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.     

The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces

In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.     

Almost periodic solutions of forced vectorial Liénard equations

We describe the set of bounded or almost periodic solutions of the following Liénard system: , where is almost periodic, is a symmetric and nonsingular linear operator, and ∇F denotes the gradient of the convex function F on R^N. Then, we state a result of existence and uniqueness of almost periodic solution.     

Qualitative behavior of global solutions to some nonlinear fourth order differential equations

We study global solutions to a fourth order semilinear ordinary differential equation. We determine sufficient conditions on the nonlinearity that ensure global continuation of the solutions. Furthermore, we discuss their qualitative behaviors such as oscillations and boundedness.     

Existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations

In this paper, under Acquistapace-Terreni conditions, we make extensive use of interpolation spaces and exponential dichotomy techniques to obtain the existence of weighted pseudo-almost periodic solutions to some classes of nonautonomous partial evolution equations. Applications include the existence of weighted pseudo-almost periodic solutions to a nonautonomous heat equation with gradient coefficients.     

Analysis of nonlinear fractional control systems in Banach spaces

In this paper, we consider the nonlinear control systems of fractional order and its optimal controls in Banach spaces. Using the fractional calculus, Hölder’s inequality, p-mean continuity, weakly singular inequality and Leray-Schauder’s fixed point theorem with compact mapping, the sufficient condition is given for the existence and uniqueness of mild solutions for a broad class of fractional nonlinear infinite dimensional control systems. Utilizing the approximately lower semicontinuity of integral functionals and weakly compactness, we extend the existence result of optimal controls for nonlinear control systems to nonlinear fractional control systems under generally mild conditions. An example is given to illustrate the effectiveness of the results obtained.     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.     

Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u^′(t)=A(t)u(t)+f(t,u(t)) in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if A(t) is time independent.     

Asymptotic equivalence of differential equations and asymptotically almost periodic solutions

In this paper we use Rab’s lemma [M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen, Czechoslovak Math. J. 83 (1958) 222–229; M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires, Czechoslovak Math. J. 91 (1966) 127–129] to obtain new sufficient conditions for the asymptotic equivalence of linear and quasilinear systems of ordinary differential equations. Yakubovich’s result [V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, New Jersey, 1966; V.A. Yakubovich, On the asymptotic behavior of systems of differential equations, Mat. Sb. 28 (1951) 217–240] on the asymptotic equivalence of a linear and a quasilinear system is developed. On the basis of the equivalence, the existence of asymptotically almost periodic solutions of the systems is investigated. The definitions of biasymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be biasymptotically equivalent and for the existence of biasymptotically almost periodic solutions are obtained. Appropriate examples are constructed.     

On monotonic solutions of systems of nonlinear second order differential equations

Bounded, monotonic, and asymptotic properties of solutions for a class of second order nonlinear differential equations are discussed. Some necessary and sufficient conditions for boundedness of all solutions are established. Moreover, all solutions are classified into four disjoint subsets which are characterized in terms of the convergence and divergence of several integrals. The obtained results extend and improve many known results of various authors.     

Almost periodic and almost automorphic solutions of linear differential/difference equations without Favard's separation condition. I

The well-known Favard's theorem states that the linear differential equation

(1)     

Weighted pseudo-almost periodic solutions to some differential equations

The paper considers some new classes of functions called weighted pseudo-almost periodic functions, which implement in a natural fashion the classical pseudo-almost periodic functions due to Zhang. Properties of these weighted pseudo-almost periodic functions are discussed, including a composition result for weighted pseudo-almost periodic functions. The results obtained are subsequently utilized to study the existence and uniqueness of a weighted pseudo-almost periodic solution to the heat equation with Dirichlet conditions.     

On the existence of periodic solutions for the third-order nonlinear ordinary differential equations

In this paper,the existence of periodic solution for the third-order nonlinear ordinarydifferential equation of the form -^x f(-^x) g(-^x) h(x)=p(t) is consldered,where f,g,h andp are the continuous functlons, and p (t T) =p (t). By using the Leray-Schauder degreemethod,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained. Some previous results are extended and improved.     

Periodic solutions of some second-order differential equations with desultorily sublinear nonlinearities

In this paper, we study the existence of periodic solutions of the Rayleigh equations

x^″+f(x^′)+g(x)=e(t).     

On the stability and boundedness of solutions of nonlinear vector differential equations of third order

In this paper, by using Liapunov’s second method, we establish some new results for stability and boundedness of solutions of nonlinear vector differential equations of third order. By constructing a Liapunov function, sufficient conditions for stability and boundedness of solutions of equations considered are obtained. Concerning to the subject, some explanatory examples are also given. Our results improve and include a result existing in the literature.     

Monotone method for a system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces

In this paper, by using a monotone iterative technique in the presence of lower and upper solutions, we discuss the existence of solutions for a new system of nonlinear mixed type implicit impulsive integro-differential equations in Banach spaces. Under wide monotonicity conditions and the noncompactness measure conditions, we also obtain the existence of extremal solutions and a unique solution between lower and upper solutions.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.     

Asymptotic properties of solutions of third-order nonlinear differential equations with deviating argument

The aim of this paper is to establish comparison principles on property A$A$, between a nonlinear differential equation of the third order with deviating argument (with delay, advanced or mixed argument) and the corresponding linear equation without deviating argument. On the basis of these comparison principles the sufficient conditions for delay, advanced and mixed equations to have property A$A$ are presented. The results obtained are compared with existing ones in the framework of the papers.     

Limit behavior of solutions of ordinary linear differential equations

A classification of classes of equivalent linear differential equations with respect to -limit sets of their canonical representations is introduced. Some consequences of this classification with respect to the oscillatory behavior of solution spaces are presented.