Positive homoclinic solutions of n‐th order difference equations with sign‐changing Green's function

In this paper we, consider an n‐th order nonlinear difference equation with parameter dependence. An exhaustive study of the related Green's function is done. The exact expression of the function is given. The range of parameter for which either it has constant sign or it changes sign is obtained. Some existence results for the nonlinear problem are deduced by using the classical Krasnosel'skii's fixed point theorem on cones and fixed point index theory.     

Existence results of periodic solutions for third‐order nonlinear singular differential equation

Using Green's function for third‐order differential equation and some fixed‐point theorems, i.e., Leray‐Schauder alternative principle and Schauder's fixed point theorem, we establish three new existence results of periodic solutions for nonlinear third‐order singular differential equation, which extend and improve significantly existing results in the literature.     

Positive solutions for fourth‐order singular nonlinear differential equation with variable‐coefficient

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation with variable‐coefficient, which extend and improve significantly existing results in the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicity results of positive solutions for fourth‐order nonlinear differential equation with singularity

By application of Green's function and some fixed‐point theorems, that is, Leray–Schauder alternative principle and Schauder's fixed‐point theorem, we establish two new existence results of positive periodic solutions for nonlinear fourth‐order singular differential equation, which extend and improve significantly existing results in the literature. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of Solutions for Impulsive Parabolic Partial Differential Equations

We discuss the propagation of heat along a homogeneous rod of length A under the influence of a nonlinear heat source and impulsive effects at fixed times. This problem is described by an initial-boundary value problem for a nonlinear parabolic partial differential equation subjected to impulsive effects at fixed times. Using Green's function, we convert the problem into a nonlinear integral equation. Sufficient conditions are provided that enable the application of fixed point theorems to prove existence and uniqueness of solutions.     

Positive periodic solution for second-order nonlinear differential equation with singularity of attractive type

This paper is devoted to investigate the following second-order nonlinear differential equation with singularity of attractive type $$ x''-a(t)x=f(t,x)+e(t), $$ where the nonlinear term $f$ has a singularity at the origin. By using the Green''s function of the linear differential equation with constant coefficient and Schauder''s fixed point theorem, we establish some existence results of positive periodic solutions.     

一类非线性chemotaxis系统的局部零能控性

本文研究了一类非线性chemotaxis系统的能控性问题.利用这类线性化结合不动点的方法,获得了非线性系统的局部零能控性结果,推广了非线性抛物-椭圆型方程能控性的结果.     

On positive solutions to fractional sum boundary value problems for nonlinear fractional difference equations

In this paper, we study a new class of 3‐point boundary value problems of nonlinear fractional difference equations. Our problems contain difference and fractional sum boundary conditions. Existence and uniqueness of solutions are proved by using the Banach fixed‐point theorem, and existence of the positive solutions is proved by using the Krasnoselskii's fixed‐point theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence and Numerical Solutions of Nonlinear Quadratic Integral Equations Defined on Unbounded Intervals

The first goal of this article is to discuss the existence of solutions of nonlinear quadratic integral equations. These equations are considered in the Banach space L ^p (?₊). The arguments used in the existence proofs are based on Schauder's and Darbo's fixed point theorems. In particular, to apply Schauder's fixed point theorem based method, a special care is devoted to the proof of the L ^p -compactness of the operators associated with our nonlinear quadratic integral equations. The second goal of this work is to study a numerical method for solving nonlinear Volterra integral equations of a fairly general type. Finally, we provide the reader with some examples that illustrate the different results of this work.     

Homoclinic solutions for nonlinear general fourth‐order differential equations

This work provides sufficient conditions for the existence of homoclinic solutions of fourth‐order nonlinear ordinary differential equations. Using Green's functions, we formulate a new modified integral equation that is equivalent to the original nonlinear equation. In an adequate function space, the corresponding nonlinear integral operator is compact, and it is proved an existence result by Schauder's fixed point theorem. Copyright © 2017 John Wiley & Sons, Ltd.     

Three positive solutions for boundary value problem for differential equation with Riemann-Liouville fractional derivative

In this paper, by using the Avery-Peterson fixed point theorem, we establish the existence result of at least three positive solutions of boundary value problem of nonlinear differential equation with Riemann-Liouville''s fractional order derivative. An example illustrating our main result is given. Our results complements and extends previous work in the area of boundary value problems of nonlinear fractional differential equations.     

Existence of positive solutions for a system of semipositone coupled discrete boundary value problems

We study the existence of positive solutions for a system of nonlinear second-order difference equations with parameters and sign-changing nonlinearities, subject to multi-point coupled boundary conditions. In the proof of our main theorems we use the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel'skii fixed point theorem.     

Fixed point theorems of operators with ppf dependence in banach spaces

In this paper the theory of fixed point theorems of nonlinear whose domain and range are different Banach spaces are considered. Also the analogues of the Contraction Mapping Principle. Krasnoselskii's fixed point theorem and a result on the convergence of quasinonexpansive mappings are dealt with. As an application, the existence and uniqueness of solutions of differential equations with retarded argument in Banach spaces are discussed     

Multiplicity results of fourth-order singular nonlinear differential equation with a parameter

In this paper, we investigate a class of fourth-order singular nonlinear differential equation with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. By applications of Green''s function and the Krasnoselskii fixed point theorem, sufficient conditions for the existence of positive periodic solutions are established.     

On the existence of positive solutions of the p‐Laplacian dynamic equations on time scales

In this paper, we investigate the existence of positive solutions for a nonlinear m‐point boundary value problem for the p‐Laplacian dynamic equations on time scales, by applying a Krasnosel'skii's fixed point theorem. As an application, an example is included to demonstrate the main results. Copyright © 2017 John Wiley & Sons, Ltd.     

Boundary value problems of singular multi‐term fractional differential equations with impulse effects

We study a class of boundary value problems for nonlinear impulsive fractional differential equations. A weighted function Banach space and a completely continuous nonlinear operator are constructed firstly. Some existence results for solutions of these problems are established continuously. Our analysis relies on the well known Schauder's fixed point theorem. Examples are given to illustrate the main results finally.     

On the theory of nonlinear singular integral equations of Cauchy type

The paper concerns the investigation of two classes of nonlinear singular integral equations with Cauchy kernel important in the applications. There are given uniqueness and existence theorems, the last by a novel application of Schauder's fixed point theorem to this type of equations.     

On a system of nonlinear PDE's for phase transitions with vector order parameter and convective effect

The paper deals with a system of nonlinear PDE's which describes a phase-field model with convection and temperature dependent constraint to the vector order parameter. Existence of solutions for the system under consideration is proved by the method of Yosida approximation and fixed point arguments.     

Existence Results for the Higher-Order Weighted Caputo-Fabrizio Fractional Derivative

By the definition of the higher-order fractional derivative, we explore the central properties of the higher-order Caputo-Fabrizio fractional derivative and integral with a weighted term. Furthermore, by dint of Schaefer''s fixed point theorem, $\alpha$-$\psi$-Contraction theorem, etc., we establish the existence of solutions for nonlinear equations. We also give three examples to make our main conclusion clear.     

Fractional boundary value problems with integral boundary conditions

In this article, we study a type of nonlinear fractional boundary value problem with integral boundary conditions. By constructing an associated Green's function, applying spectral theory and using fixed point theory on cones, we obtain criteria for the existence, multiplicity and nonexistence of positive solutions.