Differential Equations In Banach Spaces Research Articles

On the Ulam stability of fuzzy differential equations

Ulam stability problems have received considerable attention in the field of differential equations. However, how to effectively build the fuzzy model for Ulam stability problems is less attractive due to varies of differentiabilities requirements. The paper discusses the Ulam stability of fuzzy differential equations in Banach spaces. After introducing the new definitions of differentiabilities for fuzzy number-valued mappings, we give some important properties about these differentiabilities. On these bases, with different differentiabilities and conditions, we prove the Ulam stability of three kinds of fuzzy differential equations. The obtained conclusions generalize the existing results.

Asymptotic Integration of Certain Differential Equations in Banach Space

In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that the unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.

Fractional Powers of Monotone Operators in Hilbert Spaces

AbstractThe aim of this article is to provide a functional analytical framework for defining thefractional powersAs{A^{s}}for-1<s<1{-1<s<1}of maximal monotone (possibly multivalued and nonlinear) operatorsAin Hilbert spaces. We investigate the semigroup{e-As⁢t}t≥0{\{e^{-A^{s}t}\}_{t\geq 0}}generated by-As{-A^{s}}, prove comparison principles and interpolations properties of{e-As⁢t}t≥0{\{e^{-A^{s}t}\}_{t\geq 0}}in Lebesgue and Orlicz spaces. We give sufficient conditions implying thatAs{A^{s}}has a sub-differential structure. These results extend earlier ones obtained in the cases=1/2{s=1/2}for maximal monotone operators [H. Brézis, Équations d’évolution du second ordre associées à des opérateurs monotones, Israel J. Math. 12 1972, 51–60], [V. Barbu, A class of boundary problems for second order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 1972, 295–319], [V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leiden, 1976], [E. I. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 1986, 2, 514–543], and the recent advances for linear operatorsAobtained in [L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 2007, 7–9, 1245–1260], [P. R. Stinga and J. L. Torrea, Extension problem and Harnack’s inequality for some fractional operators, Comm. Partial Differential Equations 35 2010, 11, 2092–2122].

Operator cosine-functions and boundary value problems

For the first time, the theory of strongly continuous operator cosines of functions (COF) is applied to the study of the correct solvability of boundary value problems for second-order linear differential equations in Banach space (elliptic case). Usually in COF terms the correct solvability of the Cauchy problem (hyperbolic case) is formulated. The note specifies the conditions on the order of COF growth under which the Dirichlet boundary value problem is correct on the finite interval. The integral representation of the solution and its exact estimation are given.

Well-Posedness of Fractional Degenerate Differential Equations in Banach Spaces

Abstract We study the well-posedness of the fractional degenerate differential equation: Dα (Mu)(t) + cDβ(Mu)(t) = Au(t) + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces Lp(𝕋; X) and periodic Besov spaces $\begin{array}{} B_{p,q}^s \end{array}$ (𝕋; X), where A and M are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(M), c ∈ ℂ and 0 < β < α are fixed. Using known operator-valued Fourier multiplier theorems, we give necessary and sufficient conditions for Lp-well-posedness and $\begin{array}{} B_{p,q}^s \end{array}$-well-posedness of above equation.

Doss almost periodic functions, Besicovitch-Doss almost periodic functions and convolution products

In the paper under review, we analyze the invariance of Doss almost periodicity and Besicovitch-Doss almost periodicity under the actions of convolution products. We thus continue our recent research studies \cite{fedorov-novi} and \cite{NSJOM-besik} by investigating the case in which the solution operator family $(R(t))_{t>0}$ under our consideration has special growth rates at zero and infinity. In contrast to \cite{NSJOM-besik}, the results obtained in this paper can be incorporated in the qualitative analysis of solutions to abstract (degenerate) inhomogeneous fractional differential equations in Banach spaces.

Time optimal control for semilinear fractional evolution feedback control systems

In this article, we investigate a class of time feedback optimal control systems governed by Caputo fractional semilinear differential equations in Banach space. At first, we discuss the existence and uniqueness of the mild solution for the equations by using fixed point theorem. Secondly, we show that the admissible trajectories set is non-empty involving the compactness of semigroup with the help of the Cesari property and the Fillippove theorem. Moreover, we also give an existence result for time feedback optimal control. In the end, an example is given to illustrate our main results.

Remarks on Generalized Stability for Difference Equations in Banach Spaces

This note focuses on the problem of generalized exponential stability for linear time-varying discrete-time systems in Banach spaces. Characterizations for this concept are presented and connections with the classical concept of exponential stability existent in the literature is pointed out. An illustrative example clarifies the implications between these concepts.

Stability analysis of a multi-layer tumor model with free boundary

In this paper we study a multi-layer tumor model which is expressed as a free boundary problem of a system of partial differential equations. The problem consists of two elliptic equations describing the distribution of nutrient concentration and the pressure between tumor cells, respectively, in an unbounded strip-like region in which the tumor occupies. This region has two disjoint boundaries: While the lower part is fixed, the upper part, which stands for the tumor surface, can move as the tumor grows. Under certain conditions, the problem can be proved to admit a unique equilibrium which corresponds to the flat upper boundary. We first convert the model into a parabolic differential equation in certain function space. Next we compute the spectrum of the linearized problem at the equilibrium. By applying the geometric theory of parabolic differential equations in Banach spaces, we prove that if the cell-to-cell adhesiveness coefficient γ is larger than a threshold value γ∗, then the unique flat equilibrium is asymptotically stable, whereas in the case 0<γ<γ∗ the flat equilibrium is unstable.

English

We present a variation-of-constants formula for functional differential equations of the form $$\dot y = \mathcal{L}\left( t \right)y_t + f\left( {y_t,t} \right),\;y_{t_0}= \varphi $$ , where $$\mathcal{L}$$ is a bounded linear operator and φ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t $$t \mapsto f\left( {y_t,t} \right)$$ is Kurzweil integrable with t in an interval of ℝ, for each regulated function y. This means that t $$t \mapsto f\left( {y_t,t} \right)$$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J.Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type $$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x} \right],x\left( {{t_0}} \right) = \tilde x$$ and the solutions of the perturbed Cauchy problem $$\frac{{dx}}{{d\tau }} = D\left[ {A\left( t \right)x + F\left( {x,t} \right)} \right],x\left( {{t_0}} \right) = \tilde x$$ Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form $$\dot y = \mathcal{L}\left( t \right)y_t,\;y_{t_0} = \varphi$$ , where $$\mathcal{L}$$ is a bounded linear operator and φ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.

On solutions of semilinear upper diagonal infinite systems of differential equations

The goal of the paper is to investigate the existence of solutions for semilinear upper diagonal infinite systems of differential equations. We will look for solutions of the mentioned infinite systems in a Banach tempered sequence space. In our considerations we utilize the technique associated with the Hausdorff measure of noncompactness and some existence results from the theory of ordinary differential equations in abstract Banach spaces.

Caputo-Hadamard Fractional Differential Equations in Banach Spaces

Abstract This article deals with some existence results for a class of Caputo–Hadamard fractional differential equations. The results are based on the Mönch’s fixed point theorem associated with the technique of measure of noncompactness. Two illustrative examples are presented.

Existence and Uniqueness Fourth-order Differential Equations in Banach Space Having Fourier Type

The aim of this work is to study the existence of a periodic solutions of differential equations $\frac{d^{4} }{ dt^{4} }x(t) = Ax(t) + f(t)$. Our approach is based on the M-boundedness of linear operators, Fourier type, $B^{s}_{p, q}$-multipliers and Besov spaces.

Direct and inverse problems for degenerate differential equations

We are concerned with a general abstract equation that allows to handle various degenerate first and second order differential equations in Banach spaces. We indicate sufficient conditions for existence and uniqueness of a solution. Periodic conditions are assumed to improve previous approaches on the abstract problem to work on $$(-\infty ,\infty )$$ . Related inverse problems are discussed, too. All general results are applied to some systems of partial differential equations. Inverse problems for degenerate evolution integro-differential equations might be described, too.

Time Optimal Control of a System Governed by Non-instantaneous Impulsive Differential Equations

We investigate time optimal control of a system governed by a class of non-instantaneous impulsive differential equations in Banach spaces. We use an appropriate linear transformation technique to transfer the original impulsive system into an approximate one, and then we prove the existence and uniqueness of their mild solutions. Moreover, we show the existence of optimal controls for Meyer problems of the approximate. Further, in order to solve the time optimal control problem for the original system, we construct a sequence of Meyer approximations for which the desired optimal control and optimal time are well derived.

On a Bohr-Neugebauer property for some almost automorphic abstract delay equations

This paper is a continuation of the investigations done in the literature regarding the so called Bohr-Neugebauer property for almost periodic differential equations in Hilbert spaces. The aim of this work is to extend the investigation of this property to almost automorphic functional partial differential equations in Banach spaces. We use a compactness assumption which turns out to relax assumptions made in some earlier works for differential equations in Hilbert spaces. Two new integration theorems for almost automorphic functions are proven in the process. To illustrate our main results, we propose an application to a reaction-diffusion equation with continuous delay.

Characterizations of Input-to-State Stability for Infinite-Dimensional Systems

We prove characterizations of input-to-state stability (ISS) for a large class of infinite-dimensional control systems, including some classes of evolution equations over Banach spaces, time-delay systems, ordinary differential equations (ODE), and switched systems. These characterizations generalize well-known criteria of ISS, proved by Sontag and Wang for ODE systems. For the special case of differential equations in Banach spaces, we prove even broader criteria for ISS and apply these results to show that (under some mild restrictions) the existence of a noncoercive ISS Lyapunov functions implies ISS. We introduce the new notion of strong ISS (sISS) that is equivalent to ISS in the ODE case, but is strictly weaker than ISS in the infinite-dimensional setting and prove several criteria for the sISS property. At the same time, we show by means of counterexamples that many characterizations, which are valid in the ODE case, are not true for general infinite-dimensional systems.

Semilinear Conformable Fractional Differential Equations in Banach Spaces

We introduce the concept of a mild solution of conformable fractional abstract initial value problem. We establish the existence and uniqueness theorem using the contraction principle. As a regularity result for a linear problem, we show that the mild solution is in fact a strong solution. We give an example to demonstrate the applicability of the established theoretical results.

Unique strong and strict solutions for inhomogeneous hyperbolic differential equations in Banach space (revised version)

This paper is a revised version of a recent paper by the author with the same title. The purpose of the present paper is the improvement of results for solutions of the inhomogeneous differential equation: $$\begin{aligned}&\displaystyle u'(t)+A(t)u(t)+f(t)=0, \quad t\in (t_1,t_2)\\&\displaystyle u(t_1)=\varphi \end{aligned}$$ in reflexive Banach space X. For $$f(s)\in C^1\bigl ([t_1,t_2],X\bigr )$$ , Kato obtained a unique strict solution under some conditions on the operator family $$ \{A(t)\}_{t_1 \le t \le t_2} $$ to ensure the hyperbolicity of the problem. In a previous paper, the author obtained in abstract Hilbert space H a unique strong solution $$u(t)\in C^{0,1}\bigl ([t_1,t_2],H\bigr )\cap D$$ if $$f\in BV\bigl ([t_1,t_2],H \bigr )$$ and a strict solution if additionally $$f\in C^{0}\bigl ([t_1,t_2],H\bigr )$$ . Here is $$ D=D\bigl ((A(t)\bigr )$$ independent of t. In the present paper we extend these results to reflexive Banach spaces X.

Analysis of a free boundary problem modeling the growth of multicell spheroids with angiogenesis

In this paper we study a free boundary problem modeling the growth of vascularized tumors. The model is a modification to the Byrne–Chaplain tumor model that has been intensively studied during the past two decades. The modification is made by replacing the Dirichlet boundary value condition with the Robin condition, which causes some new difficulties in making rigorous analysis of the model, particularly on existence and uniqueness of a radial stationary solution. In this paper we successfully solve this problem. We prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient, whereas unstable for small surface tension coefficient. Tools used in this analysis are the geometric theory of abstract parabolic differential equations in Banach spaces and spectral analysis of the linearized operator.