Existence Results Research Articles

Periodic fractional Ambrosetti–Prodi for one-dimensional problem with drift

We prove Ambrosetti–Prodi type results for periodic solutions of some one-dimensional nonlinear problems that can have drift term whose principal operator is the fractional Laplacian of order s∈(0,1). We establish conditions for the existence and nonexistence of solutions of those problems. The proofs of the existence results are based on the sub-supersolution method combined with topological degree type arguments. We also obtain a priori bounds in order to get multiplicity results. We also prove that the solutions are C1,α under some regularity assumptions in the nonlinearities, that is, the solutions of the mentioned equations are classical. We finish the work obtaining Ambrosetti-Prodi-type results for a problem with singular nonlinearities.

On the existence of solutions to generalized Navier–Stokes–Fourier system with dissipative heating

AbstractWe consider a flow of non-Newtonian incompressible heat conducting fluids with dissipative heating. Such system can be obtained by scaling the classical Navier–Stokes–Fourier problem. As one possible singular limit may be obtained the so-called Oberbeck–Boussinesq system. However, this model is not suitable for studying the systems with high temperature gradient. These systems are described in much better way by completing the Oberbeck–Boussinesq system by an additional dissipative heating. The satisfactory existence result for such system was however not available. In this paper we show the large-data and the long-time existence of dissipative and suitable weak solution. This is the starting point for further analysis of the stability properties of such problems.

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

Existence Result and Exponential Stabilization of a Moving String with Time Dependent Delay in the Boundary Feedback

Invariant Equilibrium in Discontinuous Bayesian Games

We provide sufficient conditions on the primitives of a class of discontinuous Bayesian games such that all games in the class share equilibria. If a Bayesian game in the class also satisfies a weak efficiency condition, then we show its normal form is better-reply secure. The invariance property then provides an existence result for all Bayesian games in the class. Results are shown for both pure strategy and behavioral strategy equilibrium. We illustrate the application of the results with an example of a class of contests with bid caps.

A quasistatic electro-elastic contact problem with long memory and slip dependent coefficient of friction

In this paper we consider a mathematical model describing a quasistatic frictional contact problem between a deformable body and an obstacle, say a foundation. We assume that the behavior of the material is described by a linear electro-elastic constitutive law with long memory. The contact is modeled with a version of Coulomb's law of dry friction in which the normal stress is prescribed on the contact surface. Moreover, we consider a slip dependent coefficient of friction. We derive a variational formulation for the model, in the form of a coupled system for the displacements and the electric potential. Under a less assumption on the coefficient of friction, we prove the existence result of weak solutions of the model. We can show the uniqueness of solution by adding another condition (3.10). The proofs are based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point theorem.

On Implicit Atangana–Baleanu–Caputo Fractional Integro-Differential Equations with Delay and Impulses

In this paper, we study the existence and uniqueness of solutions for impulsive Atangana-Baleanu-Caputo ABC fractional integro-differential equations with boundary conditions. Schaefer’s fixed point theorem and Banach contraction principle are used to prove the existence and uniqueness results. An example is presented to illustrate the results.

Existence of Solution for a Katugampola Fractional Differential Equation Using Coincidence Degree Theory

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance -(Da+α,ρx)(t)=f(t,x(t),Da+α-1,ρx(t)),t∈(a,b),x(a)=0,x(b)=∫abx(t)dA(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -(D^{\\alpha ,\\rho }_{a+}x)(t)= & {} f(t,x(t),D^{\\alpha -1, \\rho }_{a+}x(t)), \\ \\ t\\in (a,b), \\\\ x(a)= & {} 0, \\ \\ x(b)=\\int _{a}^{b} x(t){\ ext {d}}A(t), \\end{aligned}$$\\end{document}where 1<α≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<\\alpha \\le 2$$\\end{document}, and Da+α,ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D^{\\alpha ,\\rho }_{a+}$$\\end{document} is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫abx(t)dA(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\int _{a}^{b} x(t){\ ext {d}}A(t)$$\\end{document} denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results.

A two-dimensional elastic contact problem with unilateral constraints

We consider a mathematical model which describes the equilibrium of two elastic membranes fixed on their boundary and attached to an adhesive body, say a glue. The variational formulation of the model is in a form of an elliptic quasivariational inequality for the displacement field. We prove the unique weak solvability of the model, and then we state and prove a convergence result, for which we provide the corresponding mechanical interpretation. Next, we consider two associated optimization problems for which we provide existence results. Finally, we the present numerical simulation which validates our convergence result. We end this paper with some concluding remarks and an Appendix, in which we present the preliminary material needed in this paper.

Modeling the dynamics of the Hepatitis B virus via a variable-order discrete system

We investigate the dynamics of the hepatitis B virus by integrating variable-order calculus and discrete analysis. Specifically, we utilize the Caputo variable-order difference operator in this study. To establish the existence and uniqueness results of the model, we employ a fixed-point technique. Furthermore, we prove that the model exhibits bounded and positive solutions. Additionally, we explore the local stability of the proposed model by determining the basic reproduction number. Finally, we present several numerical simulations to illustrate the richness of our results.

On the solutions of a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions

We consider a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions and we obtain an existence result for this problem in the case when the set-valued maps have nonconvex values.

Constructible Sandwich Cut

In mathematical measure theory, the “Ham-Sandwich” theorem states that any n objects in an n-dimensional Euclidean space can be simultaneously divided in half with a single cut by an (n-1)-dimensional hyperplane. While it guarantees its existence, the theorem does not provide a way of finding this halving hyperplane, as it is only an existence result. In this paper, we look at the problem in dimension 2, more in the style of Euclid and the antique Greeks, that is from a constructible point of view, with straight edge and compass. For two arbitrary regions in the plane, there is certainly no hope for constructing the halving line by straight edge and compass. At the opposite end, for two circles the problem is very easy, as the halving line is simply the line which passes through the centers of both circles. It turns out that for the case of two equilateral triangles the problem is already interesting and challenging from a constructible point of view, and this is the main goal of our paper. We will investigate the “deltoid” of a triangle, that is, three hyperbolas which form an “envelope” of all halving lines of a given triangle, and how to construct halving lines with certain given properties.

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions

Global existence and stability of the wave equation with boundary variable damping

AbstractIn this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities.

Exact solitary wave propagations for the stochastic Burgers’ equation under the influence of white noise and its comparison with computational scheme

In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

On a biharmonic coupled system with non-standard nonlinearity: Existence, blow up and numerics

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.

Existence and Controllability Results for Integrodifferential Equations with State-Dependent Nonlocal Conditions via Fractional Power Operators in Fréchet Spaces

Existence and Controllability Results for Integrodifferential Equations with State-Dependent Nonlocal Conditions via Fractional Power Operators in Fréchet Spaces

Existence Results and Finite-Time Stability of a Fractional (p,q)-Integro-Difference System

In this article, we mainly generalize the results in the literature for a fractional q-difference equation. Our study considers a more comprehensive type of fractional p,q-difference system of nonlinear equations. By the Banach contraction mapping principle, we obtain a unique solution. By Krasnoselskii’s fixed-point theorem, we prove the existence of solutions. In addition, finite stability has been established too. The main results in the literature have been proven to be a particular corollary of our work.

Modeling the Transmission Dynamics of Tuberculosis in Humans

Tuberculosis (TB) is a major threat to human health particularly in most of developing countries. In this article, we formulate and analyze a deterministic model for the transmission dynamics of pulmonary and extra-pulmonary tuberculosis. The next generation method is employed to find the basic reproduction number R0 which helps to determine whether TB clears or persists in the human population. Global stability of model equilibria is done through Lyapunov functions whereas the normalized forward sensitivity index method is adopted to determine parameters that drive tuberculosis. Analysis shows that both TB free and endemic equilibria exist. The TB free equilibrium is globally asymptotically stable whenever the basic reproduction number R0 &lt; 1 whereas the endemic equilibrium is globally asymptotically stable whenever R0 &gt; 1. Sensitivity analysis shows that the TB infection rate, the fraction of individuals who progress to pulmonary tuberculosis and its induced death drive TB. Numerical results indicate that when there are no interventions, susceptible humans decline significantly with time until when they are attracted to the steady state whereas latently infected, pulmonary and extra-pulmonary TB individuals increase until when they settle at the equilibrium states supporting the analytical results for existence of the endemic equilibrium. In light of these findings, we recommend treating humans infected with pulmonary TB who are carriers of the disease.

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian.