Unbounded Interval Research Articles

A New Unit Distribution Based On The Unbounded Johnson Distribution Rule: The Unit Johnson SU Distribution

This paper proposes a new probability distribution, which belongs a member of the exponential family, defined on (0,1) unit interval. The new unit model has been defined by relation of a random variable defined on unbounded interval with respect to standard logistic function. Some basic statistical properties of newly defined distribution are derived and studied. The different estimation methods and some inferences for the model parameters have been derived. We assess the performance of the estimators of these estimation methods based on the three different simulation scenarios. The analysis of three real data examples which one is related to the coronavirus data, show better fit of proposed distribution than many known distributions on the unit interval under some comparing criteria.

S-asymptotically $ \omega $-periodic mild solutions and stability analysis of Hilfer fractional evolution equations

<p style='text-indent:20px;'>In this article, we deal with the existence of S-asymptotically <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-periodic mild solutions of Hilfer fractional evolution equations. We also investigate the Ulam-Hyers and Ulam-Hyers-Rassias stability of similar solutions. These results are established in Banach space with the help of resolvent operator functions and fixed point technique on an unbounded interval. An example is also presented for the illustration of obtained results.</p>

Local stability of mappings on multi-normed spaces

First we investigate the Hyers–Ulam stability of the Cauchy functional equation for mappings from bounded (unbounded) intervals into Banach spaces. Then we study the Hyers–Ulam stability of the functional equation f(xy)=xg(y)+h(x)y for mappings from bounded (unbounded) intervals into multi-normed spaces.

Measures of noncompactness and superposition operator in the space of regulated functions on an unbounded interval

In this paper, we formulate necessary and sufficient conditions for relative compactness in the space BG({mathbb {R}}_+,E) of regulated and bounded functions defined on {mathbb R}_+ with values in the Banach space E. Moreover, we construct four new measures of noncompactness in the space BG({mathbb {R}}_+,E). We investigate their properties and we describe relations between these measures. We provide necessary and sufficient conditions so that the superposition operator (Niemytskii) maps BG({mathbb {R}}_+,E) into BG({mathbb {R}}_+,E) and, additionally, be compact.

Linear positive operators constructed by using Beta-type bases

Starting from a discrete linear approximation process that has the ability to turn polynomials into polynomials of the same degree, we introduce an integral generalization by using Beta-type bases. Some properties of this new sequence of operators are investigated in unweighted and weighted spaces of functions defined on unbounded interval. In our construction particular cases are outlined.

On Existence and Uniqueness of an Integrable Solution for a Fractional Volterra Integral Equation on 𝑹+

In this paper, by using the Banach fixed point theorem, we prove the existence and uniqueness theorem of a fractional Volterra integral equation in the space of Lebesgue integrable 𝐿1(𝑅+) on unbounded interval [0,∞).

Estimates for the Differences of Certain Positive Linear Operators

The present paper deals with estimates for differences of certain positive linear operators defined on bounded or unbounded intervals. Our approach involves Baskakov type operators, the kth order Kantorovich modification of the Baskakov operators, the discrete operators associated with Baskakov operators, Meyer–König and Zeller operators and Bleimann–Butzer–Hahn operators. Furthermore, the estimates in quantitative form of the differences of Baskakov operators and their derivatives in terms of first modulus of continuity are obtained.

Approximation properties of a family of integral type operators

In this paper we consider a general class of linear positive processes of integral type. These operators act on functions defined on unbounded interval. Among the particular cases included are Durrmeyer–Jain operators, Păltănea–Szasz–Mirakjan operators and operators using Baskakov–Szasz type bases. We focus on highlighting some approximation targeting different classes of functions. The main working tools are Bohman–Korovkin theorem, weighted K-functionals and moduli of smoothness.

Sturmian comparison theorems for completely controllable linear Hamiltonian systems in singular case

In this paper we consider two linear Hamiltonian differential systems on an open unbounded interval. We assume that the systems are completely controllable and satisfy the Sturmian majorant condition and the Legendre condition. We derive a singular Sturmian comparison theorem for the case when the minorant system has a solution, which is principal at both endpoints of the considered interval. The main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by Aharonov and Elias (2010). We illustrate our new theory by several examples.

Singular Sturmian comparison theorems for linear Hamiltonian systems

In this paper we prove singular comparison theorems on unbounded intervals for two nonoscillatory linear Hamiltonian systems satisfying the Sturmian majorant condition and the Legendre condition. At the same time we do not impose any controllability condition. The results are phrased in terms of the comparative index and the numbers of proper focal points of the (minimal) principal solutions of these systems at both endpoints of the considered interval. The main idea is based on an application of new transformation theorems for principal and antiprincipal solutions at infinity and on new limit properties of the comparative index involving these solutions. This work generalizes the recently obtained Sturmian separation theorems on unbounded intervals for one system by the authors (2019), as well as the Sturmian comparison theorems and transformation theorems on compact intervals by J. Elyseeva (2016 and 2018). We note that all the results are new even in the completely controllable case.

Numerical solution of Volterra integral equations via Szász‐Mirakyan approximation method

Szász‐Mirakyan operators are a powerful tool that can be used to approximate functions on the unbounded interval . In this contribution, we propose a numerical solution of Volterra integral equations with the help of Szász‐Mirakyan operators. In this direction, we provide both numerical scheme and estimation of error bound of solution. Numerical experiments are also presented, highlighting the performance of the new constructions of proposed algorithm in the context of one‐dimensional approximation.

Convergence analysis of Galerkin and multi-Galerkin methods on unbounded interval using Hermite polynomials

We consider in this paper the Galerkin and multi-Galerkin methods and their iterated versions to solve the linear Fredholm integral equation of the second kind on the real line with a sufficiently smooth kernels, using Hermite polynomials as basis functions. We obtain optimal convergence results in iterated Galerkin method in weighted L2−norm. We also discuss multi-Galerkin methods and we obtain the superconvergence results in both multi-Galerkin and iterated multi-Galerkin methods. Numerical results are presented to confirm the theoretical results.

Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

We study the optimal partitioning of a (possibly unbounded) interval of the real line into n subintervals in order to minimize the maximum of certain set-functions, under rather general assumptions such as continuity, monotonicity, and a Radon-Nikodym property. We prove existence and uniqueness of a solution to this minimax partition problem, showing that the values of the set-functions on the intervals of any optimal partition must coincide. We also investigate the asymptotic distribution of the optimal partitions as n tends to infinity. Several examples of set-functions fit in this framework, including measures, weighted distances and eigenvalues. We recover, in particular, some classical results of Sturm-Liouville theory: the asymptotic distribution of the zeros of the eigenfunctions, the asymptotics of the eigenvalues, and the celebrated Weyl law on the asymptotics of the counting function.

Fractional eigenvalue problems on R N

Let N ≥ 2 be an integer. For each real number s ∈ (0, 1) we denote by (−∆) s the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem (−∆) su = λV(x)u on RN, where V : RN → R is a given function. Under suitable conditions imposed on V we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional (t, p)-Laplace operator, when t ∈ (0, 1) and p ∈ (1, ∞) are such that t < s and s − N/2 = t − N/p. We show that when the function V is nonnegative on RN, the set of eigenvalues of the perturbed eigenvalue problem is exactly the unbounded interval (λ1, ∞), where λ1 stands for the first eigenvalue of the initial eigenvalue problem.

Tykhonov triples and convergence results for history-dependent variational inequalities

We deal with the Tykhonov well-posedness of a time-dependent variational inequality defined on the unbounded interval of time ℝ+= [0, +∞ ), governed by a history-dependent operator. To this end we introduce the concept of Tykhonov triple, provide three relevant examples, then we state and prove the corresponding well-posedness results. This allows us to deduce various corollaries which illustrate the continuous dependence of the solution with respect to the data. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in Mechanics, Physics and Engineering Sciences. To give an example, we consider a mathematical model which describes the equilibrium of a viscoelastic body in frictionless contact with a rigid foundation.

Solvability of an infinite system of nonlinear integral equations of Volterra-Hammerstein type

AbstractThe purpose of the paper is to study the solvability of an infinite system of integral equations of Volterra-Hammerstein type on an unbounded interval. We show that such a system of integral equations has at least one solution in the space of functions defined, continuous and bounded on the real half-axis with values in the spacel1consisting of all real sequences whose series is absolutely convergent. To prove this result we construct a suitable measure of noncompactness in the mentioned function space and we use that measure together with a fixed point theorem of Darbo type.

Allocating resources via price management systems: a dynamic programming-based approach

In this paper, a novel model for price management systems in resource allocation problems is proposed. Stochastic customer requests for resource allocations and releases are modelled as constrained parallel Birth–Death Processes (BDP). We address both instant (i.e. the customer requires a resource to be allocated immediately) and advance (i.e. the customer books a resource for future use) reservation requests, the latter with both bounded and unbounded time interval options. Algorithms based on Dynamic Programming (DP) principles are proposed for the calculation of suitable price profiles. At the core of such algorithms, there is the resolution of stochastic optimisation problems. In particular, the maximisation of the expected total revenue is formulated via a constrained Stochastic Dynamic Programming (SDP) approach, which becomes time-variant in case of advance reservation requests. Approximate Dynamic Programming (ADP) techniques are adopted in case of large state spaces. Simulations are performed to show the effectiveness of the proposed models and the related algorithms.

Estrategias y errores de conversión entre representaciones de intervalos de la recta real

En este artículo se presentan los resultados de un estudio empírico llevado a cabo a través de diferentes ciclos de investigación con alumnos de 4.º de la ESO y 1.º de Bachillerato, centrado en el concepto de intervalo de la recta real. Los objetivos son identificar las estrategias que utilizan los alumnos para realizar conversiones entre diferentes representaciones de intervalos no acotados e interpretar los posibles errores y dificultades derivados de aplicar estas estrategias. Se han detectado diferentes estrategias focalizadas en aspectos particulares, lo que ha corroborado que muchas dificultades están asociadas a la utilización de conversiones por congruencia. A partir de los resultados, se proponen varias orientaciones didácticas para facilitar el aprendizaje de este concepto a través de sus representaciones.

Ulam‐type stability for differential equations driven by measures

AbstractBased on a notion of Stieltjes derivative of a function with respect to another function, we provide Ulam–Hyers type stability results for nonlinear differential equations driven by measures on compact or on unbounded intervals, in the lack of Lipschitz continuity assumptions. In particular, one can deduce stability results for generalized differential equations, dynamic equations on time scales or impulsive differential equations (including the case of an infinite number of impulses that accumulate in the considered interval, thus allowing the study of Zeno hybrid systems).

Convergence and Optimization Results for a History-Dependent Variational Problem

We consider a mixed variational problem in real Hilbert spaces, defined on on the unbounded interval of time and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in our previous paper. Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.