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In this paper we introduce the functional framework and the necessary conditions for the well-posedness of an inverse problem arising in the mathematical modeling of disease transmission. The direct problem is given by an initial boundary value problem for a reaction diffusion system. The inverse problem consists in the determination of the disease and recovery transmission rates from observed measurement of the direct problem solution at the end time. The unknowns of the inverse problem are coefficients of the reaction term. We formulate the inverse problem as an optimization problem for an appropriate cost functional. Then, the existence of solutions of the inverse problem is deduced by proving the existence of a minimizer for the cost functional. Moreover, we establish the uniqueness up an additive constant of identification problem. The uniqueness is a consequence of the first order necessary optimality condition and a stability of the inverse problem unknowns with respect to the observations., Comment: 12 pages

In this paper we study the asymptotic behavior of nonoscillatory solutions for high order differential equations of Poincar\'e type. We introduce two new and more weak than classical hypotheses on the coefficients, which implies a well posedness result and a characterization of asymptotic behavior for the solution of the Poincar\'e equation. The proof of the main results is based on the application of the scalar method. We define a change of variable to reduce the order of Poincar\'e type equation and deduce that the new variable satisfies a nonlinear differential equation. We apply the variation of parameters method and the Banach fixed point theorem to get the well posedness and asymptotic behavior of the nonlinear equation. Then, by rewritten the results in terms of the original variable we establish the existence of a fundamental system of solutions for and we precise the formulas for the asymptotic behavior of the Poincar\'e type equation., Comment: 14 pages, 0 figures

In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, {\it PNAS} 102, 2277--2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions., Comment: 6 pages, 0 figures

In this paper, we prove some a priori estimates for a system of partial differential equations arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The unknowns of the system are the velocity field of the fluid particles, the angular velocity of rotation of the fluid particles, the mass density of the fluid and the pressure distribution. For the density functions we consider the application of the Helmholtz decomposition., Comment: 14 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1411.7023

In this paper we prove the well-posedness and we study the asymptotic behavior of nonoscillatory $L^p$-solutions for a third order nonlinear scalar differential equation. The equation consists of two parts: a linear third order with constant coefficients part and a nonlinear part represented by a polynomial of fourth order in three variables with variable coefficients. The results are obtained assuming three hypotheses: (i) the characteristic polynomial associated with the linear part has simple and real roots, (ii) the coefficients of the polynomial satisfy asymptotic integral smallness conditions, and (iii) the polynomial coefficients are in $L^p([t_0,\infty[)$. These results are applied to study a fourth order linear differential equation of Poincar\'e type and a fourth order linear differential equation with unbounded coefficients. Moreover, we give some examples where the classical theorems can not be applied., Comment: 22 pages, 0 figures

In this paper, we introduce a nonresident computer virus model and prove the existence of at least one positive periodic solution. The proposed model is based on a biological approach and is obtained by considering that all rates (rates that the computers are disconnected from the Internet, the rate that the computers are cured, etc) are time dependent real functions. Assuming that the initial condition is a positive vector and the coefficients are positive $\omega-$periodic and applying the topological degree arguments we deduce that generalized nonresident computer virus model has at least one positive $\omega-$periodic solution. The proof consists of two big parts. First, an appropriate change of variable which conserves the periodicity property and implies the positive behavior. Second, a reformulation of transformed system as an operator equation which is analyzed by applying the continuation theorem of the coincidence degree theory., Comment: 0 figures

In this paper, ordinary and exponential dichotomies are defined in differential equations with equations with piecewise constant argument of general type. We prove the asymptotic equivalence between the bounded solutions of a linear system and a perturbed system with integrable and bounded perturbations., Comment: 20 pages, 0 figures

In this paper, we prove the local uniqueness of an inverse problem arising in the nonstationary flow of a nonhomogeneous incompressible asymmetric fluid in a bounded domain with smooth boundary. The direct problem is an initial-boundary value problem for a system for the velocity field, the angular velocity of rotation of the fluid particles, the mass density and the pressure distribution. The inverse problem consists in the external force recover assuming an integral measurements on the boundary. We characterize the inverse problem solutions using an operator equation of second kind, which is deduced form the application of the Helmholtz decomposition. We introduce several estimates which implies the hypothesis of the Tikhonov fixed point theorem., Comment: 25 pages

Existence of almost automorphic solutions for abstract delayed differential equations is established. Using ergodicity, exponential dichotomy and Bi-almost automorphicity on the homogeneous part, sufficient conditions for the existence and uniqueness of almost automorphic solutions are given., Comment: 16 pages

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable satisfies a third order differential equation of Riccati type. We assume three hypothesis. The first is the following: all roots of the characteristic polynomial associated to the fourth order linear equation has distinct real part. The other two hypothesis are related with the behavior of the perturbation functions. Under this general hypothesis we obtain four main results. The first two results are related with the application of fixed point theorem to prove that the Riccati equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the Riccati equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise formulas for the asymptotic behavior of the linear fourth order differential equation., Comment: 16 pages

In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (conjectures~4.4, 2.1 and 2.2) posed by C{\^i}rtoaje in the following two works: "{\it J. Inequal. Pure Appl. Math.} 10, Article 21, 2009" and "{\it J. Nonlinear Sci. Appl.} 4:2:130-137, 2011". Moreover, we present a new proof of the inequality $a^{ra}+b^{rb}\ge a^{rb}+b^{ra}$ for all positive real numbers $a$ and $b$ and $r\in [0,e]$. In addition, three new conjectures are presented., Comment: 9 pages

In this short note the authors give answers to the three open problems formulated by Wu and Srivastava [{\it Appl. Math. Lett. 25 (2012), 1347--1353}]. We disprove the Problem 1, by showing that there exists a triangle which does not satisfies the proposed inequality. We prove the inequalities conjectured in Problems 2 and 3. Furthermore, we introduce an optimal refinement of the inequality conjectured on Problem 3., Comment: 3 pages