On global existence and attractivity results for nonlinear functional integral equations

In this paper two existence results concerning the global attractivity and global asymptotic attractivity for a certain functional nonlinear integral equation are proved. Our existence results include several existence as well as attractivity results obtained earlier by Banas and Dhage (2008) [1], Hu and Yan (2006) [3], Dhage (2009) [15] and Banas and Rzepka (2003) [7] as special cases under some weaker Lipschitz conditions. A measure theoretic fixed point theorem of Dhage (2008) [6] is used in formulating our main results and the characterizations of solutions are obtained in the space of functions defined, continuous and bounded on unbounded intervals.     

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].     

Existence for bounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains

In this paper, by using the fixed point theory, we obtain a new existence result for bounded positive solutions of the quasilinear elliptic equations in two-dimensional exterior domains.     

Existence of unbounded positive solutions of quasilinear elliptic equations in two-dimensional exterior domains

In this paper, by using fixed point theory, under quite general conditions on the nonlinear term, we obtain an existence result on unbounded positive solutions of certain quasilinear elliptic equations in two-dimensional exterior domains.     

Jean Leray and nonlinear integral equations: A partially forgotten legacy

Some applications given in Jean Leray’s thesis of 1933 seem to have been forgotten, before being rediscovered and then widely developed. The reason may be found in the fact that the Arzelà–Schmidt’s method introduced in the thesis was superseded one year later by the more general and fruitful Leray-Schauder’s method. We also show that, despite its fame and a steadily increasing number of applications, some aspects of Leray–Schauder’s paper were somewhat forgotten too. Dedicated, with admiration, to the living memory of Jean Leray     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

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.     

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.     

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.     

A class of nonlinear evolution equations subjected to nonlocal initial conditions

We prove the existence of C⁰-solutions for a class of nonlinear evolution equations subjected to nonlocal initial conditions, of the form:

     

Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching

The main aim of this paper is to discuss the almost surely asymptotic stability of the neutral stochastic differential delay equations (NSDDEs) with Markovian switching. Linear NSDDEs with Markovian switching and nonlinear examples will be discussed to illustrate the theory.     

Homeomorphism flows for non-Lipschitz stochastic differential equations with jumps

In this paper we study the continuity property as well as the homeomorphism property for the solutions of multidimensional stochastic differential equations with jumps and non-Lipschitz coefficients with respect to the initial values.     

Series expansions for functional differential equations

In this paper we prove convergence results for the series expansion of the solution to a linear functional differential equation. The results are consequences of an analysis of eigenfunction expansions for the generator of the solution map. This abstract approach unifies the treatment of retarded and neutral functional differential equations.The research of S.M. Verduyn Lunel has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

Topological structure of solution sets to asymptotic boundary value problems

Topological structure is investigated for second-order vector asymptotic boundary value problems. Because of indicated obstructions, the Rδ-structure is firstly studied for problems on compact intervals and then, by means of the inverse limit method, on non-compact intervals. The information about the structure is furthermore employed, by virtue of a fixed-point index technique in Fréchet spaces developed by ourselves earlier, for obtaining an existence result for nonlinear asymptotic problems. Some illustrating examples are supplied.     

Existence of nontrivial solutions for a nonlinear Sturm–Liouville problem with integral boundary conditions

This paper deals with the existence of nontrivial solutions for a nonlinear singular Sturm–Liouville problem with integral boundary conditions.     

Perturbation of the Drazin inverse for closed linear operators

We investigate the perturbation of the Drazin inverse of a closed linear operator recently introduced by second author and Tran, and derive explicit bounds for the perturbations under certain restrictions on the perturbing operators. We give applications to the solution of perturbed linear equations, to the asymptotic behaviour ofC ₀-semigroups of linear operators, and to perturbed differential equations. As a special case of our results we recover recent perturbation theorems of Wei and Wang.     

Fixed point theorems for mixed monotone operators and applications to integral equations

The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations.     

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.