Monotonic Research Articles

Compactness of Green operators with applications to semilinear nonlocal elliptic equations

In this paper, we consider a class of integro-differential operators L posed on a C2 bounded domain Ω⊂RN with appropriate homogeneous Dirichlet conditions where each of which admits an inverse operator commonly known as the Green operator GΩ. Under mild conditions on L and its Green operator, we establish various sharp compactness of GΩ involving weighted Lebesgue spaces and weighted measure spaces. These results are then employed to prove the solvability for semilinear elliptic equation Lu+g(u)=μ in Ω with boundary condition u=0 on ∂Ω or exterior condition u=0 in RN∖Ω if applicable, where μ is a Radon measure on Ω and g:R→R is a nondecreasing continuous function satisfying a subcriticality integral condition. When g(t)=|t|p−1t with p>1, we provide a sharp sufficient condition expressed in terms of suitable Bessel capacities for the existence of a solution. The contribution of the paper consists of (i) developing novel unified techniques which allow to treat various types of fractional operators and (ii) obtaining sharp compactness and existence results in weighted spaces, which refine and extend several related results in the literature.

Strict Faber–Krahn-type inequality for the mixed local–nonlocal operator under polarization

Abstract Let $\Omega \subset \mathbb{R}^d$ with $d\geq 2$ be a bounded domain of class ${\mathcal C}^{1,\beta }$ for some $\beta \in (0,1)$ . For $p\in (1, \infty )$ and $s\in (0,1)$ , let $\Lambda ^s_{p}(\Omega )$ be the first eigenvalue of the mixed local–nonlocal operator $-\Delta _p+(-\Delta _p)^s$ in Ω with the homogeneous nonlocal Dirichlet boundary condition. We establish a strict Faber–Krahn-type inequality for $\Lambda _{p}^s(\cdot )$ under polarization. As an application of this strict inequality, we obtain the strict monotonicity of $\Lambda _{p}^s(\cdot )$ over the annular domains and characterize the rigidity property of the balls in the classical Faber–Krahn inequality for $-\Delta _p+(-\Delta _p)^s$ .

The method of lower and upper solutions for second order periodic Stieltjes differential equations

As an application of the Schauder Fixed Point Theorem, an existence theory is developed for second order nonlinear differential equations with Stieltjes derivative related to a left-continuous, nondecreasing function. By the method of lower and upper solutions, under a Nagumo-type assumption we get a very general result which can be further applied to deduce the existence of solutions for second order nonlinear problems in the settings of impulsive differential equations, time scale analysis or generalized differential equations.

Existence of solutions for discontinuous systems of Stieltjes differential equations

We introduce new transversality conditions in terms of derivatives with respect to a nondecreasing function g : R ⟶ R , and we use them to prove the existence of solutions for discontinuous systems of Stieltjes differential equations, i.e. differential equations where usual derivatives are replaced with derivatives with respect to the derivator g . The proof of our main existence result leans on some interesting new results on g -derivatives also proven in this paper.

Investigating a Nonlinear Fractional Evolution Control Model Using W-Piecewise Hybrid Derivatives: An Application of a Breast Cancer Model

Many real-world phenomena exhibit multi-step behavior, demanding mathematical models capable of capturing complex interactions between distinct processes. While fractional-order models have been successfully applied to various systems, their inherent smoothness often limits their ability to accurately represent systems with discontinuous changes or abrupt transitions. This paper introduces a novel framework for analyzing nonlinear fractional evolution control systems using piecewise hybrid derivatives with respect to a nondecreasing function W(ι). Building upon the theoretical foundations of piecewise hybrid derivatives, we establish sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions, leveraging topological degree theory and functional analysis. Our results significantly improve upon existing theoretical understanding by providing less restrictive conditions for stability compared with standard fixed-point theorems. Furthermore, we demonstrate the applicability of our framework through a simulation of breast cancer disease dynamics, illustrating the impact of piecewise hybrid derivatives on the model’s behavior and highlighting advantages over traditional modeling approaches that fail to capture the multi-step nature of the disease. This research provides robust modeling and analysis tools for systems exhibiting multi-step behavior across diverse fields, including engineering, physics, and biology.

A non-homogeneous count process: marginalizing a Poisson driven Cox process

The paper considers a Cox process where the stochastic intensity function for the Poisson data model is itself a non-homogeneous Poisson process. We show that it is possible to obtain the marginal data process, namely a non-homogeneous count process exhibiting over-dispersion. The model generates intensity functions which are non-decreasing. This is not a restriction in practice since observed data can always be transformed which guarantees a non-decreasing intensity function. We are able, for a specific choice of Cox process, to mathematically marginalize out the latent process leaving with a directly available likelihood function. This makes inference much simpler compared to algorithms which retain the latent process.

Copies of l∞ in Quasi-normed Calderón–Lozanovskiĭ Spaces

We consider order isomorphic and order linearly isometric copies of l∞ in the quasi-normed Calderón–Lozanovskiĭ spaces Eφ. We present a number of theorems describing these copies in the natural language of suitable properties of the quasi-normed ideal space E and the non-decreasing Orlicz function φ. In particular, we will characterize quasi-normed Orlicz–Lorentz spaces having order isomorphic and order linearly isometric copies of l∞. Our studies are conducted in a full possible generality due to the measure space and the Orlicz function.

Analysis of the two-for-one swap heuristic for approximating the maximum independent set in a k-polymatroid

Let f:2N→Z+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(∅)=0). A k-polymatroid satisfies that f(e)≤k for all e∈N. We call S⊆Nindependent if f(S)=∑e∈Sf(e) and f(e)>0 for all e∈S. Such a set was also called a matching. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For k≥3, the problem is NP-hard, and a ((2/k)−ϵ)-approximation is known and is obtained by swapping as long as possible a subset of up to (1/ϵ)logk−1⁡(2k+1) elements from the current solution by a set with one more element.Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a tight (weaker) (2/(k+1))-approximation.

The first exit time of fractional Brownian motion from an unbounded domain

Consider a fractional Brownian motions starting at the interior point x0,h‖x0‖+2K∈Rd+1 with the constant K>1, for some fixed x0∈Rd, of an unbounded domain D=x,y∈Rd+1:y>h‖x‖, The function h is a nondecreasing, lower semicontinuous, and convex function on [0,∞) with h(0) being finite. Here we take h−1x=Axαlogxβwith a positive constant A for x>K. It is evident that h−1(x) exhibits monotonic behavior for sufficiently large values of x. Let τD denote the first time that the fractional Brownian motion exits from D. In most cases, we give the asymptotically equivalent estimate of logPτD>t. The proof methods are based on the earlier works of Li, Shi, Lifshits, and Aurzada.

A Filippov-Type Lemma for Stieltjes Differential Inclusions with Supremum

The aim of this paper is to generalize the classical Filippov lemma to the framework of Cauchy differential set-valued problems involving the supremum of the unknown function on a past interval and its Stieltjes derivative with respect to a left-continuous non-decreasing function. To highlight the wide spectrum of the result (coming from the fact that this setting covers the theories of ordinary differential or difference problems, impulsive problems or problems on time scales), as a consequence, a Filippov-type lemma for dynamic inclusions on time scales with supremum is obtained.

Some algebraic and analytical properties of a class of two-place functions

This article deals with the formula f(−1)(F(f(x),f(y))) generated by a one-place function f:[0,1]→[0,1] and a binary function F:[0,1]2→[0,1]. When the f is a strictly increasing function and F is a continuous, non-decreasing and associative function with neutral element in [0,1], the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the f is a non-decreasing surjective function and F is a non-decreasing associative function with neutral element in [0,1], we investigate the associativity of the formula.

An Effective and Small Sample-Size Valid Confidence Interval for Isotonic Dose–Response Curves by Inverting a Partial Likelihood Ratio Test

A dose–response curve is essential for determining the safe dosage of a drug and is widely used in bioassay and in phase 1 clinical trials. It is generally accepted that the probability of death or the probability of dose-limiting toxicity is a nondecreasing function of the dose. This article proposes and develops an effective point-wise confidence interval for an isotonic dose–response curve, a problem without a satisfactory solution so far. We show that one subset of the observations informs the lower limit while the other subset informs the upper limit. A partial likelihood ratio test using these subsets of observations is fully investigated. The resulting confidence interval is compared to three existing methods (Morris, Meyer, and Oron) in an extensive simulation study. The new method produces tight intervals while maintaining coverage. It is therefore preferred when maintaining an actual coverage probability is crucial as often in a regulatory environment. The new method extends to one-way ANOVA of a continuous response variable in a straightforward way. An R function is provided.

Strict monotonicity of stochastic process extreme distributions

Strict monotonicity is proved for the distributions of extremes of processes consisting of series of bounded function with independent random coefficients, in particular for zero mean continuous Gaussian processes over compact metric space. These results have wide applications to global inference problems on unknown functions.

On statistics which are almost sufficient from the viewpoint of the Fisher metrics

Given a statistical model, a statistic on the model is sufficient if the Fisher metric of the induced model coincides with the original Fisher metric, according to the definition by Ay-Jost-Lê-Schwachhöfer. We introduce and study its quantitative version: for 0<δ≤1, we call a statistic δ-almost sufficient if δ2g(v,v)≤g′(v,v) for every tangent vector v of the parameter space, where g and g′ are the Fisher metric of the original and the induced model, respectively. By the monotonicity theorem due to Amari-Nagaoka and Ay-Jost-Lê-Schwachhöfer, the Fisher metric g′ of the induced model for such a statistic is bi-Lipschitz equivalent to the original one g, which means that the information loss of the statistic is uniformly bounded. We characterize such statistics in terms of the conditional probability or by the existence of a certain decomposition of the density function in a way similar to the characterizations of sufficient statistics due to Ay-Jost-Lê-Schwachhöfer and Fisher-Neyman.

Stability of impulsive delayed switched systems with conformable fractional-order derivatives

In this paper, exponential stability is addressed for impulsive delayed switched systems with conformable derivatives by Halanay inequalities technique. First, based on the fractional-order monotonicity theorem and the definition of conformable fractional-order derivative, novel conformable fractional-order Halanay inequalities are developed that extend and improve the existing ones. Second, using the obtained Halanay inequalities and the estimate of conformable fractional-order exponential functions, globally exponential stability conditions that depend on the infimum and supremum of impulsive step sizes are proposed for the systems. Third, more relaxing stability conditions are presented that do not depend on the infimum and supremum of impulsive step sizes, which means that the conditions on impulses are weaker than those commonly used in the existing literature. Finally, examples are given to illustrate the validity of the theoretical results.

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

A Family of Mexican Hat Wavelet Stieltjes Transform for Unbounded Non-decreasing Functions

Robust non-monotonic Lyapunov based stability and stabilization methods for continuous-time systems: Applied on bilateral teleoperation system

This study introduces a Non-monotonic Lyapunov (NML) framework aimed at stability evaluation and controller design for continuous-time systems, particularly under conditions of uncertainty. Conventional Lyapunov techniques often exhibit a conservative nature, particularly in the context of uncertain systems, which necessitates the development of less conservative alternatives like NML. The NML methodology distinguishes itself by not imposing strict monotonicity requirements for demonstrating the decrease of a Lyapunov functional. Consequently, this paper derives new stability and stabilization criteria framed as matrix inequalities applicable to a specific class of uncertain systems. The practical applicability of the introduced approach is illustrated through controller design for uncertain systems, exemplified by a nonlinear bilateral teleoperation model. Assorted demonstrative examples and simulation outcomes support the findings, underscoring the NML approach’s efficaciousness.

Development of a total variation diminishing (TVD) sea ice transport scheme and its application in an ocean (SCHISM v5.11) and sea ice (Icepack v1.3.4) coupled model on unstructured grids

Abstract. As the demand for increased resolution and complexity in unstructured sea ice models is growing, higher demands are also placed on the sea ice transport scheme. In this study, we couple the Semi-implicit Cross-scale Hydro-science Integrated System Model (SCHISM, v5.11) with Icepack (v1.3.4), the column physics package of the Los Alamos sea ice model (CICE); a key step is to implement a total variation diminishing (TVD) transport scheme for the multi-class sea ice module in the coupled model. Compared with the second-order upwind scheme and the finite-element flux-corrected transport (FEM-FCT) scheme, the TVD transport scheme is overall superior when evaluated based on conservation, accuracy, efficiency (even with very high resolution), and strict monotonicity. Although it is slightly weaker than FEM-FCT in terms of accuracy alone, the TVD scheme still outperforms the other two schemes in comprehensive performance. The new coupled model outperforms the existing single-class ice model of SCHISM in the case of Lake Superior. For the Arctic Ocean case, it successfully reproduces the long-term changes in the sea ice extent, sea ice boundary, concentration observations from satellites, and thickness from in situ measurement.

Approximate utility

Approximating arbitrage or a perfect hedge in a limit implies approximation in utility (formally, expected utilities are continuous in L2), if and only if the utility function is bounded above and below by quadratics. We characterize when some special utility functions are continuous for all random consumptions with a common lower bound. Only linear, quadratic, and, in some cases, translated SAHARA utilities are continuous if there is no lower bound. We also characterize when a utility function defined on a convex subset can be extended to an L2-continuous nondecreasing and concave utility function on all of R.

Specification of the health production function and its behavioral implications.

The health production function of the canonical health-capital model is generalized to allow the state of health to affect the total and marginal products of health investment. If the total and marginal products of health investment are nonincreasing functions of the state of health, then the solution of the generalized model is locally qualitatively identical to that of the canonical model. Moreover, and in contrast to the canonical model, the generalized model is able to rationalize the cycling of the state of health and health investment observed in some individuals. The necessary conditions on the health production function for cyclical behavior are identified as well.