Schauder-Tychonoff Fixed Point Theorem Research Articles

Modeling and mathematical analysis of root water uptake in the rhizosphere

ABSTRACT This research focuses on elucidating a system of partial differential equations designed to model the process of water uptake by a single root within an unsaturated soil. This model extends a unidimensional root scenario by incorporating radial and axial hydraulic conductance in the root structure. The flow dynamics in the rhizosphere are represented by the Richards equation, while a diffusion equation is employed to characterize water movement within the root. The resulting system forms a strongly coupled, nonlinear parabolic-elliptic system. To establish the existence and uniqueness of a solution, we utilize the technique of semi-discretization and the Schauder-Tychonoff fixed point theorem. Additionally, we approximate the system using a mixed finite element method, and the subsequent numerical tests presented align well with the model predictions. This comprehensive approach contributes to our understanding of the intricate dynamics involved in root water uptake and provides a robust framework for further exploration in this field.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Singular semilinear elliptic problems on unbounded domains in [formula omitted

We prove the compactness of the solution operator for a class of singular semilinear elliptic problems on the exterior of a ball in Rn, n≥3. Compactness of solution operators for similar problems in Rn, n≥2 is established. Further, using these compactness results and employing Schauder-Tychonoff fixed point theorem, we prove the existence of a positive solution to classes of semipositone problems with asymptotically linear reaction terms.

Large time behavior of solutions to a system of coupled nonlinear oscillators via a generalized form of Schauder-Tychonoff fixed point theorem

Large time behavior of solutions to a system of coupled nonlinear oscillators via a generalized form of Schauder-Tychonoff fixed point theorem

Parabolic-elliptic system modeling biological ion channels

The mathematical model of the transport and diffusion of ions in biological channels is considered. It is described by the three-dimensional nonlinear evolution classical Poisson–Nernst–Planck (cPNP) system of partial differential equations with nonlinear coupled boundary conditions. In particular the Chang–Jaffé (CJ) conditions are given on the input and output of a channel. The Robin boundary conditions on a potential are taken. Theorems on the existence, uniqueness and nonnegativity of local weak solutions, in the suitable Sobolev spaces, are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

On the Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations

Using Schauder Tychonoff fixed point theorem and the technique of Kronecker product of matrices, we prove existence results for Ψ-asymptotic equivalence of the Ψ-bounded solutions of two Lyapunov matrix differential equations.

Nonoscillation of higher order mixed differential equations with distributed delays

We are concerned with a class of higher order mixed differential equations with distributed delays. Several new sufficient conditions for the existence of nonoscillatory solutions have been obtained by Schauder–Tychonoff fixed point theorem, which generalize and improve some known results. Two examples are given to explain our main results.

On bounded entire solutions of some quasilinear elliptic equations

In this paper we prove the existence of infinitely many bounded entire solutions, ground state entire solutions and entire solutions with prescribed limit at infinity to PDEs involving the ϕ-Laplace operator in the setting of Orlicz–Sobolev spaces. The main tools used are the sub-solution and super-solution technique in conjunction with the Schauder–Tychonoff fixed point theorem.

Limiting Behaviors of Nonoscillatory Solutions for Two-Dimensional Nonlinear Time Scale Systems

We consider a two-dimensional time scale system of first order dynamic equations and establish some necessary and sufficient conditions for the existence of nonoscillatory solutions for the system using Knaster fixed point theorem, the Schauder fixed point theorem and the Schauder–Tychonoff fixed point theorem. We also provide examples to underline the main results of this article.

On local weak solutions to Nernst–Planck–Poisson system

We study the one-dimensional nonlinear Nernst–Planck–Poisson system of partial differential equations with the class of nonlinear boundary conditions which cover the Chang–Jaffé conditions. The system describes certain physical and biological processes, for example ionic diffusion in porous media, electrochemical and biological membranes, as well as electrons and holes transport in semiconductors. The considered boundary conditions allow the physical system to be not only closed but also open. Theorems on existence, uniqueness, and nonnegativity of local weak solutions are proved. The main tool used in the proof of the existence result is the Schauder–Tychonoff fixed point theorem.

Fixed point theorems and the Krein-S̆mulian property in locally convex spaces

AbstractThe aim of this paper is to establish some new fixed point theorems for the superposition operators in locally convex spaces which satisfy the Krein-S̆mulian property. We employ a family of measures of noncompactness in conjunction with the Schauder-Tychonoff fixed point theorem. As an application, the existence of solutions to a quite general nonlinear Volterra type integral equation is considered in locally integrable spaces.

On the well-posedness for the compressible nematic liquid crystal flow

The Cauchy problem for the compressible flow of nematic liquid crystals in dimension ( N ≧ 2) is considered. Employing the Littlewood-Paley theory and Schauder-Tychonoff fixed point theorem, we prove the local well-posedness of the system for large initial data in critical Besov spaces based on the L p framework under the sole natural assumption that the initial density is bounded away from zero.

Infinite first order differential systems with nonlocal initial conditions

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear" growth then the system has at least one solution. The approach relies on the application, in a suitable Fr\'echet space, of the classical Schauder-Tychonoff fixed point theorem. We show that, as a special case, our approach covers the case of a system of a finite number of differential equations. An illustrative example of application is also provided.

CONVEX SOLUTIONS OF THE POLYNOMIAL-LIKE ITERATIVE EQUATION ON OPEN SET

Abstract. Because of difficulty of using Schauder’s fixed point theoremto the polynomial-like iterative equation, a lots of work are contributedto the existence of solutions for the polynomial-like iterative equation oncompact set. In this paper, by applying the Schauder-Tychonoff fixedpoint theorem we discuss monotone solutions and convex solutions of thepolynomial-like iterative equation on an open set (possibly unbounded)in R N . More concretely, by considering a partial order in R N definedby an order cone, we prove the existence of increasing and decreasingsolutions of the polynomial-like iterative equation on an open set andfurther obtain the conditions under which the solutions are convex in theorder. 1. IntroductionAs indicated in the books [7, 19] and the surveys [3, 24], the polynomial-likeiterative equation(1.1) λ 1 f(x) +λ 2 f 2 (x) +···+λ n f n (x) = F(x), x ∈ S,where S is a subset of a linear space over R, F : S → S is a given function,λ i s (i = 1,...,n) are real constants, f : S → S is the unknown functionand f

Schauder-Tychonoff Fixed-Point Theorem in Theory of Superconductivity

We study the existence of mild solutions to the time-dependent Ginzburg-Landau ((TDGL), for short) equations on an unbounded interval. The rapidity of the growth of those solutions is characterized. We investigate the local and global attractivity of solutions of TDGL equations and we describe their asymptotic behaviour. The TDGL equations model the state of a superconducting sample in a magnetic field near critical temperature. This paper is based on the theory of Banach space, Fréchet space, and Sobolew space.

An asymptotic analysis of positive solutions of generalized Thomas–Fermi differential equations — The sub-half-linear case

The existence and the asymptotics behavior for the large value of the variable of the positive solutions of generalized Thomas–Fermi equation presented in this article are proved. It is assumed that coefficient q ( t ) belongs to the class of regularly varying functions in the sense of Karamata. Properties of these functions and the Schauder–Tychonoff fixed point theorem are the main tools for the proofs.

Singular integral equations arising in draining and coating flows

We present a very general existence theorem for a nonlinear singular integral equation using the Schauder–Tychonoff fixed point theorem. Our result automatically guarantees the existence of a positive solution to a problem arising in draining and coating flows.

$L^{p}$-$L^{q}$ estimates for wave equations and the Kirchhoff equation

The aim of this paper is to derive $L^{p}$-$L^{q}$ estimates for strictly hyperbolic equations with time-dependent coefficients which are of Lipschitz class. Further-more, $L^{p}$-$L^{q}$ estimates for Kirchhoff equation can be obtained by applying the Schauder-Tychonoff fixed point theorem.

On the existence of multiple positive entire solutions for a class of quasilinear elliptic equations

Our goal is to establish the theorems of existence and multiple of positive entire solutions for a class quasilinear elliptic equations in ℝN with the Schauder‐Tychonoff fixed point theorem as the principal tool. In many articles, the theorems of existence and multiple of positive entire solutions for a class semilinear elliptic equations are established. The results of the semilinear equations are extended to the quasilinear ones and the results of semilinear equations are developed.

Fixed point theorems in locally convex spaces—the Schauder mapping method

AbstractIn the appendix to the book by F. F. Bonsal, Lectures on Some Fixed Point Theorems of Functional Analysis (Tata Institute, Bombay, 1962) a proof by Singbal of the Schauder-Tychonoff fixed point theorem, based on a locally convex variant of Schauder mapping method, is included. The aim of this note is to show that this method can be adapted to yield a proof of Kakutani fixed point theorem in the locally convex case. For the sake of completeness we include also the proof of Schauder-Tychonoff theorem based on this method. As applications, one proves a theorem of von Neumann and a minimax result in game theory.