Solution Map Research Articles

The dynamics of Wolbachia spread in mosquito population under period-2 perturbation of environments

A nonautonomous periodic discrete model is proposed to characterize the dynamics of Wolbachia spread in mosquito populations under period-2 environments, where the Wolbachia strain in the first environment is less competitive than the one in the second environment. By introducing the associated Poincaré map, the existence, exact number, and stability of periodic solutions and the long-term behavior of the discrete model are analyzed. Sufficient conditions are obtained to guarantee the bistable dynamics of the model: the model has exactly two periodic solutions, among which one is unstable, and the other is locally asymptotically stable. The origin, corresponding to the Wolbachia vanishment, of the model is locally asymptotically stable. Counting the exact number of periodic solutions of nonautonomous periodic discrete models is always challenging. In this paper, we provide three exclusive methods to prove the uniqueness of the periodic solutions, together with their stability analyses. Biologically, the unstable periodic solution serves as a threshold for Wolbachia invasion, and the stable periodic solution identifies where Wolbachia will be stabilized. Numerical simulations are provided to locate these two periodic solutions and analyze the parameter space to identify regions where periodic solutions emerge or disappear through bifurcations.

A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems

Energy Space Newton Differentiability for Solution Maps of Unilateral and Bilateral Obstacle Problems

Mapping the Gaps: A Scoping Review of Virtual Care Solutions for Caregivers of Children with Chronic Illnesses.

Background/Objectives: Caregivers of children with chronic illnesses, including chronic pain, experience high levels of distress, which impacts their own mental and physical health as well as child outcomes. Virtual care solutions offer opportunities to provide accessible support, yet most overlook caregivers' needs. We conducted a scoping review to create an interactive Evidence and Gap Map (EGM) of virtual care solutions across a stepped care continuum (i.e., from self-directed to specialized care) for caregivers of youth with chronic illnesses. Methods: The review methodology was co-designed with four caregivers. Data sources were the peer-reviewed scientific literature and a call for innovations. Records were independently coded and assessed for quality. Results: Overall, 73 studies were included. Most virtual care solutions targeted caregivers of children with cancer, neurological disorders, and complex chronic illnesses. Over half were noted at lower levels of stepped care (i.e., self-guided apps and websites), with psychological strategies being predominant (84%). However, very few addressed caregivers' physical health (15%) or provided family counseling (19%) or practical support (1%). Significant gaps were noted in interventions for managing caregiver chronic pain, despite its high prevalence and impact on child outcomes. Conclusions: Evidence and Gap Maps are innovative visual tools for knowledge synthesis, facilitating rapid, evidence-informed decision-making for patients, families, health professionals, and policymakers. This EGM highlighted high-quality virtual care solutions ready for immediate scaling and identified critical evidence gaps requiring prioritization. To address the complexities of pediatric chronic illnesses, including chronic pain, virtual care initiatives must prioritize family-centered, accessible, and equitable approaches. Engaging caregivers as partners is critical to ensure interventions align with their needs and priorities.

Global well-posedness, ill-posedness and long time behavior of solutions of the surface electromigration equation

Abstract This paper is concerned with the Cauchy problem and the long time behavior of the two-dimensional surface electromigration (SEM) equation. Firstly, we prove that the SEM equation is locally well-posed in H s ( R 2 ) with s > − 1 4 , and globally well-posed in L 2 ( R 2 ) without smallness assumption. Secondly, we prove that the SEM equation is ill-posed in H s ( R 2 ) with s < − 1 4 in the sense that the solution map is not C 2. Finally, we get a local energy decay result of the global solution in a growing unbounded in time region but not containing the soliton region. Our well-posedness results extend the latest findings in Linares et al (2021 Nonlinearity 34 5213–33), where the authors proved that the SEM equation is locally well-posed in H s ( R 2 ) with s > 1 2 . The key ingredient in this paper is to rewrite the linear and nonlinear part in the SEM equation into symmetric separate dispersive terms on the two space variables and some divergence form, respectively. After this invertible process, we use two bilinear estimates in the modified Fourier restriction space introduced in Kinoshita (2021 Ann. Inst. Henri Poincare C 38 451–505) to relax the local well-posedness index from s > 1 2 to s > − 1 4 .

CCOA‐AdaLS: Hybrid Beamforming Using Chaotic Chebyshev Aquila Optimization for mmWave Massive MIMO

ABSTRACTThis research aims to design hybrid analog and digital beamforming to improve the signal‐to‐noise ratio (SNR) and spectral efficiency (SE) of communication links. Considering the complexity and cost associated with fully connected multiple‐input multiple‐output (MIMO) communication models, a partially connected system model is adopted for the downlink millimeter‐wave (mmWave) communication model. The analog beamforming utilizes the adaptive search (AdaLS) algorithm to minimize interference and enhance user power, whereas digital beamforming is optimized using the proposed Chaotic Chebyshev Aquila Optimization (CCAO) algorithm. The CCAO algorithm integrates chaotic Chebyshev–based solution mapping with the conventional Aquila Optimization Algorithm to enhance exploration capability. The system model is illustrated, and the problem is formulated to maximize the signal‐to‐interference‐plus‐noise ratio (SINR) through the selection of the required signal at the receiver. The digital beamformer is designed using Lagrange's multiplier, and the analog beamformer is optimized using AdaLS. The proposed CCAO algorithm is detailed, incorporating chaotic dynamics to explore the solution space effectively. The research evaluates the performance of the proposed method against conventional approaches, showcasing improved normalized beam gain and SINR.

The Continuity and Convexity of a Nonlinear Scalarization Function with Applications in Set Optimization Problems Involving a Partial Order Relation

In this paper, we deal with the properties and applications of a nonlinear scalarization function for sets by using the Minkowski difference. Under some suitable assumptions, the continuity and convexity concerned with the nonlinear scalarization function for sets are presented. As applications, the path connectedness of the solution sets to set optimization problems and the continuity of the solution mappings of parametric set optimization problems are established. The results achieved do not impose the monotonicity of the set-valued objective mapping, which are obviously different from the related ones in the literature.

Continuity of the solution set mappings to parametric unified weak vector equilibrium problems via free-disposal sets

In this paper, the lower and upper semicontinuity of the solution mappings for parametric unified weak vector equilibrium problem (PUWVEP) are established, via free-disposal set and non-linear scalarization function. Moreover, example is given to illustrate the obtained results. The results improve the corresponding ones in the literature [Z.Y. Peng and S.S. Chang, Optim. Lett. (2014) 8 159–169.], [Z.Y. Peng, X.B. Li, X.J. Long and X.D. Fan, Optim. Lett. (2018) 12 1339–1356.], [Z.Y. Peng, J.W. Peng, X.J. Long and J.C. Yao, J. Global Optim. 70 (2018) 55–69.].

Unboundedness of the images of set-valued mappings having closed graphs: application to vector optimization

In this paper, we propose criteria for unboundedness of the images of set-valued mappings having closed graphs in Euclidean spaces. We focus on mappings whose domains are non-closed or whose values are connected. These criteria allow us to see structural properties of solutions in vector optimization, where solution sets can be considered as the images of solution mappings associated to specific scalarization methods. In particular, we prove that if the domain of a certain solution mapping is non-closed, then the weak Pareto solution set is unbounded. Furthermore, for a quasi-convex problem, we demonstrate two criteria to ensure that if the weak Pareto solution set is disconnected then each connected component is unbounded.

A remark on randomization of a general function of negative regularity

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier–Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside L 2 ( T d ) L^2(\mathbb {T}^d) such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier–Lebesgue γ \gamma -radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.

Error analysis of kernel/GP methods for nonlinear and parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear, and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Similarity joins and clustering for SPARQL

The SPARQL standard provides operators to retrieve exact matches on data, such as graph patterns, filters and grouping. This work proposes and evaluates two new algebraic operators for SPARQL 1.1 that return similarity-based results instead of exact results. First, a similarity join operator is presented, which brings together similar mappings from two sets of solution mappings. Second, a clustering solution modifier is introduced, which instead of grouping solution mappings according to exact values, brings them together by using similarity criteria. For both cases, a variety of algorithms are proposed and analysed, and use-case queries that showcase the relevance and usefulness of the novel operators are presented. For similarity joins, experimental results are provided by comparing different physical operators over a set of real world queries, as well as comparing our implementation to the closest work found in the literature, DBSimJoin, a PostgreSQL extension that supports similarity joins. For clustering, synthetic queries are designed in order to measure the performance of the different algorithms implemented.

On semi-continuity and continuity of solution maps of parametric generalized multiobjective games in fuzzy environments

The purpose of this article is to study new results on the continuity of solution maps for parametric generalized multiobjective games in fuzzy environments. Firstly, we revisit parametric fuzzy generalized multiobjective games. Secondly, we establish some sufficient conditions for upper semi-continuity, Hausdorff upper semi-continuity, closedness and compactness of solution maps for such problem under suitable conditions. Thirdly, we introduce the auxiliary problem of parametric fuzzy generalized multiobjective games and the concept of strong Ki-convexity of objective functions for these problems. Finally, results on the lower semi-continuity, Hausdorff lower semi-continuity, continuity and Hausdorff continuity of solution maps to parametric generalized multiobjective games in fuzzy environments are established and studied. Many examples are given to illustrate our results.

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.

Fixed points of regular set-valued mappings in quasi-metric spaces

Metric fixed point theory is becoming increasingly significant across various fields, including data science and iterative methods for solving optimization problems. This paper aims to introduce new fixed point theorems for set-valued mappings under novel regularity conditions, such as orbital regularity and orbital pseudo-Lipschitzness. Instead of traditional metric spaces, we adopt the framework of quasi-metric spaces, motivated by the need to address problems in spaces that are not necessarily metric, such as function spaces of homogeneous type. We also explore the stability of the set of fixed points under variations of the set-valued mapping. Additionally, we provide estimates for the distances from a given point to the set of fixed points and between two sets of fixed points. Building on these findings, we extend the discussion to similar problems involving fixed, coincidence, and cyclic/double fixed points within this framework. Our results generalize recent findings from the literature, including those in Ait Mansour M, Bahraoui MA, El Bekkali A. [Metric regularity and Lyusternik-Graves theorem via approximate fixed points of set-valued maps in noncomplete metric spaces. Set-Valued Var Anal. 2022;30(1):233–256. doi: 10.1007/s11228-020-00553-1], Dontchev AL, Rockafellar RT. [Implicit functions and solution mappings. a view from variational analysis. Dordrecht: Springer; 2009. Springer Monographs in Mathematics], Ioffe AD. [Variational analysis of regular mappings. Springer, Cham; 2017. Springer Monographs in Mathematics; theory and applications. doi: 10.1007/978-3-319-64277-2], Lim TC. [On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl. 1985;110(2):436–441. doi: 10.1016/0022-247X(85)90306-3] and Tron NH. [Coincidence and fixed points of set-valued mappings via regularity in metric spaces. Set-Valued Var Anal. 2023;31(2):22. Paper No. 17. doi: 10.1007/s11228-023-00680-5].

On the existence and Lipschitz-like property of the solution map to parametric equilibrium problems governed by trifunction with applications

In this paper, we study a class of parametric equilibrium problems governed by a trifunction. We apply the Fan-KKM theorem under new type of monotonicity and coercivity conditions for trifunctions to obtain the existence result of a solution for the equilibrium problem. For the solution map of parametric equilibrium problems, we obtain the Hölder-type Lipschitz-like property relative to a closed and convex set, which generalizes some existing results in the literature. We illustrate the application of our abstract results in the study of a new model of a fluid flow problem with slip and leak boundary conditions of frictional type.

Critical local well-posedness of the nonlinear Schrödinger equation on the torus

In this paper, we study the local well-posedness of nonlinear Schrödinger equations on tori \mathbb{T}^{d} at the critical regularity. We focus on cases where the nonlinearity |u|^{a}u is nonalgebraic with small a>0 . We prove the local well-posedness for a wide range covering the mass-supercritical regime. Moreover, we supplementarily investigate the regularity of the solution map. In pursuit of lowering a , we prove a bilinear estimate for the Schrödinger operator on tori \mathbb{T}^{d} , which enhances previously known multilinear estimates. We design a function space adapted to the new bilinear estimate and a package of Strichartz estimates, which is not based on conventional atomic spaces.

Local convergence of augmented Lagrangian methods for composite optimization

In this paper, we propose an augmented Lagrangian method for composite optimization with the outer function satisfying the second order epi-regular property. The local convergence and rate of convergence of this algorithm will be proved under the assumptions of the second order sufficient condition via the second order epi-derivative of augmented Lagrangian and the semi-isolated calmness of the solution mapping of perturbed generalized equation formulated from KKT system. For the penalty parameters large enough, we obtain the primal-dual Q-linear convergence rate.

Global dynamics and threshold behavior of an SEIR epidemic model with nonlocal diffusion

This paper studies the global dynamics of an SEIR (Susceptible–Exposed–Infectious–Recovered) model with nonlocal diffusion. We show the model’s well-posedness, proving the solutions’ existence, uniqueness, and positivity, along with a disease-free equilibrium. Next, we prove that the model admits the global threshold dynamics in terms of the basic reproduction number R0, defined as the spectral radius of the next-generation operator. We show that the solution map has a global compact attractor, offering insights into long-term dynamics. In particular, the analysis shows that for R0<1, the disease-free equilibrium is globally stable. Using the persistence theory, we show that there is an endemic equilibrium point for R0>1. Moreover, by constructing an appropriate Lyapunov function, we establish the global stability of the unique endemic equilibrium in two distinct scenarios.

Motivations and Potential Solutions in Developing a Knowledge Management System for Organization at Higher Education: A Systematic Literature Review

Background: Amidst a rapidly evolving digital landscape that accelerates the flow of information, higher education institutions face the unique challenge of managing vast and dynamic knowledge resources. This research delves into the motivations and innovative solutions for developing Knowledge Management Systems (KMS), which is key to optimizing knowledge resource utilization and enhancing academic collaboration. Objective: This research provides a comprehensive mapping of problems and solutions for developing university knowledge management systems based on previous research. Not only that, but the results of this study also suggest three future research studies that can be adopted. Methods: This study used the Kitchenham systematic literature review method. The author uses literature in the form of journals and conference proceedings published from 2019 to 2023. Twenty-three articles were used for this study from 5 databases, such as ACM, ProQuest, Scopus, Taylor &amp; Francis, and IEEE Xplore. Results: This study reveals research trends in knowledge management systems within higher education, examining aspects such as country, data collection methods, research methodologies, and theoretical frameworks. The main problems motivating the development of KMS are identified and categorized based on the people, process, and technology framework. In overcoming these problems in the university business process, there are several alternative solutions, both in the form of requirements and systems. Thus, the results of this study seek to provide guidelines for future research to adopt alternative solutions from this research and develop KMS to provide new solutions. Conclusion: This study advances knowledge about various trends, motivations, requirements, and system solutions to address KMS problems in higher education. The authors' research results can add valuable insights to improve our understanding of the development of KMS in universities in various countries. Future research can identify new potential in KMS in business processes currently running in a university with appropriate methodologies. Keywords: Knowledge management system, higher education, systematic literature review, problem, solution