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In this paper the authors present three different Lyapunov-type inequalities for a higher-order Caputo fractional differential equation with identical boundary conditions marking the inaugural instance of such an approach in the existing literature. Their findings extend and complement certain prior results in the literature.

In this article we use coincidence degree theory to study the existence of a positive periodic solutions to the following bioeconomic model in fishery dynamics \begin{equation*}\label{eq1.3} \begin{cases} \frac{dn}{dt} = n \left(r(t) \left(1-\frac{n}{K}\right)-\frac{q(t)E}{n+D}\right),\\ \frac{dE}{dt} = E\left(\frac{A(t)q(t)}{\alpha(t)} \frac{n}{n+D}-\frac{q^2(t)}{\alpha(t)} \frac{n^2E}{(n+D)^2}-c(t)\right), \end{cases} \end{equation*} where the functions $r,q,A,c$ and $\alpha$ are continuous positive $T$-periodic functions. This is the model of a coastal fishery represented as a single site with $n(t)$ is the fish stock biomass, and $E(t)$ is the fishing effort. Examples are given to strengthen our results.

In this paper, we consider an integro-differential model of nerve conduction which presents the propagation of impulses in the nerve’s membranes. First, we approximate the original problem via cellular nonlinear networks (CNNs). The dynamics of the CNN model is investigated by means of local activity theory. The edge of chaos domain of the parameter set is determined in the low-dimensional case. Computer simulations show the bifurcation diagram of the model and the dynamic behavior in the edge of chaos region. Moreover, stabilizing control is applied in order to stabilize the chaotic behavior of the model under consideration to the solutions related to the original behavior of the system.

We propose necessary and sufficient conditions for the negativity of the two-point boundary value problem in the form of the Vallée-Poussin theorem about differential inequalities for the Hadamard fractional functional differential problem \[ \begin{cases} \left({}^{\rm H}{D}{^{\alpha}}x\right)(t) + (Tx)(t)=f(t),\\ x(1)=x(e)=0. \end{cases} \] Here, the operator $ T:C\rightarrow L_{\infty } $ can be an operator with deviation (of delayed or advanced type), an integral operator or various linear combinations and superpositions. For example, the operator can be of the forms $ (Tx)(t)=q(t)x(t-\tau (t)) $ , $ (Tx)(t)=\int _{1}^{e} Q(t,s)x(\theta (s)){\rm d}s $ or $ (Tx)(t)=\int _{1}^{e} x(s){\rm d}_{s}Q(t,s) $ . We obtain explicit tests of negativity of Green's function in the form of algebraic inequalities. Our paper is the first one where a general form of the operator is considered with Hadamard fractional derivatives.

A version of the Floquet theory for first order delay differential equations is proposed. Formula of solutions representation is obtained. On this basis, the stability of first order delay differential equations is studied. An analogue of the classical integral Lyapunov–Zhukovskii test of stability is proved. New, in comparison with all known, tests of the exponential stability are obtained on the basis of the Floquet theory. A possibility to achieve the exponential stability is connected with oscillation of solutions. [ABSTRACT FROM AUTHOR]

This research paper explores the existence of solution to a higher-order fractional differential equation with a general boundary condition, shedding light on novel extensions beyond existing literature. The equation, characterized by a Caputo fractional derivative exhibits nonlinearity and resonance, making it a compelling subject of study. The investigation employs coincidence degree theory, a robust tool for the examination of differential equations and the identification of solution. Notably, this paper delves into nonlinear growth patterns of function. The main results of the research are accompanied by an illustrative example to clarify the concepts discussed. [ABSTRACT FROM AUTHOR]

The model of the spread of the coronavirus pandemic in the form of a system of integro-differential equations is studied. We focus our consideration on the number of hospitalized patients, i.e., on the needs of the system regarding hospital beds that can be provided for hospitalization and the corresponding medical personnel. Traditionally, in such models, the number of places needed was defined as a certain percentage of the number of infected at the moment. This is not quite adequate, since it takes a certain period of time for the development of the disease to the stage at which hospitalization is required. This will be especially evident at the start of new waves of the epidemic, when there is a large surge in the number of infected people, but the need for hospitalization places and additional medical personnel will appear later. Taking this circumstance into account using integral terms in the model allows us to conclude in corresponding additional to existing cases that the wave of disease will attenuate after some time. In others, it will relieve unnecessary panic, because the healthcare system has a certain period to create additional hospitalization places, order medicines and mobilize the necessary medical personnel. We obtain estimates of reproduction number in the case of the model described by a system of integro-differential equations. Results on the exponential stability of this integro-differential system are obtained. It is demonstrated that the condition of the exponential stability coincides with the fact that the reproduction number of the spread of the pandemic is less than one.

Abstract In this paper we propose a method for stability studies of functional differential systems. The idea of our method is to reduce the analysis of an n-dimensional system to one for an ( n + m ) $(n+m)$ -dimensional system, where m is a natural number, to obtain stability and then to come back and make conclusions on the stability of the given n-dimensional system. As an example, a model describing testosterone by distributed inputs feedback control is considered. The aim of the regulation is to hold testosterone concentration above an appropriate level. The feedback control with integral term is proposed. We have to increase the testosterone level to the normal one. The control we proposed could destroy the stability of the model. That is why we have to choose the parameters of our distributed control, namely a dosage or intensity of assimilation of a medicine in a human body in such a form that the stability of our system is preserved. Thus the problem of regulation of testosterone level leads us to the stability analysis of the functional differential system describing a connection between the concentrations of hormones (GnRH), (LH), and testosterone (Te). Constructing the system, we discard the connections which seem nonessential. To estimate the effect of these connections is an important problem. We construct the Cauchy matrix of integro-differential system to estimate this influence.

Abstract We consider the following second order impulsive differential equation with delays: {(Lx)(t)≡x″(t)+∑j=1paj(t)x′(t−τj(t))+∑j=1pbj(t)x(t−θj(t))=f(t),t∈[0,ω],x(tk)=γkx(tk−0),x′(tk)=δkx′(tk−0),k=1,2,…,r. $$ \textstyle\begin{cases} (Lx)(t)\equiv x''(t)+\sum_{j=1}^{p} a_{j}(t) x'(t-\tau _{j}(t)) + \sum_{j=1}^{p} b_{j}(t) x(t-\theta _{j}(t)) = f(t), \quad t \in [0, \omega ], \\ x(t_{k})=\gamma _{k} x(t_{k}-0), \quad\quad x'(t_{k})=\delta _{k} x'(t_{k}-0), \quad k=1,2,\ldots,r. \end{cases} $$ In this paper we consider sufficient conditions of nonpositivity of Green’s function for impulsive differential equation with nonlocal boundary conditions.

We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained.