Existence Of Solutions Research Articles

Dynamics Behavior of a Predatorg-Prey Diffusion Model Incorporating Hunting Cooperation and Predator-Taxis

In this paper, we consider a predator-prey diffusion model incorporating hunting cooperation and predator-taxis. Firstly, we establish the global existence of a classical solution for the model in any spatial dimension. Secondly, we analyze the stability/instability caused by predator-taxis, and we observe that predator-taxis play a key role in inducing stability changes. Specifically, if the positive equilibrium is stable for the corresponding reaction-diffusion model, the attractive predator-taxis can further stabilize the system, while the repulsive predator-taxis may lead to a change in spatial stability, if the positive equilibrium is unstable for the corresponding reaction-diffusion model, the attractive predator-taxis makes the model remain unstable, while the repulsive predator-taxis has a stabilizing effect. Finally, numerical simulations are employed to validate the obtained results.

Minimization principle for hemivariational–variational inequality driven by uniformly monotone operators with application to problems in contact mechanics

In this paper, we consider hemivariational–variational inequalities driven by uniformly monotone or d-monotone operators in Banach spaces. We establish related minimization principles leading to the existence and uniqueness of solutions to the inequality considered as well as we suggest the Ritz type numerical approximations. The theoretical results obtained are next applied to some problems inspired by models from contact mechanics.

New results concerning a singular biharmonic equations with p-Laplacian and Hardy potential

This paper deals with a singular biharmonic equations with p-Laplacian and Hardy potential. Using Mountain Pass Theorem and Fountain Theorem with Cerami condition, we obtain the existence and multiplicity of sign-changing high-energy solutions under some suitable conditions on the nonlinear term f ( x , u ) . In addition, by applying an abstract critical point theorem of Kajikiya in [A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations. J Funct Anal. 2005;225(2):352–370], a sequence of sign-changing solutions converging to zero for the biharmonic equations with sublinear nonlinearities is also obtained.

Nonstandard Nearly Exact Analysis of the FitzHugh–Nagumo Model

The FitzHugh–Nagumo model has been used empirically to model certain types of neuronal activities. It is also a non-linear dynamical system applicable to chemical kinetics, population dynamics, epidemiology and pattern formation. In the literature, many approaches have been proposed to study its dynamics. In this paper, initially, we have employed cutting-edge tools from discrete dynamics for discretization and fixed points. It has been proven that an exact discrete scheme exists for this paradigm. This project also considers the phase space and integral surfaces of these evolutionary equations. In addition, it carries out a thorough symmetry analysis of this reaction diffusion system to find equivalent systems. Moreover, steady-state solutions are obtained using ansatzes for traveling wave solutions. The existence of infinite traveling wave solutions has also been proven. Yet again, this investigation establishes the potential of symmetry methods to unravel non-linearity. Finally, singular perturbation theory has been employed to obtain analytical approximations and to study stability in different parameter regimes.

Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.

Traveling waves in a model of calcium ions influx via mechanically stimulated membrane channels

We consider the problem of existence and properties of pulse solutions to a system of equations modeling fast calcium waves in long cells. These waves have the speed up to 1000 m/s. They propagate via the inflow of calcium ions from the extracellular space through the mechanically stimulated membrane channels. The channels open due to mechano‐chemical interaction, in which stretching of the cell's membrane at a point opens the calcium channels at neighboring points due to the forces exerted by the actomyosin network. The existence of homoclinic solutions is based on the celebrated exchange lemma, which cannot be applied straightforwardly due to some specificities of the model equations.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

AbstractThe aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

Some new results on the negative polynomial Pell’s equation

We consider the negative polynomial Pell's equation $P^2(X)-D(X)Q^2(X)=-1$, where $D(X)\in \mathbb{Z}[X]$ be some fixed, monic, square-free, even degree polynomials. In this paper, we investigate the existence of polynomial solutions $P(X), Q(X)$ with integer coefficients.

Fixed Point Results for Compatible Mappings in Extended Parametric Sb-Metric Spaces

This study aims to establish common fixed point theorems for a pair of compatible self-mappings within the framework of extended parametric Sb-metric spaces. To support our assertions, we provide corollaries and examples accompanied with graphical representations. Moreover, we leverage our principal outcome to guarantee the existence of a common solution to a system of integral equations.

Nonlinear parabolic double phase variable exponent systems with applications in image noise removal

In this paper, a novel parabolic system involving nonlinear and nonhomogeneous partial differential operator with variable growth structure is introduced for investigating the image denoising and restoration. More precisely, our model is based on regularizing the classical models involving variable exponent operators by considering a nonlinear operator with double phase flux. We begin by investigating theoretically the solvability to the parabolic system under consideration. Under the setting of Musielak-Orlicz spaces, we build a suitable functional framework to study the proposed system. Therefore, we develop the Faedo-Galerkin approach to demonstrate the existence and uniqueness of a weak solution to our model. To illustrate our theoretical results in the context of image noise removal, we present various numerical implementations on some grayscale images. To enrich these simulations, we test the robustness efficiency of the proposed model in the so-called Magnetic Resonance Images (MRI). The obtained numerical results claim that our model is more efficient and robust against noise, in comparison (visually and quantitatively) to some existing state-of-the-art methods.

Existence and multiplicity of solutions for a Steklov eigenvalue problem involving the p(x)-Laplacian-like operator

Using the variational method, we prove theexistence and multiplicity of solutions for a Steklov problem involving the $p(x)$-Laplacian-like operator, originated from a capillary phenomena. Especially, an existence criterion for infinite many pairs of solutions for the problem is obtained.

Stochastic evolution equations with Wick-analytic nonlinearities

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher–KPP equations, stochastic Allen–Cahn, stochastic Newell–Whitehead–Segel, and stochastic Fujita–Gelfand equations. By implementing the theory of C 0 − semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove the existence and uniqueness of solutions for this class of stochastic partial differential equations.

Semilinear elliptic problems via the nonlinear Rayleigh quotient with two nonlocal nonlinearities

It is establish existence and multiplicity of solutions for nonlocal elliptic problems where the nonlinearity is driven by two convolutions terms. More specifically, we shall consider the following Choquard type problem: { − Δ u + V ( x ) u = μ ( I α 1 ∗ | u | q ) | u | q − 2 u − λ ( I α 2 ∗ | u | p ) | u | p − 2 u in R N , u ∈ H 1 ( R N ) , where p > q , λ , μ > 0 , α 1 ≤ α 2 ; α 1 , α 2 ∈ ( 0 , N ) , N ≥ 3 ; p ∈ ( 2 α 2 , 2 α 2 ∗ ) ; q ∈ ( 2 α 1 , 2 α 1 ∗ ) , 2 α j = ( N + α j ) / N and 2 α j ∗ = ( N + α j ) / ( N − 2 ) , j = 1 , 2 . Here we employ some variational arguments together with the Nehari method and the nonlinear Rayleigh quotient. The main feature in the present work is to find a sharp μ n > 0 and λ ∗ , λ ∗ > 0 such that our main problem admits at least two solutions for each μ > μ n where λ ∈ ( 0 , min ( λ ∗ , λ ∗ ) ) . The main difficulty here is to prove that the infimum associated to the energy functional restricted to the Nehari set is a weak solution for our main problem. This phenomenon occurs since the fibering maps for the associated energy functional have inflection points. Furthermore, we prove a nonexistence result for our main problem for each μ < μ n and λ > 0 .

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.

Discussion on optimal feedback control for stochastic fractional differential system by hemivariational inequalities

This study focuses on optimal feedback control for stochastic Caputo fractional evolution systems in separable Hilbert spaces. Initially, we deal with the existence of a mild solution, establishing it through the application of the generalised Clarke's subdifferential and a fixed point theorem for multivalued maps. Then, we utilise the Cesari property and the Filippov theorem to demonstrate the existence of feasible pairs. Additionally, we provide pairs of suitable feedback controls under specific conditions. Finally, we present a simple illustration to support our theoretical findings.

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.

Global existence and blowup of solutions for nonlinear Love equation

In this paper, we are concerned with the nonlinear Love equation, which describes some sort of “propagation process” in physics. Firstly, we establish the local existence and uniqueness of the solution by using the contraction mapping principle and the regularity theory for the elliptic problem. Secondly, we construct a family of potential wells to obtain a threshold for the existence of global solutions and blowup in a finite time of the solution in both cases with subcritical and critical initial energy. Additionally, we provide the decay rate of the global solution and estimate the lifespan of a blow‐up solution. Moreover, at the supercritical initial energy level, we provide a sufficient condition for initial data leading to blow‐up results. This paper, along with the previous work of A. N. Carvalho, J. Cholewa, Local well‐posedness, asymptotic behavior, and asymptotic bootstrapping for a class of semilinear evolution equations of the second order in time, Trans. Amer. Math. Soc. 361 (5) (2009) 2567‐2586, provides us with a comprehensive and systematic study of the dynamic behavior of the solution to the nonlinear Love equation.

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

Optimal control results for second‐order semilinear integro‐differential systems via resolvent operators

AbstractIn the framework of a second‐order semilinear integro‐differential control system in Hilbert spaces, the paper provides sufficient conditions for proving the existence of optimal control. The Banach fixed point theorem is used to investigate the existence and uniqueness of mild solutions for the proposed problem. Additionally, it is shown that, under specific assumptions, there exists at least one optimal control pair for the Lagrange's problem as presented in the article. An example for validation is included in the paper to further support the theoretical findings.