Schauder Fixed Point Theorem Research Articles

Approximate controllability of neutral fractional stochastic differential systems with control on the boundary

In this paper, the approximate boundary controllability results for neutral fractional stochastic differential system with fractional Brownian motion in Hilbert space is discussed. The existence mild solution of neutral fractional stochastic differential system with fractional Brownian motion is proved by using fractional calculus, compact semigroup, Schauder fixed-point theorem and stochastic analysis. The sufficient conditions for approximate boundary controllability of this system is established under the corresponding linear system is approximately controllable. Finally, an example is given to show the application of our results.

Optimal control of a parabolic equation with memory

An optimal control problem for a semilinear parabolic partial differential equation with memory is considered. The well-posedness as well as the first and the second order differentiability of the state equation is established by means of Schauder fixed point theorem and the implicity function theorem. For the corresponding optimal control problem with the quadratic cost functional, the existence of optimal control is proved. The first and the second order necessary conditions are presented, including the investigation of the adjoint equations which are linear parabolic equations with a measure as a coefficient of the operator. Finally, the sufficiency of the second order optimality condition for the local optimal control is proved.

Existence of traveling waves by means of fixed point theory for an epidemic model with Hattaf-Yousfi incidence rate and temporary immunity acquired by vaccination

The main aim of this work is to investigate the existence of traveling waves of an epidemic model with temporary immunity acquired by vaccination. The incidence rate of the disease used in the epidemic model is of the form Hattaf-Yousfi that includes many types existing in the literature. By means of Schauder fixed point theorem and construction of a pair of upper and lower solutions, the existence of traveling wave solution that connects the disease-free equilibrium and the endemic equilibrium is obtained and characterized by two parameters that are the basic reproduction number and the minimal wave speed.

On periodic solutions of a recruitment model with iterative terms and a nonlinear harvesting

We consider a first-order delay differential equation involving iterative terms. We prove the existence of positive periodic and bounded solutions by utilizing the Schauder's fixed point theorem combined with the Green's functions method. Furthermore, by virtue of the Banach contraction principle, the uniqueness and stability of the solution are also analyzed. Our new results are illustrated with two examples that show the feasibility of our main findings.

On Implicit Coupled Hadamard Fractional Differential Equations with Generalized Hadamard Fractional Integro-Differential Boundary Conditions

This study is devoted to studying the existence and uniqueness of solutions for Hadamard implicit fractional differential equations with generalized Hadamard fractional integro-differential boundary conditions by utilizing the contraction principle of the Banach and Leray–Schauder fixed point theorems. Moreover, with two different approaches, the Hyers–Ulam stabilities are also discussed. Different ordinary differential equations of the third order with different boundary conditions (e.g., initial, anti periodic and integro-differential) can be obtained as a special case for our proposed model. Finally, for verification, an example is presented, and some graphs for the particular variables and particular functions are drawn using MATLAB.

Multiple solutions of the Dirichlet problem in multidimensional billiard spaces

Dirichlet problem in an n-dimensional billiard space is investigated. In particular, the system of ODEs ddot{x}(t) = f(t,x(t)) together with Dirichlet boundary conditions x(0) = A, x(T) = B in an n-dimensional interval K with elastic impact on the boundary of K is considered. The existence of multiple solutions having prescribed number of impacts with the boundary is proved. As a consequence the existence of infinitely many solutions is proved, too. The problem is solved by reformulating it into non-impulsive problem with a discontinuous right-hand side. This auxiliary problem is regularized and the Schauder Fixed Point Theorem is used.

ON SOLVABILITY OF QUADRATIC HAMMERSTEIN INTEGRAL EQUATIONS IN HÖLDER SPACES

Using Schauder's fixed point theorem we consider the solvability of a quadratic Hammerstein integral equation in the space of functions satisfying a H\"{o}lder condition. An example is included to illustrate our results.

Hemivariational inequality for contaminant reaction–diffusion model of recovered fracturing fluid in the wellbore of shale gas reservoir

The goal of this paper is to study the dynamic behavior of recovered fracturing fluid and the reaction–diffusion phenomenon of contaminants in the wellbore of shale gas reservoir during the early stage of fracturing fluid flowback. First, by applying various physical constitutive laws, a complicated recovered fracturing fluid model which consists of a stationary incompressible Stokes equation with a nonmonotone and multivalued friction law and a reaction–diffusion equation with the Neumann boundary condition, is established. Then, we use the variational techniques and the theory of multivalued analysis to formulate a new recovered fracturing fluid model. The latter is a variational system which consists of a hemivariational inequality with constraint driven by an elliptic variational equation. The system involves mutually coupled velocity and the concentration fields. Finally, under some mild assumptions, we employ a surjectivity theorem for pseudomonotone operators, monotonicity arguments and Schauder fixed point theorem to prove the existence of a weak solution to the recovered fracturing fluid model.

The Existence of Solution to Fractional Boundary Value Problem with Riemann-Liouville Type History-State-Based Variable-Order Derivative

The paper is devoted to studying the solutions of boundary value problem for nonlinear fractional differential equation with Riemann-Liouville type history-state-based variable-order derivative. Using Schauder fixed point theorem and Banach fixed point thoerem, we prove the existence and uniqueness of solutions in the Hölder space. Lastly, two examples are given to show the applicability of the existence theorems.

Existence Solutions for Implicit Fractional Relaxation Differential Equations with Impulsive Delay Boundary Conditions

The aim of this paper is to study the existence and uniqueness of solutions for nonlinear fractional relaxation impulsive implicit delay differential equations with boundary conditions. Some findings are established by applying the Banach contraction mapping principle and the Schauder fixed-point theorem. An example is provided that illustrates the theoretical results.

A class of piecewise fractional functional differential equations with impulsive

Abstract In this paper, we study a class of piecewise fractional functional differential equations with impulsive and integral boundary conditions. By using Schauder fixed point theorem and contraction mapping principle, the results for existence and uniqueness of solutions for the piecewise fractional functional differential equations are established. And by using cone stretching and cone contraction fixed point theorems in norm form, the existence of positive solutions for the equations are also obtained. Finally, an example is given to illustrate the effectiveness of the conclusion.

Traveling waves for a three-species competition system with two weak aboriginal competitors

In this paper, we study the traveling wave solutions for a three species competition system with two weak aboriginal competitors and one strong alien competitor. We are concerned with the existence of traveling waves such that these two co-existence aboriginal competitors are wiped out by the invading alien strong competitor. First, we derive the existence of wave profiles based on an application of Schauder's fixed point theorem with the help of constructing suitable generalized upper-lower solutions to capture the unstable wave tail limit. Then a new method for deriving the stable wave tail limit is introduced. Finally, the minimal invading speed is characterized.

Generalized Hukuhara Weak Solutions for a Class of Coupled Systems of Fuzzy Fractional Order Partial Differential Equations without Lipschitz Conditions

As is known to all, Lipschitz condition, which is very important to guarantee existence and uniqueness of solution for differential equations, is not frequently satisfied in real-world problems. In this paper, without the Lipschitz condition, we intend to explore a kind of novel coupled systems of fuzzy Caputo Generalized Hukuhara type (in short, gH-type) fractional partial differential equations. First and foremost, based on a series of notions of relative compactness in fuzzy number spaces, and using Schauder fixed point theorem in Banach semilinear spaces, it is naturally to prove existence of two classes of gH-weak solutions for the coupled systems of fuzzy fractional partial differential equations. We then give an example to illustrate our main conclusions vividly and intuitively. As applications, combining with the relevant definitions of fuzzy projection operators, and under some suitable conditions, existence results of two categories of gH-weak solutions for a class of fire-new fuzzy fractional partial differential coupled projection neural network systems are also proposed, which are different from those already published work. Finally, we present some work for future research.

Existence and stability of traveling wavefronts for a nonlocal delay Belousov–Zhabotinskii system

This paper is concerned with the existence and stability of traveling wavefronts for a nonlocal delay Belousov–Zhabotinskii system. We first introduce the new variable and change the two-dimensional system with nonlocal delay into a three-dimensional system without delay. Then the existence of traveling wavefronts is obtained by constructing a pair of suitable super- and sub-solutions and applying Schauder's fixed point theorem. Finally, we establish the exponential stability of traveling wavefronts with large speed by the weighted energy method together with the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as , but can be arbitrarily large in other locations.

Smooth traveling waves for doubly nonlinear degenerate diffusion equations with time delay

ABSTRACT This paper deals with the existence of traveling waves for doubly nonlinear degenerate diffusion equations with Nicholson's blowflies-type source term and time delay. We prove the existence of monotone and non-monotone traveling waves by the monotone iteration and Schauder's fixed point theorem respectively. Furthermore, the non-critical traveling waves are proven to be globally -stable. The adopted approach for the proof is the weighted energy method combining the compactness analysis.

Qualitative Properties of Positive Solutions of a Kind for Fractional Pantograph Problems using Technique Fixed Point Theory

The current paper intends to report the existence and uniqueness of positive solutions for nonlinear pantograph Caputo–Hadamard fractional differential equations. As part of a procedure, we transform the specified pantograph fractional differential equation into an equivalent integral equation. We show that this equation has a positive solution by utilising the Schauder fixed point theorem (SFPT) and the upper and lower solutions method. Another method for proving the existence of a singular positive solution is the Banach fixed point theorem (BFPT). Finally, we provide an example that illustrates and explains our conclusions.

Solvability Criterion for Fractional q-Integro-Difference System with Riemann-Stieltjes Integrals Conditions

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system with the nonlocal boundary conditions involving diverse fractional q-derivatives and Riemann-Stieltjes q-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii’s fixed point theorem, Schaefer’s fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results.

Mathematical study of neural feedback roles in small target motion detection.

Building an efficient and reliable small target motion detection visual system is challenging for artificial intelligence robotics because a small target only occupies few pixels and hardly displays visual features in images. Biological visual systems that have evolved over millions of years could be ideal templates for designing artificial visual systems. Insects benefit from a class of specialized neurons, called small target motion detectors (STMDs), which endow them with an excellent ability to detect small moving targets against a cluttered dynamic environment. Some bio-inspired models featured in feed-forward information processing architectures have been proposed to imitate the functions of the STMD neurons. However, feedback, a crucial mechanism for visual system regulation, has not been investigated deeply in the STMD-based neural circuits and its roles in small target motion detection remain unclear. In this paper, we propose a time-delay feedback STMD model for small target motion detection in complex backgrounds. The main contributions of this study are as follows. First, a feedback pathway is designed by transmitting information from output-layer neurons to lower-layer interneurons in the STMD pathway and the role of the feedback is analyzed from the view of mathematical analysis. Second, to estimate the feedback constant, the existence and uniqueness of solutions for nonlinear dynamical systems formed by feedback loop are analyzed via Schauder's fixed point theorem and contraction mapping theorem. Finally, an iterative algorithm is designed to solve the nonlinear problem and the performance of the proposed model is tested by experiments. Experimental results demonstrate that the feedback is able to weaken background false positives while maintaining a minor effect on small targets. It outperforms existing STMD-based models regarding the accuracy of fast-moving small target detection in visual clutter. The proposed feedback approach could inspire the relevant modeling of robust motion perception robotics visual systems.

Existence of at least one positive continuing solution of Urysohn quadratic integral equation by Schauder fixed-point theorem.

We employ Schauder fixed-point Theorem to prove the existence of at least one positive con-tinuous solution of the quadratic integral equation
 Moreover, the maximal and the minimal solutions of the last equation are also proved.

Existence of solution of a coupled system of differential equation with the nonlocal two-point boundary conditions

2 The aim of the paper is to study the existence of at least one solution of a nonlocal problem of coupled system of differential equations with nonlocal two-point boundary conditions. By means of Schauder fixed point theorem, an existence result of a solution is proved for the nonlocal problem considered.