L_p-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

Existence of solutions for some functional integrodifferential equations with nonlocal conditions

In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and Sadovskii-Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory.     

Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations

In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.     

Optimal Secant-Type Methods for Operator Equations

We prove an existence and uniqueness theorem for operator equations in Banach spaces with (generally non-differentiable) operators whose divided differences are Lipschitz continuous on operator's domain. The theorem makes possible to apply the concept of entropy optimality of iterative methods introduced in our earlier work to the class of secant-type methods. Using this concept, we show that it is feasible to get a method that needs the same information (one value of the operator) per iteration but exhibits a faster convergence than the secant and secant-update methods.     

Banach空间中非线性脉冲Volterra型积分方程组的可解性

在较弱的条件下，利用锥理论和单调迭代方法首先建立了Banach空间中一类非线性算子方程组最小最大解的存在性定理；然后作为应用，利用一个新的比较结果，得到了Banach空间中非线性脉冲Volterra型积分方程组的整体解，改进了最近的许多结果．     

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.     

抽象空间中非线性算子方程变号解的存在性研究

利用Banach空间中的锥理论和不动点定理讨论了非线性算子方程变号解的存在性,给出了E_u_0空间下非线性算子方程变号解至少有一个变号解、一个正解和一个负解的条件,并讨论了仅通过一个上解条件得出非线性算子方程变号解的存在性定理.     

Existence of positive continuous solutions of perturbed hammerstein equations

We improve previous existence results for a class of perturbed Hammerstein integral equations, where the relevant Hammerstein operator is decreasing on positive functions. We strongly weaken the assumptions on the nonlinearities involved, and obtain existence of positive continuous solutions, even on noncompact domains, applying the Schauder-Tychonoff fixed-point theorem, via the generalized Ascoli theorem and the regularizing effect of the integral operator. An application to perturbed quadratic integral equations of interest in transport theory is also given     

Existence theorems and Hyers-Ulam stability for a class of Hybrid fractional differential equations with $p$-Laplacian operator

In this paper, we prove necessary conditions for existence and uniqueness of solution (EUS) as well Hyers-Ulam stability for a class of hybrid fractional differential equations (HFDEs) with $p$-Laplacian operator. For these aims, we take help from topological degree theory and Leray Schauder-type fixed point theorem. An example is provided to illustrate the results.     

无限滞后测度泛函微分方程的平均化

本文研究了无限滞后测度泛函微分方程的平均化.利用广义常微分方程的平均化方法,在无限滞后测度泛函微分方程可以转化为广义常微分方程的基础上,获得了这类方程的周期和非周期平均化定理,推广了一些相关的结果.     

The solution existence and convergence analysis for linear and nonlinear differential-operator equations in Banach spaces within the Calogero type projection-algebraic scheme of discrete approximations

The projection-algebraic approach of the Calogero type for discrete approximations of linear and nonlinear differential operator equations in Banach spaces is studied. The solution convergence and realizability properties of the related approximating schemes are analyzed. For the limiting-dense approximating scheme of linear differential operator equations a new convergence theorem is stated. In the case of nonlinear differential operator equations the effective convergence conditions for the approximated solution sets, based on a Leray-Schauder type fixed point theorem, are obtained.     

Multiple solutions for discrete boundary value problems

A recent multiplicity theorem for the critical points of a functional defined on a finite-dimensional Hilbert space, established by Ricceri, is extended. An application to Dirichlet boundary value problems for difference equations involving the discrete p-Laplacian operator is presented.     

Heisenberg evolution of quantum observables represented by unbounded operators

This paper deals with open quantum systems. In particular, we focus on the adjoint quantum master equations with initial conditions given by unbounded operators. Examples of this type of initial data are the position and momentum operators of quantum oscillators and the occupation number operator in many-body particle systems. The article establishes the existence and uniqueness of solutions of the operator equations governing the motion of unbounded observables under the Born-Markov approximations. To this end, we develop the relation between operator evolution equations arising in quantum mechanics and stochastic evolutions equations of Schrödinger type. Furthermore, we examine quantum dynamical semigroup properties of the Heisenberg evolutions of general classes of observables.     

拟增算子的多个不动点

该文应用变分方法和 Banach 空间的常微分方程理论证明了关于拟增算子的一个三不动点定理.     

一个减算子不动点定理及其对非线性Volterra型积分方程的应用

利用锥理论,在不需要上下解的条件下,获得一个新的减算子不动点定理,推广了最近文献的结果;并给出Banach空间中非线性Volterra型积分方程解的唯一性.     

A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional ε(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h→0) by applying suitably chosen extension operators p_h. A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of ε(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided.Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples.     

Implicit function theorem via the DSM

Sufficient conditions are given for an implicit function theorem to hold. The result is established by an application of the Dynamical Systems Method (DSM). It allows one to solve a class of nonlinear operator equations in the case when the Fréchet derivative of the nonlinear operator is a smoothing operator, so that its inverse is an unbounded operator.     

Krein Orthogonal Entire Matrix Functions and Related Lyapunov Equations: A State Space Approach

For a class of entire matrix valued functions of exponential type new necessary and sufficient conditions are derived in order that these functions are Krein orthogonal functions. The conditions are stated in terms of certain operator Lyapunov equations. These equations arise by using infinite dimensional state space representations of the entire matrix functions involved. As a corollary, using a recent operator inertia theorem, we give a new proof of the Ellis-Gohberg-Lay theorem which relates the number of zeros of a Krein orthogonal function in the open upper half plane to the number of negative eigenvalues of the corresponding selfadjoint convolution operator.     

A unification theory of Krasnoselskii for differential equations

The theory of differential equations is very broad and contains many seemingly unrelated types of problems with markedly different methods of solution. It is very difficult to discern any unity in the theory. Yet, sixty years ago one of the foremost investigators, Krasnoselskii, suggested the possibility of finding unity. He claimed that the inversion of a perturbed differential operator yields the sum of a contraction and compact map. Accordingly, he proved a general fixed point theorem to cover this situation. In this paper we begin a long study with a view to putting his idea to the test. We begin with fractional differential equations of Caputo type, continue to neutral functional differential equations, and conclude with a study of an old problem of Volterra which continues to describe many important real-world problems. For these problems there is the perfect unity predicted by Krasnoselskii. It is an invitation to continue the study by examining other important real-world problems.