Existence Of Solutions Research Articles

ON THE EXISTENCE OF PERIODIC SOLUTIONS OF A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER WITH QUASI-HOMOGENEOUS NON-LINEARITY

In the paper was investigated the a priori estimate and the existence of periodic solutions of a fixed period for a system of second-order ordinary differential equations with the main quasi-homogeneous non-linearity. It is proved that an a priori estimate of periodic solutions takes place if the corresponding unperturbed system does not have non-zero bounded solutions. Under the conditions of an a priori estimate, using methods for calculating the mapping degree of vector fields, a criterion for the existence of periodic solutions under any perturbation from a given class is formulated and proven. The results obtained differ from earlier results in that the set of zeros of the main non-linear part is not taken into account.

THE CAUCHY PROBLEM FOR AN NONLINEAR WAVE EQUATION

A heat-electric (1+ 1)-dimensional model of semiconductor heating in an electric field is considered. For the corresponding Cauchy problem, the existence of a classical solution that is short-lived in time is proved, a global a priori estimate is obtained in time, and a result is obtained about the absence of even a classical solution local in time.

EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS TO MIXED-TYPE STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY THE FRACTIONAL BROWNIAN MOTIONS WITH HURST INDICES 𝐻 > 1/4

In this paper there is investigated the problem of unique solvability of the Caushy problem for the mixed type stochastic differential equation driven by the standard Brownian motion and fractional Brownian motions with Hurst indices 𝐻 > 1/4. We prove a theorem on the existence and uniqueness of strong solutions to the mentioned mixed type stochastic differential equations.

ASYMPTOTIC BEHAVIOR OF THE SOLUTION OF THE CAUCHY PROBLEM FOR A NONLINEAR EQUATION

For a nonlinear partial differential equation generalizing a damped sixth-order Boussinesq equation with double dispersion and the equation of transverse oscillations of a viscoelastic Voigt–Kelvin beam under the action of external and internal friction and whose deformation is considered taking into account the correction for the inertia of section rotation, sufficient conditions for the existence and exponential decay of a global solution of the Cauchy problem are found.

Dynamics and optimal control of a stochastic delayed human papillomavirus epidemic model with media coverage

In this work, we propose a stochastic Human Papillomavirus (HPV) epidemic model with two kinds of delays and media influences. These two time delays are the delay time caused by media receiving the disease information and the delay time of public feedback after the media coverage. In addition, media coverage not only has a negative impact on the infection rate, but it also has a positive impact on the vaccination rate of disease. We discuss the existence and uniqueness of the positive solution for the HPV epidemic model, and then put forward a positively invariant set. The sufficient conditions of the extinction and persistence for the HPV epidemic are given. For the optimal control problem of the HPV epidemic, we obtain an optimal strategy. Our numerical simulations validate the theoretical results of this paper, showing that appropriate media coverage can help control the development of the disease.

Positivity results to iterative system of higher order boundary value problems

The present research explores the existence of positive solutions for the iterative system of higher-order differential equations with integral boundary conditions that include a non-homogeneous term. To address the boundary value problem, the solution is expressed as a solution of an equivalent integral equation involving kernels. Subsequently, bounds for these kernels are determined to facilitate further analysis. The primary tool employed in this study is the Guo-Krasnosel’skii fixed-point theorem, which is utilized to establish the existence of positive solutions within a cone of a Banach space. This approach enables a rigorous exploration of the existence of at least one positive solution and provides insights into the behavior of the differential equation under the given boundary conditions.

Existence and Uniqueness Results for a Coupled System of Nonlinear Hadamard Fractional Differential Equations

In this paper, we examine the existence and uniqueness of solutions to a system of four-point non-separated Hadamard fractional differential equations. Applying Leray-Schauder’s alternative yields the existence of solutions; Banach’s contraction principle establishes the uniqueness of the solution. Lastly, we provide two examples to demonstrate our results.

Existence and Uniqueness of Solutions for Fractional Dynamic Equations with Impulse Effects

Abstract The aim of the study is to establish sufficient conditions to ensure the existence and uniqueness of solutions for nonlinear Riemann-Liouville fractional dynamic equations under impulse effects. The current state of the literature reveals a visible gap in the investigation of the existence-uniqueness aspects of such equations, and this research makes a significant contribution to filling this gap. To highlight the practical implications of our results, we present an illustrative example that exemplifies the applicability of the established conditions.

Long time dynamics of the 𝑢^{𝑝} Klein-Gordon equation on general rectangular tori

We address two interesting questions in the theory of the u p u^p Klein-Gordon equation defined on the general rectangular tori. First, we show the existence and uniqueness of solutions using an explicit combinatorial analysis under the exponential decay assumption in the Fourier space. The second question is the long time behavior of such a solution. In a weakly nonlinear setting, we can prove that the nonlinear solution is asymptotic to the linear solution.

On a periodic problem for super-linear second-order ODEs

Abstract The present paper concerns the periodic problem u ″ = p ( t ) u − q ( t , u ) u + f ( t ) ; u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) , $$\begin{array}{} \displaystyle u''=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\, u'(0)=u'(\omega), \end{array}$$ where p, f : [0, ω] → ℝ are Lebesgue integrable functions and q : [0, ω] × ℝ → ℝ is a Carathéodory function. We assume that the anti-maximum principle holds for the corresponding linear problem and provide sufficient conditions guaranteeing the existence and uniqueness of a positive solution to the given non-linear problem. The general results obtained are applied to the non-autonomous Duffing type equation with a super-linear power non-linearity.

Periodic Solutions for Conformable Non-autonomous Non-instantaneous Impulsive Differential Equations

Abstract This paper studies a new type of conformable non-autonomous non-instantaneous impulsive differential equations. We present the solution by a new kinds of conformable Cauchy matrix. Also, we present its some properties. Next, we respectively discuss about the existence and uniqueness of 𝓒-periodic solutions of linear homogeneous and nonhomogeneous problems. Further, we study the nonlinear problem via fixed point theorem. Examples are also given to verify theory results.

Optimal controls for multi-term fractional stochastic integro-differential equations with impulses and infinite delay

This paper aims to investigate a new class of functional fractional stochastic impulsive integro-differential equations with infinite delay in Hilbert space with two Riemann–Liouville fractional derivatives. To demonstrate the existence of mild solutions for the considered systems, we use the fixed point approach, stochastic analysis, fractional calculus and ( α , β , λ ) resolvent family. Moreover, by utilising the Marzur lemma, we derived the optimal control results for the proposed problem. Finally, we provide an example to illustrate the obtained results.

Analysis and numerical treatment of a thermo-electro-visoplastic frictional contact problem with adhesion and internal states variable

We consider a frictional contact problem between a thermo-electro-viscoplastic body and a conductive foundation. The contact is frictional; it is modeled with normal compliance, and the adhesion of the contact surfaces is taken into account. The material has internal state variables. We derive a variational formulation for the model and prove the existence of a unique weak solution to the problem. The proof is based on results for elliptic variational inequalities, differential equations, and fixed-point theory. Finally, we present a finite element algorithm to approximate solutions to our problem and provide an error estimate for these approximations.

Existence and uniqueness of solution for a non-local singular boundary value problem

The main goal of this work is the examination of a mixed singular problem, which is characterized by a singularity of order n at the origin. This problem incorporates a Dirichlet boundary condition alongside an integral with variable bounds. We start by outlining the functional framework for analyzing the posed problem. Initially, we establish a bilateral a priori estimate using energy inequalities. To demonstrate the existence of a solution, we illustrate the density of the image of the operator associated with the problem, relying on regularization operators for our analysis. This approach seeks to demonstrate the existence of a solution to the mixed singular problem, ensuring that the solution adheres to the Dirichlet boundary conditions and the integral with variable bounds. Central to this process are the regularization operators and energy inequalities, which help manage the singularities and establish the existence of a well-defined solution.

Numerical solution of fractional integro-differential equations using ultraspherical polynomials

In this work we will discuss the fractional integro-differential (FIDE) equations of the Fredholm and Volterra type, where we begin by presenting some special concepts about the definition of the fractional integral and the fractional derivative of Riemann-Liouville and Caputo, while addressing the Gegenbauer (Ultraspherical) polynomials and its special differential equation, mentioning the primary polynomials, in addition to presenting some theories such as the Krasnoselskii fixed point theorem, which guarantees the existence and uniqueness of solutions to fractional integro-differential equations (FIDE). To find the approximate solution of fractional integro-differential equations, we relied on orthogonal polynomials of the ultraspherical type (Gegenbauer), which have a great advantage in solving integral equations and integro-differential equations. These polynomials, with the application of fractional Caputo derivatives with the series method, in addition to fixing the values and changing the degree of the polynomials used with fixing the values of , lead to the equation becoming a linear algebraic system of equations. By solving this linear system, the approximate solution is obtained in the form of a Gegenbauer (ultraspherical) series. The change in the degree of the polynomials used will inevitably show the effectiveness and speed of the method through the examples that will be presented, where the approximate solutions obtained are compared with the exact solutions.

Ground state solutions and multiple positive solutions for nonhomogeneous Kirchhoff equation with Berestycki-Lions type conditions

Abstract This article is concerned with the following Kirchhoff equation: − a + b ∫ R 3 ∣ ∇ u ∣ 2 d x Δ u = g ( u ) + h ( x ) in R 3 , -\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}{| \nabla u| }^{2}{\rm{d}}x\right)\Delta u=g\left(u)+h\left(x)\hspace{1em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{3}, where a a and b b are positive constants and h ≠ 0 h\ne 0 . Under the Berestycki-Lions type conditions on g g , we prove that the equation has at least two positive solutions by using variational methods. Furthermore, we obtain the existence of ground state solutions.

OPTIMIZATION APPROACH FOR SOLVING THE CONTINUATION PROBLEM IN FREQUENCY DOMAIN

The paper presents analytical expressions for solving the problem of continuation of the electromagnetic field in the frequency domain for a horizontally layered medium. The numerical solution algorithm uses layerwise recalculation of the required quantities, for which analytical expressions are presented in a form that allows calculations and layerwise recalculation without the accumulation of rounding errors. The solution to the continuation problem is obtained as a solution to the cost functional minimization problem. The strong convexity of the functional is proven, which implies the existence of a unique solution to the problem posed. Before solving the continuation problem, it is proposed to apply the Butterworth filter to smooth the radargrams and the Hilbert transform technique to clarify the coordinates of the discontinuity points of the medium.

WELL-POSEDNESS AND CONVERGENCE ANALYSIS OF THE MULTI-LEVEL MONTE CARLO METHOD FOR A SYSTEM OF PDEs WITH RANDOM COEFFICIENTS MODELING DRUG TRANSPORT IN TUMORS

In this paper, we consider a mathematical model of drug transport in tumors given by a system of PDEs with random coefficients and initial data. The existence of a strong solution to this model in any dimension is proved under the assumption that the random coefficients are uniformly bounded. Along with the finite difference method (FD), we applied a multilevel Monte Carlo method to simulate these random PDEs. We also derived the overall convergence rate and estimated the total computation cost. Finally, some numerical results are presented to confirm our theoretical results. Additionally, we provide a comparison between the stochastic and deterministic approaches for solving the drug transport equation, demonstrating the efficiency of the method we adopted.

On modified Mittag–Leffler coupled hybrid fractional system constrained by Dhage hybrid fixed point in Banach algebra

This research investigates the dynamics of nonlinear coupled hybrid systems using a modified Mittag–Leffler fractional derivative. The primary objective is to establish criteria for the existence and uniqueness of solutions through the implementation of Dhage’s hybrid fixed-point theorem. The study further analyzes the stability of the proposed model. To demonstrate the practical application of this framework, we utilize a modified Mittag–Leffler operator to model the transmission of the Ebola virus, known for its complex and diverse dynamics. The analysis is conducted using a combination of theoretical and numerical methods, including transforming the system of equations into an equivalent integral form, applying the fixed-point theorem, and developing a numerical scheme based on Lagrange’s interpolation for simulating the Ebola virus model. This study aims to enhance our understanding of Ebola virus dynamics and provide valuable insights for developing effective control strategies.

Theoretical advances in the smagorinsky model for turbulence simulations

This article proposes significant theoretical enhancements to the Smagorinsky model for turbulence simulations, presenting new theorems with complete mathematical proofs to increase the precision and efficiency of these simulations. Acknowledging the limitations of the original model, this work addresses these challenges through a rigorous analysis in Sobolev spaces and the filtered Navier-Stokes equations, which are essential for subgrid-scale simulation development. The study begins by introducing conditions of existence and uniqueness for weak solutions in the modified model, based on methods such as the Galerkin technique, which facilitates the analysis of behavior and stability under small initial perturbations. The application of these techniques enables a detailed verification of the model's accuracy in different turbulent flow scenarios. To validate the theoretical advancements, computational simulations are implemented to compare the performance of the original and modified Smagorinsky models. The results of these simulations demonstrate a significant improvement in both accuracy and computational efficiency, with the enhanced model achieving greater precision with less processing demand. These theoretical and numerical advancements in the Smagorinsky model open new perspectives for the use of turbulence simulations in various fields of engineering and applied sciences, allowing for more detailed analyses and more reliable predictions of complex turbulent phenomena. Thus, this work contributes to advancing the state of the art in turbulence modeling, providing a solid foundation for future research and practical evelopments in the field.