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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]

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]

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