Positive solutions for a system of fourth-order differential equations with integral boundary conditions and two parameters

In this work, we investigate a class of nonlinear fourth-order systems with coupled integral boundary conditions and two parameters. We give the Green's functions for the system with boundary conditions, and then obtain some useful properties of the Green's functions. By using the Guo–Krasnosel'skii fixed-point theorem and the Green's functions, some sufficient conditions for the existence of positive solutions are presented. As applications, two examples are presented to illustrate the application of our main results

On α-convex operators

AbstractIn this paper, the existence and uniqueness of positive fixed points for a class of convex operators is obtained by means of the properties of cone, concave operators and the monotonicity of set-valued maps. In the end, we give a simple application to certain integral equations

Existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions

The purpose of this paper is to investigate the existence and uniqueness of positive solutions for a class of fractional differential equation with integral boundary conditions. Our analysis relies on two fixed point theorems of a sum operator in partial ordering Banach space. The main results obtained can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it

Some existence, uniqueness results on positive solutions for a fractional differential equation with infinite-point boundary conditions

We investigate a class of Riemann–Liouville's fractional differential equation with infinite-point boundary conditions. We give some new properties of the Green's function associated with the fractional differential equation boundary value problem. Based upon these new properties and by using Schauder's fixed point theorem, we establish some existence results on positive solutions for the boundary value problem. Further, by using a fixed point theorem of general concave operators, we also present an existence and uniqueness result on positive solutions for the boundary value problem

Existence and uniqueness of convex monotone positive solutions for boundary value problems of an elastic beam equation with a parameter

The purpose of this paper is to investigate the existence and uniqueness of convex monotone positive solutions for a boundary value problem of an elastic beam equation with a parameter. The proofs of the main results rely on a fixed point theorem and some properties of eigenvalue problems for a class of general mixed monotone operators. The results can guarantee the existence of a unique convex monotone positive solution and can be applied to construct two iterative sequences for approximating it. Moreover, we present some pleasant properties of convex monotone positive solutions for the boundary value problem dependent on the parameter. Finally, an example is given to illustrate the main results

Existence and uniqueness of positive solutions for Neumann problems of second order impulsive differential equations

This work is concerned with the existence and uniqueness of positive solutions for Neumann boundary value problems of second order impulsive differential equations. The result is obtained by using a fixed point theorem of generalized concave operators

Nonlocal q-fractional boundary value problem with Stieltjes integral conditions

In this paper, we are dedicated to investigating a new class of one-dimensional lower-order fractional q-differential equations involving integral boundary conditions supplemented with Stieltjes integral. This condition is more general as it contains an arbitrary order derivative. It should be pointed out that the problem discussed in the current setting provides further insight into the research on nonlocal and integral boundary value problems. We first give the Green's functions of the boundary value problem and then develop some properties of the Green's functions that are conductive to our main results. Our main aim is to present two results: one considering the uniqueness of nontrivial solutions is given by virtue of contraction mapping principle associated with properties of u0-positive linear operator in which Lipschitz constant is associated with the first eigenvalue corresponding to related linear operator, while the other one aims to obtain the existence of multiple positive solutions under some appropriate conditions via standard fixed point theorems due to Krasnoselskii and Leggett–Williams. Finally, we give an example to illustrate the main results. &nbsp

Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section