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.), (2008), 29.   Google Scholar [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), 1.   Google Scholar [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.   Google Scholar [4] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision,, Mem. Amer. Math. Soc., 93 (1991).   Google Scholar [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.   Google Scholar [6] I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992).   Google Scholar [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.   Google Scholar [8] G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukrainian Math. J., 41 (1989), 1137.   Google Scholar [9] G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes,, J. Approx. Theory, 80 (1995), 272.   Google Scholar [10] G. Derfel and A. Iserles, The pantograph equation in the complex plane,, J. Math. Anal. Appl., 213 (1997), 117.   Google Scholar [11] G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling,, J. Phys. A, 29 (1996), 4537.   Google Scholar [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.   Google Scholar [13] J. E. Hutchinson, Fractals and self similarlity,, Indiana Univ. Math. J., 30 (1981), 713.   Google Scholar [14] A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1.   Google Scholar [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.   Google Scholar [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.   Google Scholar [17] D. Revuz, Markov Chains, $2^{nd}$ edition,, North-Holland, (1984).   Google Scholar [18] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications,, Russian Math. Surveys, 45 (1990), 87.   Google Scholar [19] R. Schilling, Spatially chaotic structures,, in Nonlinear Dynamics in Solids (ed. H. Thomas), (1992), 213.   Google Scholar [20] A. N. Shiryaev, Probability, $2^{nd}$ edition,, Springer-Verlag, (1996).   Google Scholar [21] B. Solomyak, Notes on Bernoulli convolutions,, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, (2004), 207.   Google Scholar [22] N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions],, Aequationes Math., 27 (1984), 87.   Google Scholar [23] G. Strang, Wavelets and dilation equations: A brief introduction,, SIAM Rev., 31 (1989), 614.   Google Scholar

show all references

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.), (2008), 29.   Google Scholar [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), 1.   Google Scholar [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.   Google Scholar [4] A. S. Cavaretta, W. Dahmen and C. A. Micchelli, Stationary subdivision,, Mem. Amer. Math. Soc., 93 (1991).   Google Scholar [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.   Google Scholar [6] I. Daubechies, Ten Lectures on Wavelets,, SIAM, (1992).   Google Scholar [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.   Google Scholar [8] G. A. Derfel, Probabilistic method for a class of functional-differential equations,, Ukrainian Math. J., 41 (1989), 1137.   Google Scholar [9] G. Derfel, N. Dyn and D. Levin, Generalized refinement equations and subdivision processes,, J. Approx. Theory, 80 (1995), 272.   Google Scholar [10] G. Derfel and A. Iserles, The pantograph equation in the complex plane,, J. Math. Anal. Appl., 213 (1997), 117.   Google Scholar [11] G. Derfel and R. Schilling, Spatially chaotic configurations and functional equations with rescaling,, J. Phys. A, 29 (1996), 4537.   Google Scholar [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.   Google Scholar [13] J. E. Hutchinson, Fractals and self similarlity,, Indiana Univ. Math. J., 30 (1981), 713.   Google Scholar [14] A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1.   Google Scholar [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.   Google Scholar [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.   Google Scholar [17] D. Revuz, Markov Chains, $2^{nd}$ edition,, North-Holland, (1984).   Google Scholar [18] V. A. Rvachev, Compactly supported solutions of functional-differential equations and their applications,, Russian Math. Surveys, 45 (1990), 87.   Google Scholar [19] R. Schilling, Spatially chaotic structures,, in Nonlinear Dynamics in Solids (ed. H. Thomas), (1992), 213.   Google Scholar [20] A. N. Shiryaev, Probability, $2^{nd}$ edition,, Springer-Verlag, (1996).   Google Scholar [21] B. Solomyak, Notes on Bernoulli convolutions,, in Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, (2004), 207.   Google Scholar [22] N. Steinmetz and P. Volkmann, Funktionalgleichungen für konstante Funktionen, (German) [Functional equations for constant functions],, Aequationes Math., 27 (1984), 87.   Google Scholar [23] G. Strang, Wavelets and dilation equations: A brief introduction,, SIAM Rev., 31 (1989), 614.   Google Scholar
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] 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 [15] 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 [16] 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 [17] 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 [18] 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 [19] 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 [20] 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: