Existence Of Solutions Research Articles

An approach of equilibrium problems based on selection theorems

In the study of the existence of solutions to equilibrium problems, the intersection theorems are useful tools of investigation. In this paper, the proofs combine selection results with intersection theorems. An important feature of our results is that they are obtained without the standard equilibrium condition.

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

A novel approach on the sequential type ψ-Hilfer pantograph fractional differential equation with boundary conditions

This article investigates sufficient conditions for the existence and uniqueness of solutions to the ψ-Hilfer sequential type pantograph fractional boundary value problem. Considering the system depends on a lower-order fractional derivative of an unknown function, the study is carried out in a special working space. Standard fixed point theorems such as the Banach contraction principle and Krasnosel’skii’s fixed point theorem are applied to prove the uniqueness and the existence of a solution, respectively. Finally, an example demonstrating our results with numerical simulations is presented.

The Global existence and blow-up of solutions for a class of Steklov parabolic problems

In this paper, we study weak solutions to the following Steklov parabolic problem: We prove theorems of existence of weak solutions, via Galerkin approximation. Moreover, we show the existence of solutions which blow up in a finite time.

Existence of solutions for elliptic systems involving the fractional p(x)-Laplacian

In this paper, we consider a class of non-homogeneous fractional (p (.), q (.))−Laplacian systems. By using variational methods and combining with the theory of Lebesgue and fractional Sobolev spaces with variable exponents, we prove the existence of solutions to the system.

Threshold for existence, non-existence and multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball

Threshold for existence, non-existence and multiplicity of positive solutions with prescribed mass for an NLS with a pure power nonlinearity in the exterior of a ball

Existence and regularity of solutions for semilinear fractional Rayleigh–Stokes equations

Existence and regularity of solutions for semilinear fractional Rayleigh–Stokes equations

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

Periodic solutions in a 2D-symmetric Hamiltonian system through reduction and averaging method

We study a type of perturbed polynomial Hamiltonian system in 1:1 resonance. The perturbation consists of a homogeneous quartic potential invariant by rotations of π / 2 radians. The existence of periodic solutions is established using reduction and averaging theories. The different types of periodic solutions, linear stability, and bifurcation curves are characterized in terms of the parameters. Finally, some choreography of bifurcations are obtained, showing in detail the evolution of the phase flow.

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.

Fixed Point Results in Complex Valued Neutrosophic b-Metric Spaces with Application

In this manuscript, we introduce the idea of complex-valued Neutrosophic b-metric spaces along with numerous significant illustrations. We provide fixed-point results for contraction maps. To support the main result, we establish the existence and uniqueness of solutions for nonlinear integral equations after the work. 

Magnetoacoustic waves in spin-1/2 dense quantum degenerate plasma: nonlinear dynamics and dissipative effects

Abstract Within the confines of a two-fluid quantum magnetohydrodynamic model, the investigation of magnetoacoustic shock and solitary waves is conducted in an electron-ion magnetoplasma that considers electrons of spin 1/2. When the plasma system is nonlinearly investigated using the reductive perturbation approach, the Korteweg de Vries-Burgers (KdVB) equation is produced. Sagdeev’s potential is created, revealing the presence of solitary solutions. However, when dissipative terms are included, intriguing physical solutions can be obtained. The KdVB equation is further investigated using the phase plane theory of a planar dynamical system to demonstrate the existence of periodic and solitary wave solutions. Predicting several classes of traveling wave solutions is advantageous due to various phase orbits, which manifest as soliton-shock waves, and oscillatory shock waves. The presence of a magnetic field, the density of electrons and ions, and the kinematic viscosity significantly alter the properties of magnetoacoustic solitary and shock waves. Additionally, electric fields have been identified. The outcomes obtained here can be applied to studying the nature of magnetoacoustic waves that are observed in compact astrophysical environments, where the influence of quantum spin phenomena remains significant, and also in controlled laboratory plasma experiments.

Ground state solutions for generalized quasilinear Schrödinger equations

In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

Qualitative analysis of solutions for a degenerate partial differential equations model of epidemic spread dynamics

Compartmental models are widely used in mathematical epidemiology to describe the dynamics of infectious diseases or in mathematical models of population genetics. In this study, we study a time-dependent Susceptible-Infectious-Susceptible (SIS) Partial Differential Equation (PDE) model that is based on a diffusion-drift approximation of a probability density from a well-known discrete-time Markov chain model. This SIS-PDE model is conservative due to the degeneracy of the diffusion term at the origin. The main results of this article are the qualitative behavior of weak solutions, the dependence of the local asymptotic property of these solutions on initial data, and the existence of Dirac delta function type solutions. Moreover, we study the long-term behavior of solutions and confirm our analysis with numerical computations.

Strong and Weak Convergence Theorems for the Split Feasibility Problem of (β,k)-Enriched Strict Pseudocontractive Mappings with an Application in Hilbert Spaces

The concept of symmetry has played a major role in Hilbert space setting owing to the structure of a complete inner product space. Subsequently, different studies pertaining to symmetry, including symmetric operators, have investigated real Hilbert spaces. In this paper, we study the solutions to multiple-set split feasibility problems for a pair of finite families of β-enriched, strictly pseudocontractive mappings in the setup of a real Hilbert space. In view of this, we constructed an iterative scheme that properly included these two mappings into the formula. Under this iterative scheme, an appropriate condition for the existence of solutions and strong and weak convergent results are presented. No sum condition is imposed on the countably finite family of the iteration parameters in obtaining our results unlike for several other results in this direction. In addition, we prove that a slight modification of our iterative scheme could be applied in studying hierarchical variational inequality problems in a real Hilbert space. Our results improve, extend and generalize several results currently existing in the literature.

A note on Kirchhoff type boundary value problem involving Riemann-Liouville fractional derivative

In this paper, we study some nonlinear Kirchhoff boundary value problems of fractional differential equations involving Riemann Liouville operator. Under appropriate assumptions on the functions in the given problem, we establish the existence of solutions using variational methods combined with the mountain pass theorem. Moreover, an illustrative example is presented to prove the validity of the main result.

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

Existence of a ground-state solution for a quasilinear Schrödinger system

In this paper, we consider the following quasilinear Schrödinger system.−Δu+u+k2Δ|u|2u=2αα+β|u|α−2u|v|β,x∈RN,−Δv+v+k2Δ|v|2v=2βα+β|u|α|v|β−2v,x∈RN,where k < 0 is a real constant, α > 1, β > 1, and α + β < 2*. We take advantage of the critical point theorem developed by Jeanjean (Proc. R. Soc. Edinburgh Sect A., 1999, 129: 787–809) and combine it with Pohožaev identity to obtain the existence of a ground-state solution, which is the non-trivial solution with the least possible energy.

Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth

Existence and Multiplicity of Solutions for a Class of Kirchhoff–Boussinesq-Type Problems with Logarithmic Growth

Modeling of implicit multi term fractional delay differential equation: Application in pollutant dispersion problem

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances.