Differential Equations In Banach Spaces Research Articles

Well-Posedness and Approximation for Nonhomogeneous Fractional Differential Equations

In this paper, we consider the well-posedness and approximation for nonhomogeneous fractional differential equations in Banach spaces E. Firstly, we get the necessary and sufficient condition for the well-posedness of nonhomogeneous fractional Cauchy problems in the spaces Secondly, by using implicit difference scheme and explicit difference scheme, we deal with the full discretization of the solutions of nonhomogeneous fractional differential equations in time variables, get the stability of the schemes and the order of convergence.

Almost Periodic Solutions of Differential Equations

We construct the Favard–Amerio theory for almost periodic differential equations in Banach spaces without using the ℋ-classes of these equations. For linear equations, we present the first examples of almost periodic operators that have no analogs in the classical Favard–Amerio theory.

Hölder regularity for abstract semi-linear fractional differential equations in Banach spaces

In the present work the optimal regularity, in the sense of Hölder continuity, of linear and semi-linear abstract fractional differential equations is investigated in the framework of complex Banach spaces. This framework has been considered by the authors as the most convenient to provide a posteriori error estimates for the time discretizations of such a kind of abstract differential equations. In the spirit of the classical a posteriori error estimates, under certain assumptions, the error is bounded in terms of computable quantities, in our case measured in the norm of Hölder continuous and weighted Hölder continuous functions.

S-Asymptotically Bloch Type Periodic Solutions to Some Semi-Linear Evolution Equations in Banach Spaces

This paper is mainly concerned with the S-asymptotically Bloch type periodicity. Firstly, we introduce a new notion of S-asymptotically Bloch type periodic functions, which can be seen as a generalization of concepts of S-asymptotically ω-periodic functions and S-asymptotically ω-anti-periodic functions. Secondly, we establish some fundamental properties on S-asymptotically Bloch type periodic functions. Finally, we apply the results obtained to investigate the existence and uniqueness of S-asymptotically Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces.

Periodic solutions and attractiveness for some partial functional differential equations with lack of compactness

This paper deals with the existence of periodic solutions and attractiveness for some partial functional differential equations in Banach spaces. We assume that the first linear part generates a strongly continuous semigroup, while the delayed part is periodic with respect to the first argument. We prove that the existence of a bounded solution implies the existence of a periodic solution. Several results regarding uniqueness and global attractiveness of periodic solutions are also established. The analysis relies on a fixed point theorem of Chow and Hale’s type and uses some arguments of weak topology. Our theorems extend in a broad sense some new and classical related results. An application to a transport equation with delay is also presented.

Reduction principle at work

Abstract The purpose of this informal paper is three-fold: First, filling a gap in the literature, we provide a (necessary and sufficient) principle of linearized stability for nonautonomous difference equations in Banach spaces based on the dichotomy spectrum. Second, complementing the above, we survey and exemplify an ambient nonautonomous and infinite-dimensional center manifold reduction, that is Pliss’s reduction principle suitable for critical stability situations. Third, these results are applied to integrodifference equations of Hammerstein- and Urysohn-type both in C- and Lp -spaces. Specific features of the nonautonomous case are underlined. Yet, for the simpler situation of periodic time-dependence even explicit computations are feasible.

Dynamics and stability for Katugampola random fractional differential equations

<abstract> This paper deals with some existence of random solutions and the Ulam stability for a class of Katugampola random fractional differential equations in Banach spaces. A random fixed point theorem is used for the existence of random solutions, and we prove that our problem is generalized Ulam-Hyers-Rassias stable. An illustrative example is presented in the last section. </abstract>

Asymptotic behavior of solutions to difference equations in Banach spaces

We investigate the asymptotic properties of solutions to higher order nonlinear difference equations in Banach spaces. We introduce a new technique based on a vector version of discrete L'Hospital's rule, remainder operator, and the regional topology on the space of all sequences on a given Banach space. We establish sufficient conditions for the existence of solutions with prescribed asymptotic behavior. Moreover, we are dealing with the problem of approximation of solutions. Our technique allows us to control the degree of approximation of solutions.

Existence of solutions for Hadamard fractional differential equations in Banach spaces

In this work, we investigate the existence of solutions for Hadamard fractional differential equations with integral boundary conditions in a Banach space. We will make use the measure of noncompactness and the Monch fixed point theorem to prove the ¨ main results. An example is given to illustrate our results.

On the controllability of Hilfer-Katugampola fractional differential equations

By employing Kuratowski's measure of noncompactness together with Sadovskii's fixed point theorem, sufficient conditions for controllability results of Hilfer-Katugampola fractional differential equations in Banach spaces are derived.

Iterative Solution for Systems of a Class of Abstract Operator Equations in Banach Spaces and Application

In this paper, by using the partial order method, the existence and uniqueness of a solution for systems of a class of abstract operator equations in Banach spaces are discussed. The result obtained in this paper improves and unifies many recent results. Two applications to the system of nonlinear differential equations and the systems of nonlinear differential equations in Banach spaces are given, and the unique solution and interactive sequences which converge the unique solution and the error estimation are obtained.

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

Nonlocal Fractional Differential Equations and Applications

The regularity properties of nonlocal fractional differential equations in Banach spaces are studied. Uniform $$L_{p}$$ -separability properties and sharp resolvent estimates are obtained for abstract elliptic operator in terms of fractional derivatives. Particularly, it is proven that the fractional elliptic operator generated by these equations is sectorial and also is a generator of an analytic semigroup. Moreover, maximal regularity properties of nonlocal fractional abstract parabolic equation are established. As an application, the nonlocal anisotropic fractional differential equations and the system of nonlocal fractional differential equations are studied.

On the stability and stabilization of some semilinear fractional differential equations in Banach spaces

In this paper, we prove the existence and uniqueness of a mild solution to a class of semilinear fractional differential equation in an infinite Banach space with Caputo derivative order 0 < α ≤ 1. Furthermore, we establish the stability conditions and then prove that the considered initial value problem is exponentially stabilizable when the stabilizer acts linearly on the control system.

Asymptotic stability of a perturbed abstract differential equations in Banach spaces

Asymptotic stability of a perturbed abstract differential equations in Banach spaces

Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces

&lt;abstract&gt; &lt;p&gt;The aim of the reported results in this manuscript is to handle the existence, uniqueness, extremal solutions, and Ulam-Hyers stability of solutions for a class of $ \Psi $-Caputo fractional relaxation differential equations and a coupled system of $ \Psi $-Caputo fractional relaxation differential equations in Banach spaces. The obtained results are derived by different methods of nonlinear analysis like the method of upper and lower solutions along with monotone iterative technique, Banach contraction principle, and Mönch's fixed point theorem concerted with the measures of noncompactness. Furthermore, the Ulam-Hyers stability of the proposed system is studied. Finally, two examples are presented to illustrate our theoretical findings. Our acquired results are recent in the frame of a $ \Psi $-Caputo derivative with initial conditions in Banach spaces via the monotone iterative technique. As a results, we aim to fill this gap in the literature and contribute to enriching this academic area.&lt;/p&gt; &lt;/abstract&gt;

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&lt;s&lt;1{-1&lt;s&lt;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.