Regularity properties of functional equations

AbstractIn this paper, we apply the collocation methods to a class of Volterra integral functional equations with multiple proportional delays (VIFEMPDs). We shall present the existence, uniqueness and regularity properties of analytic solutions for this type of equations, and then analyze the convergence orders of the collocation solutions and give corresponding error estimates. The numerical results verify our theoretical analysis.

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On Diophantine approximations of the solutions of q-functional equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210502000331 ◽

2002 ◽

Vol 132 (03) ◽

pp. 639

Keyword(s):

Functional Equations ◽

Diophantine Approximations

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On some functional equations related to derivations and bicircular projections in rings

Glasnik Matematicki ◽

10.3336/gm.49.2.06 ◽

2014 ◽

Vol 49 (2) ◽

pp. 313-331

Author(s):

Maja Fošner ◽

◽

Benjamin Marcen ◽

Nejc Širovnik ◽

Joso Vukman ◽

...

Keyword(s):

Functional Equations

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Functional equations related to weightable quasi-metrics

Hacettepe Journal of Mathematics and Statistics ◽

10.15672/hjms.2015449441 ◽

2015 ◽

Vol 4 (1047) ◽

Author(s):

M.J. Campion ◽

E. Indurain ◽

G. Ochoa

Keyword(s):

Functional Equations

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Degenerate Diffusion Operators Arising in Population Biology (AM-185)

10.23943/princeton/9780691157122.001.0001 ◽

2017 ◽

Author(s):

Charles L. Epstein ◽

Rafe Mazzeo

Keyword(s):

Adjoint Operator ◽

Integral Kernel ◽

Mathematical Finance ◽

Elliptic Operators ◽

Complete Analysis ◽

Time Behavior ◽

Degenerate Elliptic Operators ◽

Regularity Properties ◽

Degenerate Elliptic ◽

Mathematical Foundations

This book provides the mathematical foundations for the analysis of a class of degenerate elliptic operators defined on manifolds with corners, which arise in a variety of applications such as population genetics, mathematical finance, and economics. The results discussed in this book prove the uniqueness of the solution to the martingale problem and therefore the existence of the associated Markov process. The book uses an “integral kernel method” to develop mathematical foundations for the study of such degenerate elliptic operators and the stochastic processes they define. The precise nature of the degeneracies of the principal symbol for these operators leads to solutions of the parabolic and elliptic problems that display novel regularity properties. Dually, the adjoint operator allows for rather dramatic singularities, such as measures supported on high codimensional strata of the boundary. The book establishes the uniqueness, existence, and sharp regularity properties for solutions to the homogeneous and inhomogeneous heat equations, as well as a complete analysis of the resolvent operator acting on Hölder spaces. It shows that the semigroups defined by these operators have holomorphic extensions to the right half plane. The book also demonstrates precise asymptotic results for the long-time behavior of solutions to both the forward and backward Kolmogorov equations.

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