New fixed point theorems for mixed monotone operators and local existence–uniqueness of positive solutions for nonlinear boundary value problems

Abstract

In this article we present a new fixed point theorem for a class of general mixed monotone operators, which extends the existing corresponding results. Moreover, we establish some pleasant properties of nonlinear eigenvalue problems for mixed monotone operators. Based on them the local existence-uniqueness of positive solutions for nonlinear boundary value problems which include Neumann boundary value problems, three-point boundary value problems and elliptic boundary value problems for Lane-Emden-Fowler equations is proved. The theorems for nonlinear boundary value problems obtained here are very general.

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

“…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.…”

“…These studies not only have theoretical significance but also have a wide range of applications in engineering, nuclear physics, biology, chemistry, technology, etc. Because of the crucial role played by nonlinear equations in applied science as well as mathematics, nonlinear functional analysis has been an active area of research, and nonlinear operators with connection to nonlinear (fractional) differential and integral equations have been extensively studied over the past several decades (see [5,6,12,14,15,19,25,29,30,31,32,35,36,37]). …”

Positive solutions of fractional differential equations involving the Riemann–Stieltjes integral boundary condition

“…It is easy to see that P h ⊂ P is convex and λP h = P h for all λ > 0. We refer the readers to the references [9] and [19] for details. Definition 2.1 ([19]) T : P × P → P is said to be a mixed monotone operator if T(x, y) is increasing in x and decreasing in y, i.e., u i ,…”

“…In [2], by means of a fixed point theorem, Bai is the Riemann-Liouville fractional derivative, f is a Carathédory function, and f (t, x, y, z) is singular at the value 0 of its arguments x, y, z. Some recent contributions of fractional differential equations can be seen in [1][2][3][4][5][6][7][8][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”