Differential Equations In Banach Spaces Research Articles

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results.

Differential Equations in Banach Spaces with an Noninvertible Operator in the Principal Part and Nonclassical Initial Conditions

Differential Equations in Banach Spaces with an Noninvertible Operator in the Principal Part and Nonclassical Initial Conditions

Hausdorff and fractal dimensions of attractors for functional differential equations in Banach spaces

The main objective of this paper is to obtain estimations of Hausdorff dimension as well as fractal dimension of global attractors and pullback attractors for both autonomous and nonautonomous functional differential equations (FDEs) in Banach spaces. New criterions for the finite Hausdorff dimension and fractal dimension of attractors in Banach spaces are firslty proposed by combining the squeezing property and the covering of finite subspace of Banach spaces, which generalize the method established in Hilbert spaces. In order to surmount the barrier caused by the lack of orthogonal projectors with finite rank, which is the key tool for proving the squeezing property of partial differential equations in Hilbert spaces, we adopt the state decomposition of phase space based on the exponential dichotomy of the studied FDEs to obtain similar squeezing property. The theoretical results are applied to a retarded nonlinear reaction-diffusion equation and a non-autonomous retarded functional differential equation in the natural phase space, for which explicit bounds of dimensions that do not depend on the entropy number but only depend on the spectrum of the linear parts and Lipschitz constants of the nonlinear parts are obtained.

Existence and Regularity of solutions in alpha norm for some second order partial neutral functional differential equations in Banach Spaces

The purpose of this paper is to study the existence and regularity of solutions in the $\alpha$-norm for  some second order partial neutral functional differential equations with finite delay in Banach spaces. To do this, we use the cosine family theory and Schauder's fixed point Theorem to establish the local existence of solutions and we give some sufficients conditions to enssure the regularity of solutions. Finally, we give an example to illustrate the theoretical results.

Theoretical analysis of a class of $ \varphi $-Caputo fractional differential equations in Banach space

<abstract><p>A study of a class of nonlinear differential equations involving the $ \varphi $-Caputo type derivative in a Banach space framework is presented. Weissinger's and Meir-Keeler's fixed-point theorems are used to achieve some quantitative results. Two illustrative examples are provided to justify the theoretical results.</p></abstract>

A Study of an IBVP of Fractional Differential Equations in Banach Space via the Measure of Noncompactness

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, ς and τ, such that ς∈1,2 and τ∈0,ς. Our paper ends with a conclusion.

Solution Stability of Delay Differential Equations in Banach Spaces

Solution Stability of Delay Differential Equations in Banach Spaces

Existence of mild solutions of second-order impulsive differential equations in Banach spaces

We discuss the existence of solutions for second-order impulsive differential equation with nonlocal conditions in Banach spaces. Our approach is based on the generalization of Schauder fixed pointprinciple that is Darbo fixed point theorem. An example is also presented to for illustration.

NEW RESULTS ON K- STABILITY AND BOUNDEDNESS OF SOLUTIONS OF A CERTAIN DIFFERENTIAL EQUATIONS IN BANACH SPACES INTERACTING WITH WEAKLY SUBSYSTEMS

The article gives new results of K- Stability and Boundedness of solutions of certain differential equations in Banach Spaces interacting with weakly subsystems. Some results from the theory of semigroups, and the description of the general method of solution of the posed problem were also given.  The application of the matrix – valued Lyapunov function is discussed and the main theorem of the method is formulated for differential equations in a Banach Space. The vector Lyapunov function was the main tool applied for the analysis of K – Stability and Boundedness of a system in Banach Spaces. All the results of this Article for the ‘equation (2.2)’ are essentially new.

On uniform polynomial splitting of variational nonautonomous difference equations in Banach spaces

Abstract In this paper we consider a concept of uniform polynomial splitting for a discrete cocycle over a discrete semiflow in Banach spaces. We obtain some characterizations of Datko type and also in terms of Lyapunov functions. The study is made from the point of view of the invariant projectors and also strongly invariant projectors.

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.

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.

Generalized Bloch type periodicity and applications to semi-linear differential equations in banach spaces

AbstractIn this paper, we mainly introduce some new notions of generalized Bloch type periodic functions namely pseudo Bloch type periodic functions and weighted pseudo Bloch type periodic functions. A Bloch type periodic function may not be Bloch type periodic under certain small perturbations while it can be quasi Bloch type periodic in sense of generalized Bloch type periodic functions. We firstly show the completeness of spaces of generalized Bloch type periodic functions and establish some further properties such as composition and convolution theorems of such functions. We then apply these results to investigate existence results for generalized Bloch type periodic mild solutions to some semi-linear differential equations in Banach spaces. The obtained results show that for each generalized Bloch type periodic input forcing disturbance, the output mild solutions to reference evolution equations remain generalized Bloch type periodic.

Approximate Fixed Point Theory for Countably Condensing Maps and Multimaps and Applications

In this paper, we establish an approximate fixed point theorem for Γ-τ-sequentially continuous countably -condensing mappings. As application, we obtain an approximate fixed point result for demicontinuous mappings and use this to prove new results in asymptotic fixed point theory. Moreover, we obtain new fixed point results for some countably -condensing mappings and multivalued mappings and fixed point results of Krasnoselskii-Daher type for the sum of two τ-sequentially continuous mappings in non-separable Hausdorff topological vector spaces. We also present a multivalued version of an approximation result of Ky Fan for τ-Γ-s.l.sc multivalued mappings. Apart from that, we show the applicability of our results to the theory of Volterra integral equations in Banach spaces and we prove the existence of limiting-weak solutions for differential equations in Banach spaces not necessarily reflexive. Our results extend the results of Banaś and Ben Amar, Barroso and Seda.

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

Periodic Solutions to Second-Order Nonlinear Differential Equations in Banach Spaces

In this article, we deal with the existence of solutions for the following second-order differential equation: u′′(t)=f(t,u(t))+h(t)u(a)-u(b)=u′(a)-u′(b)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{aligned}&u''(t)=f(t,u(t))+h(t)\\\\&u(a)-u(b)= u'(a)-u'(b)=0, \\end{aligned}\\right. \\end{aligned}$$\\end{document}where {mathbb {B}} is a reflexive real Banach space, f:[a,b]times {mathbb {B}}rightarrow {mathbb {B}} is a sequentially weak–strong continuous mapping, and h:[a,b]rightarrow {mathbb {B}} is a continuous function on {mathbb {B}}. The proofs are obtained using a recent generalization of the well-known Bolzano–Poincaré–Miranda theorem to infinite-dimensional Banach spaces. In the last section, we present three examples of application of the general result.

Boundary Value Problem for ψ-Caputo Fractional Differential Equations in Banach Spaces via Densifiability Techniques

A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the ψ-Caputo fractional derivative in Banach spaces. Besides that, an example is included to verify our main results. Moreover, the outcomes obtained in this research paper ameliorate and expand some previous findings in this area.

Stability Analysis of Multistage One-Step Multiderivative Methods for Nonlinear Impulsive Differential Equations in Banach Spaces

In this paper, we first introduce the problem class K p μ , λ , ζ with respect to the initial value problems of nonlinear impulsive differential equations in Banach spaces. The stability and asymptotic stability results of the analytic solution of the problem class K p μ , λ , ζ are obtained. Then, the numerical stability and asymptotic stability conditions of multistage one-step multiderivative methods are also given. Two numerical experiments are given to confirm the theoretical results in the end.

Asymptotic stability of fractional order (1,2] stochastic delay differential equations in Banach spaces

• In this article we have studied the asymptotic stability and mean square stability of fractional-order (1,2] stochastic differential equations (SDEs). • We have considered the family of SDEs with variable delay in state. • For obtaining main results we used Banach fixed point theorem and imposed Lipschitz condition on nonlinearity. • Results are advanced and weighted enough as a contribution to stability theory of fractional SDEs. In this article, we discuss the asymptotic stability and mean square stability of stochastic differential equations of fractional-order 1 < α ≤ 2 . We have considered the family of stochastic differential equations with variable delay in the state. For proving our main results, we apply the Banach fixed point theorem and imposed the Lipschitz condition on nonlinearity. Finally, we present an example to illustrate the obtained theory.

Global attractivity of delay difference equations in Banach spaces via fixed-point theory

We formulate initial value problems for delay difference equations in Banach spaces as fixed-point problems in sequence spaces. By choosing appropriate sequence spaces various types of attractivity can be described. This allows us to establish global attractivity by means of fixed-point results. Finally, we provide an application to delay integrodifference equations in the space of continuous functions over a compact domain.