Functional Equations Research Articles

On Ulam type stability of the solution to a ψ-Hilfer abstract fractional functional differential equation

Abstract This article explores the stability of the solutions to a ψ-Hilfer abstract fractional functional differential equation under feasible hypotheses. By utilizing the Banach fixed point theorem and generalized Grönwall’s inequality, the existence, uniqueness, and stability of the solutions are rigorously established. The analysis distinguishes between Ulam-Hyers stability, which enures bounded deviations under constant perturbations, and Ulam-Hyers-Rassias stability, which accounts for state-dependent perturbations, offering greater adaptability for dynamic systems. To contextualize the problem, we highlight the significance of fractional-order systems in capturing memory effects and hereditary dynamics, which are essential for modeling complex real-world phenomena in biological, physical, and engineering domains. Numerical experiments are performed to examine solution trajectories under varying fractional orders and weight functions, demonstrating the flexibility and robustness of the fractional framework. The examples and the plots authenticate the theoretical findings and emphasize the applicability of the proposed model in addressing practical challenges.

Hyers-Ulam Stability of Quartic Functional Equation in IFN-Spaces and 2-Banach Spaces by Classical Methods

Hyers-Ulam Stability of Quartic Functional Equation in IFN-Spaces and 2-Banach Spaces by Classical Methods

Convex Solutions of an Iterative Functional Differential Equation

Abstract In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.

Enumeration of three quadrant walks with small steps and walks on other 𝑀-quadrant cones

We address the enumeration of walks with small steps confined to a two-dimensional cone, for example the quarter plane, three-quarter plane or the slit plane. In the quarter plane case, the solutions for unweighted step-sets are already well understood, in the sense that it is known precisely for which cases the generating function is algebraic, D-finite or D-algebraic, and exact integral expressions are known in all cases. We derive similar results in a much more general setting: we enumerate walks on an M M -quadrant cone for any positive integer M M , with weighted steps starting at any point. The main breakthrough in this work is the derivation of an analytic functional equation which characterises the generating function of these walks, which is analogous to one now used widely for quarter-plane walks. In the case M = 3 M=3 , which corresponds to walks avoiding a quadrant, we provide exact integral-expression solutions for walks with weighted small steps which determine the generating function C ( x , y ; t ) \mathsf {C}(x,y;t) counting these walks. Moreover, for each step-set and starting point of the walk we determine whether the generating function C ( x , y ; t ) \mathsf {C}(x,y;t) is algebraic, D-finite or D-algebraic as a function of x x and y y . In fact we provide results of this type for any M M -quadrant cone, showing that this nature is the same for any odd M M . For M M even we find that the generating functions counting these walks are D-finite in x x and y y , and algebraic if and only if the starting point of the walk is on the same axis as the boundaries of the cone.

On the sum form functional equation related to diversity index

Abstract The focal point of this article is to study a functional equation of sum form involving four unknown mappings. We primarily concentrate on exploring all possible solutions of the equation. An attempt is made to delve into the intricacies of obtaining these solutions without assuming any regularity condition on the mappings. Our study further aims to examine the stability of these general solutions, thus shedding light on the behaviour of sum form functional equations under perturbations. The article concludes by reflecting upon the relevance of the general solutions in information theory and diversity index.

Fixed Points of New Hybrid Contractions with Applications to Nonlinear Economic Models

One of the active researchs in metric fixed point theory involves the development of more unifying Lipstchitz-type inequalities so as to fine-tune their applications in various areas of nonlinear functional equations and related domains. In this direction, the goal of this paper is to offer a new hybrid contraction that is a combination of Meir-Keeler and Zamfirescu contractions in the structure of a complete metric space. The existence and uniqueness conditions of a fixed point of such contractions are examined. Mathematical economics is an abstract and applicable science in which economic objects, processes, and phenomena are modeled using mathematically formulated functional equations, either as a differential or an integral equation. In this direction, we study a class of integral equations that can be considered as a form of economic model and apply one of our obtained results to develop new solvability conditions for a unique solution of the integral equation. Additionally, nontrivial examples are formulated to support our proposed results.

Global second‐order estimates in anisotropic elliptic problems

AbstractThis work deals with boundary value problems for second‐order nonlinear elliptic equations in divergence form, which emerge as Euler–Lagrange equations of integral functionals of the Calculus of Variations built upon possibly anisotropic norms of the gradient of trial functions. Integrands with nonpolynomial growth are included in our discussion. The ‐regularity of the stress‐field associated with solutions, namely, the nonlinear expression of the gradient subject to the divergence operator, is established under the weakest possible assumption that the datum on the right‐hand side of the equation is a merely ‐function. Global regularity estimates are offered in domains enjoying minimal assumptions on the boundary. They depend on the weak curvatures of the boundary via either their degree of integrability or an isocapacitary inequality. By contrast, none of these assumptions is needed in the case of convex domains. An explicit estimate for the constants appearing in the relevant estimates is exhibited in terms of the Lipschitz characteristic of the domains, when their boundary is endowed with Hölder continuous curvatures.

A new approach to approximate the solution of two general functional equations in quasi-Banach spaces

A new approach to approximate the solution of two general functional equations in quasi-Banach spaces

On nonoscillation of fractional order functional differential equations with forcing term and distributed delays

Abstract The paper deals with the nonoscillatory solutions of arbitrary noninteger-order neutral equations with distributed delays. Through the use of the LFD (Liouville Fractional Derivative) of order α ≥ 0 on the half-axis and BCP (Banach Contraction Principle), we are able to get the nonoscillation criteria. The obtained results are emphasized with some appropriate examples.

A cubic functional equation of the Euler–Lagrange type on restricted domains

A cubic functional equation of the Euler–Lagrange type on restricted domains

Analysis of a threshold-based priority queue

This paper considers a discrete-time single-server queueing system, with two classes of customers, named class 1 and class 2. We propose and analyze a novel threshold-based priority scheduling scheme that works as follows. Whenever the number of class-1 customers in the system exceeds a given thresholdm≥0, the server of the system gives priority to class-1 customers; otherwise, it gives priority to class-2 customers. Consequently, for m=0, the system is equivalent to a classical priority queue with absolute priority for class-1 customers, whereby the (mean) delay of class-1 customers is lowered as much as possible at the expense of longer (mean) delays for class-2 customers. On the other hand, for m→∞, the system is equivalent to a priority queue with absolute priority for class-2 customers, with the opposite effect on the class-specific (mean) delays. By choosing 0<m<∞, we aim at a more gradual delay differentiation between the two customer classes of the system. The queueing analysis of the model turns out to be quite challenging. We first establish a kernel-type functional equation for the steady-state joint probability generating function U(z1,z2) of the numbers of customers in the two queues, from which U(z1,z2) can be solved in terms of a finite number of unknown boundary functions. Next, we develop a method to determine these boundary functions in principle and discuss the main practical obstacles in deriving explicit results from this. We show that the difficulty of a full analysis depends heavily on the value of m and the precise form of the arrival process. For the special case m=1, we derive explicit formulas for U(z1,z2). We also develop a mean-value analysis technique, applicable for any m, to compute closed-from expressions for the class-specific mean customer delays. Abundant numerical results demonstrate the impact of the threshold m and the traffic mix (proportion of class-1 and class-2 traffic in the arrival process) on the delay-differentiating capabilities of the proposed scheduling discipline.

A Tapestry of Ideas with Ramanujan’s Formula Woven In

Zeta-functions play a fundamental role in many fields where there is a norm or a means to measure distance. They are usually given in the forms of Dirichlet series (additive), and they sometimes possess the Euler product (multiplicative) when the domain in question has a unique factorization property. In applied disciplines, those zeta-functions which satisfy the functional equation but do not have Euler products often appear as a lattice zeta-function or an Epstein zeta-function. In this paper, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta-functions by considering some generalizations of the Eisenstein series of level 2ϰ to the complex variable level s. Naturally, generalized Eisenstein series and Barnes multiple zeta-functions arise, which have affinity to dissections, as they are (semi-) lattice functions. The method of Lewittes (and Chapman) and Kurokawa leads to some limit formulas without absolute value due to Tsukada and others. On the other hand, Komori, Matsumoto and Tsumura make use of the Barnes multiple zeta-functions, proving their modular relation, and they give rise to generalizations of Ramanujan’s formula as the generating zeta-function of σs(n), the sum-of-divisors function. Lewittes proves similar results for the 2-dimensional case, which holds for all values of s. This in turn implies the eta-transformation formula as the extreme case, and most of the results of Chapman. We shall unify most of these as a tapestry of ideas arising from the merging of additive entity (Dirichlet series) and multiplicative entity (Euler product), especially in the case of limit formulas.

On the Formulation of Boundary-Value Problems for Binomial Functional Equations

On the Formulation of Boundary-Value Problems for Binomial Functional Equations

Cauchy–Schwarz-type inequalities for solutions of Levi-Civita-type functional equations

Abstract The main goal of this paper is to show that if a real-valued function defined on a groupoid satisfies a certain Levi-Civita-type functional equation, then it also fulfills a Cauchy–Schwarz-type functional inequality. In particular, if the groupoid is the multiplicative structure of a commutative ring, then we can establish the existence of nontrivial additive functions satisfying inequalities connected to the multiplicative structure.

Generalized Stability of a General Septic Functional Equation

In this paper, we generalize previous results on the generalized stability of the general septic functional equation Δy8f(x)=0 for all x,y∈V.

Transport processes and coalescence of two entrapping bubbles during upward solidification.

Transport processes and coalescence of two entrapping bubbles during upward solidification.

Controllability Results for Multi-Order Impulsive Neutral Fuzzy Functional Integro-Differential Equations with Finite Delay

Controllability Results for Multi-Order Impulsive Neutral Fuzzy Functional Integro-Differential Equations with Finite Delay

Clique Collocation Method to Solve the Third-Order Multisingular (MS) Functional Differential Equations

In this paper, a Clique collocation method is presented to numerically solve the third-order multisingular (MS) functional differential equation. This method convert this equation to a system of the algebraic equations via the collocation points and the matrix relations. Also, the error estimation technique is constituted for the third-order multisingular (MS) functional differential equation. Applications of the Clique collocation method and the error estimation technique are made for three examples. In addition, the comparison is made with another method in the literature. The obtained results are tabulated and visualized to demonstrate the effectiveness of the presented method. Applications of the method and graphics are made by using MATLAB. According to the applications, it is observed that the results have quite decent errors.

On a New Stochastic Space with Applications to Nonlinear Economic Models

This article will utilize a weighted regular matrix composed of Fibonacci numbers and variable exponent sequence spaces to create a novel stochastic space with certain geometric and topological properties. This area demonstrates the new form of the Kannan contraction operator with a fixed point. In mathematical economics, we represent economic entities, processes, and phenomena by mathematically structured functional equations, either as summable equations or integral equations. We investigate a category of Volterra-type non-linear dynamical systems, similar to an economic model. We employ our acquired results to formulate new solvability criteria for a unique solution of these non-linear discrete economic dynamical systems. Ultimately, we illustrate our findings with specific instances and applications related to the presence of solutions in non-linear dynamical systems of Volterra-type.

Hyperstability of Functional Equation Deriving from Quintic Mapping in Banach Space by Fixed Point Method

Hyperstability of Functional Equation Deriving from Quintic Mapping in Banach Space by Fixed Point Method