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The paper focuses on establishing sufficient conditions for the existence of the solutions in some functional q-integral equations, particularly in Banach spaces. In this method, the technique of measures of noncompactness and Petryshyn's fixed point theorem in Banach space is used. We provide some examples of equations, which confirm that our result is applicable to a wide class of integral equations. [ABSTRACT FROM AUTHOR]

We present some remarks on [Carpathian J. Math. 39 (2023), No 2, 541-551] in order to obtain a unique non trivial solution. [ABSTRACT FROM AUTHOR]

Given a rational number α ≠ 2, we establish a criterion for the existence of the general solution of an alternative quadratic functional equation of the form f (x y) + f (x y-1) = 2f (x) + 2f (y) or f (x y) + f (x y-1) = αf (x) + 2f (y), where f is a mapping from an abelian group (...) to a uniquely divisible abelian group (...). We also find the general solution in the cases when G is a 6-divisible abelian group and G is a cyclic group. [ABSTRACT FROM AUTHOR]

We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. Our main results are that we describe the complex-valued solutions of the following functional equations g (x σ (y)) = g (x) g (y) + f (x) f (y) , x , y ∈ S , f (x σ (y)) = f (x) g (y) + f (y) g (x) , x , y ∈ S , and f (x σ (y)) = f (x) g (y) - f (y) g (x) , x , y ∈ S , where σ : S → S is an automorphism that need not be involutive. As a consequence we show that the first two equations are equivalent to their variants. We also give some applications. [ABSTRACT FROM AUTHOR]

The relationship is studied here between the 3D incompressible Brinkman–Forchheimer problem with delay and its generalized steady state. First, with some restrictive condition on the delay term, the global well-posedness of 3D Brinkman–Forchheimer problem and its steady state problem are obtained by compactness method and Brouwer fixed point method respectively. Then the global L p (2 ≤ p < ∞) decay estimates are established for weak solution of non-autonomous Brinkman–Forchheimer equations with delay by using a retarded integral inequality. The global decay estimates can be proved for strong solution as well. Finally, the exponential stability property is investigated for weak solution of the 3D non-autonomous Brinkman–Forchheimer problem by a direct approach and also for the autonomous system by using a retarded integral inequality. Furthermore, the Razumikhin approach is utilized to achieve the asymptotic stability for strong solution of autonomous system under a relaxed restriction. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying the non-probability Gibbs measures for a Hard-Core (HC) model on a Cayley tree of order k ≥ 2, where the set of integers ℤ is the set of spin values. It is well known that each Gibbs measure, whether it be a gradient or non-probability measure, of this model corresponds to a boundary law. A boundary law can be thought of as an infinite-dimensional vector function (with strictly positive coordinates) defined at the vertices of the Cayley tree, which satisfies a nonlinear functional equation. Furthermore, every normalizable boundary law corresponds to a Gibbs measure. However, a non-normalizable boundary law can define the gradient or non-probability Gibbs measures. In this paper, we investigate the conditions for uniqueness and non-uniqueness of translation-invariant and periodic non-probability Gibbs measures for the HC model on a Cayley tree of any order k ≥ 2. [ABSTRACT FROM AUTHOR]

A problem of identification of the set of a thermal-conductivity coefficients for the nonlinear heat equation in the case of non-uniqueness is considered. Classes of inverse heat conduction problems (IHCP) with a non-unique solution are defined. Explicit descriptions of sets of thermal-conductivity coefficients for these classes are obtained. For solving the identification problem the functional identification approach is used. Unlike traditional methods, the proposed algorithm does not utilize approximations of the coefficient with a finite system of basis functions. The results of computational experiments are presented. It is shown that the functional identification approach makes it possible to numerically identify the non-uniqueness of the solution of the inverse problem of heat conduction. [ABSTRACT FROM AUTHOR]

The spatial Three-Dimensional (3D) edge network is one of the typical rank-lossless networks. The current network adjustment usually uses Least Squares (LS) algorithm, which has the complexity of linearization derivation, computational volume and other problems. It is based on high-precision ranging values. This study aims to minimize the sum of the difference between the inverse distance of the control point coordinates and the observation distance, the composition of the non-linear system of equations to build a functional model. Considering the advantages of the intelligent optimization algorithm in the non-linear equation system solving method, such as no demand derivation and simple formula derivation, the Particle Swarm Optimization (PSO) algorithm is introduced and the improved PSO algorithm is constructed; at the same time, the improved Gauss-Newton (G-N) algorithm is studied for the calculation of the 3D control network adjustment function model to solve the problems of computational volume and poor convergence performance of the algorithm with large residuals of the unknown parameters. The results show that the improved PSO algorithm and the improved G-N algorithm can guarantee the accuracy of the solution results. Compared with the traditional PSO algorithm, the improved PSO algorithm has a faster optimization speed. When the residuals of the unknown parameters are too large, the improved G-N algorithm is more stable than the improved PSO algorithm, which not only provides a new way to solve the spatial 3D network, but also provides theoretical support for the establishment of the spatial 3D network. [ABSTRACT FROM AUTHOR]

Imposing some conditions on derivatives of the known functions, using the Fiber Contraction Theorem we prove the existence of C 1 solutions of a class of iterative functional equations which involves iterates of the unknown functions and a nonlinear term. [ABSTRACT FROM AUTHOR]

The note is concerned with the functional equation λ 1 H 1 (f (x)) + λ 2 H 2 (f 2 (x)) + ⋯ + λ n H n (f n (x)) = F (x) , which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability. [ABSTRACT FROM AUTHOR]

How to free a road from vehicle traffic as efficiently as possible and in a given time, in order to allow for example the passage of emergency vehicles? We are interested in this question which we reformulate as an optimal control problem. We consider a macroscopic road traffic model on networks, semi-discretized in space and decide to give ourselves the possibility to control the flow at junctions. Our target is to smooth the traffic along a given path within a fixed time. A parsimony constraint is imposed on the controls, in order to ensure that the optimal strategies are feasible in practice. We perform an analysis of the resulting optimal control problem, proving the existence of an optimal control and deriving optimality conditions, which we rewrite as a single functional equation. We then use this formulation to derive a new mixed algorithm interpreting it as a mix between two methods: a descent method combined with a fixed point method allowing global perturbations. We verify with numerical experiments the efficiency of this method on examples of graphs, first simple, then more complex. We highlight the efficiency of our approach by comparing it to standard methods. We propose an open source code implementing this approach in the Julia language. [ABSTRACT FROM AUTHOR]

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin's Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner's 6j symbols, in details. The later are rather well known in the theory of 3 D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state. [ABSTRACT FROM AUTHOR]

For any N ≥ 3 , we study invariant measures of the dynamics of N hard spheres whose centres are constrained to lie on a line. In particular, we study the invariant submanifold M of the tangent bundle of the hard sphere billiard table comprising initial data that lead to the simultaneous collision of all N hard spheres. Firstly, we obtain a characterisation of those continuously-differentiable N-body scattering maps which generate a billiard dynamics on M admitting a canonical weighted Hausdorff measure on M (that we term the Liouville measure on M ) as an invariant measure. We do this by deriving a second boundary-value problem for a fully nonlinear PDE that all such scattering maps satisfy by necessity. Secondly, by solving a family of functional equations, we find sufficient conditions on measures which are absolutely continuous with respect to the Hausdorff measure in order that they be invariant for billiard flows that conserve momentum and energy. Finally, we show that the unique momentum- and energy-conserving linearN-body scattering map yields a billiard dynamics which admits the Liouville measure on M as an invariant measure. [ABSTRACT FROM AUTHOR]

We examine the fundamental Kazakov–Migdal (FKM) model on a generic graph, whose partition function is represented by the Ihara zeta function weighted by unitary matrices. The FKM model becomes unstable in the critical strip of the Ihara zeta function. We discover a duality between small and large couplings, associated with the functional equation of the Ihara zeta function for regular graphs. Although the duality is not precise for irregular graphs, we show that the effective action in the large coupling region can be represented by a summation of all possible Wilson loops on a graph similar to that in the small coupling region. We estimate the phase structure of the FKM model in both the small and large coupling regions by comparing it with the Gross–Witten–Wadia model. We further validate the theoretical analysis through detailed numerical simulations. [ABSTRACT FROM AUTHOR]

Some properties of the constant in asymptotic expansion of iterates of a cubic function were investigated. This paper analyzed the monotonicity, differentiability of the constant with respect to the initial value and the functional equation that is satisfied. [ABSTRACT FROM AUTHOR]

In this article, we study the structure of a multiple variable mapping. Indeed, we reduce the system of several mixed additive-quadratic equations defining a multivariable mapping to obtain a single functional equation, say, the multimixed additive-quadratic equation. We also show that such mappings under some conditions can be multi-additive, multi-quadratic and multi-additive-quadratic. Moreover, we establish the Hyers–Ulam stability of the multimixed additive-quadratic equation, using the so-called direct (Hyers) method. Additionally, we present a concrete example (the numerical approximation) regarding the stability of some two variable mappings into real numbers. Applying some characterization results, we indicate two examples for the case that a multimixed additive-quadratic mapping (in the special cases) cannot be stable. [ABSTRACT FROM AUTHOR]

In the process of solving the problems of synthesis of antennas with given amplitude characteristics, it is often necessary to apply nonlinear spectral theory. The practical statement of the problems of synthesis is based on the use of the amplitudes of desired functions. A standard procedure of optimization is to deduce the Euler equation for the functional used as an optimization criterion. As a rule, the indicated equation is integral and nonlinear due to the specific features of the problem. This equation is characterized by the nonuniqueness of solutions and their branching or bifurcations. Finding branched solutions requires the investigation of the corresponding homogeneous equations and the corresponding eigenvalue problem. The analysis of the problem makes it possible to determine the set of points of the input parameters at which the corresponding eigenvalues are equal to one, which determines the points of branching of the solutions. These calculations demonstrate the ability of the proposed approach to numerically determine the solutions of nonlinear equations, their properties, and branching points of the solutions with relatively low computational costs. [ABSTRACT FROM AUTHOR]

For the inner product space, we have Appolonius' identity. From this identity, Park and Th. M. Rassias induced and investigated the quadratic functional equation of the Apollonius type. And Park and Th. M. Rassias first introduced an Apollonius-type additive functional equation. In this work, we investigate an Apollonius-type additive functional equation in 2-normed spaces. We first investigate the stability of an Apollonius-type additive functional equation in 2-Banach spaces by using Hyers' direct method. Then, we consider the instability of an Apollonius-type additive functional equation in 2-Banach spaces. [ABSTRACT FROM AUTHOR]

The motion of the polymer center of mass (CM) is driven by two stochastic terms that are Gaussian white noise generated by standard thermal stirring and chain polymerization processes, respectively. It can be described by the Langevin equation and is Brownian non-Gaussian by calculating the kurtosis. We derive the forward Fokker–Planck equation governing the joint distribution of the motion of CM and the chain polymerization process. The backward Fokker–Planck equation governing only the probability density function (PDF) of CM position for a given number of monomers is also derived. We derive the forward and backward Feynman–Kac equations for the functional distribution of the motion of the CM, respectively, and present some of their applications, which are validated by a deep learning method based on backward stochastic differential equations (BSDEs), i.e. the deep BSDE method. [ABSTRACT FROM AUTHOR]

In this study, we significantly extend the concept of modular metric-like spaces to introduce the notion of b-metric-like spaces. Furthermore, by incorporating a binary relation R , we develop the framework of R -modular b-metric-like spaces. We establish a groundbreaking fixed point theorem for certain extensions of Geraghty-type contraction mappings, incorporating both simulation function and E -type contraction within this innovative structure. Moreover, we present several novel outcomes that stem from our newly defined notations. Afterwards, we introduce an unprecedented concept, the graphical modular b-metric-like space, which is derived from the binary relation R. Finally, we examine the existence of solutions for a class of functional equations that are pivotal in dynamic programming and in solving initial value problems related to the electric current in an RLC parallel circuit. [ABSTRACT FROM AUTHOR]

Over the past two decades, significant advancements have been made in understanding the stability according to Hyers–Ulam involving different functional equations (FEs). This study investigates the generalized stability of norm-based (norm-additive) FEs within the framework of arbitrary (noncommutative) groups and p-uniformly convex spaces. Specifically, we analyze two key functional equations, ∥ η (g h) ∥ = ∥ η (g) + η (h) ∥ and ∥ η (g h − 1) ∥ = ∥ η (g) − η (h) ∥ for every g , h ∈ G , where (G , ·) denotes an arbitrary group and B is considered to be a p-uniformly convex space. The surjectivity of the function η : G → B is a critical assumption in our analysis. Drawing upon the foundational works of L. Cheng and M. Sarfraz, this paper applies the large perturbation method tailored for p-uniformly convex spaces, where p ≥ 1 . This study extends previous research by offering a deeper exploration of the conditions under which these functional equations demonstrate Hyers–Ulam stability. In this study, the additive functional equation demonstrates a fundamental form of symmetry, where the order of operands does not affect the results. This symmetry under permutation of arguments is crucial for the analysis of stability. In the context of norm-additive FEs, this stability criterion investigates how small changes in the inputs of a functional equation affect the outputs, especially when the function is expected to follow an additive form. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Mahani Mathematical Research Center is the property of Shahid Bahonar University of Kerman, Department of Pure Mathematics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In 1940, the stability problem of functional equations was arose due to a question of Stanisaw Ulam concerning the stability of group homomorphisms. Significant work was done by Donald H. Hyers about HYERS-ULAM STABILITY and obtained a partial affirmative answer to the question of Ulam in the context of banach spaces in the case of additive mappings. In 1978, T. M. Rassias expanded Hyers's theorem for mappings between Banach spaces by considering an unbounded Cauchy difference subject to a continuity condition upon the mapping. After that, Many Researchers had studied about Hyers-Ulam stability of an additive quadratic type functional equations. In this research article, the Hyers-Ulam stability of an additive quadratic type functional equation was discussed and obtained the generalization of Hyers-Ulam stability of an additive quadratic type functional equation f(x + ay) + af(x - y) = f(x - ay) + af(x + y) for any integer a with a ≠ 1; 0; 1 in 2-Banach space. [ABSTRACT FROM AUTHOR]

In this article, the solvability of the pantograph equation of fractional orders under a fractal-fractional feedback control was investigated. This investigation was located in the class of all continuous functions. The necessary conditions for the solvability of that problem and the continuous dependence of the solution on some parameters and the control variable were established with the help of some fixed point theorems. Additionally, the Hyers-Ulam stability of the issue was explored. Finally, some specific problems extended to the corresponding problem with integer orders were illustrated. The theoretical results were supported by numerical simulations and comparisons with existing results in the literature. [ABSTRACT FROM AUTHOR]

We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on graphs induced by finite-state reversible continuous time Markov chains. In these graphs, each edge is weighted by the transition probability and invariant measure of the original process. Using these controlled dynamics on the graph and the dynamic programming principle for the value function, we derive several key components: the mean field game systems, the functional Hamilton-Jacobi equations, and the master equations on a finite probability space for potential mean field games. The existence and uniqueness of solutions to the potential mean field game system are ensured through a convex optimization reformulation in terms of the density-flux pair. We also derive variational principles for the master equations of both non-potential games and mixed games on a continuous state space. Finally, we offer several concrete examples of discrete mean field game dynamics on a two-point space, complete with closed-formula solutions. These examples include discrete Wasserstein distances, mean field planning, and potential mean field games. [ABSTRACT FROM AUTHOR]

This research studies the stabilization of the linear KdV equation with time-delay on boundary feedback in the presence of a saturated source term. Under certain hypotheses, the proof of well-posedness is established. The result of exponential stability is demonstrated using an appropriate Lyapunov functional. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Tea Science is the property of Journal of Tea Science Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This inquire about ponder is committed to investigating a few properties in connection to behaviors of solutions to an extended fractional structure of the standard jerk equation. Here, we define the scheme of the general fractional jerk problem using the generalized G operators. The existence result of such a new model is derived and analyzed based on some inequalities and fixed point tools. Furthermore, analysis of its Ulam–Hyers–Rassias type stability is performed and finally, we give numerical simulations for the existing parameters of the mentioned fractional G-jerk system in the Katugampola, Caputo–Hadamard and Caputo settings under different arbitrary orders. [ABSTRACT FROM AUTHOR]

In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function σ α (n) : = ∑ d | n d α. We obtain an exact identity relating the Dirichlet series ζ (s) ζ (s − α) and a segment of the Euler product attached to it. Specifically, our main theorems are valid in the critical strip. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. In the present paper, we determine the complex-valued solutions (f, g) of the functional equation g(xσ(y)) = g(x)g(y) − f(x)f(y) + αf(xσ(y)), x, y ∈ S, where σ : S → S is an automorphism that need not be involutive, and α ∈ C is a fixed constant. Our results generalize and extend the ones by Stetkær in The cosine addition law with an additional term. Aequat Math., no. 6, 90, 1147-1168 (2016), and also the ones by Aserrar and Elqorachi in A generalization of the cosine addition law on semigroups. Aequat Math. 97, 787–804 (2023). Some consequences of our results are presented. [ABSTRACT FROM AUTHOR]

We derive functional equations for distributions of six classical statistics (ascents, descents, left-to-right maxima, right-to-left maxima, left-to-right minima, and right-to-left minima) on separable and irreducible separable permutations. The equations are used to find a third degree equation for joint distribution of ascents and descents on separable permutations that generalizes the respective known result for the descent distribution. Moreover, our general functional equations allow us to derive explicitly (joint) distribution of any subset of maxima and minima statistics on irreducible, reducible and all separable permutations. In particular, there are two equivalence classes of distributions of a pair of maxima or minima statistics. Finally, we present three unimodality conjectures about distributions of statistics on separable permutations. [ABSTRACT FROM AUTHOR]

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay. [ABSTRACT FROM AUTHOR]

The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. Special attention is given to modelling such functions by systems of functional equations. [ABSTRACT FROM AUTHOR]

Let G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We find the solutions f , g , h ∈ C (G) of the Levi-Civita equation f (x y) = f (x) h (y) + g (x) f (y) , x , y ∈ G , which is an extension of the sine addition law. Representations of G on C 2 play an important role. As a corollary we get the solutions f , g ∈ C (G) of the sine subtraction law f (x y ∗) = f (x) g (y) - g (x) f (y) , x , y ∈ G , in which x ↦ x ∗ is a continuous involution, meaning that (x y) ∗ = y ∗ x ∗ and x ∗ ∗ = x for all x , y ∈ G . [ABSTRACT FROM AUTHOR]

We present a linear stability analysis of the Richtmyer-Meshkov instability that develops when a shock wave reaches a sinusoidally perturbed premixed flame from behind. In the hydrodynamic regime, when acoustic contributions dominate the flame growth rate, the problem is analytically addressed by the direct integration of the sound wave equations at both sides of the flame, which are bounded by the reflected and transmitted shock waves and the flame front that acts as a contact surface in the hydrodynamic limit. The resolution involves: a hyperbolic change of variables to modify the triangular spatio-temporal domain, a transformation in the Laplace variable, the resolution of the functional equations in the frequency domain, and the final inverse Laplace transform. The latter involves a novel resolution method that is proven beneficial for long-time dynamics. Asymptotic analysis is also carried out to describe the early time and late time hydrodynamic response. The nonuniform flow field resulting from distorted oscillating shocks is characterized by acoustic, rotational, and entropic disturbances, each of which exerts a substantial influence on flame dynamics. These disturbances contribute to the intricate interplay of factors shaping the behavior of the flame in response to the nonuniform flow. In particular, the sensitivity of local flame propagation to temperature disturbances is investigated. This work contributes to a deeper understanding of Richtmyer-Meshkov instability dynamics and offers insights into instability reduction through the modulation of temperature disturbances generated by the transmitted shock. • Introduction of a self-consistent analytical model for Richtmyer-Meshkov instability within shock-flame interactions. • Development of an enhanced resolution method for the functional equation in the Laplace variable. • Introduction of a new resolution method for the transient evolution via the inverse Laplace transform. • Provision of a comprehensive mechanism regarding Richtmyer-Meshkov instability reduction. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to show that in certain classes of real (or complex) functions, Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not related to Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich. [ABSTRACT FROM AUTHOR]

An open problem posed by G. Maksa and Z. Páles is to find the general solution of the functional equation { ‖ f (x) - β f (y) ‖ : β ∈ T n } = { ‖ x - β y ‖ : β ∈ T n } (x , y ∈ H) where f : H → K is between two complex normed spaces and T n : = { e i 2 k π n : k = 1 , ⋯ , n } is the set of the nth roots of unity. With the aid of the celebrated Wigner's unitary-antiunitary theorem, we show that if n ≥ 3 and H and K are complex inner product spaces, then f satisfies the above equation if and only if there exists a phase function σ : H → T n such that σ · f is a linear or anti-linear isometry. Moreover, if the solution f is continuous, then f is a linear or anti-linear isometry. [ABSTRACT FROM AUTHOR]

The functional equation f (x + y) = g (x y) is investigated with unknown functions f : A + A → Y , g : A · A → Y in the following cases: A : = α , β ⊆ F + where F is an Archimedean ordered field; A is the set of all positive integers; A is the set of all positive dyadic rational numbers. The set Y is an arbitrarily fixed (infinite) set. The main result of the paper shows that there exists a set A ⊆ R + that is closed under addition and multiplication and there exist functions f, g : A → Y which satisfy the equation f (x + y) = g (x y) for all x , y ∈ A such that the range of the function f is infinite. Finally, some application of the above results is also given. [ABSTRACT FROM AUTHOR]

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. [ABSTRACT FROM AUTHOR]

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this work, we mainly focus on the local existence and regularity of integral solutions for a class of nondensely defined partial neutral functional integrodifferential equations with unbounded delay. We use the theory of integrated resolvent operators introduced by Oka [H. Oka, Integrated resolvent operators, J. Integral Equations Appl. 7 1995, 2, 193–232]. Finally, we provide an example to demonstrate the basic findings of our work. [ABSTRACT FROM AUTHOR]

We find all nonnegative solutions f of the equation f x = ∏ j = 1 n f s j x p j , defined in a one-sided vicinity of 0 and having a prescribed asymptotic at 0. The main theorem extends a result obtained by J. A. Baker [Proc. Amer. Math. Soc., 121, 767 (1994)]. [ABSTRACT FROM AUTHOR]

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of SL 2 over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of π. We obtain an expression of the gamma factor, which shows its entireness on C. Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on PGL 2 ≅ SO 3 associated with the prehomogenous vector space of symmetric matrices of degree 2. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem with the boundary functional for a Schrödinger equation describing the motion of a charged particle. By using the existence of an optimal solution, we search the necessary optimality conditions for the examined control problem. First, we constitute an adjoint problem by a Lagrange multiplier that is related to constraints of theory on symmetries and conservation laws. The adjoint problem obtained is a boundary value problem with a nonhomogeneous boundary condition. We prove the existence and uniqueness of the solution of the adjoint problem. Then, we demonstrate the differentiability of the objective functional in the sense of Frechet and get a formula for its gradient. Finally, we give a necessary optimality condition in the form of a variational inequality. [ABSTRACT FROM AUTHOR]

Clonoids are sets of finitary functions from an algebra A to an algebra B that are closed under composition with term functions of A on the domain side and with term functions of B on the codomain side. For A , B (polynomially equivalent to) finite modules we show: If A , B have coprime order and the congruence lattice of A is distributive, then there are only finitely many clonoids from A to B. This is proved by establishing for every natural number k a particular linear equation that all k-ary functions from A to B satisfy. Else if A , B do not have coprime order, then there exist infinite ascending chains of clonoids from A to B ordered by inclusion. Consequently any extension of A by B has countably infinitely many 2-nilpotent expansions up to term equivalence. [ABSTRACT FROM AUTHOR]