Existence Of Solutions Research Articles

Nonlocal boundary value problems for functional hybrid differential equations involving generalized w-Caputo fractional operator

In this manuscript , we establish the existence and uniqueness of solutions for boundary value problems of nonlinear hybrid fractional differential equations involving generalized $\omega-$Caputo fractional derivatives. The proofs are based on Krasnoselskii fixed point theorem and some basic concept of $\omega-$Caputo fractional analysis. As application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.

Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

Mathematical analysis of a within-host dengue virus dynamics model with adaptive immunity using Caputo fractional-order derivatives

AbstractDengue fever poses a significant global health threat, with over 50 million annual infections spanning more than 100 countries. Given the absence of a specific treatment, medical intervention primarily targets symptom alleviation. The present study utilizes a Caputo-type fractional-order derivative operator to investigate and analyze the dynamics of dengue virus spread within a host with adaptive immune responses. The developed model describes and analyzes the dynamics of immune cells, free dengue particles, infected monocytes, and susceptible monocytes in the presence of cytotoxic T-Lymphocytes. A range of analytical methods is employed to probe the fractional-order within-host model. The application of the generalized mean value theorem aids in investigating the model’s solutions, employing positivity and boundedness theory. Furthermore, the Banach fixed-point approach is utilized to establish the existence and uniqueness of solutions. Employing the normalized forward sensitivity approach, the fractional-order system’s response to various model parameters is scrutinized. The study reveals that the dynamics of the viral model are significantly influenced by the transmission rate and parameters representing adaptive immune responses. Numerical simulations underscore the critical role of transmission rates and adaptive immune responses in the model. Additionally, the study examines the impact of memory on the density of susceptible monocytes, infected monocytes, free dengue particles, and immune cells to optimize immune responses. Through simulations, the study illustrates the influence of memory on immune dynamics.

Existence and Ulam stability of mild solutions for nonlinear fractional inegro-differential equations in a Banach space

Existence and Ulam stability of mild solutions for nonlinear fractional inegro-differential equations in a Banach space

Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.

Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces

AbstractIn this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic–elliptic Keller–Segel system on whole spaces detailized by Euclidean space and real hyperbolic space . We work in framework of critical spaces such as on weak‐Lorentz space to obtain the results for the Keller–Segel system on and on for to obtain those on . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviors of periodic mild solutions of the Keller–Segel system obtained in and the one in .

Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media

Existence of a weak solution to a regularized moving boundary fluid-structure interaction problem with poroelastic media

On solutions of a semilinear measure‐driven evolution equation with nonlocal conditions on infinite interval

AbstractThis paper studies the existence and asymptotic properties of solutions for a semilinear measure‐driven evolution equation with nonlocal conditions on an infinite interval. The existence result of the solutions for the considered equation is established by Schauder's fixed point theorem. Then, the asymptotic stability of solutions is further proved to show that all the solutions may converge to the unique solution of the corresponding Cauchy problem. In addition, under some conditions the existence of global attracting sets and quasi‐invariant sets of mild solutions is investigated as well. Finally, an example is provided to illustrate the applications of the obtained results.

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

This article aims to establish the existence of traveling waves for a predator-prey system with Beddington-DeAngelis functional response, reproductive Allee effect, and time delay. We investigate the existence of solutions for a system with two special delay kernels by geometric singular perturbation theory, invariant manifold theory, and Fredholm orthogonality theory. In addition, we discuss the asymptotic behaviors of traveling waves with the aid of the asymptotic theory. For more information see https://ejde.math.txstate.edu/Volumes/2024/33/abstr.html

Existence of solutions for nonlinear elliptic variational inequalities with L 1 data

This article addresses the regularizing effect of the interaction between the coefficient of the zero-order term and the datum f in some elliptic variational inequalities. Even if the datum f only belongs to L 1 , the existence of bounded weak solutions to such problems are proven. This generalizes, to some extent, the previous results in the literature.

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

Existence of semi-nodal solutions for elliptic systems related to Gross-Pitaevskii equations

In this work we consider existence of semi-nodal solutions, i.e., solutions of the form \((u, v)\) with \(u&gt;0\) and \(v^\pm:=\max\{0,\pm v\}\not\equiv0\) for a class of elliptic systems related to the Gross-Pitaevskii equation. For more information see https://ejde.math.txstate.edu/Volumes/2024/32/abstr.html

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

Existence of positive solutions to a class of boundary value problems with derivative dependence on the half-line

Determination of the Speed and Surface Temperature of Aircraft Using the Second Approximation of the System of Moment Equations

This study focuses on solving the initial-boundary value problem for a one-dimensional non-stationary nonlinear system of moment equations in the second approximation. We prove the existence of a unique local in-time solution of the problem under consideration in the space of functions that are continuous in time and square integrable over the spatial variable x. To solve the direct and inverse problems for this system, we develop an iterative numerical method and reduce the system of moment equations to canonical form. Furthermore, we present the software implementation of the algorithms and their application for solving inverse problems related to determining the speed and surface temperature of an aircraft, as well as atmospheric parameters. We provide the results of numerical experiments conducted to validate our approach.

Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations

In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s gH-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s gH-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s gH-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform.

Solving the cohomological equation for locally Hamiltonian flows, part I - local obstructions

We study the cohomological equation Xu=f for smooth locally Hamiltonian flows on compact surfaces. The main novelty of the proposed approach is that it is used to study the regularity of the solution u when the flow has saddle loops, which has not been systematically studied before. Then we need to limit the flow to its minimal components. We show the existence and (optimal) regularity of solutions regarding the relations with the associated cohomological equations for interval exchange transformations (IETs). Our main theorems state that the regularity of solutions depends not only on the vanishing of the so-called Forni's distributions (cf. [9,11]), but also on the vanishing of families of new invariant distributions (local obstructions) reflecting the behavior of f around the saddles. Our main results provide some key ingredient for the complete solution to the regularity problem of solutions (in cohomological equations) for a.a. locally Hamiltonian flows (with or without saddle loops) to be shown in [15].The main contribution of this article is to define the aforementioned new families of invariant distributions dσ,jk, Cσ,lk and analyze their effect on the regularity of u and on the regularity of the associated cohomological equations for IETs. To prove this new phenomenon, we further develop local analysis of f near degenerate singularities inspired by tools from [14] and [17]. We develop new tools of handling functions whose higher derivatives have polynomial singularities over IETs.

A New Model Making the Transition From Oil to Solar Energy

Today, the use of renewable energy sources such as sun, wind and hot water to generate electricity is increasingly becoming a global priority. In this paper, we propose a deterministic dynamical system reporting the transition from oil to solar energy, in which we then add the randomness of this phenomenon. First, we study the positivity of the deterministic oil-solar model and show that there is a maximum solution of the deterministic model and that the system is controllable at any time T &gt; 0 from an initial point. We also prove the existence of an equilibrium state, study its nature and prove that there is no limit cycle between the quantity of oil available and the solar production capacity installed. Secondly, we formulate the stochastic solar energy model from the deterministic model, for which we proved the existence of a unique solution. Finally, using a model extracted from the model performed and examining the transition from the amount of available oil to the installed solar generation capacity, we associate a linear optimal control problem, from which we prove that the optimal control components are of bang-bang type, and we characterize them.

Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

Existence of non-trivial solutions to a class of fractional p-Laplacian equations of Schrödinger-type with a combined nonlinearity

Fractional nonlinear heat equations and characterizations of some function spaces in terms of fractional Gauss–Weierstrass semi–groups

AbstractWe present a new proof of the caloric smoothing related to the fractional Gauss–Weierstrass semi–group in Triebel-Lizorkin spaces. This property will be used to prove existence and uniqueness of mild and strong solutions of the Cauchy problem for a fractional nonlinear heat equation.

A note on a supercritical [formula omitted]-Laplacian equation with logarithmic perturbation

In this short note, we study the existence of ground state solutions for p-Laplacian elliptic problems with supercritical growth and logarithmic nonlinearity, both within the interior and on the boundary of a domain.