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"Functional equations"
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Recent trends in formal and analytic solutions of diff. equations : Virtual Conference Formal and Analytic Solutions of Diff. Equations, June 28-July 2, 2021, University of Alcalá, Alcalá de Henares, Spain
This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.
eBook
Spectral expansions of non-self-adjoint generalized Laguerre semigroups
We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local
Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a
subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the
framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between
this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As
a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our
methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to
equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert
space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are
led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these
norms.
eBook
Fuzzy stability of a generalized 4-dimensional AQCQ functional equation
Functional equations are widely used in various fields for solving practical examples, exploring theoretical ideas, and modeling complex relationships and the study of their stability is essential for understanding how small changes in the inputs or functional form affect the solutions. This has both theoretical significances and practical applications across mathematics, science, and engineering. For this purpose, in this paper, we explore the Ulam-Hyers stability of
Λ
(
z
1
+
z
2
+
λ
(
z
3
+
z
4
)
)
+
Λ
(
z
1
+
z
2
+
λ
(
z
3
+
z
4
)
)
=
λ
2
{
Λ
(
z
1
+
z
2
+
z
3
+
z
4
)
+
Λ
(
z
1
+
z
2
−
z
3
−
z
4
)
}
+
2
(
1
−
λ
2
)
Λ
(
z
1
+
z
2
)
+
(
λ
4
−
λ
2
)
12
{
Λ
(
2
(
z
3
+
z
4
)
)
+
Λ
(
−
2
(
z
3
+
z
4
)
)
−
4
(
Λ
(
z
3
+
z
4
)
+
Λ
(
−
z
3
−
z
4
)
)
}
, a generalized 4-dimensional AQCQ functional equation in fuzzy normed spaces using two different methods.
Journal Article
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of
Brunn-Minkowski type for a nonlinear capacity,
In the first part of this article, we prove the Brunn-Minkowski inequality for this
capacity:
In the second part of this article we study a Minkowski problem for a certain measure associated with a compact
convex set
eBook
On Ulam Stability of Functional Equations in Non-Archimedean Spaces
We present a survey of outcomes on Ulam stability of functional equations in non-Archimedean normed spaces. We focus mainly on functional equations in several variables (including the Cauchy equation, the Jordan–von Neumann equation, the Jensen equation, and their generalizations), but we also report a result on a general equation in a single variable, which can be applied to the very important linear functional equation. Let us note that one can observe the symmetry between the presented results and the analogous ones obtained for both classical and two-normed spaces.
Journal Article
Riemann zeta fractional derivative—functional equation and link with primes
This paper outlines further properties concerning the fractional derivative of the Riemann ζ function. The functional equation, computed by the introduction of the Grünwald–Letnikov fractional derivative, is rewritten in a simplified form that reduces the computational cost. Additionally, a quasisymmetric form of the aforementioned functional equation is derived (symmetric up to one complex multiplicative constant). The second part of the paper examines the link with the distribution of prime numbers. The Dirichlet η function suggests the introduction of a complex strip as a fractional counterpart of the critical strip. Analytic properties are shown, particularly that a Dirichlet series can be linked with this strip and expressed as a sum of the fractional derivatives of ζ. Finally, Theorem 4.3 links the fractional derivative of ζ with the distribution of prime numbers in the left half-plane.
Journal Article
The solution of Drygas functional equations with additional conditions
We determine the solution of the Drygas functional equation that satisfies the additional condition
(
y
2
+
y
)
f
(
x
)
=
(
x
2
+
x
)
f
(
y
)
on a restricted domain. Also, some other properties of Drygas functions are given as well.
Journal Article
Functional equations on discrete sets
Let Y (+) be a group, D ⊆ ℤ2 where ℤ(+, ⩽) denotes the ordered group of all integers, and ℤ2 := ℤ×ℤ. We shall use the notations Dx := {u ∈ ℤ | ∃v ∈ X : (u, v) ∈ D}, Dy := {v ∈ ℤ | ∃u ∈ ℤ : (u, v) ∈ D}, Dx+y := {z ∈ ℤ | ∃(u, v) ∈ D : z = u + v}. The main purpose of the article is to find sets D ⊆ ℤ2 that the general solution of the functional equation f (x+y) = g(x)+h(y) for all (x, y) ∈ D with unknown functions f : Dx+y → Y, g : Dx → Y, h : Dy → Y is in the form of f (u) = a(u) + C1 + C2 for all u ∈ Dx+y, g(v) = a(v) + C1 for all v ∈ Dx, h(z) = a(z) + C2 for all z ∈ Dy where a : ℤ → Y is an additive function, C1, C2 ∈ Y are constants.
Journal Article