Abstract
This paper investigates the linear functional equation with constant coefficients , where both and are constants, f is a given continuous function on , and is unknown. We present all continuous solutions of this functional equation. We show that (i) if , then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if , then the equation has a unique continuous solution; and (iii) if , then the equation has a continuous solution depending on a single parameter under a suitable condition on f.
1. Introduction
Recently, Mickens [1] considered a linear functional equationwhere is a positive integer, m is a positive real number, and φ is an unknown function with the domain . When , this equation is called pomeron functional equation. The functional equation comes from some phenomena in physics (cf. [2, 3, 4]). Mickens gives an exact solution of (1) in [1].where c is any positive constant. Mickens mistakenly thought solution (2) as the general solution. We recall that the general solution of functional equations, which in general depends on arbitrary functions, are quite different from the one of differential equations, which in general depends on arbitrary constants.
This paper considers the generalized pomeron functional equation:where and , f is a given continuous function on , and is unknown. Note that equation (3) is a linear, inhomogeneous, and functional equation. Bessenyei in [5] and Shi and Gong in [6] considered cyclic cases for some linear functional equations. Brydak in [7] showed the stability of the linear equation . Czerwik in [8] studied the continuous dependence on given functions for solutions. We use the recurrence method to present the general solution of equation (3). Compared with Kuczma’s monograph ([9], Chapter II), our method follows more directly and easily.
2. The General Solution to Equation (3)
Note that, if and φ satisfies (3), then . Furthermore, if and (3) has a solution, then .
According to the range of κ, we consider the following three cases to investigate continuous solutions of equation (3).
2.1. The Case
For this case, equation (3) has infinitely many continuous solutions, which depend on arbitrary functions.
Theorem 1. Choose arbitrarily and . Let and be any continuous function defined in and , respectively, such thatDefine on bywhereDefine on bywhereThen,is the general continuous solution on of equation (3).
Proof. One can check that φ defined above satisfies equation (3).
Firstly, we prove that is continuous in . By the continuity of in , we have AndThus, is continuous for . Now . Assume that is continuous for , . We have for ,Thus, is continuous for . Similarly, we can show that is continuous for for .
Secondly, we shall prove thatLet . Then, for some and . Hence, we obtainLet . Since f is continuous, there exists such that, for every with , we have . Therefore, we obtainTogether with (13) and (14), we have Therefore, is continuous in . Similarly, one can prove that is continuous in .
On the other hand, every continuous solution can be obtained in this manner.
2.2. The Case
The corresponding homogeneous equation of (3) is given by
According to the superposition principle for the linear equation, the general solution of (3) is given by the additions of the general solution to the corresponding homogeneous equation (16) and a particular solution of the original inhomogeneous equation (3), cf. [9].
Lemma 1. Suppose . Then, is the only continuous solution of equation (16).
Proof. We use the proof by contradiction.
Assume . Then, in . If fact, if there exists a such that , then by ,which contradicts the assumption. Thus, it follows from thatThis is a contradiction. Therefore, .
Assume that there exists an interval such thatGiven an , we can find a such thatThus, there exists an such that for , which implies the inequality for and . Hence, we haveThis is a contradiction. Therefore, on .
Theorem 2. Suppose . Then, equation (3) has a unique continuous solution:
Proof. We can obtain the uniqueness from Lemma 1. It suffices to prove that (22) is a continuous solution of equation (3).
Choose an arbitrary . Put . One can see that for . LetWe haveConsequently, the series in (22) converges uniformly in for every . Therefore, in (22) is a continuous function in . Further,Thus, is the only continuous solution of equation (3).
2.3. The Case
If , then equation (3) becomes the following form:
Lemma 2. Suppose is a continuous solution of equation (26). Then,
Proof. Let φ be a continuous solution of equation (26). Then,Substituting into equation (26), we have . Consequently, .
Note that the series in (27) may not converge. Some additional assumption on f should be made to guarantee the existence of continuous solutions of equation (26).
We present one sufficient condition on f in the following theorem.
Theorem 3. Suppose that there exists positive constants δ, σ, and L such that Then, equation (26) has a continuous solution.
Proof. Sincethe series (27) converges uniformly in the interval for any fixed . It follows from Lemma 2 that in (27) is a continuous solution.
3. Conclusion
This paper investigates one kind of linear functional equation with constant coefficients , where and , f is a given continuous function on , and is unknown. If , then the equation has infinite many continuous solutions, which depend on arbitrary functions. If , then the equation has the unique continuous solution:
If and the equation has a continuous solution, then and
For the last case, we only give one sufficient condition for the existence of continuous solutions. The problem for the necessary and sufficient condition still remains open.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This project was supported by the Scientific Research Fund of Sichuan Provincial Education Department (18ZA0274).