Existence Results Research Articles

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.

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 < 1 whereas the endemic equilibrium is globally asymptotically stable whenever R0 > 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.

Evolution of dispersal and the analysis of a resource flourished population model with harvesting

This study explores a spatially distributed harvesting model that signifies the outcome of the competition of two species in a heterogeneous environment. The model is controlled by reaction-diffusion equations with resource-based diffusion strategies. Two different situations are maintained by the harvesting effects: when the harvesting rates are independent in space and do not exceed the intrinsic growth rate; and when they are proportional to the time-independent intrinsic growth rate. In particular, the competition between both species differs only by their corresponding migration strategy and harvesting intensity. We have computed the main results for the global existence of solutions that represent either coexistence or competitive exclusion of two competing species depending on the harvesting levels and different imposed diffusion strategies. We also established some estimates on harvesting efforts for which coexistence is apparent. Also, some numerical results are exhibited in one and two spatial dimensions, which shed some light on the ecological implementation of the model.

On the dynamic Coulomb's frictional contact problem for thermo‐piezoelectric materials

AbstractIn this work, we study, from a variational point of view, a dynamic contact problem between a thermo‐piezoelectric body and a thermally conductive foundation. The normal compliance contact condition and Coulomb's friction law are employed to model the contact. We provide existence and uniqueness results of a weak solution to the model using adequate auxiliary problems, an abstract result on nonlinear first‐order evolution inequalities and Banach fixed point argument. Finally, the continuous dependence of the solution on the surface traction force and the surface electrical charge is studied.

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.

A Copositivity-type Existence Result for Weakly Homogeneous Variational Inequalities

A Copositivity-type Existence Result for Weakly Homogeneous Variational Inequalities

Mixed Hilfer and Caputo Fractional Riemann–Stieltjes Integro-Differential Equations with Non-Separated Boundary Conditions

In this paper, we investigate a sequential fractional boundary value problem which contains a combination of Hilfer and Caputo fractional derivative operators and non-separated boundary conditions. We establish the existence of a unique solution via Banach’s fixed point theorem, while by applying Leray–Schauder’s nonlinear alternative, we prove an existence result. Finally, examples are provided to demonstrate the results obtained.

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad &amp; x\in \Omega,\quad t&gt;0,\\ v_t=\Delta v-uv,\quad &amp; x\in \Omega,\quad t&gt;0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m&gt;\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m&gt;\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m&gt;\frac {3N}{2N+2}$ when $N\geq 2$ .

Flow of Bingham fluid through a three dimensional thin layer

The paper is devoted to the study of asymptotic behavior of the solution of three dimensional steady flow of Bingham fluid through a three dimensional thin layer with Dirichlet boundary conditions. We are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. The limit problem obtained after transforming the original problem into one posed over a fixed reference domain and the parameter representing the thickness of the layer tend to zero is studied. The lower-dimensional constitutive law and the differential equation satisfied by the limit variables in the non rigid zone are obtained. In addition, the uniqueness of limit solution has been also established. Existence and uniqueness results and a lower-dimensional constitutive law are obtained. An identical study of a two-dimensional problem yields a one dimensional model prevalent in engineering literature.

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.

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.

Critical points with prescribed energy for a class of functionals depending on a parameter: existence, multiplicity and bifurcation results

We look for critical points with prescribed energy for the family of even functionals Φμ=I1−μI2 , where I1,I2 are C 1 functionals on a Banach space X, and μ∈R . For a given c∈R and several classes of Φμ , we prove the existence of infinitely many couples (μn,c,un,c) such that Φμn,c′(±un,c)=0andΦμn,c(±un,c)=c∀n∈N. More generally, we analyse the structure of the solution set of the problem Φμ′(u)=0,Φμ(u)=c with respect to µ and c. In particular, we show that the maps c↦μn,c are continuous, which gives rise to a family of energy curves for this problem. The analysis of these curves provide us with several bifurcation and multiplicity type results, which are then applied to some elliptic problems. Our approach is based on the nonlinear generalized Rayleigh quotient method developed in Il’yasov (2017 Topol. Methods Nonlinear Anal. 49 683–714).

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

The 3-D time-dependent Navier–Stokes equations on multi-connected domains with inhomogeneous boundary conditions and spectral hyperviscosity

We consider the time-dependent 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω⊂R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset \\mathbb {R^{\ extrm{3}}}$$\\end{document} with inhomogeneous boundary data β∈H1/2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in H^{1/2}(\\Gamma )$$\\end{document} on ∂Ω=Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Omega =\\Gamma $$\\end{document}, where Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is a union of Lipschitz continuous surfaces Γ0,Γ1,⋯,Γl\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{0},\\Gamma _{1},\\dots ,\\Gamma _{l}$$\\end{document}. This assumption includes the particular case when the Γi\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma _{i}$$\\end{document} are disjoint, the stationary version of which is classically known as Leray’s problem. Existence results for Leray’s problem have either assumed flux conditions beyond the general flux condition necessitated by compatibility constraints, or required size restrictions on the data. Here we incorporate a spectral hyperviscosity term in the time-dependent case and obtain existence and foundational results, assuming only the general flux condition and without imposing size restrictions on the boundary data β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}. For any such β∈H1/2(Γ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta \\in H^{1/2}(\\Gamma )$$\\end{document} we establish global existence and uniqueness of mild solutions. Then on any interval [0, T] on which these solutions and the corresponding NSE solution share a common H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{1}$$\\end{document}-bound (as is present on local intervals of existence of strong solutions, in certain special cases, or as is commonly assumed in achieving strong convergence results in numerical studies) we show for slightly more regular β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document} that the spectrally-hyperviscous solutions converge strongly and uniformly in H1(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^{1}(\\Omega )$$\\end{document} to the NSE solution as the spectral hyperviscosity term vanishes in the limit of key parameters. To achieve this robust sense of approximation of the NSE system, an involved setup and specially-adapted semigroup techniques assume essential roles, and the exposition is new for the case β=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =0$$\\end{document} as well. Our final results adapt the NSE reformulation in Liu et al. (J Comput Phys 229(9):3428–3453, 2010) to recast our approximating system in a form potentially more adaptable to computation.

Scalar BSDEs of iterated-logarithmically sub-linear generators with integrable terminal values

We establish a general existence and uniqueness result of integrable adapted solutions to scalar backward stochastic differential equations with integrable parameters, where the generator g has an iterated-logarithmically uniform continuity in the second unknown variable z. The result improves our previous one in Fan et al. (2023), and can be applied to the (conditional) g-expectation and the star-shaped risk measure.

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations