Equations In Banach Spaces Research Articles

Abstract fractional linear pseudo-parabolic equations in Banach spaces: well-posedness, regularity, and asymptotic behavior

In this paper we study the well-posedness, regularity, and asymptotic behavior of the solutions to the abstract pseudo-parabolic equation $$\begin{aligned} \partial _t^\alpha u(t) = A u(t) + B\partial _t^\beta u(t) + f(t), \end{aligned}$$where A, B are closed linear operators in a Banach space, and \(\partial _t^\gamma u\) denotes the Caputo or Riemann–Liouville fractional derivative of order \(\gamma >0\).

ON THE CONVERGENCE OF THE SIXTH ORDER HOMEIER LIKE METHOD IN BANACH SPACES

A sixth order Homeier-like method is introduced for approximating a solution of the non-linear equation in Banach space. Assumptions only on first and second derivatives are used to obtain a sixth order convergence. Our proof does not depend on Taylor series expansions as in the earlier studies for the similar methods.

Stability analysis of implicit–explicit [formula omitted]-methods for composite stiff neutral functional differential equations in Banach space

This paper is concerned with the stability and asymptotic stability of implicit–explicit (IMEX) θ-methods for composite stiff neutral functional differential equations in Banach space. The stability and asymptotic stability conditions of the methods are obtained. The given numerical experiments confirmed the accuracy of the obtained theoretical results and showed that the IMEX θ-methods have a higher computational efficiency compared to the θ-methods in the end.

Ball Convergence of Modified Homeier-Like's Method in Banach Spaces under Weak Continuity Condition

The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. $\omega$-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution.

On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral

Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.

Direct and Fixed-Point Stability–Instability of Additive Functional Equation in Banach and Quasi-Beta Normed Spaces

Over the last few decades, a certain interesting class of functional equations were developed while obtaining the generating functions of many system distributions. This class of equations has numerous applications in many modern disciplines such as wireless networks and communications. The Ulam stability theorem can be applied to numerous functional equations in investigating the stability when approximated in Banach spaces, Banach algebra, and so on. The main focus of this study is to analyse the relationship between functional equations, Hyers–Ulam–Rassias stability, Banach space, quasi-beta normed spaces, and fixed-point theory in depth. The significance of this work is the incorporation of the stability of the generalised additive functional equation in Banach space and quasi-beta normed spaces by employing concrete techniques like direct and fixed-point theory methods. They are powerful tools for narrowing down the mathematical models that describe a wide range of events. Some classes of functional equations, in particular, have lately emerged from a variety of applications, such as Fourier transforms and the Laplace transforms. This study uses linear transformation to explain our functional equations while providing suitable examples.

Hyperstability of cubic functional equation in banach space

Hyperstability of cubic functional equation in banach space

Approximate controllability of semi-linear stochastic integrodifferential system with multiple delays and Poisson jumps in control

The objective of this paper is to interpret the approximate controllability of a semi-linear stochastic integrodifferential system with multiple delays and Poisson jumps in control in infinite-dimensional spaces. Sufficient conditions for the approximate controllability of semi-linear control system have been established. The results are obtained using the Banach fixed point theorem and the theory of resolvent operator developed in Grimmer [Resolvent operator for integral equations in Banach spaces, Trans. Am. Math. Soc. 273 (1982), pp. 333– 349.]. An example is introduced to show the effectiveness of the result.

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of C0-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results.

Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.

Controllability and Hyers-Ulam Stability of Impulsive Integro-differential Equations in Banach Spaces via Iterative Methods

We investigate Hyers-Ulam stability  and controllability of a system governed by  impulsive  integro-differential equations.  Sufficient conditions for controllability and Hyers-Ulam stability are simultaneously established  by using iterative methods.  Moreover, we provide some examples for demonstrating  the main result.

Extended Convergence of Three Step Iterative Methods for Solving Equations in Banach Space with Applications

Symmetries are vital in the study of physical phenomena such as quantum physics and the micro-world, among others. Then, these phenomena reduce to solving nonlinear equations in abstract spaces. These equations in turn are mostly solved iteratively. That is why the objective of this paper was to obtain a uniform way to study three-step iterative methods to solve equations defined on Banach spaces. The convergence is established by using information appearing in these methods. This is in contrast to earlier works which relied on derivatives of the higher order to establish the convergence. The numerical example completes this paper.

Permanence of nonuniform nonautonomous hyperbolicity for infinite-dimensional differential equations

In this paper, we study stability properties of nonuniform hyperbolicity for evolution processes associated with differential equations in Banach spaces. We prove a robustness result of nonuniform hyperbolicity for linear evolution processes, that is, we show that the property of admitting a nonuniform exponential dichotomy is stable under perturbation. Moreover, we provide conditions to obtain uniqueness and continuous dependence of projections associated with nonuniform exponential dichotomies. We also present an example of evolution process in a Banach space that admits nonuniform exponential dichotomy and study the permanence of the nonuniform hyperbolicity under perturbation. Finally, we prove persistence of nonuniform hyperbolic solutions for nonlinear evolution processes under perturbations.

Integro-differential equations with bounded operators in Banach spaces

The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.

Optimal controls for nonlocal Cauchy problems of multi-term fractional evolution equations

Abstract This paper is mainly concerned with a controlled multi-term fractional evolution equation in Banach spaces. Firstly, we give formula of its mild solutions and show the existence result for the problem via $\omega $-sectorial operator technique. Secondly, we establish the Lagrange optimal control and time optimal control for the system invoked by the nonlocal Cauchy problems of multi-term fractional evolution equation by properties of resolvent operators.

On Asymptotic Behavior of a 2-Linear Functional Equation

In this paper, we deal with a 2-linear functional equation. The Hyers-Ulam stability of this functional equation is shown on some restricted unbounded domains, and the obtained results are applied to get several hyperstability consequences. Moreover, some asymptotic behaviors of 2-linear functions are investigated. We also study the Hyers-Ulam stability and superstability of the 2-linear functional equation in 2-Banach spaces.

Local Convergence of a Seventh Order Derivative-Free Method for Solving Nonlinear Equations in Banach Spaces

Local Convergence of a Seventh Order Derivative-Free Method for Solving Nonlinear Equations in Banach Spaces

Qualitative Analyses of ψ-Caputo Type Fractional Integrodifferential Equations in Banach Spaces

In this research paper, we develop and extend some qualitative analyses of a class of a nonlinear fractional integro-differential equation involving ψ-Caputo fractional derivative (ψ-CFD) and ψ-Riemann-Liouville fractional integral (ψ-RLFI). The existence and uniqueness theorems are obtained in Banach spaces via an equivalent fractional integral equation with the help of Banach’s fixed point theorem (B’sFPT) and Schaefer’s fixed point theorem (S’sFPT). An example explaining the main results is also constructed.

Approximation of nonlinear infinite delay equations in Banach spaces by finite delay equations

Approximation of nonlinear infinite delay equations in Banach spaces by finite delay equations

Well-posedness of a mathematical model of diabetic atherosclerosis with advanced glycation end-products

Atherosclerosis is a leading cause of death worldwide; it emerges as a result of multiple dynamical cell processes, including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. The pathophysiology of diabetic vascular disease is generally understood. Dyslipidemia with increased levels of atherogenic low-density lipoprotein, hyperglycemia, oxidative stress and increased inflammation are factors that increase the risk and accelerate development of atherosclerosis. In a recent paper, Xie [Well-posedness of a mathematical model of diabetic atherosclerosis. J Math Anal Appl. 2022;505(2):125606], we have developed a mathematical model that includes the effect of hyperglycemia and insulin resistance on plaque growth. In this paper, we propose a more comprehensive mathematical model for diabetic atherosclerosis which include more variables; in particular, it includes the variable for Advanced Glycation End-Products (AGEs) concentration. Hyperglycemia trigger vascular damage by forming AGEs, which are not easily metabolized and may accelerate the progression of vascular disease in diabetic patients. The model is given by a system of partial differential equations (PDEs) with a free boundary. We also establish the local existence and uniqueness of solution to the model. The methodology is to reduce the free boundary to a fixed boundary and the system of PDEs to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.