Existence Of Solutions Research Articles

Analyzing election trends incorporating memory effect through a fractional-order mathematical modeling

This study proposes a new mathematical model to analyze and predict the results of a political election. In general, we predict or analyze the results using statistical methods; however, to minimize the effort of the study, we propose a fractional-order modeling approach. This study proposes a model to analyze and predict general election result trends in India, focusing on the state of West Bengal. To incorporate memory into the model, we consider the Caputo fractional derivative. The model solution’s positivity, boundedness, existence, and uniqueness were tested analytically. Numerical simulations were carried out to investigate the impact of the parameters and evaluate the model’s performance by incorporating the implications of the previous election for realistic situations. Following this, a qualitative analysis of the performance of political parties is discussed, and a prediction of the electoral victory is obtained.

A study on variable-order delay fractional differential equations: existence, uniqueness, and numerical simulation via a predictor corrector algorithm

In this study, we adapted a predictor-corrector technique to simulate delay differential equations incorporating variable-order Caputo-type fractional derivatives. We addressed the existence and uniqueness of solutions for the studied models. Then, we presented numerical simulation of some delay differential equations with variable-order fractional derivatives to demonstrate the efficiency of the used technique. Various periodic and chaotic characteristics of the studied models are observed for some variable-orders from the performed graphical simulations. The used technique can be modified and extended to handle different classes of initial value problems which involve variable-order fractional derivatives.

Existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions

Abstract In this article, we investigate the existence and regularity of solutions for non-autonomous integrodifferential evolution equations involving nonlocal conditions. Using the theory of resolvent operators, some fixed point theorems, and an estimation technique of Kuratowski measure of noncompactness, we first establish some existence results of mild solutions for the proposed equation. Subsequently, we show by applying a newly established lemma that these solutions have regularity property under some conditions. Finally, as a sample of application, the obtained results are applied to a class of non-autonomous nonlocal partial integrodifferential equations.

The Existence and Uniqueness of Radial Solutions for Biharmonic Elliptic Equations in an Annulus

This paper concerns with the existence of radial solutions of the biharmonic elliptic equation ▵2u=f(|x|,u,|∇u|,▵u) in an annular domain Ω={x∈RN:r1<|x|<r2}(N≥2) with the boundary conditions u|∂Ω=0 and ▵u|∂Ω=0, where f:[r1,r2]×R×R+×R→R is continuous. Under certain inequality conditions on f involving the principal eigenvalue λ1 of the Laplace operator −▵ with boundary condition u|∂Ω=0, an existence result and a uniqueness result are obtained. The inequality conditions allow for f(r,ξ,ζ,η) to be a superlinear growth on ξ,ζ,η as |(ξ,ζ,η)|→∞. Our discussion is based on the Leray–Schauder fixed point theorem, spectral theory of linear operators and technique of prior estimates.

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

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

Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

Global existence of solutions for the drift–diffusion system with large initial data in [formula omitted] ([formula omitted])

In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in Ḃ∞,∞−2(Rd).

Existence of solution for a second-order differential inclusion with a deviated nonlocal condition

In this paper, we study the existence of solution for the second-order differential inclusion With the local deviated nonlocal condition. Where and φ is deviated given function. As an application the problem with local and integral condition Will be considered.

A mathematical theoretical study of Atangana-Baleanu fractional Burgers’ equations

In this paper, the Burgers’ equations using the fractional derivative of Atangana-Baleanu sense are investigated and discussed. A Laplace variational iteration approach is used to demonstrate the fractional model's mathematical solution. The solution's existence and uniqueness are examined using fixed point theory. Several numerical simulations that enhance the efficacy of the employed derivative are presented and discussed.

Application of Fixed Point Theorems to Differential Equation in b-Multiplicative Metric Spaces

Objectives: In this manuscript, some fixed point theorems have been provided in b-multiplicative metric spaces. Methods: We proved the unique fixed point theorems using the Banach contraction principle and generalized Lipschitz contractive mappings. Findings: We established the P Property and the T-Stability of Picard’s iteration in b-multiplicative metric spaces. We also provide an example to demonstrate the result. Novelty: Using our results, we obtained the existence and uniqueness of solutions for ordinary multiplicative differential equations with initial value problems. 2020 Mathematics Subject Classification: 47H10, 54H25. Keywords: Fixed Point, Generalized Lipschitz Contractive, Picard’s iteration, b-Multiplicative Metric Space (b − MMS), T -Stability, Differential equation

Positive solutions for nonhomogenous Choquard equations involving doubly critical exponents

This paper is concerned with nonhomogenous Choquard equations involving doubly critical exponents. First, we use a fixed point theorem to prove the existence of positive solutions to the above equation under certain assumptions. Then, on the positive solution of the equation, we get the regularity in a bounded domain with estimating the ‐bounds by Moser's iterative technique.

Stochastic exponential stabilization and optimal control results for a class of fractional order equations

The object of this study is to define existence, controllability and exponential stability results within a Sobolev-type fractional stochastic neutral equations of order 1<ω<2 with sectorial operator and optimal control. To do this, first, this study relies on the confluence of stochastic analysis, fractional calculus, sectorial operator, and the applicability of Sadovskii’s fixed point method. First, we highlight the existence of mild solutions to the fractional stochastic control equation and then introduce the concept of approximate controllability. Next, employing an impulsive Poisson system yields sufficient conditions for guaranteeing the exponential stability of the mild solution in the mean square moment. Further, our inquiry expands into the existence of optimal control. Finally, an example is provided to illustrate the obtained theory.

Existence of positive solutions to the biharmonic equations in $$\mathbb {R}^{N}$$

Existence of positive solutions to the biharmonic equations in $$\mathbb {R}^{N}$$

Local and global well-posedness for fractional porous medium equation in critical Fourier-Besov spaces

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rn for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 β ≤ 2, the local solution v ∈ XT := LT ∞(FBp,r (2 − 2m −β + n/p' )(Rn))∩ LTρ1(FBp,r s1(Rn))∩ LTρ2(FBp,r s2 ( Rn)) with 1 ≤ p &lt; ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FBp,r (2 − 2m −β + n/p' )(Rn) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB∞,1 (2 − 2m −β + n )(Rn) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FBp,1 (2 − 2m + n/p' )(Rn) with 1 ≤ p &lt; ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

Existence and Hyers–Ulam Stability of Stochastic Delay Systems Governed by the Rosenblatt Process

Under the effect of the Rosenblatt process, time-delay systems of nonlinear stochastic delay differential equations are considered. Utilizing the delayed matrix functions and exact solutions for these systems, the existence and Hyers–Ulam stability results are derived. First, depending on the fixed point theory, the existence and uniqueness of solutions are proven. Next, sufficient criteria for the Hyers–Ulam stability are established. Ultimately, to illustrate the importance of the results, an example is provided.

A New Reduced-Dimension Iteration Two-Grid Crank–Nicolson Finite-Element Method for Unsaturated Soil Water Flow Problem

The main objective of this paper is to reduce the dimensionality of unknown coefficient vectors of finite-element (FE) solutions in two-grid (CN) FE (TGCNFE) format for the nonlinear unsaturated soil water flow problem by using a proper orthogonal decomposition (POD) and to design a new reduced-dimension iteration TGCNFE (RDITGCNFE). For this objective, a new time semi-discrete CN (TSDCN) scheme for the nonlinear unsaturated soil water flow problem is first designed and the existence, stability, and error estimates of TSDCN solutions are demonstrated. Subsequently, a new TGCNFE format for the nonlinear unsaturated soil water flow problem is designed and the existence, unconditional stability, and error estimates of TGCNFE solutions are demonstrated. Next, a new RDITGCNFE format with the same FE basis functions as the TGCNFE format is built by the POD method and the existence, unconditional stability, and error estimates of RDITGCNFE solutions are discussed. Ultimately, the rightness of theory results and the superiority of the RDITGCNFE format are verified by two sets of numerical tests. It is worth noting that the RDITGCNFE format differs completely from all previous reduced-dimension methods, including the authors’ previous works. Therefore, the study of this paper can not only provide a new theoretical method for the dimensionality reduction of numerical models for nonlinear problems but also provide an algorithm implementation technology for the numerical simulation of practical engineering problems.

Existence of Solutions for Impulsive Neutral Semilinear Evolution Equations with Nonlocal Conditions

Existence of Solutions for Impulsive Neutral Semilinear Evolution Equations with Nonlocal Conditions

Existence of solutions to elliptic equations on compact Riemannian manifolds

The aim of this paper is to investigate the existence of weak solutions of a nonlinear elliptic problem with Dirichlet boundary value condition, in the framework of Sobolev spaces on compact Riemannian manifolds

Mild solution for (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem

Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and (k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(k,\\Psi )$\\end{document}-Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional integral and (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the (ρ,α,β,k,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,\\alpha ,\\beta ,k,r)$\\end{document}-resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the (ρ,k,Ψ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(\\rho ,k,\\Psi )$\\end{document}-proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.

Multiplicity of solutions for anisotropic Dirichlet problem with variable exponent

We establish some results on the existence of multiple nontrivial solutions for a general anisotropic elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces,combined with adequate variational methods and a variant of the Mountain Pass lemma.