    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:
  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.), (2008), 29. Google Scholar  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), 1.   Google Scholar  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. Google Scholar  A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision,, Mem. Amer. Math. Soc., 93 (1991). Google Scholar  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. Google Scholar  I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992). Google Scholar  I. Daubechies and J. C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions,, SIAM J. Math. Anal., 22 (1991), 1388. Google Scholar  G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukrainian Math. J., 41 (1989), 1137. Google Scholar  G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes,, J. Approx. Theory, 80 (1995), 272. Google Scholar  G. Derfel and A. Iserles, The pantograph equation in the complex plane,, J. Math. Anal. Appl., 213 (1997), 117. Google Scholar  G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling,, J. Phys. A, 29 (1996), 4537. Google Scholar  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. Google Scholar  J. E. Hutchinson, Fractals and self similarlity,, Indiana Univ. Math. J., 30 (1981), 713. Google Scholar  A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1. Google Scholar  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.   Google Scholar  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.   Google Scholar  D. Revuz, Markov Chains, $2^{nd}$ edition,, North-Holland, (1984). Google Scholar  V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications,, Russian Math. Surveys, 45 (1990), 87. Google Scholar  R. Schilling, Spatially chaotic structures,, in Nonlinear Dynamics in Solids (ed. H. Thomas), (1992), 213.   Google Scholar  A. N. Shiryaev, Probability, $2^{nd}$ edition,, Springer-Verlag, (1996). Google Scholar  B. Solomyak, Notes on Bernoulli convolutions,, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, (2004), 207. Google Scholar  N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions],, Aequationes Math., 27 (1984), 87. Google Scholar  G. Strang, Wavelets and dilation equations: A brief introduction,, SIAM Rev., 31 (1989), 614. Google Scholar

show all references

##### References:
  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.), (2008), 29. Google Scholar  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), 1.   Google Scholar  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. Google Scholar  A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision,, Mem. Amer. Math. Soc., 93 (1991). Google Scholar  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. Google Scholar  I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992). Google Scholar  I. Daubechies and J. C. Lagarias, Two-scale difference equations. I. Existence and global regularity of solutions,, SIAM J. Math. Anal., 22 (1991), 1388. Google Scholar  G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukrainian Math. J., 41 (1989), 1137. Google Scholar  G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes,, J. Approx. Theory, 80 (1995), 272. Google Scholar  G. Derfel and A. Iserles, The pantograph equation in the complex plane,, J. Math. Anal. Appl., 213 (1997), 117. Google Scholar  G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling,, J. Phys. A, 29 (1996), 4537. Google Scholar  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. Google Scholar  J. E. Hutchinson, Fractals and self similarlity,, Indiana Univ. Math. J., 30 (1981), 713. Google Scholar  A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1. Google Scholar  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.   Google Scholar  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.   Google Scholar  D. Revuz, Markov Chains, $2^{nd}$ edition,, North-Holland, (1984). Google Scholar  V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications,, Russian Math. Surveys, 45 (1990), 87. Google Scholar  R. Schilling, Spatially chaotic structures,, in Nonlinear Dynamics in Solids (ed. H. Thomas), (1992), 213.   Google Scholar  A. N. Shiryaev, Probability, $2^{nd}$ edition,, Springer-Verlag, (1996). Google Scholar  B. Solomyak, Notes on Bernoulli convolutions,, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, (2004), 207. Google Scholar  N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions],, Aequationes Math., 27 (1984), 87. Google Scholar  G. Strang, Wavelets and dilation equations: A brief introduction,, SIAM Rev., 31 (1989), 614. Google Scholar
  Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303  Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049  Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, 2021, 20 (1) : 319-338. doi: 10.3934/cpaa.2020268  Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032  Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264  Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050  Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443  Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317  Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047  Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321  Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440  Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468  Sushil Kumar Dey, Bibhas C. Giri. Coordination of a sustainable reverse supply chain with revenue sharing contract. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020165  Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296  Zonghong Cao, Jie Min. Selection and impact of decision mode of encroachment and retail service in a dual-channel supply chain. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020167  Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117  Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107  Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070  Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017  Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

Impact Factor: