# American Institute of Mathematical Sciences

2015, 2015(special): 132-141. doi: 10.3934/proc.2015.0132

## Analysis of the archetypal functional equation in the non-critical case

 1 Department of Statistics, School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom 2 Department of Mathematics, Ben-Gurion University of the Negev, Be'er Sheva 84105, Israel 3 Department of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, United States

Received  September 2014 Revised  December 2014 Published  November 2015

We study the archetypal functional equation of the form $y(x)=\iint_{\mathbb{R}^2} y(a(x-b))\,\mu(da,db)$ ($x\in\mathbb{R}$), where $\mu$ is a probability measure on $\mathbb{R}^2$; equivalently, $y(x)=\mathbb{E}\{y(\alpha(x-\beta))\}$, where $\mathbb{E}$ is expectation with respect to the distribution $\mu$ of random coefficients $(\alpha,\beta)$. Existence of non-trivial (i.e. non-constant) bounded continuous solutions is governed by the value $K:=\iint_{\mathbb{R}^2}\ln|a|\,\mu(da,db) =\mathbb{E}\{\ln|\alpha|\}$; namely, under mild technical conditions no such solutions exist whenever $K<0$, whereas if $K>0$ (and $\alpha>0$) then there is a non-trivial solution constructed as the distribution function of a certain random series representing a self-similar measure associated with $(\alpha,\beta)$. Further results are obtained in the supercritical case $K>0$, including existence, uniqueness and a maximum principle. The case with $\mathbb{P}(\alpha<0)>0$ is drastically different from that with $\alpha>0$; in particular, we prove that a bounded solution $y(\cdot)$ possessing limits at $\pm\infty$ must be constant. The proofs employ martingale techniques applied to the martingale $y(X_n)$, where $(X_n)$ is an associated Markov chain with jumps of the form $x ⇝ \alpha(x-\beta)$.
Citation: Leonid V. Bogachev, Gregory Derfel, Stanislav A. Molchanov. Analysis of the archetypal functional equation in the non-critical case. Conference Publications, 2015, 2015 (special) : 132-141. doi: 10.3934/proc.2015.0132
##### References:
 [1] L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation, in Topics in Stochastic Analysis and Nonparametric Estimation (eds. P.-L. Chow et al.), Springer-Verlag, New York, 2008, pp. 29-49. [2] L. V. Bogachev, G. Derfel and S. A. Molchanov, On bounded continuous solutions of the archetypal equation with rescaling, Proc. Royal Soc. A, 471 (2015), 20150351, 1-19. [3] B. van Brunt and G. C. Wake, A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model, European J. Appl. Math., 22 (2011), 151-168. [4] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93 (1991), no. 453. [5] G. Choquet and J. Deny, Sur l'équation de convolution $\mu=\mu\star\sigma$, (French) [On the convolution equation $\mu=\mu\star\sigma$], C. R. Acad. Sci. Paris, 250 (1960), 799-801. [6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [7] I. Daubechies and J. C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), 1388-1410. [8] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrainian Math. J., 41 (1989), 1137-1141 (1990). [9] G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes, J. Approx. Theory, 80 (1995), 272-297. [10] G. Derfel and A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl., 213 (1997), 117-132. [11] G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling, J. Phys. A, 29 (1996), 4537-4547. [12] A. K. Grintsevichyus, On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines, Theor. Probab. Appl., 19 (1974), 163-168. [13] J. E. Hutchinson, Fractals and self similarlity, Indiana Univ. Math. J., 30 (1981), 713-747. [14] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. [15] T. Kato and J. B. McLeod, The functional-differential equation $y'(x)=a y(\lambda x)+b y(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937. [16] J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London A, 322 (1971), 447-468. [17] D. Revuz, Markov Chains, $2^{nd}$ edition, North-Holland, Amsterdam, 1984. [18] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications, Russian Math. Surveys, 45 (1) (1990), 87-120. [19] R. Schilling, Spatially chaotic structures, in Nonlinear Dynamics in Solids (ed. H. Thomas), Springer-Verlag, Berlin, 1992, pp. 213-241. [20] A. N. Shiryaev, Probability, $2^{nd}$ edition, Springer-Verlag, New York, 1996. [21] B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1 (eds. M. L. Lapidus and M. van Frankenhuijsen), Amer. Math. Soc., Providence, RI, 2004, pp. 207-230. [22] N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions], Aequationes Math., 27 (1984), 87-96. [23] G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Rev., 31 (1989), 614-627.

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##### References:
 [1] L. Bogachev, G. Derfel, S. Molchanov and J. Ockendon, On bounded solutions of the balanced generalized pantograph equation, in Topics in Stochastic Analysis and Nonparametric Estimation (eds. P.-L. Chow et al.), Springer-Verlag, New York, 2008, pp. 29-49. [2] L. V. Bogachev, G. Derfel and S. A. Molchanov, On bounded continuous solutions of the archetypal equation with rescaling, Proc. Royal Soc. A, 471 (2015), 20150351, 1-19. [3] B. van Brunt and G. C. Wake, A Mellin transform solution to a second-order pantograph equation with linear dispersion arising in a cell growth model, European J. Appl. Math., 22 (2011), 151-168. [4] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc., 93 (1991), no. 453. [5] G. Choquet and J. Deny, Sur l'équation de convolution $\mu=\mu\star\sigma$, (French) [On the convolution equation $\mu=\mu\star\sigma$], C. R. Acad. Sci. Paris, 250 (1960), 799-801. [6] I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992. [7] I. Daubechies and J. C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions, SIAM J. Math. Anal., 22 (1991), 1388-1410. [8] G. A. Derfel, Probabilistic method for a class of functional-differential equations, Ukrainian Math. J., 41 (1989), 1137-1141 (1990). [9] G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes, J. Approx. Theory, 80 (1995), 272-297. [10] G. Derfel and A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl., 213 (1997), 117-132. [11] G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling, J. Phys. A, 29 (1996), 4537-4547. [12] A. K. Grintsevichyus, On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines, Theor. Probab. Appl., 19 (1974), 163-168. [13] J. E. Hutchinson, Fractals and self similarlity, Indiana Univ. Math. J., 30 (1981), 713-747. [14] A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. [15] T. Kato and J. B. McLeod, The functional-differential equation $y'(x)=a y(\lambda x)+b y(x)$, Bull. Amer. Math. Soc., 77 (1971), 891-937. [16] J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. London A, 322 (1971), 447-468. [17] D. Revuz, Markov Chains, $2^{nd}$ edition, North-Holland, Amsterdam, 1984. [18] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications, Russian Math. Surveys, 45 (1) (1990), 87-120. [19] R. Schilling, Spatially chaotic structures, in Nonlinear Dynamics in Solids (ed. H. Thomas), Springer-Verlag, Berlin, 1992, pp. 213-241. [20] A. N. Shiryaev, Probability, $2^{nd}$ edition, Springer-Verlag, New York, 1996. [21] B. Solomyak, Notes on Bernoulli convolutions, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 1 (eds. M. L. Lapidus and M. van Frankenhuijsen), Amer. Math. Soc., Providence, RI, 2004, pp. 207-230. [22] N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions], Aequationes Math., 27 (1984), 87-96. [23] G. Strang, Wavelets and dilation equations: A brief introduction, SIAM Rev., 31 (1989), 614-627.
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