American Institute of Mathematical Sciences

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

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

Leonid V. Bogachev 1, , Gregory Derfel 2, and  Stanislav A. Molchanov 3,

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 $x ⇝ \alpha(x-\beta)$.
Keywords: Functional & functional-differential equations, harmonic function, martingale., pantograph equation, Markov chain.
Mathematics Subject Classification: Primary: 39B05; Secondary: 34K06, 39A22, 60G42, 60J0.
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

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