Nonlinear Functional Analysis Research Articles

Localization and multiplicity in the homogenization of nonlinear problems

Abstract We 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.

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.

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.

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.

Hyers‐Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions

In this article, we deal with the existence and Hyers‐Ulam stability of solution to a class of implicit fractional differential equations (FDEs), having some initial and impulsive conditions. Some adequate conditions for the required results are obtained by utilizing fixed point theory and nonlinear functional analysis. At the end, we provide an illustrative example to demonstrate the applications of our obtained results.

Establishing the existence of Hilfer fractional pantograph equations with impulses

In [1], the authors established the existence of a class of fractional differential equations of a complex order. In this note, we derive some sufficient conditions for the existence of solutions to a class of Hilfer fractional pantograph equations with impulsive effect. Further, using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss various kinds of Ulam-Hyers stability.

A mixed virtual element method for a nonlinear Brinkman model of porous media flow

In this work we introduce and analyze a mixed virtual element method for the two-dimensional nonlinear Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. For the continuous formulation we consider a dual-mixed approach in which the main unknowns are given by the gradient of the velocity and the pseudostress, whereas the velocity itself and the pressure are computed via simple postprocessing formulae. In addition, because of analysis reasons we add a redundant term arising from the constitutive equation relating the pseudostress and the velocity, so that the well-posedness of the resulting augmented formulation is established by using known results from nonlinear functional analysis. Then, we introduce the main features of the mixed virtual element method, which employs an explicit piecewise polynomial subspace and a virtual element subspace for approximating the aforementioned main unknowns, respectively. In turn, the associated computable discrete nonlinear operator is defined in terms of the $$\mathbb {L}^2$$ -orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. Next, we show the well-posedness of the discrete scheme and derive the associated a priori error estimates for the virtual element solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken $$\mathbb {H}(\mathbf {div})$$ -norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

The construction and comparison of damage detection index based on the nonlinear output frequency response function and experimental analysis

When the structures or parts are subjected to fatigue damage, its nonlinear characteristics will be characterized sensitively. The nonlinear characteristics of the system can be well described by the nonlinear output frequency response function deduced by the Volterra series. The nonlinear output frequency response function is the collection of frequency functions of various orders which contain different nonlinear information about the system. In other words, it is not intuitive when it is used directly for the detection of fatigue damage to structures or parts. Therefore, how to build an intuitive detection index will be the key to the application of the nonlinear output frequency response function in engineering practice. In this paper, based on the concept of information entropy, complexity and divergence, 4 detection indexes were proposed and the definition and estimation methods were given. Fatigue tests were carried out on the specimens by a fatigue test machine, and a series of specimens with different fatigue cycle times were obtained. The nonlinear output frequency response function was estimated by using the data of the hammer excitation measurement. The detection effect of 4 detection indexes and traditional detection index Fe was analyzed and compared. The results show that the proposed detection indexes are more sensitive and effective for detecting fatigue damage before microcracks formed.

Ulam stability results for the solutions of nonlinear implicit fractional order differential equations

In this manuscript, we study the existence and uniqueness of solution for a class of fractional order boundary value problem (FBVP) for implicit fractional differential equations with Riemann-Liouville derivative. Furthermore, we investigate different kinds of Ulam stability such as Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability for the proposed problem. The concerned analysis is carried out through using classical technique of nonlinear functional analysis. The main results are illustrated by providing a couple of examples

On the Controllability of Linear and Semilinear Impulsive Systems

ABSTRACTIn this paper, we establish some sufficient conditions for the complete controllability of linear and semilinear impulsive systems. For the semilinear systems, we assume that nonlinearities are Lipschitzian type. We invoke the tools of nonlinear functional analysis and fixed point theorems to establish the results. The results are also compared with the existing results in the literature in order to show the effectiveness of the results obtained. Few illustrative examples are provided to substantiate the results.

Mathematical analysis of HIV/AIDS infection model with Caputo-Fabrizio fractional derivative

This manuscript is concerned to the existence and stability of HIV/AIDS infection model with fractional order derivative. The corresponding derivative is taken in Caputo-Fabrizio sense, which is a new approach for such type of biological models. With the help of Sumudu transform, some new results are handled. Further for the corresponding results, existence theory and uniqueness for the equilibrium solution are provided via using nonlinear functional analysis and fixed point theory due to Banach.

A prescribed anisotropic mean curvature equation modeling the corneal shape: A paradigm of nonlinear analysis

In this paper we survey, complete and refine some recent results concerning the Dirichlet problem for the prescribed anisotropic mean curvature equation$\begin{equation*}{\rm{ -div}}\left({\nabla u}/{\sqrt{1 + |\nabla u|^2}}\right) = -au + {b}/{\sqrt{1 + |\nabla u|^2}},\end{equation*}$in a bounded Lipschitz domain $Ω \subset \mathbb{R}^N$ , with $a,b>0$ parameters. This equation appears in the description of the geometry of the human cornea, as well as in the modeling theory of capillarity phenomena for compressible fluids. Here we show how various techniques of nonlinear functional analysis can successfully be applied to derive a complete picture of the solvability patterns of the problem.

Some Remarks on Perov Type Mappings in Cone Metric Spaces

Let \(\left( E,C,t\right) \) be a real ordered topological vector space and let (X, d) be a tvs-cone metric space over cone C. Using Proposition 19.9 of Deimling (Nonlinear functional analysis, Springer, Berlin, 1985), we show that E can be equipped with a norm such that C is a normal monotone solid cone. Hence, a tvs-cone metric space \(\left( X,d\right) \) over a solid cone C is a normal cone metric space over the same cone C. This assures that tvs-cone metric spaces are not a genuine generalization of cone metric spaces introduced by Huang and Zhang, recently. Further, if the cone C is solid then we have only cone metric spaces over normal solid cone (with coefficient of normality \(K=1\)). Here, we introduce also the notion of Sehgal–Guseman–Perov type mappings and we establish a result of existence and uniqueness of fixed points for this class of mappings.