First-order Functional Differential Equations Research Articles

Some remarks on the solution of the cell growth equation

A process of growth and division of cells is modelled by an initial boundary value problem that involves a first-order linear functional partial differential equation, the so-called sell growth equation. The analytical solution to this problem was given in the paper Zaidi et al. (Zaidi et al . 2015 Solutions to an advanced functional partial differential equation of the pantograph type ( Proc. R. Soc. A 471 , 20140947 ( doi:10.1098/rspa.2014.0947 )). In this note, we simplify the arguments given in the paper mentioned above by using the theory of operator semigroups. This theory enables us to prove the existence and uniqueness of the solution and to express this solution in terms of Dyson–Phillips series. The asymptotics of the solution is also discussed from the point of view of the theory of operator semigroups.

Uniqueness and existence of positive periodic solutions of functional differential equations

<abstract><p>In this paper, some new findings on the uniqueness and existence of positive periodic solutions to first-order functional differential equations are presented. These equations have wide applications in a variety of fields. The most important feature of our argument is that we use the theory of Hilbert's metric to prove the uniqueness of the positive periodic solution when $ q=-1 $ and $ -1 < q < 0 $. In addition, we also investigate the existence results of positive periodic solutions by applying a fixed point theorem for completely continuous maps in a cone. Two examples demonstrate our findings.</p></abstract>

"Positive periodic solutions for a class of first-order iterative differential equations with an application to a hematopoiesis model"

"In this paper, a first-order iterative functional differential equation is investigated. With the help of the Schauder’s fixed point theorem, we established some sufficient criteria that ensure the existence of positive periodic solutions. In addition, an application to three hematopoiesis models is also provided to corroborate the effeteness of our main findings. These last ones substantially enrich and complement some earlier works."

Transversality conditions for the existence of solutions of first-order discontinuous functional differential equations

We are concerned with the existence of extremal solutions to a large class of first-order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well-known models: a generalized logistic equation or second-order problems in the presence of dry friction.

On Periodic Boundary-Value Problems for Systems of Functional-Differential Equations

We obtain necessary and sufficient conditions for the existence of a unique solution to a periodic boundary-value problem for all systems of first-order functional-differential equations from a given family of systems. Families of systems of functional-differential equations are detrmined by the norms of positive functional operators of equations of the system. The verification of necessary and sufficient conditions of the existence of a unique periodic solution for all systems from a given family consists of the verification of positivity of a finite number of a real-valued functions defined on a finite-dimensional set.

On Anti-Periodic Solutions for a Class of Impulsive Retarded Functional Differential Equations

In this paper, some new results on the existence and uniqueness of anti-periodic solution for a first-order impulsive functional differential equation with delay are obtained. The main tool to be employed here is the coincidence degree theory.

Multi-parameter, impulsive effects and positive periodic solutions of first-order functional differential equations

The authors consider the existence of positive periodic solution for the impulsive functional differential equation $u'(t)=-a(t)g(u(h_{1}(t)))u(t)+\lambda b(t)f (t, u(h_{2}(t)), \int_{-\zeta}^{0}e(s)u(t+s)\, ds )$ , $t\in\mathbf{R}$ , $t\neq t_{k}$ , $u(t_{k}^{+})-u(t_{k})=\mu I_{k}(t_{k},u(t_{k}))$ , $k\in\mathbf{Z}$ , where $\lambda>0 $ and $\mu>0$ are two parameters. Several new and more general existence and multiplicity results are derived in terms of different values of $\lambda>0$ and $\mu>0$ . Here we not only consider the case that g is bounded, but the case that g is not necessarily bounded is also considered. Our results improve those in (Ma et al. in J. Math. Anal. Appl. 384:527-535, 2011; Li et al. in Comput. Math. Appl. 56:2556-2560, 2008). Moreover, the parameter dependence of the periodic solution is also studied.

Existence and uniqueness of positive periodic solutions for a first-order functional differential equation

In this article, we establish sufficient conditions for the existence and uniqueness of positive periodic solutions for a class of first-order functional differential equations. Our analysis relies on some fixed point theorems for mixed monotone operators. Our results can not only guarantee the existence of unique positive periodic solutions, but also be applied to construct an iterative scheme for approximating them. Some examples are given to illustrate our main results.

On estimates of solutions of the periodic boundary value problem for first-order functional differential equations

Inequalities for periodic solutions of first-order functional differential equations are obtained. These inequalities are best possible in a certain sense.MSC:34K06, 34K10, 34K13.

A Well-Posedness Framework for Inpainting Based on Coherence Transport

Image inpainting is the process of touching up damaged or unwanted portions of a picture and is an important task in image processing. For this purpose Bornemann and Marz (J Math Imaging Vision, 28:259---278, 2007) introduced a very efficient method, called image inpainting based on coherence transport, that fills the missing region by advecting the image information along integral curves of a coherence vector field from the boundary toward the interior of the hole. The mathematical model behind this method is a first-order functional advection partial differential equation posed on a compact domain with all inflow boundaries. We show that this problem is well posed under certain conditions.

Periodic Boundary Value Problems for a Class of Impulsive Functional Differential Equations of Hybrid Type

This paper is concerned with the existence of extreme solutions of periodic boundary value problems for a class of first-order impulsive functional differential equations of hybrid type. We obtain the sufficient conditions for existence of extreme solutions by using upper and lower solutions method coupled with monotone iterative technique.

On positiveness of the Cauchy function of a singular linear functional differential equation

We study first-order functional differential equations which are singular in the independent variable, establish criteria for the constant sign property of the Cauchy function, and prove an assertion analogous to the Vallee-Poussin theorem.

非自治中立型无穷时滞泛函微分方程的加权伪概自守解

We first introduce the notion of Stepanov-like weighted pseudo almost automorphic functions of class h and the notion of Stepanov-like weighted pseudo almost automorphic functions of class ∞, and then establish some basic results not only on the completeness of the space that consists of the Sp -weighted pseudo almost automorphic functions of class h and class ∞ but also on the compositions of such functions. Finally, we apply the abstract results to prove the existence and uniqueness of weighted pseudo almost automorphic solutions with Sp -weighted pseudo almost automorphic coefficients to a non-autonomous first-order abstract partial neutral functional differential equation with infinite delay.

Stepanov-like square-mean pseudo almost automorphic solutions to partial neutral stochastic functional differential equations

We obtain the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to first-order partial neutral stochastic functional differential equations with Stepanov-like almost automorphic coefficients in a real separable Hilbert space

Conditions for the solvability of the Cauchy problem for linear first-order functional differential equations

Conditions for the unique solvability of the Cauchy problem for a certain family of scalar functional dierential equations are obtained. These conditions are sucient for the solvability of the Cauchy problem for every equation from the given family. Moreover, obtained conditions are optimal and cannot be weakened. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions.

One class of singular linear functional differential equations

We consider one class of first-order functional differential equations with a singularity in the independent variable. We obtain conditions for the Fredholm property and the solvability of the mentioned equations.

Positive solutions of impulsive boundary value problems for first-order nonlinear functional differential equations

In this paper, we study the problem on the existence of positive solutions for a class of impulsive periodic boundary value problems of first-order nonlinear functional differential equations. By using the fixed point theorem in cones and some analysis techniques, we present some sufficient conditions which guarantee the existence of one and multiple positive solutions for the impulsive periodic boundary value problems. Our results generalize and improve some previous results. Moreover, our results show that positive solutions for the impulsive periodic boundary value problems may be yielded completely by some proper impulsive conditions (see Example 4.1 and Remark 4.2 in Sect. 4), and also implies that proper impulsive conditions are of great significance to simulate processes, optimal control, population model and so on.

Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments

We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays.

Weak solutions of functional differential inequalities with first-order partial derivatives

The article deals with functional differential inequalities generated by the Cauchy problem for nonlinear first-order partial functional differential equations. The unknown function is the functional variable in equation and inequalities, and the partial derivatives appear in a classical sense. Theorems on weak solutions to functional differential inequalities are presented. Moreover, a comparison theorem gives an estimate for functions of several variables by means of functions of one variable which are solutions of ordinary differential equations or inequalities. It is shown that there are solutions of initial problems defined on the Haar pyramid.Mathematics Subject Classification: 35R10, 35R45.

Eigenfunctions arising from a first-order functional differential equation in a cell growth model

A boundary-value problem for cell growth leads to an eigenvalue problem. In this paper some properties of the eigenfunctions are studied. The first eigenfunction is a probability density function and is of importance in the cell growth model. We sharpen an earlier uniqueness result and show that the distribution is unimodal. We then show that the higher eigenfunctions have nested zeros. We show that the eigenfunctions are not mutually orthogonal, but that there are certain orthogonality relations that effectively partition the set of eigenfunctions into two sets. doi:10.1017/S1446181111000575