Unbounded Interval Research Articles

Generalizations of Rolle’s Theorem

The classical Rolle’s theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line.

On some generalizations of Darbo- and Sadovskii- type fixed point theorems in Fréchet spaces

Integral equations form an essential part of nonlinear analysis. In many situations, when seeking solutions to such equations, we utilize fixed-point theorems. Among the most well-known and useful in this regard are the theorems of Darbo and Sadovskii, which generalize the classical results of Schauder and Tikhonov. These theorems have seen many generalizations. The aim of the paper is to strengthen (by weakening the assumptions) several known Darbo- and Sadovskii-type fixed-point theorems in the so-called power version for Fréchet spaces, which are particularly useful for studying the solvability of equations defined on an unbounded interval. A theorem on the existence of solutions to a nonlinear integral equation, illustrating the application of our main result, is also provided.

Hyers–Ulam–Rassias stability of fractional delay differential equations with Caputo derivative

This paper is devoted to the study of Hyers–Ulam–Rassias (HUR) stability of a nonlinear Caputo fractional delay differential equation (CFrDDE) with multiple variable time delays. We obtain two new theorems with regard to HUR stability of the CFrDDE on bounded and unbounded intervals. The method of the proofs is based on the fixed point approach. The HUR stability results of this paper have indispensable contributions to theory of Ulam stabilities of CFrDDEs and some earlier results in the literature.

Note on singular Sturm comparison theorem and strict majorant condition

In this note we present a singular Sturm comparison theorem for two linear Hamiltonian systems satisfying a standard majorant condition and the identical normality assumption. Both endpoints of the considered interval may be singular. We identify the exact form of the strict majorant condition, which is necessary and sufficient for the property that every solution (conjoined basis) of the majorant system has more focal points than the solutions of the minorant system. We provide a formula for the exact number of focal points of any solution of the majorant system, depending on the number of focal points of solutions of the minorant system and on the number of right focal points of a solution of a certain transformed linear Hamiltonian system. This transformed system may be in general abnormal. Our result extends the previous Sturm comparison theorems for linear Hamiltonian systems by Kratz (1995) [18] on a compact interval and by the authors (2020) [35,36] on an open or unbounded interval. The main result is also new for the second order differential equations, where it extends the singular comparison theorem by Aharonov and Elias (2010) [1].

Total positivity and high relative accuracy for several classes of Hankel matrices

SummaryGramian matrices with respect to inner products defined for Hilbert spaces supported on bounded and unbounded intervals are represented through a bidiagonal factorization. It is proved that the considered matrices are strictly totally positive Hankel matrices and their catalecticant determinants are also calculated. Using the proposed representation, the numerical resolution of linear algebra problems with these matrices can be achieved to high relative accuracy. Numerical experiments are provided, and they illustrate the excellent results obtained when applying the theoretical results.

On extensibility and qualitative properties of solutions to Riccati's equation

We consider Riccati's equation on the real axis with continuous coefficients and non-negative discriminant of the right-hand side. We study the extensibility of its solutions to unbounded intervals. We obtain asymptotic formulae for its solutions in their dependence on the initial values and the properties of the functions representing roots of the right-hand side of the equation. We obtain results on the asymptotical behaviour of solutions defined near $\pm\infty$. We study the structure of the set of bounded solutions in the case when the roots of the right-hand side of the equation are $C^1$-functions which are different on the whole of their domain and tend monotonically to some limits as $x\to\pm\infty$. We extend, improve, or refine some well-known results. Bibliography: 47 titles.

Approximation of the Hilbert transform in the Lebesgue spaces

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory because of its relevance to such problems as envelope detection and demodulation, as well as its use in relating the real and imaginary components, and the magnitude and phase components of spectra. The Hilbert transform is a multiplier operator and is widely used in the theory of Fourier transforms. The Hilbert transform is the main part of the singular integral equations on the real line. Therefore, approximations of the Hilbert transform are of great interest. Many papers have dealt with the numerical approximation of the singular integrals in the case of bounded intervals. On the other hand, the literature concerning the numerical integration on unbounded intervals is by far poorer than the one on bounded intervals. The case of the Hilbert Transform has been considered very little. This article is devoted to the approximation of the Hilbert transform in Lebesgue spaces by operators which introduced by V.R.Kress and E.Mortensen to approximate the Hilbert transform of analytic functions in a strip. In this paper, we prove that the approximating operators are bounded maps in Lebesgue spaces and strongly converges to the Hilbert transform in these spaces.

An Investigation of a Nonlinear Delay Functional Equation with a Quadratic Functional Integral Constraint

This research paper focuses on investigating the solvability of a constrained problem involving a nonlinear delay functional equation subject to a quadratic functional integral constraint, in two significant cases: firstly, the existence of nondecreasing solutions in a bounded interval L1[0,T] and, secondly, the existence of nonincreasing solutions in unbounded interval L1(R+). Moreover, the paper explores various qualitative properties associated with these solutions for the given problem. To establish the validity of our claims, we employ the De Blasi measure of noncompactness (MNC) technique as a basic tool for our proofs. In the first case, we provide sufficient conditions for the uniqueness of the solution ψ∈L1[0,T] and rigorously demonstrate its continuous dependence on some parameters. Additionally, we establish the equivalence between the constrained problem and an implicit hybrid functional integral equation (IHFIE). Furthermore, we delve into the study of Hyers–Ulam stability. In the second case, we examine both the asymptotic stability and continuous dependence of the solution ψ∈L1(R+) on some parameters. Finally, some examples are provided to verify our investigation.

Oscillation criterion for linear equations with coefficients containing powers of natural logarithm

Applying an averaging technique for the adapted Prüfer angle, we obtain an oscillation criterion for linear second order differential equations whose coefficients consist of products of powers of natural logarithm and general (bounded or unbounded) continuous functions. The presented criterion is illustrated by new corollaries and examples. The novelty is caused by the used averaging technique over unbounded intervals.

Dynamics of a rational map: unbounded cycles, unbounded chaotic intervals and organizing centres

A one-dimensional rational map f ( x ) = ( x 2 − a ) / ( x 2 − b ) depending on the two parameters a and b is considered. Sequences of bifurcations peculiar of rational maps are evidenced, as those occurring due to unbounded cycles (that is, periodic orbits having one point at infinity, related to the vertical asymptotes) that are superstable, as well as to unbounded chaotic intervals. Moreover, two particular bifurcation points, having the role of organizing centres in the ( a , b ) -parameter plane, are studied. Each point is related to a pair of conditions, which allow us to consider them as the bifurcation points of codimension-2, as it is usual for this kind of organizing centres. However, the two conditions are related not to bifurcations but to degeneracies in the graph of the function. The sequences of bifurcations leading to attracting cycles associated with these particular points are investigated, analytically and numerically, making use of particular properties of the rational map.

Isolating all the real roots of a mixed trigonometric-polynomial

Mixed trigonometric-polynomials (MTPs) are functions of the form f(x,sin⁡x,cos⁡x) where f is a trivariate polynomial with rational coefficients, and the argument x ranges over the reals. In this paper, an algorithm “isolating” all the real roots of an MTP is provided and implemented. It automatically divides the real roots into two parts: one consists of finitely many roots in an interval [μ−,μ+] while the other consists of countably many roots in R﹨[μ−,μ+]. For the roots in [μ−,μ+], the algorithm returns isolating intervals and corresponding multiplicities while for those greater than μ+, it returns finitely many mutually disjoint small intervals Ii⊂[−π,π], integers ci>0 and multisets of root multiplicity {mj,i}j=1ci such that any root >μ+ is in the set (∪i∪k∈N(Ii+2kπ)) and any interval Ii+2kπ⊂(μ+,∞) contains exactly ci distinct roots with multiplicities m1,i,...,mci,i, respectively. The efficiency of the algorithm is shown by experiments. The method used to isolate the roots in [μ−,μ+] is applicable to any other bounded interval as well. The algorithm takes advantages of the weak Fourier sequence technique and deals with the intervals period-by-period without scaling the coordinate so to keep the length of the sequence short. The new approaches can easily be modified to decide whether there is any root, or whether there are infinitely many roots in unbounded intervals of the form (−∞,a) or (a,∞) with a∈Q.

Rasyonel Üslü Cebirsel ve Üstel Eşleme Yaklaşımı ile Thomas-Fermi Denklemi için İkinci Derece Doğruluklu Sonlu Farklar Yöntemi

Many problems based on natural sciences need to be solved by the scientists and engineers to serve the humanity. One of the well-known model in atomic universe is condensed into an equation, and called the Thomas-Fermi equation. It is a second order differential equation, which describes charge distributions of heavy, neutral atoms. No exact analytical solution has been found for the equation yet. In fact, strong nonlinearity, singular character and unbounded interval of the problem causes great difficulty to obtain an approximate numerical solution as well. In this paper, the Thomas-Fermi equation is solved using a second order finite difference method along with application of quasi-linearization method. Semi-infinite interval of the problem is converted into [0, 1) using two different coordinate transformations, namely algebraic and exponential mapping. Numerical order of accuracy has been checked using systematic mesh refinements and comparing the calculated initial slope y'(0). Calculated results for initial slope is found in good agreement with the results available in the literature. Lastly, accuracy is improved by the application of the Richardson extrapolation.

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

Definable Tietze extension property in o-minimal expansions of ordered groups

The following two assertions are equivalent for an o-minimal expansion of an ordered group $\mathcal M=(M,<,+,0,\ldots)$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $f:A \rightarrow M$ defined on a definable closed subset of $M^n$ has a definable continuous extension $F:M^n \rightarrow M$.

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

Ulam–Hyers–Rassias Stability of Neutral Functional Integrodifferential Evolution Equations with Non-instantaneous Impulses on an Unbounded Interval

MOVER-R and Penalized MOVER-R Confidence Intervals for the Ratio of Two Quantities

Developing a confidence interval for the ratio of two quantities is an important task in statistics because of its omnipresence in real world applications. For such a problem, the MOVER-R (method of variance recovery for the ratio) technique, which is based on the recovery of variance estimates from confidence limits of the numerator and the denominator separately, was proposed as a useful and efficient approach. However, this method implicitly assumes that the confidence interval for the denominator never includes zero, which might be violated in practice. In this article, we first use a new framework to derive the MOVER-R confidence interval, which does not require the above assumption and covers the whole parameter space. We find that MOVER-R can produce an unbounded confidence interval, just like the well-known Fieller method. To overcome this issue, we further propose the penalized MOVER-R. We prove that the new method differs from MOVER-R only at the second order. It, however, always gives a bounded and analytic confidence interval. Through simulation studies and a real data application, we show that the penalized MOVER-R generally provides a better confidence interval than MOVER-R in terms of controlling the coverage probability and the median width.

Fuzzy Differential Privacy Theory and Its Applications in Subgraph Counting

Transportation networks are essential to the operation of societies and economies. Protecting the privacy of sensitive information is a meaningful conception in sustainable transport when mining the transportation data. In data mining, differential privacy (DP) has provable privacy guarantees for releasing sensitive data by introducing randomness into query results. However, it suffers from significant accuracy loss of outputs when the query has high sensitivity (e.g., triangle counting). The reason is that the range of random perturbation to each query result in DP is too large. It consists of all possible output values for a query that forms a large or even unbounded interval. However, when impose perturbation only in a small neighborhood of the true query result, the similarity measure based on randomness in DP fails. Thereupon, we introduce fuzziness into DP to formulate new models which have smaller disturbance via fuzzy similarity measures. In this article, we establish a novel and general theory of private data analysis, fuzzy differential privacy (FDP). The new theory FDP aims to acquire a more flexible tradeoff between the accuracy of outputs and the privacy-preserving level of data. FDP combines DP with fuzzy set theory by introducing fuzziness into the query results and characterizing similarities between outputs via multiple fuzzy similarity measures. From this perspective, DP can be viewed as a special case of FDP with probabilistic similarity measure. Compared with DP, FDP has three superiorities: 1) most fuzzy similarity measures in FDP support sliding window perturbation strategies we proposed, which refer to perturbation in a small neighborhood of the query results; 2) FDP adds noise to the query results only according to a fraction of all possible neighboring datasets; and 3) the fuzzy similarity with valued in [0,1] quantifies the privacy protection level intuitively. These three points enable more accurate outputs while providing provable and intuitive privacy guarantees. As for subgraph counting, the state-of-the-art method is ladder framework in DP. We illustrate FDP mechanisms by applying them to a common application in subgraph counting–triangle/4-cliques counting. Experiments show that FDP is effective and efficient with smaller output errors than DP.

Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients

&lt;p style='text-indent:20px;'&gt;This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large ones in the case that the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. If the degeneracy of the equations at the lateral boundary point is strong enough, it is shown that any nontrivial solution must blow up in a finite time. If the degeneracy of the equations at the lateral boundary point is not strong, it is proved that the critical Fujita curve is determined by the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is also considered.&lt;/p&gt;

Complete Classification of Gradient Blow-Up and Recovery of Boundary Condition for the Viscous Hamilton–Jacobi Equation

It is known that the Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation $$u_t-\Delta u=|\nabla u|^p$$ with $$p>2$$ , which has important applications in stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after the appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. The solutions eventually become classical again for large time, but in-between they may undergo losses and recoveries of boundary conditions at multiple times (as well as GBU at multiple times), hence forming a quite complex dynamics. In this paper, with the help of braid group theory combined with inner/outer region analysis, we give a complete classification of all viscosity solutions in one-dimensional case. Namely, we fully describe the (countably many, type II) rates and space–time profiles of gradient blow-up (GBU) or recovery of boundary condition (RBC), we characterize them at any time when such a phenomenon occurs, and we moreover determine whether the space–time profiles are stable or unstable. These results can be modified in radial domains in general dimensions. Even for type II blow-up in other PDEs, as far as we know, there has been no complete classification except Mizoguchi (Commun Pure Appl Math 75:1870–1886, 2022), in which the argument relies on features peculiar to chemotaxis system. Whereas there are many results on construction of special type II blow-up solutions of PDEs with investigation of stability/instability of bubble, determination of stability/instability of space–time profile for general solutions has not been done. A key in our proofs is to focus on braid group algebraic structure with respect to vanishing intersections with the singular steady state, as a GBU or RBC time is approached. In turn, the GBU and RBC rates and profiles, as well as their stability/instability, can be given a complete geometric characterization by the number of vanishing intersections. We construct special solutions in bounded and unbounded intervals in both GBU and RBC cases, based on methods from Herrero and Velázquez (Preprint, 1994), and then we apply braid group theory to get upper and lower estimates of the rates. After that, we rule out oscillation of the rates, which leads us to the complete space–time profile. In the process, careful construction of special solutions with specific behaviors in intermediate and outer regions, which is far from bubble and the RBC point, plays an essential role. The application of such techniques to viscosity solutions is completely new.

Existence Results for Fractional Integral Equations in Frechet Spaces

The objective of this paper is to present results on the existence of solutions for a class of fractional integral equations in Fr\'{e}chet spaces of Banach space-valued functions on the unbounded interval. Our main tool is the technique of measures of noncompactness and fixed points theorems.