Abstract
This paper investigates the linear functional equation with constant coefficients φt=κφλt+ft, where both κ>0 and 1>λ>0 are constants, f is a given continuous function on ℝ, and φ:ℝ⟶ℝ is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ>1, then the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0<κ<1, then the equation has a unique continuous solution; and (iii) if κ=1, then the equation has a continuous solution depending on a single parameter φ0 under a suitable condition on f.
Highlights
Mickens [1] considered a linear functional equation t φ(t) NφNm − (N − 1), (1)where N ≥ 1 is a positive integer, m is a positive real number, and φ is an unknown function with the domain R
When m 2, this equation is called pomeron functional equation. e functional equation comes from some phenomena in physics
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
Summary
Numerical Simulation Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang, Sichuan 641112, China. Is paper investigates the linear functional equation with constant coefficients φ(t) κφ(λt) + f(t), where both κ > 0 and 1 > λ > 0 are constants, f is a given continuous function on R, and φ: R ⟶ R is unknown. We present all continuous solutions of this functional equation. We show that (i) if κ > 1, the equation has infinite many continuous solutions, which depends on arbitrary functions; (ii) if 0 < κ < 1, the equation has a unique continuous solution; and (iii) if κ 1, the equation has a continuous solution depending on a single parameter φ(0) under a suitable condition on f
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