorem 3.

Let p, q, r be real constants such that the cubic equation
(8)x3+px2-qx+r=0
has the following properties:

α1 and α2 are two distinct nonzero real roots of the cubic equation (8);

it holds true that either (αi+p)2+4r/αi>0 for i∈{1,2} or (αi+p)2+4r/αi<0 for i∈{1,2}.

Let X be either a real vector space if (αi+p)2+4r/αi>0 for i∈{1,2} or a complex vector space if (αi+p)2+4r/αi<0 for i∈{1,2}. Then, a function f:R→X is a solution of the functional equation (6) if and only if there exist functions h1,h2:[-1,1)→X such that
(9)f(x)=α1α1-α2V[x]+1h2(x-[x])+α1rα2(α1-α2)V[x]h2(x-[x]-1)-α2α1-α2U[x]+1h1(x-[x])-α2rα1(α1-α2)U[x]h1(x-[x]-1),
where [x] denotes the largest integer not exceeding x, and Un, Vn are defined in (13) and (23).
Proof.

Assume that f:R→X is a solution of (6). If we define an auxiliary function g1:R→X by
(10)g1(x):=f(x)+α1f(x-1),
then it follows from (6) that g1 satisfies
(11)g1(x)=(α1+p)g1(x-1)+rα1g1(x-2)
for any x∈R. According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function h1:[-1,1)→X such that
(12)g1(x)=f(x)+α1f(x-1)=U[x]+1h1(x-[x])+rα1U[x]h1(x-[x]-1)
for all x∈R, where
(13)Un=an-bna-b(n∈Z)
and a, b are the distinct roots of the quadratic equation
(14)x2-(α1+p)x-rα1=0,
that is,
(15)a=α1+p2+(α1+p2)2+rα1,b=α1+p2-(α1+p2)2+rα1.

Since a is a root of the quadratic equation (14), we have
(16)a2=(α1+p)a+rα1.
We multiply both sides of (16) with a and make use of (16) and (i) to get
(17)a3=pa2+α1a2+rα1a=pa2+α1((α1+p)a+rα1)+rα1a=pa2+aα1(α13+pα12+r)+r=pa2+qa+r.
Similarly, we also obtain
(18)b3=pb2+qb+r.
Using (13), (17), and (18), we have
(19)pUn-1+qUn-2+rUn-3=(pa2+qa+r)an-3-(pb2+qb+r)bn-3a-b=an-bna-b=Un
for all n∈Z.

If we define an auxiliary function g2:R→X by
(20)g2(x):=f(x)+α2f(x-1),
then it follows from (6) that g2 satisfies
(21)g2(x)=(α2+p)g2(x-1)+rα2g2(x-2)
for any x∈R. According to [16, Theorem 1.1] or [3, Theorem 2.1], there exists a function h2:[-1,1)→X such that
(22)g2(x)=f(x)+α2f(x-1)=V[x]+1h2(x-[x])+rα2V[x]h2(x-[x]-1)
for all x∈R, where
(23)Vn=cn-dnc-d(n∈Z)
and c, d are the distinct roots of the quadratic equation
(24)x2-(α2+p)x-rα2=0,
that is,
(25)c=α2+p2+(α2+p2)2+rα2,d=α2+p2-(α2+p2)2+rα2.
As in the first part, we verify that
(26)Vn=pVn-1+qVn-2+rVn-3
for all n∈Z.

We now multiply (12) with α2 and (22) with α1, we subtract the former from the latter, and we then divide the resulting equation by (α1-α2) to get (9).

We assume that a function f:R→X is given by (9), where h1,h2:[-1,1)→X are arbitrarily given functions and Un, Vn are given by (13) and (23), respectively. Then, by (9), (19), and (26), we have
(27)pf(x-1)+qf(x-2)+rf(x-3)=α1α1-α2(pV[x]+qV[x]-1+rV[x]-2)h2(x-[x])+α1rα2(α1-α2)(pV[x]-1+qV[x]-2+rV[x]-3)×h2(x-[x]-1)-α2α1-α2(pU[x]+qU[x]-1+rU[x]-2)h1(x-[x])-α2rα1(α1-α2)(pU[x]-1+qU[x]-2+rU[x]-3)×h1(x-[x]-1)=α1α1-α2V[x]+1h2(x-[x])+α1rα2(α1-α2)V[x]h2(x-[x]-1)-α2α1-α2U[x]+1h1(x-[x])-α2rα1(α1-α2)U[x]h1(x-[x]-1)=f(x)
for all x∈R, which implies that f is a solution of (6).

According to [17, p. 92], the Fibonacci numbers Fn satisfy the identity
(28)Fn2=2Fn-12+2Fn-22-Fn-32
for all integers n>3. We can easily notice that the linear equation of third order
(29)f(x)=2f(x-1)+2f(x-2)-f(x-3)
is strongly related to identity (28).

Corollary 4.

Let X be a real vector space. A function f:R→X is a solution of the functional equation (29) if and only if there exist functions h1,h2:[-1,1)→X such that
(30)f(x)=5+3510U[x]+1h1(x-[x])+15+7510U[x]h1(x-[x]-1)+5-3510V[x]+1h2(x-[x])+15-7510V[x]h2(x-[x]-1),
where Un and Vn are defined in (33).

Proof.

If we set p=2, q=2, and r=-1 in (8), then the cubic equation
(31)x3+2x2-2x-1=0
has three distinct nonzero roots including
(32)α1=-32+52,α2=-32-52.
Moreover, it holds that (α1+p)2+4r/α1>0 and (α2+p)2+4r/α2>0. By (13), (15), (23), and (25), we have
(33)Un=an-bna-b,Vn=cn-dnc-d,
where we make use of (15) and (25) to calculate
(34)a=3+52,b=-1,c=3-52,d=-1.

Finally, in view of Theorem 3, we conclude that the assertion of our corollary is true.

Corollary 5.

If a function f:R→R is a solution of functional equation (29), then there exist real constants μ1, μ2, ν1, and ν2 such that
(35)f(n)=5+3510μ1Un+1+15+7510μ2Un+5-3510ν1Vn+1+15-7510ν2Vn
for all n∈Z, where Un and Vn are defined in (33).

3. Hyers-Ulam Stability

We apply the classical direct method to the proof of the following theorem. The classical direct method was first proposed by Hyers [2].

Theorem 6.

Let p, q, r be real constants with r≠0, let α be a nonzero root of the cubic equation (8), and let a, b be the roots of the quadratic equation x2-(α+p)x-r/α=0 with |a|>1 and 0<|b|<1. Let X be either a real Banach space if (α+p)2+4r/α>0 or a complex Banach space if (α+p)2+4r/α<0. If a function f:R→X satisfies the inequality
(36)∥f(x)-pf(x-1)-qf(x-2)-rf(x-3)∥≤ɛ
for all x∈R and for some ɛ≥0, then there exists a solution G:R→X of (6) such that
(37)∥f(x)+αf(x-1)-G(x)∥≤|a|-|b||a-b|ɛ(|a|-1)(1-|b|)
for all x∈R.

Proof.

If we define an auxiliary function g:R→X by
(38)g(x):=f(x)+αf(x-1),
then, as we did in (11), it follows from (36) that g satisfies the inequality
(39)∥g(x)-(α+p)g(x-1)-rαg(x-2)∥≤ɛ
or
(40)∥g(x)-ag(x-1)-b[g(x-1)-ag(x-2)]∥≤ɛ
for any x∈R.

If we replace x with x-k in the last inequality, then we have
(41)∥[g(x-k-1)-ag(x-k-2)]g(x-k)-ag(x-k-1)-b[g(x-k-1)-ag(x-k-2)]∥≤ɛ
for all x∈R. Furthermore, we get
(42)∥bk[g(x-k)-ag(x-k-1)]-bk+1[g(x-k-1)-ag(x-k-2)]bk[g(x-k)-ag(x-k-1)]∥≤|b|kɛ
for all x∈R and k∈Z. By (42), we obviously have
(43)∥g(x)-ag(x-1)-bn[g(x-n)-ag(x-n-1)]∥≤∑k=0n-1∥bk[g(x-k)-ag(x-k-1)]h-bk+1[g(x-k-1)-ag(x-k-2)]∥≤∑k=0n-1|b|kɛ
for x∈R and n∈N.

For any x∈R, (42) implies that the sequence {bn[g(x-n)-ag(x-n-1)]} is a Cauchy sequence (note that 0<|b|<1). Therefore, we can define a function G1:R→X by
(44)G1(x):=limn→∞bn[g(x-n)-ag(x-n-1)],
since X is complete. In view of the definition of G1 and using the relations, a+b=α+p and ab=-r/α, we obtain
(45)(α+p)G1(x-1)+rαG1(x-2)=(a+b)G1(x-1)-abG1(x-2)=a+bblimn→∞bn+1[g(x-(n+1))-ag(x-(n+1)-1)]-abb2limn→∞bn+2[g(x-(n+2))-ag(x-(n+2)-1)]=a+bbG1(x)-abG1(x)=G1(x)
for all x∈R. Since α is a nonzero root of the cubic equation (8), it follows from (45) that
(46)G1(x)-pG1(x-1)-qG1(x-2)-rG1(x-3)=(α+p)G1(x-1)+rαG1(x-2)-pG1(x-1)-qG1(x-2)-rG1(x-3)=αG1(x-1)+(-q+rα)G1(x-2)-rG1(x-3)=αG1(x-1)+(-α2-pα)G1(x-2)-rG1(x-3)=α((α+p)G1(x-2)+rαG1(x-3))-α(α+p)G1(x-2)-rG1(x-3)=0
for all x∈R. Hence, we conclude that G1 is a solution of (6).

If n tends to infinity, then (43) yields that
(47)∥g(x)-ag(x-1)-G1(x)∥≤ɛ1-|b|
for every x∈R.

On the other hand, it also follows from (36) that
(48)∥g(x)-bg(x-1)-a[g(x-1)-bg(x-2)]∥≤ɛ
for all x∈R. Analogously to (42), replacing x by x+k in the last inequality and then dividing by |a|k both sides of the resulting inequality, then we have
(49)∥a-k[g(x+k)-bg(x+k-1)]-a-k+1[g(x+k-1)-bg(x+k-2)]∥≤|a|-kɛ
for all x∈R and k∈Z. By using (49), we further obtain
(50)∥a-n[g(x+n)-bg(x+n-1)]-[g(x)-bg(x-1)]∥≤∑k=1n∥a-k[g(x+k)-bg(x+k-1)]h-a-k+1[g(x+k-1)-bg(x+k-2)]∥≤∑k=1n|a|-kɛ
for x∈R and n∈N.

On account of (49), we see that the sequence {a-n[g(x+n)-bg(x+n-1)]} is a Cauchy sequence for any fixed x∈R (note that |a|>1). Hence, we can define a function G2:R→X by
(51)G2(x):=limn→∞a-n[g(x+n)-bg(x+n-1)].
Due to the definition of G2 and the relations, a+b=α+p and ab=-r/α, we get
(52)(α+p)G2(x-1)+rαG2(x-2)=(a+b)G2(x-1)-abG2(x-2)=a+balimn→∞a-(n-1)[g(x+n-1)-bg(x+n-2)]-aba2limn→∞a-(n-2)[g(x+n-2)-bg(x+n-3)]=a+baG2(x)-baG2(x)=G2(x)
for any x∈R. Similarly as in the first part, we can show that G2 is a solution of (6).

If we let n tend to infinity, then it follows from (50) that
(53)∥G2(x)-g(x)+bg(x-1)∥≤ɛ|a|-1
for x∈R.

It follows from (47) and (53) that
(54)∥g(x-1)-1a-bG2(x)+1a-bG1(x)∥≤∥1a-bG1(x)-1a-bg(x)+aa-bg(x-1)∥+∥1a-bg(x)-ba-bg(x-1)-1a-bG2(x)∥≤|a|-|b||a-b|ɛ(|a|-1)(1-|b|)
for any x∈R.

Finally, if we define a function G:R→X by
(55)G(x):=1a-bG2(x+1)-1a-bG1(x+1)
for all x∈R, then G is also a solution of (6). Moreover, the validity of (37) follows from the last inequality.

The following theorem is the main theorem of this paper.

Theorem 7.

Given real constants p, q, r with r≠0, let α1 and α2 be distinct nonzero roots of cubic equation (8) and let ai, bi be the roots of the quadratic equation x2-(αi+p)x-r/αi=0 with |ai|>1 and 0<|bi|<1 for i∈{1,2}. Assume that either (αi+p)2+4r/αi>0 for all i∈{1,2} or (αi+p)2+4r/αi<0 for all i∈{1,2}. Let X be either a real Banach space if (αi+p)2+4r/αi>0 or a complex Banach space if (αi+p)2+4r/αi<0. If a function f:R→X satisfies inequality (36) for all x∈R and for some ɛ≥0, then there exists a solution F:R→X of (6) such that
(56)∥f(x)-F(x)∥≤|a1|-|b1||a1-b1||α2||α1-α2|ɛ(|a1|-1)(1-|b1|)+|a2|-|b2||a2-b2||α1||α1-α2|ɛ(|a2|-1)(1-|b2|)
for all x∈R.

Proof.

According to Theorem 6, there exists a solution Fi:R→X of (6) such that
(57)∥f(x)+αif(x-1)-Fi(x)∥≤|ai|-|bi||ai-bi|ɛ(|ai|-1)(1-|bi|)
for any x∈R and i∈{1,2}. In view of the last inequalities, we have
(58)∥f(x)-α1α1-α2F2(x)+α2α1-α2F1(x)∥≤∥α2α1-α2F1(x)-α2α1-α2f(x)-α1α2α1-α2f(x-1)∥+∥α1α1-α2f(x)+α1α2α1-α2f(x-1)-α1α1-α2F2(x)∥≤|a1|-|b1||a1-b1||α2||α1-α2|ɛ(|a1|-1)(1-|b1|)+|a2|-|b2||a2-b2||α1||α1-α2|ɛ(|a2|-1)(1-|b2|)
for all x∈R.

If we define a function F:R→X by
(59)F(x):=α1α1-α2F2(x)-α2α1-α2F1(x)
for each x∈R, then F is also a solution of (6), and inequality (56) follows from the last inequality.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research paper was completed while Soon-Mo Jung was a visiting scholar at National Technical University of Athens during February 2014. He would like to express his cordial thanks to Professor Themistocles M. Rassias for his hospitality and kindness. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1A1A2005557). The authors would like to express their cordial thanks to the referees for useful remarks.

UlamS. M.HyersD. H.On the stability of the linear functional equationJungS.-M.Functional equation fx=pfx-1-qf(x-2) and its Hyers-Ulam stabilityJungS.-M.JungS.-M.PopaD.RassiasM. T.On the stability of the linear functional equation in a single variable on complete metric groupsMosznerZ.On the stability of functional equationsBrillouet-BelluotN.BrzdekJ.CieplinskiK.On some recent developments in Ulam's type stabilityJungS.-M.Hyers-Ulam stability of Fibonacci functional equationRibenboimP.BaronK.JarczykW.Recent results on functional equations in a single variable, perspectives and open problemsKuczmaM.KuczmaM.ChoczewskiB.GerR.PilyuginS. Y.BrzdekJ.PopaD.XuB.Hyers-Ulam stability for linear equations of higher ordersTrifT.Hyers-Ulam-Rassias stability of a linear functional equation with constant coefficientsBrzdekJ.JungS.-M.A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequencesKoshyT.