Fixed Point Principle Research Articles

Pseudo almost automorphic solutions of hematopoiesis model with mixed delays

This paper is concerned with a hematopoiesis model with mixed delays. Under new conditions, we studied the existence, uniqueness and global exponential stability of pseudo almost automorphic solutions for the suggested model. Our approach is mainly based on the exponential dichotomy of linear differential equation, Banach’s fixed-point principle and suitable Lyapunov functional. At the end, some numerical examples are presented to demonstrate the effectiveness of our findings.

Evolutionary variational\u2013hemivariational inequalities with applications to dynamic viscoelastic contact mechanics

The purpose of this work is to introduce and investigate a complicated variational–hemivariational inequality of parabolic type with history-dependent operators. First, we establish an existence and uniqueness theorem for a first-order nonlinear evolution inclusion problem, which is driven by a convex subdifferential operator for a proper convex function and a generalized Clarke subdifferential operator for a locally Lipschitz superpotential. Then, we employ the fixed point principle for history-dependent operators to deliver the unique solvability of the parabolic variational–hemivariational inequality. Finally, a dynamic viscoelastic contact problem with the nonlinear constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, where normal contact and friction are described, respectively, by two nonconvex and nonsmooth multi-valued terms.

Voltage Stability Online Monitoring and Analysis Method Based on the Fixed-Point Principle

ABSTRACT Continuous assessment of voltage stability is vital to ensure the secure operation of the power system. Based on this concern, a number of online voltage stability analysis methods have been proposed. Among these indices, the impedance match concept based on Thévenin equivalent has attracted a lot of interest. However, the traditional impedance match is reported not to perform as expected under multi-load power systems. This paper proposes a new approach for voltage stability assessment based on the Banach fixed-point theorem. This leads to a real-time voltage stability monitoring scheme without the need to estimate Thévenin parameters. The derived method was tested on a 5-generators and 18-bus system by time-domain dynamic simulations and simulated on an actual power system using real phasor measurement unit.

Stability analysis for first-order nonlinear differential equations with three-point boundary conditions

Abstract In the present paper, we study a system of nonlinear differential equations with three-point boundary conditions. The given original problem is reduced to the equivalent integral equations using Green function. Several theorems are proved concerning the existence and uniqueness of solutions to the boundary value problems for the first order nonlinear system of ordinary differential equations with three-point boundary conditions. The uniqueness theorem is proved by Banach fixed point principle, and the existence theorem is based on Schafer’s theorem. Then, we describe different types of Ulam stability: Ulam-Hyers stability, generalized Ulam-Hyers stability. We discuss the stability results providing suitable example.

History-dependent differential variational-hemivariational inequalities with applications to contact mechanics

<p style='text-indent:20px;'>The primary objective of this paper is to explore a complicated differential variational-hemivariational inequality involving a history-dependent operator in Banach spaces. A well-posedness result for the inequality, including the existence, uniqueness, and continuous dependence on the initial data of the solution is established by using a fixed point principle for history-dependent operators. Moreover, to illustrate the applicability of the theoretical results, an elastic contact problem with wear and long time dependent effort is explored.

Fully history-dependent evolution hemivariational inequalities with constraints

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability.

Existence and uniqueness of solutions to nonlocal-impulsive fractional Cauchy semilinear conformable differential equations

This paper is devoted to study the existence and uniqueness of solutions to fractional semilinear differential equations with conformable fractional derivative. We prove our results for local and nonlocal-impulsive conditions. This will be accomplished by using Schaefer’s fixed-point theorem and Banach’s fixed-point principle. Some examples are given to illustrate the main theorems.

A class of history-dependent differential variational inequalities with application to contact problems

ABSTRACTIn this paper a class of generalized differential variational inequalities with constraints involving history-dependent operators in Banach spaces is investigated. The unique solvability and regularity results are obtained via surjectivity of multivalued pseudomonotone operators combined with a fixed point principle. From abstract results, a theorem concerning existence, uniqueness and regularity of weak solution to a frictional viscoelastic contact problem with adhesion and history-dependent operator is established. Further, a theoretical analysis of a penalty method for history-dependent differential variational inequality is provided. The unique solvability of a penalized problem is shown, as well as the convergence of its solution to the solution of the original history-dependent differential variational inequality, as a penalty parameter tends to zero. Finally, results on a penalty method are applied to another contact problem, history-dependent frictional viscoelastic contact problem with a generalized normal compliance condition instead of a generalized Signorini contact condition.

Multi-valued graph contraction principle with applications

ABSTRACT The aim of this paper is to present a generalization of Nadler's fixed point principle for the case of multi-valued graph contractions. Also, a strict fixed point theorem is obtained. Connections to some variational analysis concepts are discussed and applications to generalized coupled fixed point problems are given.

Local and blowing‐up solutions for a space‐time fractional evolution system with nonlinearities of exponential growth

Local and blowing‐up solutions for the Cauchy problem for a system of space and time fractional evolution equations with time‐nonlocal nonlinearities of exponential growth are considered. The existence and uniqueness of the local mild solution is assured by the Banach fixed point principle. Then, we establish a blow‐up result by Pokhozhaev capacity method. Finally, under some suitable conditions, an estimate of the life span of blowing‐up solutions is established.

Robust ISS of uncertain discrete-time singularly perturbed systems with disturbances

ABSTRACTThis paper is concerned with robustly input-to-state stable (ISS) and Robust ISS by feedback of uncertain discrete-time singularly perturbed systems (SPSs) with disturbances. Meanwhile, robust stability and stabilisation of uncertain discrete-time SPSs are also obtained as the particular cases of robust ISS and robust ISS by feedback. We first find a sufficient condition by using the fixed-point principle in terms of linear matrix inequalities (LMIs) to guarantee that the considered system is always standard discrete-time SPSs subject to uncertainty and disturbances. Then, the full systems could decompose into the continuous-time uncertain slow subsystem with disturbance and discrete-time uncertain fast subsystems with disturbance, respectively. Based on the two-time-scale decomposition technique, sufficient condition in terms of LMIs is given such that the full systems are uniformly standard and robust ISS simultaneously. In addition, a state feedback controller is constructed by using the LMI approach such that the resulting closed-loop systems are robust ISS. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.

System of fuzzy fractional differential equations in generalized metric space

In this paper, we study the existence of integral solutions of fuzzy fractional differential systems with nonlocal conditions under Caputo generalized Hukuhara derivatives. These models are considered in the framework of completegeneralized metric spaces in the sense of Perov. The novel feature of our approach is the combination of the convergentmatrix technique with Schauder fixed point principle of vector valued operators in semilinear Banach spaces. Somecomputational examples are represented to demonstrate our theoretical results.

Unique solvability and exponential stability of differential hemivariational inequalities

ABSTRACT In this paper, we study a differential hemivariational inequality (DHVI, for short) in the framework of reflexive Banach spaces. Our aim is three fold. The first one is to investigate the existence and the uniqueness of mild solution, by applying a general fixed-point principle. The second one is to study its exponential stability, by employing the formula for the variation of parameters and inequality techniques. Finally, the third aim is to illustrate an application of our abstract results in the study of an initial and boundary value problem which describes the contact of an elastic rod with an obstacle.

Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations

Coupled and mixed coupled hybrid fixed point principles in a partially ordered Banach algebra and PBVPs of nonlinear coupled quadratic differential equations

A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

Studying the Equilibrium State Stability of the Biocenosis Dynamics System under the Conditions of Interspecies Interaction

Introduction. The article considers the problem of stability of the mathematical model of the trivial equilibrium. The model describes the biocenosis dynamics with the predatorprey type interspecific interaction, which is a nonlinear system of ordinary differential equations with perturbations in the form of vector polynoms. The examined system is considered provided that the birth rate of biological species does not exceed mortality rate. Materials and Methods. The article states the sufficient conditions for asymptomatic stability. The proof is based on the construction of an operator equation in a Banach space, which connects the solution of the nonlinear system and its linear approximation. The existence of the operator equation solution is proved through using the Schauder fixed point principle. It is shown that there is Brauer local asymptotic equivalence between the solutions of the investigated system and its linear approximation and the differences between the components of the solutions of the nonlinear system and its linear approximation tends to zero evenly with respect to the initial values. Results. As a case in point, the authors consider the model of the predator-prey type in the case when two species feed on the third one. The conditions for stability and asymptotic stability for a part of the variables of the trivial equilibrium of the abundance dynamics of two predator populations and one prey population under different fertility rates of biological species are given. The graphs of a number of populations with different vaues of the difference between the birth rate and the mortality rate of partucular species are constructed. Conclusions. Depending on the difference between fertility and mortality of biological species, the population dynamics of two populations of “predators” and one population of “preys” is analyzed over time.

Pseudo Almost Automorphic Solutions for Multidirectional Associative Memory Neural Network with Mixed Delays

In the paper, under some sufficient conditions, the existence and the global exponential stability of a pseudo almost automorphic solution to multidirectional associative memory (MAM) neural network with time-varying coefficients and mixed delays is obtained by combining both differential inequality techniques and the classical Banach’s fixed-point principle. As you will see, our results improve and generalize many previous known results in MAM. Finally, an example together with its numeric simulations show the feasibility and effectiveness of our main results.

Neutral stochastic functional differential evolution with varying-time delays driven by Rosenblatt process in Hilbert spaces

In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process in a real separable Hilbert space. We prove an existence and uniqueness result of mild solution by means of the Banach fixed point principle. A practical example is provided to illustrate the viability of the abstract result of this work.

Infinite horizon reflected backward stochastic differential equations with Markov chains

ABSTRACTWe first give the existence and uniqueness results for infinite horizon backward stochastic differential equations with Markov chains, taking advantage of the martingale representation theorem and fixed point principle. Then we prove the well-posedness results for infinite horizon reflected backward stochastic differential equations with Markov chains, by virtue of the Snell envelope theory and contraction mapping method. Comparison theorems for the above two kinds of equations are also obtained, via the linearization approach or properties of reflected backward stochastic differential equations, respectively.

THEORY AND APPROXIMATION OF SOLUTIONS TO A HARVESTED HIERARCHICAL AGE-STRUCTURED POPULATION MODEL

This article is concerned with theoretic analysis and numerical approximation of solutions to a hierarchical age-structured population model, in which the vital rates of an individual depend more on the number of older individuals. The well-posedness of the model is rigorously treated by means of fixed point principle, and an algorithm and convergence analysis are presented. An example is used to show the effectiveness of the numerical method.