Characterizations of reproducing cones and uniqueness of fixed points

The purpose of this paper is to discuss the existence and uniqueness of fixed point in a partially ordered Banach space. Based on the characterizations of reproducing cones, some fixed point theorems for nonlinear operators are proved. As an application, the existence and uniqueness of periodical solution for a first order differential equation is discussed.     

Existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations

By using the Schauder fixed point theorem, we establish a result for the existence of solutions of a boundary value problem on the half-line to second order nonlinear delay differential equations. We also present the application of our result to the special case of second order nonlinear ordinary differential equations as well as to a specific class of second order nonlinear delay differential equations. Moreover, we give a general example which demonstrates the applicability of our result. Received: 10 May 2004     

Nontrivial application of Nielsen theory to differential systems

In reply to a problem of Jean Leray concerning application of the Nielsen theory to differential systems for obtaining multiplicity results, we present a nontrivial example of such an application. The emphasis is on the parameter space in order to ensure that no subdomain becomes subinvariant under the related Hammerstein solution operator. To achieve this goal, we develop a general method applicable also for ordinary differential equations with or without uniqueness as well as for upper-Carathéodory differential inclusions. We are not aware that any alternative approach can be employed, even in the single-valued case.     

Systems of first order differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of first order differential inclusions with maximal monotone terms satisfying the periodic boundary condition. Our proofs rely on the theory of maximal monotone operators, and the Schauder and the Kakutani fixed point theorems. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of first order differential equations.     

Limit cycles for generalized Kukles polynomial differential systems

We study the limit cycles of generalized Kukles polynomial differential systems using averaging theory of first and second order.     

Averaging methods for finding periodic orbits via Brouwer degree

We consider the problem of finding T-periodic solutions for a differential system whose vector field depend on a small parameter ε. An answer to this problem can be given using the averaging method. Our main results are in this direction, but our approach is new. We use topological methods based on Brouwer degree theory to solve operator equations equivalent to this problem. The regularity assumptions are weaker then in the known results (up to second order in ε). A result for third order averaging method is also given.As an application we provide a way to study bifurcations of limit cycles from the period annulus of a planar system and notice relations with the displacement function. A concrete example is given.     

增算子的不动点和耦合不动点

Under some general continuous and compact conditions, the existence problems of fiked points andd coupled fixed points for increasing operators are studied. an application, we utilize the results obtained to study the existence of solutions for differential inclusions in Banach spaces.     

Existence and continuous dependence results for nonlinear differential inclusions with infinite delay

This paper is concerned with nonlinear functional differential inclusions with infinite delay in Banach spaces. Using tools involving the measure of noncompactness and multi-valued fixed point theory, existence and continuous dependence results are obtained, for integral solutions, without the assumption of compactness on the associated nonlinear semigroup.     

Fixed point theorems for generalized contractions in partially ordered metric spaces with semi-monotone metric

In this paper, we study the existence and uniqueness of (coupled) fixed points for mixed monotone mappings in partially ordered metric spaces with semi-monotone metric. As an application, we prove the existence and uniqueness of the solution for a first-order differential equation with periodic boundary conditions.     

Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems

Eigenvalue intervals and the existence of finitely many positive eigenfunctions for semi-positone Hammerstein integral equations are obtained. The positive characteristic values and their upper and lower bounds of the corresponding linear Hammerstein integral operators are studied. Applications of the results are given to third-order differential equations with three-point boundary conditions.     

Weakly coupled oscillators and topological degree

By using the topological degree of Brouwer for mappings along with averaging method, we study the existence of forced periodic solutions for certain weakly coupled periodically perturbed ordinary differential equations.     

On certain classes of functional inclusions with causal operators in Banach spaces

We introduce the notion of a multivalued causal operator and consider an abstract Cauchy problem in a Banach space for various classes of functional inclusions with causal operators. The methods of the topological degree theory for condensing maps are applied to obtain local and global existence results for this problem and to study the continuous dependence of a solution set on initial data. As application we generalize some existence results for semilinear functional differential inclusions and Volterra integro-differential inclusions with delay.     

Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces

In this paper we give the existence of integral solutions for nonlinear differential equations with nonlocal initial conditions under the assumptions of the Hausdorff measure of noncompactness in separable and uniformly smooth Banach spaces.     

On existence of positive periodic solutions

We present the geometric method for detecting periodic solutions of time periodic nonautonomous differential equations in interior of convex subset of euclidean space. The method is based on the Lefschetz fixed point theorem and the topological principle of Waewski. Two applications to the existence of positive periodic solutions are considered.Research supported by the KBN grant 2 P03A 040 10     

Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

Contributions to coincidence degree theory of asymptotically homogeneous operators

In this paper we give contributions to the coincidence degree theory of asymptotically homogeneous operators. Applications are given to the periodic problem for second-order functional differential equations.     

A class of function spaces and its application in solving second-order nonlinear differential equations

In this paper, we prove an existence theorem for time global monotone positive solutions of nonlinear second-order ordinary differential equations by applying the Schauder-Tikhonov fixed point theorem. This result generalizes the result of existence on a half-line given in Yin (2003) [8].     

Periodic solutions for a nonautonomous ordinary differential equation

We consider the nonautonomous differential equation of second order x^″+a(t)x−b(t)x²+c(t)x³=0, where a(t),b(t),c(t) are T-periodic functions. This is a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of at least two positive T-periodic solutions for this equation, using coincidence degree theories.     

Study the stability of solutions of functional differential equations via fixed points

In this paper, we study the stability properties of solutions of a class of functional differential equations with variable delay. By using the fixed point theory under an exponentially weighted metric, we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable.     

Controllability for systems governed by semilinear differential inclusions in a Banach space with a noncompact semigroup

We study the controllability problem for a system governed by a semilinear differential inclusion in a Banach space not assuming that the semigroup generated by the linear part of inclusion is compact. Instead we suppose that the multivalued nonlinearity satisfies the regularity condition expressed in terms of the Hausdorff measure of noncompactness. It allows us to apply the topological degree theory for condensing operators and to obtain the controllability results for both upper Carathéodory and almost lower semicontinuous types of nonlinearity. As application we consider the controllability for a system governed by a perturbed wave equation.