Existence Of Solutions Research Articles

Fractional-order analysis of temperature- and rainfall-dependent mathematical model for malaria transmission dynamics

Malaria remains a substantial public health challenge and economic burden globally. Currently, malaria has been declared as endemic in 85 countries. In this study, we developed and analyzed a fractional-order mathematical model for malaria transmission dynamics that incorporates variability of temperature and rainfall using Caputo-type AB operators. The existence and uniqueness of the model's solutions were established using the Banach fixed-point theorem. The model system's equilibria (both disease-free and endemic) were identified, and lemmas and theorems were developed to prove their stability. Furthermore, we used different temperature ranges and rainfall data, validating them against existing literature. Numerical simulations using the Toufik-Atangana schemes with various fractional-order alpha values revealed that as the value of alpha approaches 1, the behavior of the fractional-order model converges to that of the classical model. The numerical results are promising and are expected to be valuable for future research related to fractional-order models.

On weak and strong solutions of time inhomogeneous Itô’s equations with VMO diffusion and Morrey drift

We prove the existence of weak solutions of Itô’s stochastic time dependent equations with irregular diffusion and drift terms of Morrey spaces. Weak uniqueness (generally conditional) and a conjecture pertaining to strong solutions are also discussed. Our results are new even if the drift term vanishes.

Positive solutions for a Kirchhoff problem of Brezis–Nirenberg type in dimension four

We consider a Kirchhoff problem of Brezis–Nirenberg type in a smooth bounded domain of R4 with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters b and λ coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.

Weakly coupled systems of semilinear σ$$ \sigma $$‐evolution equations with friction and visco‐elastic type damping

In the present paper, we are interested to study global (in time) existence of small data Sobolev solutions to the Cauchy problem for a weakly coupled system of two semilinear ‐evolution equations with friction and visco‐elastic type damping, where for . We study model (1.1) (see below) in several cases with respect to the regularity for the data: First, we assume data . By using estimates of Sobolev solutions to related linear models with vanishing right‐hand side, we explain the admissible range of exponents in (1.1) which allow to prove the global (in time) existence of small data Sobolev solutions. Then we suppose that the data belong to , where now , . So the data have additional regularity to the first assumed integrability. We restrict ourselves to data from energy space (on the basis of ) and from energy space with additional suitable higher regularity. We compare in our statements the admissible set of exponents and in the power nonlinearities with the so‐called modified Fujita exponent. Finally, some blow‐up results complete the paper.

Fractional Sobolev space: Study of Kirchhoff-Schrödinger systems with singular nonlinearity

This study extensively investigates a specific category of Kirchhoff-Schrödinger systems in fractional Sobolev space with Dirichlet boundary conditions. The main focus is on exploring the existence and multiplicity of non-negative solutions. The non-linearity of the problem generally exhibits singularity. By employing minimization arguments involving the Nehari manifold and a variational approach, we establish the existence and multiplicity of positive solutions for our problem with respect to the parameters \(\eta\) and \(\zeta\) in suitable fractional Sobolev spaces. Our key findings are novel and contribute significantly to the literature on coupled systems of Kirchhoff-Schrödinger system with Dirichlet boundary conditions.

Existence of solutions to k‐Wave models of nonlinear ultrasound propagation in biological tissue

AbstractWe investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k‐Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. We first consider a system that incorporates additional viscosity in the equation for momentum conservation. By constructing a Galerkin approximation procedure, we prove the local existence of its solutions. In view of inverse problems arising from imaging tasks, the theory allows for the variable background mass density, speed of sound, and the nonlinearity parameter in the systems. Second, under stronger conditions on the data, we take the vanishing viscosity limit of the problem, thereby rigorously establishing the existence of solutions for the limiting system as well.

Global Existence in the Critical Space for the Space-Time Monopole Equations

We prove the global existence of solutions in the critical space \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} for space-time monopole equations under the Lorenz gauge condition. The existence of a global solution was showed in our previous paper. Here we adapt a different function space to allow for a simplified proof. We first prove the local existence in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1({\\mathbb {R}})$$\\end{document} and extend the local solution to the global one.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

Existence, Uniqueness, and Stability of Solutions for Nabla Fractional Difference Equations

In this paper, we study a class of nabla fractional difference equations with multipoint summation boundary conditions. We obtain the exact expression of the corresponding Green’s function and deduce some of its properties. Then, we impose some sufficient conditions in order to ensure existence and uniqueness results. Also, we establish some conditions under which the solution to the considered problem is generalized Ulam–Hyers–Rassias stable. In the end, some examples are included in order to illustrate our main results.

Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations*

Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations.

An advection–diffusion equation with a generalized advection term: Well-posedness analysis and examples

We consider a Cauchy–Dirichlet problem for a semilinear advection–diffusion equation with a generalized advection term. Specific examples include an incision–diffusion landscape evolution model and a viscous Hamilton–Jacobi equation with an absorbing gradient term. We establish existence and uniqueness of a weak solution and its continuous dependence on initial data and a source term.

Time-optimal ergodic search: Multiscale coverage in minimum time

Search and exploration capabilities are essential for robots to inspect hazardous areas, support scientific expeditions in extreme environments, and potentially save human lives in natural disasters. The variability of scale in these problems requires robots to reason about time alongside their dynamics and sensor capabilities to effectively assess and explore for information. Recent advances in ergodic search methods have shown promise in supporting trajectory planning for exploration in continuous, multiscale environments with dynamics consideration. However, these methods are still limited by their inability to effectively reason about and adapt the time to explore in response to their environment. This ability is crucial for adapting exploration to variable-resolution information-gathering tasks. To address this limitation, this paper poses the time-optimal ergodic search problem and investigates solutions for fast, multiscale, and adaptive robotic exploration trajectories. The problem is formulated as a minimum-time problem with an ergodic inequality constraint whose upper bound specifies the amount of coverage needed. We show the existence of optimal solutions using Pontryagin’s conditions of optimality, and we demonstrate effective, minimum-time coverage numerically through a direct transcription optimization approach. The efficacy of the approach in generating time-optimal search trajectories is demonstrated in simulation under several nonlinear dynamic constraints, and in a physical experiment using a drone in a cluttered environment. We find that constraints such as obstacle avoidance are readily integrated into our formulation, and we show through an ablation study the flexibility of search capabilities at various scales. Last, we contribute a receding-horizon formulation of time-optimal ergodic search for sensor-driven information-gathering and demonstrate improved adaptive sampling capabilities in localization tasks.

Single transition layer in mass-conserving reaction-diffusion systems with bistable nonlinearity

Abstract Mass-conserving reaction-diffusion systems with bistable nonlinearity are useful models for studying cell polarity formation, which is a key process in cell division and differentiation. We rigorously show the existence and stability of stationary solutions with a single internal transition layer in such reaction-diffusion systems under general assumptions by the singular perturbation theory. Moreover, we present a meaningful model for understanding the existence of an unstable transition layer solution; our numerical simulations show that the unstable solution is a separatrix of the dynamics of the model.

Existence of solutions for systems of conformable fractional dynamic equations on time scales

Existence of solutions for systems of conformable fractional dynamic equations on time scales

On Average Optimality for Non-Stationary Markov Decision Processes in Borel Spaces

This paper studies Borel models of nonstationary Markov decision processes (MDPs) with average criteria. We establish a suitable fixed point theorem, which is used to show the existence of solutions to the average optimality equation (AOE), without using contractions. The existence of optimal policies follows from the obtained solutions to the AOE. Furthermore, we show that versions of the rolling horizon algorithm can be used to produce an optimal policy or an ϵ-optimal policy. Finally, we compare the optimality conditions imposed in this paper with the existing ones in the literature, and demonstrate that they can be satisfied while the previous ones are not. Funding: This research was supported by National Key Research and Development Program of China [2022YFA1004600], National Natural Science Foundation of China [No. 12301170], Guangdong Basic and Applied Research Foundation [2023A1515012644], and the Fundamental Research Funds for the Central Universities, Sun Yat-Sen University [No. 23qnpy39].

Existence and Uniqueness of Homogeneous Linear Equation Solutions in Supertropical Algebra

This research investigates the existence and uniqueness of solutions to homogeneous linear equations in supertropical algebra. We analyze the structure of supertropical matrices to identify the conditions in which nontrivial solutions exist for the system of equations A⊗𝒙 ⊨ 𝜀, where A is a matrix over a supertropical semiring and x is a vector. By applying determinant-based criteria, we demonstrate how tropical and supertropical values influence the solution space. The research applies theorems that determine the presence of trivial and nontrivial solutions and uses examples to illustrate practical methods for solving homogeneous matrix systems. This highlights the distinct characteristics of supertropical algebra compared to classical linear algebra. Our findings provide a deeper insight into solution behaviors in supertropical systems, paving the way for further research in tropical mathematics.

Existence and concentration of positive solutions to generalized Chern-Simons-Schrödinger system with critical exponential growth

We are concerned with a class of generalized Chern-Simons-Schrödinger systems{−Δu+λV(x)u+A0u+∑j=12Aj2u=f(u),∂1A2−∂2A1=−12|u|2,∂1A1+∂2A2=0,∂1A0=A2|u|2,∂2A0=−A1|u|2, where λ>0 denotes a sufficiently large parameter, V:R2→R admits a potential well Ω≜intV−1(0) and the nonlinearity f fulfills the critical exponential growth in the Trudinger-Moser sense at infinity. Under some suitable assumptions on V and f, based on variational method together with some new technical analysis, we are able to get the existence of positive solutions for some large λ>0, and the asymptotic behavior of the obtained solutions is also investigated when λ→+∞.

A novel dimensionality reduction iterative method for the unknown coefficient vectors in TGFECN solutions of unsaturated soil water flow problem

The main purpose of this paper is to reduce the dimensionality of unknown coefficient vectors in finite element (FE) solutions of two-grid FE Crank-Nicolson (CN) (TGFECN) format for the unsaturated soil water flow (USWF) problem with two strong nonlinear terms by using proper orthogonal decomposition (POD). For this purpose, our first step involves designing a time semi-discrete CN (TSDCN) scheme for the USWF problem and demonstrate the existence, boundedness, and error estimations of TSDCN solutions. Thereafter, we discretize the TSDCN scheme using the two-grid FE method to create a new TGFECN format and prove the existence, boundedness, and error estimations of TGFECN solutions. The primary focus should be on reducing the dimension of unknown coefficient vectors of TGFECN solutions through the utilization of the POD technique in creating a novel format, referred to as dimension reduction iterative TGFECN (DRITGFECN) format, while establishing the existence, boundedness, and error estimations for DRITGFECN solutions. Lastly, we use two sets of numerical tests to exhibit the advantage of the DRITGFECN format. Due to the presence of two highly nonlinear terms in the unsaturated soil flow problem, the development and analysis of DRITGFECN format pose greater challenges and necessitates more advanced technical skills compared to previous studies. However, the significance and broad applications of this research make it a valuable subject for investigation.

Existence and Uniqueness of Solution of Fractional Differential Equation Using Non-local Operator

Objectives : To investigate the existence and uniqueness of a solution to the non-local Cauchy problem of order using the Atangana Baleanu (AB) fractional derivative operator. Methods : Using Schauder's fixed point theorem and the Arzela-Ascoli theorem, the study proved the existence of a solution to the given problem. Further, it obtains results for the uniqueness of the solution. Findings : This study proves the existence of a solution to the non-local Cauchy problem under the given conditions. Results are also provided for the uniqueness of the solution. Novelty : A novel approach to fractional differential equations is represented by applying Atangana-Baleanu fractional derivative operator to the non-local Cauchy problem. Schauder's fixed point theorem and Arzela-Ascoli's theorem, are used to show existence and uniqueness. Further, one detailed example has been solved. Keywords: Existence, Uniqueness, Non-local operator, Schauder fixed point theorem, Arzela-Ascoli theorem, Cauchy problem

Existence of solutions and positivity of the infimum eigenvalue for elliptic equations driving by p(x)-triharmonic operator

Existence of solutions and positivity of the infimum eigenvalue for elliptic equations driving by p(x)-triharmonic operator