Existence of a positive solution with concave and convex components for a system of boundary value problems

We prove the existence of at least one positive solution for a system of two nonlinear second-order differential equations with nonlocal boundary conditions. One component of the solution is a concave function, and the other one is a convex function. A recent hybrid Krasnosel’skiĭ–Schauder fixed point theorem is used to prove the existence of a positive solution. To illustrate the applicability of the obtained result, an example is considered

On stability and convergence of difference schemes for one class of parabolic equations with nonlocal condition

In this paper, we construct and analyze the finite-difference method for a two-dimensional nonlinear parabolic equation with nonlocal boundary condition. The main objective of this paper is to investigate the stability and convergence of the difference scheme in the maximum norm. We provide some approaches for estimating the error of the solution. In our approach, the assumption of the validity of the maximum principle is not required. The assumption is changed to a weaker one: the difference problem’s matrix is the M-matrix. We present numerical experiments to illustrate and supplement theoretical results

Effect of gravity on the pattern formation in aqueous suspensions of luminous Escherichia coli

This paper presents a nonlinear two-dimensional-in-space mathematical model of self-organization of aqueous bacterial suspensions. The reaction–diffusion–chemotaxis model is coupled with the incompressible Navier–Stokes equations, which are subject to a gravitational force proportional to the relative bacteria density and include a cut-off mechanism. The bacterial pattern formation of luminous Escherichia coli is modelled near the inner lateral surface of a circular microcontainer, as detected by bioluminescence imaging. The simulated plume-like patterns are analysed to determine the values of the dimensionless model parameters, the Schmidt number, Rayleigh number and oxygen cut-off threshold, that closely match the patterns observed experimentally in a luminous E. coli colony. The numerical simulation at the transient conditions was carried out using the finite difference technique

On k-fuzzy metric spaces with applications

With application point of view, Gopal et al. [D. Gopal, W. Sintunavarat, A.S. Ranadive, S. Shukla, The investigation of k-fuzzy metric spaces with the first contraction principle in such spaces, Soft Comput., 27:11081–11089, 2023] generalized the conceptions of a fuzzy metric space and introduced the definition of k-fuzzy metric space. Here a fuzzy set defined in k-fuzzy metric space is a membership function FY : X × X × (0, +∞)k -> [0; 1], that is, the fuzzy distance between two points of the set depends on more than one parameter, and then also introduced first contraction principle in this space. In this sequel, we extend the work on k-fuzzy metric spaces by generalizing Banach contraction principle by introducing various type of inequalities. Here we introduce Tirado-type k-fuzzy contraction condition and prove fixed point theorem for Tirado-type contractive mapping. We also discuss the k-fuzzy ψ-contractive mapping, where ψ ∈ Ψ, and Ψ is a class of mappings defined from ψ : [0; 1] -> [0; 1] that has certain properties, and also obtained fixed point for such class of mappings. Later, we define Ćirić-type contraction inequalities to prove fixed point results by restricting ourselves on l-natural property of the fuzzy space to ensure the existence of fixed point. Between all results, a set of supportive examples are also produced to validate the results. In application section, we discuss the solutions of Volterra-type integral equations and second-order nonlinear ordinary differential equation

New formulation of Lyapunov direct method for nonautonomous real-order systems

Lyapunov stability analysis of nonautonomous real-order systems is put forward here in the sense of Caputo in a new and different way. We introduce new theorems and inequalities that give stability of constant solutions in the domain of attraction to such systems when attached with random initial time placed on the real axis. We give some examples including an advanced nonlinear Lorenz system to illustrate the results

Codimension-two bifurcation analysis of a discrete predator–prey system with fear effect and Allee effect

In this paper, we study the dynamic behavior of a discrete predator–prey model with fear effect and Allee effect by theoretical analysis and numerical simulation. Firstly, the existence and stability of the equilibrium points of the model are proved. Secondly, the existence of codimension-2 bifurcations (1 : 2, 1 : 3, and 1 : 4 strong resonances) in the case of two parameters is verified by bifurcation theory. In order to illustrate the complexity of the dynamic behavior of the model in the two-parameter space, we simulate the bifurcation diagrams, phase diagrams, maximum Lyapunov exponent diagrams, and isoperiodic diagram, and we verify the influence of model parameters on the population size

On a novel type of generalized simulation functions with fixed point results for wide Ws-contractions

One of the most significant hypotheses in fixed point theory is the nonexpansivity condition of contractive mappings. This property is crucial as operators that do not satisfy this criterion may lack fixed points. In this paper, we propose a novel condition that, within the appropriate framework, can obviate the necessity of imposing the nonexpansivity requirement in the initial hypotheses. By employing this new condition, we illustrate how innovative results can be derived in this area. Finally, we examine the existence and uniqueness of a solution for an elastic beam equation with nonlinear boundary conditions grounded in the introduced fixed point results

Existence of solution for a fractional differential system on the chemical graph of glycerol

In this paper, we study the chemical graph for an important polyalcoholic compound with the molecular formula C3H8O3 by using 0 or 1 to label the elements of its molecular structure graph and formulating the corresponding fractional boundary value problem on each edge of the graph. Under the sense of Caputo’s fractional derivatives, the existence of solutions of the fractional boundary value problem on the glycerol graph is investigated by introducing some suitable growth conditions and combing with some fixed point theorems. A specific example is given to verify our results

On a sublinear nonlocal fractional problem

This paper deals with existence results of nonnegative solutions for a one-parameter sublinear elliptic boundary-value problem driven by the classical fractional Laplacian operator. The existence of a weak solution for any parameter λ beyond the first resonance has been proved by using a slight variation of the classical Mountain Pass result due to Ambrosetti and Rabinowitz

Bifurcation in a Leslie–Gower system with fear in predators and strong Allee effect in prey

In this paper, we consider a modified Leslie–Gower predator–prey model with Allee effect on prey and fear effect on predators. Results show complex dynamical behaviors in the model with these factors. Existence of equilibrium points and their stability of the model are first given. Then it is found that, with the Allee and fear effects, the model exhibits various and different bifurcations, such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov–Takens bifurcation. Theoretical analysis is verified through some numerical simulations