A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

Abstract

This paper is concerned with an operator equation Ax + Bx + C x = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane-Emden-Fowler equations.

The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems

“…Theorem 3.1 extends the main results in [7,21,27,28]. In fact, in Theorem 3.1 of [21], A : P → P is an increasing sub-homogeneous operator and B : P → P is a decreasing operator which satisfies B(t −1 y) ≥ tBy, for all t ∈ (0, 1), x, y ∈ P.…”

“…Our results in this paper will extend and improve many known results in the field and in particular those in [2,7,21,27,28,32]. The rest of the paper is organized as follows.…”

“…Thereafter, many authors have investigated various kinds of nonlinear mixed monotone operators in Banach spaces such as nonlinear operators with concave-convex (see [4]), mixed monotone operators with α-concave-convex (see [33,34]), nonlinear operators with φ-concave-convex (see [10,26]), nonlinear operators with e-concave-convex (see [39]), and also obtained a lot of important results on mixed monotone operators (see [1,2,7,8,11,13,16,17,21,22,23,24,27,28,29,31,32,38]). These studies not only have theoretical significance but also have a wide range of applications in engineering, nuclear physics, biology, chemistry, technology, etc.…”

“…In [27], Zhai and Anderson considered the existence and uniqueness of positive solutions to the following operator equation in ordered Banach spaces E Ax + Bx + Cx = x, x ∈ P, where P is a cone in E and A : P → E is an increasing α-concave operator, B : P → E is an increasing hypo-homogeneous operator and C : P → E is an homogeneous operator.…”

Existence results to certain functional equations in probabilistic Banach spaces with an application to integral equations

“…It is very important to know if such equations admit solutions and if a solution exists, it is unique. Many authors studied such questions and considered various classes of operator equations posed in a Banach space (see, for examples, [1,2,3,7,9,12,13,14,15,16]). In [8], we studied a class of operator equations posed in a probabilistic Banach space and involving decreasing and convex operators.…”