Nonlinear Functional Analysis Research Articles

Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects

In this article, we establish some sufficient conditions for total controllability of a neutral fractional differential system with impulsive conditions in the finite-dimensional spaces. This type of controllability concerns the controllability problem not only at the final time but also at the impulse time. We use Mittag-Leffler matrix function, nonlinear functional analysis, controllability Grammian type matrices and fixed point theorem due to Krasnoselskii’s to establish these results. Also, we extend these results for the studied problem with the integro term. At last, some simulated examples are given to validate the obtained results.

Nonlinear functional canonical correlation analysis via distance covariance

Functional canonical correlation analysis (FCCA) is a tool for exploring the associations between a pair of functional data. However, when the association is nonlinear or even nonmonotone, FCCA can fail to discover any meaningful relationship between the pair. In this paper, nonlinear FCCA estimators are constructed based on some popular measures of dependence — distance covariance and distance correlation. Consistency of the estimators is shown. Numerical studies are presented that demonstrate nonlinear FCCA can uncover new association patterns between functional covariates.

Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations.

Controllability results of fractional integro-differential equation with non-instantaneous impulses on time scales

Abstract In this work, we investigate the controllability results of a fractional integro-differential equation with non-instantaneous impulses on time scales. Banach contraction theorem and the non-linear functional analysis have been used to establish these results. In support, a numerical example with simulation for different time scales is given to validate the obtained analytical outcomes.

Stratified periodic water waves with singular density gradients

We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.

Observability of Fractional-Order Impulsive Control Integro-Differential System with Nonlocal Initial Condition

The article describes a new concept for initial and exactly observability of nonlocal fractional-order impulsive control integro-differential system. This is based on the concepts of the abstract Cauchy problem, which depended on some necessary and sufficient conditions. These conditions established on the semigroup theory of bounded operators as a dynamical operator system, which generated by bounded linear operators. Moreover, invertible operators play a primary role, and we presented a necessary condition for some nonlinear multi variables functions. Thus, all these operators were treated in nonlinear functional analysis to guaranty the initial observable and exactly observability. Therefore, from the mild solution of the system and exactly homogenous part, we proved the equivalent concepts between the initial observability and exactly the observability. Thus, our approach in this article is to prove the uniqueness of initial nonlocal values with admissible control, which belongs to the second-order Lebesgue integrable. The interest of observability results in this article lies by proving a unique fixed point, which is nonlocal initial values that are described in the proposal formula by using Banach’s fixed point theory. The processing observability for complexly systems (such as this system) with all components and properties was established and can be used for many control system applications.

Existence, Stability and Controllability Results of Coupled Fractional Dynamical System on Time Scales

In this work, we establish existence, uniqueness, Hyer-Ulam (HU) stability and controllability results for a coupled fractional dynamical system on time scales. Some fixed point theorems and nonlinear functional analysis have been used to establish these results. Also, we have given an example for different time scales, to show the applications of these obtained analytical results.

Existence for (p,q) critical systems in the Heisenberg group

AbstractThis paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p,q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p,q) systems.

Asymptotic Galerkin convergence and dynamical system results for the 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains

The spectrally-hyperviscous Navier–Stokes equations (SHNSE) represent a subgrid-scale model of turbulence for which previous studies were limited to periodic-box domains. Then in Avrin (The 3-D spectrally-hyperviscous Navier–Stokes equations on bounded domains with zero boundary conditions. arXiv:1908.11005 ) the SHNSE was adapted to general bounded domains with zero boundary conditions. Here we extend to this new setting the convergence and dynamical-system results in Avrin (J Dyn Differ Equ 20(2):479–518, 2008) and Avrin and Xiao (J Differ Equ 247(10):2778–2798, 2009), obtaining clear and straightforward Galerkin-convergence estimates, and in the case of decaying turbulence new convergence results featuring asymptotic decay rates in time. In extending the attractor-dimension results in Avrin (2008) our new degrees-of-freedom estimates stay strictly within the Landau–Lifschitz estimates (Landau and Lifshitz in Fluid mechanics, Addison-Wesley, Reading, 1959) for most computationally-relevant parameter values and exhibit a reduction in the number of degrees of freedom in calculations. The foundational properties of our bounded-domain setting also allow us to adapt the quadratic-form machinery of Temam (in: Brezis, Lions (eds) Nonlinear partial differential equations and their applications, Pitman, Boston, 1985; Browder (ed) Nonlinear functional analysis and its applications, American Mathematical Society, Providence, 1986) to carry over the main inertial-manifold results of Avrin (2008).

Localization and multiplicity in the homogenization of nonlinear problems

AbstractWe propose a method for the localization of solutions for a class of nonlinear problems arising in the homogenization theory. The method combines concepts and results from the linear theory of PDEs, linear periodic homogenization theory, and nonlinear functional analysis. Particularly, we use the Moser-Harnack inequality, arguments of fixed point theory and Ekeland's variational principle. A significant gain in the homogenization theory of nonlinear problems is that our method makes possible the emergence of finitely or infinitely many solutions.

Investigation of Ulam Stability Results of a Coupled System of Nonlinear Implicit Fractional Differential Equations

This manuscript deals with the existence theory, uniqueness, and various kinds of Ulam–Hyers stability of solutions for a class and coupled system of fractional order differential equations involving Caputo derivatives. Applying Schaefer and Banach’s fixed point approaches, existence and uniqueness results are obtained for the proposed problems. Stability results are investigated by using the classical technique of nonlinear functional analysis. Examples are given with each problem to illustrate the main results.

Ulam\u2013Hyers stability analysis to a class of nonlinear implicit impulsive fractional differential equations with three point boundary conditions

In this article, we discuss the sufficient conditions for the existence, uniqueness and stability of solutions to a class of nonlinear impulsive boundary value problem of fractional order differential equations. Using classical fixed point theorems, we develop the required conditions. Further, using the techniques of nonlinear functional analysis, we investigate Ulam–Hyers stability results to the proposed problem. For applications of our derived results, we present two numerical examples.

Ulam stability results to a class of nonlinear implicit boundary value problems of impulsive fractional differential equations

In this paper, we derive some sufficient conditions which ensure the existence and uniqueness of a solution for a class of nonlinear three point boundary value problems of fractional order implicit differential equations (FOIDEs) with some boundary and impulsive conditions. Also we investigate various types of Hyers–Ulam stability (HUS) for our concerned problem. Using classical fixed point theory and nonlinear functional analysis, we obtain the required conditions. In the last section we give an example to show the applicability of our obtained results.

Existence and stability of impulsive coupled system of fractional integrodifferential equations

Abstract In this manuscript, we deal with a class and coupled system of implicit fractional differential equations, having some initial and impulsive conditions. Existence and uniqueness results are obtained by means of Banach’s contraction mapping principle and Krasnoselskii’s fixed point theorem. Hyers–Ulam stability is investigated by using classical technique of nonlinear functional analysis. Finally, we provide illustrative examples to support our obtained results.

Existence theory and numerical solutions to smoking model under Caputo-Fabrizio fractional derivative.

In this paper, taking fractional derivative due to Caputo and Fabrizo, we have investigated a biological model of smoking type. By using Sumudu transform and Picard successive iterative technique, we develop the iterative solutions for the considered model. Furthermore, some results related to uniqueness of the equilibrium solution and its stability are discussed utilizing the techniques of nonlinear functional analysis. The dynamics of iterative solutions for various compartments of the model are plotted with the help of Matlab.

Analysis of the Tailored Coupled-Cluster Method in Quantum Chemistry

In quantum chemistry, one of the most important challenges is the static correlation problem when solving the electronic Schrödinger equation for molecules in the Born--Oppenheimer approximation. In this article, we analyze the tailored coupled-cluster method (TCC), one particular and promising method for treating molecular electronic-structure problems with static correlation. The TCC method combines the single-reference coupled-cluster (CC) approach with an approximate reference calculation in a subspace (complete active space (CAS)) of the considered Hilbert space that covers the static correlation. A one-particle spectral gap assumption is introduced, separating the CAS from the remaining Hilbert space. This replaces the nonexisting or nearly nonexisting gap between the highest occupied molecular orbital and the lowest unoccupied molecular orbital usually encountered in standard single-reference quantum chemistry. The analysis covers, in particular, CC methods tailored by tensor-network states (TNS-TCC methods). The problem is formulated in a nonlinear functional analysis framework, and, under certain conditions such as the aforementioned gap, local uniqueness and existence are proved using Zarantonello's lemma. From the Aubin--Nitsche-duality method, a quadratic error bound valid for TNS-TCC methods is derived, e.g., for linear-tensor-network TCC schemes using the density matrix renormalization group method.

The work of Norman Dancer

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions. The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work.

Математичне моделювання коливних процесів у смузі

The boundary-value periodic problem for differential equations in partial derivatives, including hyperbolic equations, are complicated and controversial subject to study. Boundary-value problems with data throughout the border region as well as the problem of non-local (including integrated) conditions for hyperbolic equations in limited areas are, generally speaking, relatively correct. Many authors link the solvability of such problems with the problem of small denominators and use the methods of nonlinear functional analysis, the theory of implicit functions, variation methods. Another authors use the analytical methods in the research of periodic boundary-value problems for the second order hyperbolic equations. They construct integral operators and search the solutions in specially defined spaces of continuously differentiated functions for specific cases of periodicity.In this paper we find an analytic formula of the function v(x, t), which is a solution of the boundary-value 2π-periodic time-varying problem in the class of odd functions for which f (t) = – f(π – t). The properties of this function are established and the estimates of the solution of the boundary-value 2π-periodic problem are given.The results of the study are used for mathematical modeling of oscillating processes described by the second order hyperbolic equations.On the basis of the found function v(x, t) we can draw conclusions about the behavior of the solution of the undisturbed equation (ε = 0, ε is a small parameter) in the study of the general nonlinear the second order hyperbolic equation by the asymptotic methods.

Ulam stability to a toppled systems of nonlinear implicit fractional order boundary value problem

In this manuscript, we give some sufficient conditions for existence, uniqueness and various kinds of Ulam stability for a toppled system of fractional order boundary value problems involving the Riemann–Liouville fractional derivative. Applying the Banach contraction principle and the Leray–Schauder result of cone type, uniqueness and existence results are proved for the proposed toppled system. Stability is investigated by using the classical technique of nonlinear functional analysis. The results obtained are well illustrated with the aid of an example.

Existence and stability of fractional integro differential equation with non-instantaneous integrable impulses and periodic boundary condition on time scales

The present paper is devoted to the study of existence and stability of fractional integro differential equation with non-instantaneous impulses and periodic boundary condition on time scales. This paper consists of two segments: the first segment of the work is concerned with the theory of existence, uniqueness and the other segment is to Hyer’s-Ulam type’s stability analysis. The tools for study include the Banach fixed point theorem and nonlinear functional analysis. Finally, in support, an example is presented to validate the obtained results.