Equations In Banach Spaces Research Articles

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.

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

Asymptotic Behavior of Mild Solutions for Nonlinear Fractional Difference Equations

AbstractIn this paper, we establish some sufficient criteria for the existence, uniqueness of discrete weighted pseudo asymptotically periodic mild solutions and asymptotic behavior for nonlinear fractional difference equations in Banach space, where the nonlinear perturbation is Lipschitz type, or non-Lipschitz type. The results are a consequence of application of different fixed point theorems, namely, the Banach contraction mapping principle, Leray-Schauder alternative theorem and Matkowski’s fixed point technique.

Local convergence for a Chebyshev-type method in Banach space free of derivatives

This paper is devoted to the study of a Chebyshev-type method free of derivatives for solving nonlinear equations in Banach spaces. Using the idea of restricted convergence domain, we extended the applicability of the Chebyshev-type methods. Our convergence conditions are weaker than the conditions used in earlier studies. Therefore the applicability of the method is extended. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

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.

Unique strong and strict solutions for inhomogeneous hyperbolic differential equations in Banach space

The purpose of the present paper is the improvement of results for solutions of the inhomogeneous differential equation: $$\begin{aligned} u'(t)+A(t)u(t)+f(t)= & {} 0, \quad t\in (t_1,t_2)\\ u(t_1)= & {} \varphi \end{aligned}$$ 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.

Approximate solutions of impulsive integro-differential equations

In this paper, we consider impulsive integro-differential equations in Banach space and we establish the bound on the difference between two approximate solutions. We also discuss nearness and convergence of solutions of the problem under consideration. The impulsive integral inequality of Grownwall type is used to obtain results.

The generalized hyperstability of general linear equations in quasi-Banach spaces

In this paper, we study the hyperstability for the general linear equation in the setting of quasi-Banach spaces. We first extend the fixed point result of Brzdek et al. [5, Theorem 1] in metric spaces to b-metric spaces, in particular to quasi-Banach spaces. Then we use this result to generalize the main results on the hyperstability for the general linear equation in Banach spaces to quasi-Banach spaces. We also show that we can not omit the assumption of completeness in [5, Theorem 1]. As a consequence, we conclude that we need more explanations to replace a normed space by its completion in the proofs of some results in the literature.

Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

ABSTRACTThe aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.

On Existence and uniqueness to solutions nonlocal integrodifferential equations

The Study aims in this paper to give and investigate the existence and uniqueness of mild solutions to nonlinear functional integrodifferential equations in Banach Spaces. the fixed point theorem, according to Sadovskii and sutible necessary conditions, are concepts consulted to obtain the results in the work

Dichotomous solutions of linear impulsive differential equations

The notion of Lp(h,k) solutions of linear impulsive differential equations in Banach spaces is introduced. Sufficient conditions for existence of such solutions are derived. Possible applications to linear control systems with impulses are considered. An illustrative example is given.

A STABLE METHOD FOR LINEAR EQUATION IN BANACH SPACES WITH SMOOTH NORMS

A stable method for numerical solution of a linear operator equation in reflexive Banach spaces is proposed. The operator and the right-hand side of the equation are assumed to be known approximately. The corresponding error levels may remain unknown. Approximate operators and their conjugate ones must possess the property of strong pointwise convergence. The exact normal solution is assumed to be sourcewise representable and some upper estimate for the norm of its source element must be known. The norm in the Banach space of solutions is supposed to satisfy the following smoothness-type condition: some function of the norm must be differentiable. Under these conditions a stability of the method with respect to nonuniform perturbations in operator is shown and the strong convergence to the normal solution is proved. A boundary control problem for the one-dimensional wave equation is considered as an example of possible application. The results of the model numerical experiments are presented.

Stability for Linear Volterra Difference Equations in Banach Spaces

This paper is devoted to studying the existence and stability of implicit Volterra difference equations in Banach spaces. The proofs of our results are carried out by using an appropriate extension of the freezing method to Volterra difference equations in Banach spaces. Besides, sharp explicit stability conditions are derived.

Optimal Control Existence for Degenerate Infinite Dimensional Systems of Fractional Order

Abstract Optimal control problems solvability are researched for infinite dimensional control systems, described by semilinear evolution equations in Banach spaces with degenerate linear operator at the Caputo fractional derivative. The pair of linear operators in the equation is relatively bounded and the nonlinear operator satisfies some smoothness conditions, in particular the condition of uniform Lipschitz continuity, and one of two types additional conditions: independence of degeneracy subspace elements or non-belonging of the operator image to the degeneracy subspace. The control system is endowed by the generalized Showalter — Sidorov initial conditions, which are natural for degenerate evolution equations. Optimal control have to belong to a convex closed set of admissible controls and to minimize a convex, bounded from below, lower semicontinuous and coercive cost functional. Solvability conditions are found for the optimal control problem of this class. If the existence of the initial problem solution with an admissible control is obvious, it is shown that the local Lipschitz continuity in phase variables that uniform with respect to time is sufficient for the optimal control existence. Abstract results are illustrated by optimal control problem for the equations system of the fractional viscoelastic Kelvin — Voigt fluid dynamics.

О ФОРМАЛЬНОМ ПРЕДСТАВЛЕНИИ РЕШЕНИЙ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ДРОБНОГО ПОРЯДКА

The paper presents a formal representation of solutions of non-scalar semilinear differential equations in Banach spaces by means of the Mattag-Leffler function.

Semilocal convergence analysis of a fifth-order method using recurrence relations in Banach space under weak conditions

In the present paper, we consider a fifth-order method considered in Singh et al. (2016) to solve equations in Banach space under weaker assumptions. Using the idea of restricted convergence domains we extend the applicability of the method considered by

Existence results for second order linear differential equations in Banach spaces

In this paper we are concerned with the existence of solutions for certain classes of second order differential equations. First we deal with an infinite system of second order linear differential equations, which is reduced to an ordinary differential equation posed in the space of convergent sequences. Next we investigate the problem of existence for a second order differential equation posed on an arbitrary Banach space. The used approach is based on the measures of noncompactness concept, the use of Darbo's fixed point theorem and Kamke comparison functions.

Lavrentiev's regularization method for nonlinear ill-posed equations in Banach spaces

In this paper, we deal with nonlinear ill-posed problems involving m-accretive mappings in Banach spaces. We consider a derivative and inverse free method for the implementation of Lavrentiev regularization method. Using general Hölder type source condition we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

Solution and Generalized Ulam-Hyers Stability of a n-Dimensional Additive Functional Equation in Banach Space and Banach Algebra: Direct and Fixed Point Methods

Solution and Generalized Ulam-Hyers Stability of a n-Dimensional Additive Functional Equation in Banach Space and Banach Algebra: Direct and Fixed Point Methods

On approximate solutions for a class of semilinear fractional-order differential equations in Banach spaces

We apply the topological degree theory for condensing maps to study approximation of solutions to a fractional-order semilinear differential equation in a Banach space. We assume that the linear part of the equation is a closed unbounded generator of a C_{0}-semigroup. We also suppose that the nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to obtain a result on the approximation of solutions to the problem.