We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two example...

In this paper, we establish existence of positive solutions of the nonlinear problems of one dimensional p-Laplacian with nonlinear parameter φp(u ′(t))′ + a(t)f(λ, u) = 0, t ∈ (0, 1), u(0) = u(1) = 0. where a : Ω→ R is continuous and may change sign, λ > 0 is a parameter, f(λ, 0) > 0 for all λ > 0. By applying Leray-Schauder fixed point theorem we obtain the existence of positive solutions.

We give a new proof of the Komlós-Major-Tusnády embedding theorem for the simple random walk. The only external tool that we use is the Schauder-Tychonoff fixed point theorem for locally convex spaces. Besides that, the proof is almost entirely based on a series of soft arguments and easy inequalities, and no hard computations (implicit or explicit) are involved. This provides the first genuine...

In this paper, inspired from the study on denoising, segmentation and reconstruction in image processing, and combining with the theories of two phase flows, we introduce one class of initial-boundary value problem of the Cahn-Hilliard equation with nonlocal terms. Then, by using the Schauder fixed point theorem, we obtain the existence of weak solutions to this initial boundary value problem f...

We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under conideration include the kinetic Cucker-Smale equation [6, 7] with possibly non-symmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [14]. The main tools employed in the analysis are the velocity...

where 1 < α < 2, 0 < βi < 1, i = 1, 2, . . . ,m – 2, 0 < η1 < η2 < · · · < ηm–2 < 1, ∑m–2 i=1 βiη α–1 i < 1, D α 0+ is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green’s function, we give some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray–Schauder nonlinear a...

This paper introduces the concept of square-mean piecewise almost periodic for impulsive stochastic processes. The existence of square-mean piecewise almost periodic solutions for linear and nonlinear impulsive stochastic differential equations is established by using the theory of the semigroups of the operators and Schauder fixed point theorem. The stability of the square-mean piecewise almos...

In this paper we investigate the existence of solutions to a kind of fourth-order impulsive differential equations with integral boundary value conditions. By employing the Schauder fixed point theorem, we obtain sufficient conditions which ensure the system has at lease one solution. Also by using the contraction mapping theorem we get the uniqueness result. Finally an example is given to illu...

In this paper, with the help of the Leray-Schauder fixed point theorem, differential inequality techniques and suitable Lyapunov functional, several novel sufficient conditions on the existence and global exponential stability of periodic solutions for the generalized shunting Inhibitory cellular neural networks are developed, which improve some published results. Particularly, the precise conv...

This paper is devoted to the study of a mixed problem for a nonlinear parabolic integro-differential equation which mainly arise from a one dimensional quasistatic contact problem. We prove the existence and uniqueness of solutions in a weighted Sobolev space. Proofs are based on some a priori estimates and on the Schauder fixed point theorem. we also give a result which helps to establish the ...