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Using the coincidence degree theory of Mawhin and constructing appropriate operators, we investigate the existence of solutions to Hadamard fractional differential equations (FRDEs) at resonance ... where 1 < γ < 2, f: [1, e]×R² → R satisfies Carathéodory conditions, ∫e1 u(t)dA(t) is the Riemann-Stieltjes integration, and (HDγu) is the Hadamard fractional derivation of u of order γ. An example is included to illustrate our result. [ABSTRACT FROM AUTHOR]

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 ... Here, the operator T:C→L∞ 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-τ(t)), (Tx)(t)=∫e1Q(t,s)x(θ(s))ds or (Tx)(t)=∫e1x(s)dsQ(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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of at least one solution to the following higher order Riemann–Liouville fractional differential equation with Riemann–Stieltjes integral boundary condition at resonance: −(D0+αx)(t)=f(t,x(t),D0+α−1x(t)),n−1<α≤n,t∈[0,1],x(0)=x′(0)=...=x(n−2)(0)=0,x(k)(1)=∫01x(k)(t)dA(t),$$ {\displaystyle \begin{array}{cc}\hfill -\left({D}_{0+}^{\alpha }x\right)(t)=& f\left(t,x(t),{D}_{0+}^{\alpha -1}x(t)\right),n-1<\alpha \le n,t\in \left[0,1\right],\hfill \\ {}\hfill x(0)=& {x}^{\prime }(0)=\dots ={x}^{\left(n-2\right)}(0)=0,{x}^{(k)}(1)=\int_0^1{x}^{(k)}(t) dA(t),\hfill \end{array}} $$by using Mawhin's coincidence degree theory. Here, D0+α$$ {D}_{0+}^{\alpha } $$ is the standard Riemann–Liouville fractional derivative of order α,f:[0,1]×ℝ2→ℝ$$ \alpha, f:\left[0,1\right]\times {\mathbb{R}}^2\to \mathbb{R} $$, and ∫01x(k)(t)dA(t)$$ {\int}_0^1{x}^{(k)}(t) dA(t) $$ is the Riemann–Stieltjes integral of x(k)$$ {x}^{(k)} $$ with respect to A$$ A $$. Our choice of k$$ k $$ in the boundary condition can be any integer between 0 and n−1$$ n-1 $$, which supplements many boundary conditions assumed in the literature. Several examples are given to strengthen our result. [ABSTRACT FROM AUTHOR]

In this paper, the functional differential equation (D C a + α x) (t) + ∑ i = 0 m (T i x (i)) (t) = f (t) , t ∈ [ a , b ] , with Caputo fractional derivative D C a + α is studied. The operators Ti act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green's functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green's functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (D C a + α ν) (t) + ∑ i = 0 m (T i v (i)) (t) < 0 , t ∈ [ a , b ] . Choosing the function v, we obtain explicit sufficient tests of sign-constancy of Green's functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for "small" deviations. [ABSTRACT FROM AUTHOR]

We propose explicit tests of unique solvability of two-point and focal boundary value problems for fractional functional differential equations with Riemann-Liouville derivative. [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 FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

An analog of the classical Vallée-Poussin theorem about differential inequality in the theory of ordinary differential equations is developed for fractional functional differential equations. The main results are obtained in a form of a theorem about several equivalent assertions. Among them solvability of two-point boundary value problems with fractional functional differential equation, negativity of Green's function, and its derivatives and existence of a function v(t) satisfying a corresponding differential inequality. Thus the Vallée-Poussin theorem presents one of the possible "entrances" to assertions on nonoscillating properties and assertions about the negativity of Green's functions and their derivatives for various two-point problems. Choosing the function v(t) in the condition, we obtain explicit tests of sign-constancy of Green's functions and their derivatives. It can be stressed that a choice of a corresponding function in the Vallée-Poussin theorem leads to explicit criteria in the form of algebraic inequalities, which, as we demonstrate with examples, cannot be improved. Replacing strict inequalities with non-strict ones will already lead to incorrect statements. In some cases, the well-known results obtained in the form of the Lyapunov inequalities for fractional differential equations can be improved based on ours. Another development, we propose, is a concept to consider fractional functional differential equations. The basis of this concept is to reduce boundary value problems for fractional functional differential equations to operator equations in the space of essentially bounded functions. Fractional functional differential equations can appear in various applications and in the process of construction of equations for a corresponding component of a solution-vector of a system of fractional equations. [ABSTRACT FROM AUTHOR]

For the following scalar functional–differential equation x ̈ (t) + ∫ t 0 t G (t , s) x ̇ (s) d s + ∫ t 0 t H (t , s) x (s) d s = f (t) , t ≥ t 0 , we obtain sufficient conditions for boundedness of all solutions and uniform exponential stability for the homogeneous equation. As examples, equations with singular kernels and equations with bounded delays x ̈ (t) + ∫ g (t) t G (t , s) x ̇ (s) d s + ∫ h (t) t H (t , s) x (s) d s = f (t) , t ≥ t 0 , are considered. [ABSTRACT FROM AUTHOR]

There are almost no results on the exponential stability of differential equations with unbounded memory in mathematical literature. This article aimes to partially fill this gap. We propose a new approach to the study of stability of integro-differential equations with unbounded memory of the following forms x ‴ (t) + ∑ i = 1 m ∫ t − τ i (t) t b i (t) e − α i (t − s) x (s) d s = 0 , x ‴ (t) + ∑ i = 1 m ∫ 0 t − τ i (t) b i (t) e − α i (t − s) x (s) d s = 0 , $$\begin{array}{} \begin{split} \displaystyle x'''(t)+\sum_{i=1}^{m}\int\limits_{t-\tau_{i}(t)}^{t}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &=0, \\ x'''(t)+\sum_{i=1}^{m}\int\limits_{0}^{t-\tau _{i}(t)}b_{i}(t)\text{e}^{-\alpha _{i}(t-s) }x(s)\text{d} s &= 0, \end{split} \end{array}$$ with measurable essentially bounded bi(t) and τi(t), i = 1, ..., m. We demonstrate that, under certain conditions on the coefficients, integro-differential equations of these forms are exponentially stable if the delays τi(t), i = 1, ..., m, are small enough. This opens new possibilities for stabilization by distributed input control. According to common belief this sort of stabilization requires first and second derivatives of x. Results obtained in this paper prove that this is not the case. [ABSTRACT FROM AUTHOR]

Abstract There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained. [ABSTRACT FROM AUTHOR]

In 1950, Wazewski obtained the following necessary and sufficient condition for positivity of linear systems of ordinary differential equations: the matrix of coefficients is Metzler (i.e. off-diagonal elements in the matrix of coefficients are nonnegative). No results on the positivity of solutions to delay systems in the case where the matrix is non-Metzler were expected to be obtained. It was demonstrated that for delay systems the Wazewski condition is not a necessary one. We prove results on positivity of solutions for non-Metzler systems of delay differential equations. New explicit tests for exponential stability are obtained as applications of results on positivity for non-Metzler systems. Examples demonstrate possible applications of our theorems to stabilization. For instance, in view of our results, the implicit requirement on dominance of the main diagonal can be skipped. Our approach is based on nonoscillation of solutions and positivity of the Cauchy functions of scalar diagonal delay differential equations. [ABSTRACT FROM AUTHOR]

There exists a well-developed stability theory for many classes of functional-differential equations and only a few results on their instability. One of the aims of this paper is to fill this gap. Explicit tests for instability of linear delay differential equations of the second order with damping terms are obtained. [ABSTRACT FROM AUTHOR]

For a delay differential equation... a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation. [ABSTRACT FROM AUTHOR]

Efficient conditions guaranteeing the solvability of multi-point boundary value problems for linear functional-differential equations are established in this paper. The results are proved using the theorems on functional-differential inequalities. [ABSTRACT FROM AUTHOR]

We consider the following second order differential equation with delay: In this paper we use focal problems to analyze the sign constancy of Green's functions. [ABSTRACT FROM AUTHOR]

In this paper a method for studying stability of the equation $x^{\prime \prime }(t)+\sum_{i=1}^{m}a_{i}(t)x(t- \tau_{i}(t))=0$ not including explicitly the first derivative is proposed. We demonstrate that although the corresponding ordinary differential equation $x^{\prime \prime}(t)+\sum_{i=1}^{m}a_{i}(t)x(t)=0$ is not exponentially stable, the delay equation can be exponentially stable. [ABSTRACT FROM AUTHOR]

In this paper we propose new explicit tests for positivity and exponential stability of systems with large time-varying delays. Our approach is based on nonoscillation of solutions of the corresponding diagonal scalar delay differential equations. Numerical examples illustrate the efficiency of the results. [ABSTRACT FROM AUTHOR]

For the delay differential equation x ¨ ( t ) + a ( t ) x ( h ( t ) ) - b ( t ) x ( g ( t ) ) = 0 , g ( t ) ⩽ t , h ( t ) ⩽ t , without damping term, explicit exponential stability conditions are obtained. [ABSTRACT FROM AUTHOR]

Delays, arising in nonoscillatory and stable ordinary differential equations, can induce oscillation and instability of their solutions. That is why the traditional direction in the study of nonoscillation and stability of delay equations is to establish a smallness of delay, allowing delay differential equations to preserve these convenient properties of ordinary differential equations with the same coefficients. In this paper, we find cases in which delays, arising in oscillatory and asymptotically unstable ordinary differential equations, induce nonoscillation and stability of delay equations. We demonstrate that, although the ordinary differential equation x''(t) + c(t)x(t) = 0 can be oscillating and asymptotically unstable, the delay equation x''(t) + a(t)x(t - h(t)) - b(t)x(t - g(t)) = 0, where c(t) = a(t) - b(t), can be nonoscillating and exponentially stable. Results on nonoscillation and exponential stability of delay differential equations are obtained. On the basis of these results on nonoscillation and stability, the new possibilities of non-invasive (non-evasive) control, which allow us to stabilize a motion of single mass point, are proposed. Stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper refute this delusion. [ABSTRACT FROM AUTHOR]

We consider the following second order differential equation with delay {(Lx)(t) ≡ x″(t) + ∑j=1pj=1 bj (t)x(t - ϑj (t)) = f(t); t ∈ [0; ω], x(tj) = γj x(tj - 0), x‘( tj) = δ jx′(t‘ - 0), j = 1, 2, . . ., r. In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality ∑i=1p bi(t) (1/4 + r) 2/ω² is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case 0 < γi ≤ 1, 0 < δi ≤ 1 for i = 1, . . ., p. [ABSTRACT FROM AUTHOR]

The classical Wazewski theorem established that nonpositivity of all nondiagonal elements pij (i ≠ j, i, j ≠ 1,..., n) is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equations xi'(t) + ∑j=1n Pij (t) xj (t) = fi (t), i = 1,..., n. Results on nonnegativity of the Cauchy matrix for system of delay differential equations xi'(t) + ∑j=1n Pij (t) xj (t) xj (hij(t)) = fi(t), i = 1,..., n, which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficients pij is not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained. [ABSTRACT FROM AUTHOR]

The following differential equation u(n)(t) + p(t)|u(σ(t))|µ(t) sign u(σ(t)) = 0 is considered. Here p ∈ Lloc(R+;R+), µ ∈ C(R+;(0, + ∞)), σ ∈ C(R+;R+), σ(t) ≤ t, and limt→+∞σ(t) = +∞. We say that the equation is almost linear if the condition limt→+∞µ(t) = 1 is fulfilled, while if lim supt→+∞µl(t) ≠ 1 or lim inft→+∞µ(t) ≠ 1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying Property A for delay Emden-Fowler equations are obtained. [ABSTRACT FROM AUTHOR]

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 ∫dn/dt = n(r(t) - 1 - n/K) - q(t)E/n+D), dE/dt = E (A(t)q(t)/α(t) n/n+D - q²(t)/α(t) n²E/(n+D)² - c(t)), where the functions r, q, A, c and α 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. Keywords: periodic solution, coincidence degree theory, existence of solutions. [ABSTRACT FROM AUTHOR]

Theorems on the unique solvability and nonnegativity of solutions to the characteristic initial value problem u(1,1)(t, x) = l0(u)(t, x) + l1(u(1,0)(t, x) + l2(u(0,1)(t, x) + q(t, x), u(t, c) = α(t) for t ∈ [a, b], u(a, x) = β(x) for x ∈ [c, d] given on the rectangle [a, b] x [c, d] are established, where the linear operators l0, l1, l2 map suitable function spaces into the space of essentially bounded functions. General results are applied to the hyperbolic equations with essentially bounded coefficients and argument deviations. [ABSTRACT FROM AUTHOR]

We obtain the maximum principles for the first-order neutral functional differential equation (Mx)(t) = x'(t) - (Sx')(t) - (Ax)(t) + (Bx)(t) - f(t), t ∈ [0, ω], where A : C[0,ω] → L∞ [0,ω], B : C[0,ω] → L∞[0,ω], and S : L∞[0,ω] → L∞[0,ω] are linear continuous operators, A and B are positive operators, C[0,ω] is the space of continuous functions, and L∞[0,ω] is the space of essentially bounded functions defined on [0,ω]. New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles. [ABSTRACT FROM AUTHOR]

Abstract: In this paper the problems of unique solvability of boundary value problems for infinite functional differential systems and positivity of their solutions are reduced to analysis of scalar functional equations in the space of measurable essentially bounded functions . Efficient tests of the solvability and the positivity are obtained. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the maximum principles for functional equations in the space of measurable essentially bounded functions. The necessary and sufficient conditions for validity of corresponding maximum principles are obtained in a form of theorems about functional inequalities similar to the classical theorems about differential inequalities of the Vallee Poussin type. Assertions about the strong maximum principle are proposed. All results are also true for difference equations, which can be considered as a particular case of functional equations. The problems of validity of the maximum principles are reduced to nonoscillation properties and disconjugacy of functional equations. Note that zeros and nonoscillation of a solution in a space of discontinuous functions are defined in this paper. It is demonstrated that nonoscillation properties of functional equations are connected with the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Simple sufficient conditions of nonoscillation, disconjugacy and validity of the maximum principles are proposed. The known nonoscillation results for equation in space of functions of one variable follow as a particular cases of these assertions. It should be noted that corresponding coefficient tests obtained on this basis cannot be improved. Various applications to nonoscillation, disconjugacy and the maximum principles for partial differential equations are proposed. [Copyright &y& Elsevier]

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

Abstract: The state-dependent delay differential equation with state-dependent impulses is under consideration. Sufficient conditions for positivity of solutions to the Cauchy and periodic problems as well as conditions for positivity of solutions to the problem with a condition on the right end of the interval are given. Sufficient conditions for nonoscillation of solutions to the homogeneous equation (, ) on the half-line are formulated. [Copyright &y& Elsevier]

Abstract: In this paper we shall demonstrate that problems of chaos stabilization can be reduced to the analysis of a corresponding stability of differential equations with unbounded memory. In order to obtain stability results a special monotone technique, based on the positivity of the Cauchy function, is used. New results on stabilization of linear and nonlinear systems are proposed. [Copyright &y& Elsevier]

Abstract: In this paper distribution of zeros of solutions of functional equations in the space of functions of two variables is studied. A zero of a solution in the space of noncontinuous functions is defined. It is demonstrated that oscillatory properties of functional equations are determined by the spectral radius of a corresponding operator acting in the space of essentially bounded functions. Zones of solution positivity are estimated. Various exact oscillation and non-oscillation tests are proposed. A necessary and sufficient condition of oscillation is obtained. [Copyright &y& Elsevier]

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) -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 FROM AUTHOR]

In this paper the model of infection diseases by Marchuk is considered. Mathematical questions which are important in its study are discussed. Among them there are stability of stationary points, construction of the Cauchy matrices of linearized models, estimates of solutions. The novelty we propose is in a distributed feedback control which affects the antibody concentration. We use this control in the form of an integral term and come to the analysis of nonlinear integro-differential systems. New methods for the study of stability of linearized integro–differential systems describing the model of infection diseases are proposed. Explicit conditions of the exponential stability of the stationary points characterizing the state of the healthy body are obtained. The method of the paper is based on the symmetry properties of the Cauchy matrices which allow us their construction. [ABSTRACT FROM AUTHOR]

There are almost no results in mathematical literature on the exponential stability of third-order delay differential equations. One of the main purposes of the paper is to fill this gap. We propose an approach to the study of stability for third-order delay differential equations.On the basis of these results, new possibilities of stabilization by delay feedback input control are proposed. [ABSTRACT FROM AUTHOR]

In this a Lyapunov‐type inequality is obtained for the fractional differential equation with Caputo derivative CDaγx(t)+q(t)x(t)=0,a

Abstract: Nonoscillation plays an important role in the theory of ordinary differential equations, but for functional differential equations and their important class such as delay differential equations, nonoscillation is defined only as the existence of an eventually positive solution on the semiaxis and cannot be used in the analysis of boundary value problems. The use of Azbelev’s definition of homogeneous equations allows us to deal with the standard notion of the nonoscillation interval and to obtain results about the existence and uniqueness of the solutions for the interpolation boundary value problems and sign behavior of their Green’s functions. [Copyright &y& Elsevier]

One of the classical topics in the qualitative theory of differential equations is the Floquet theory. It provides a means to represent solutions and helps in particular for stability analysis. In this paper first we shall study Floquet theory for integro-differential equations (IDE), and then employ it to address stability problems for linear and nonlinear equations. [ABSTRACT FROM AUTHOR]