﻿ 高阶非线性分数次边值问题正解的唯一性 The Uniqueness of Positive Solutions for Higher Order Boundary Value Problem

Vol.05 No.03(2016), Article ID:18407,5 pages
10.12677/AAM.2016.53052

The Uniqueness of Positive Solutions for Higher Order Boundary Value Problem

Jie Wang, Chenxing Zhou

College of Mathematics, Changchun Normal University, Changchun Jilin

Received: Jul. 28th, 2016; accepted: Aug. 19th, 2016; published: Aug. 25th, 2016

ABSTRACT

In this paper, using the fixed-point theorem in partially ordered sets and the properties of the Green function, the uniqueness of positive solutions to the higher order boundary value problem is obtained. Our results make up the related conclusion in the existing literature.

Keywords:Higher Order Equation, Fixed-Point Theorem, Positive Solution, Uniqueness

1. 引言

(1)

(2)

2. 相关引理

， (5)

(6)

(7)

(i)上是一个连续函数，并且

(ii)关于第一个变量是严格的递增函数；

(iii)

3. 问题(1)~(2)非减正解的存在性和唯一性

(a) 当，其中 (指的是测度)时，那么是关于第二个变量的单调非增的连续函数；

(b) 当，且时，存在，使得

，显然在上是连续函数且单调非减，在上有。此外，当时，我们有

The Uniqueness of Positive Solutions for Higher Order Boundary Value Problem[J]. 应用数学进展, 2016, 05(03): 416-420. http://dx.doi.org/10.12677/AAM.2016.53052

1. 1. Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Vol. 204. Elsevier, Amsterdam.

2. 2. Wang, L. and Zhang, X. (2014) Existence of Positive Solutions for a Class of Higher-Order Nonlinear Fractional Differential Equations with Integral Boundary Conditions and a Parameter. Journal of Applied Mathematics and Computing, 44, 293-316. http://dx.doi.org/10.1007/s12190-013-0694-9

3. 3. Ji, Y., Guo, Y., Qiu, J. and Yang, L. (2015) Existence of Positive Solutions for a Boundary Value Problem of Nonlinear Fractional Differential Equations. Advances in Difference Equations, 13, 1-11.

4. 4. Han, Z., Lu, H. and Zhang, C. (2015) Positive Solutions for Eigenvalue Problems of Fractional Differential Equation with Generalized p-Laplacian. Applied Mathematics and Computation, 257, 526-536. http://dx.doi.org/10.1016/j.amc.2015.01.013

5. 5. Henderson, J. and Luca, R. (2015) Nonexistence of Positive Solutions for a System of Coupled Fractional Boundary Value Problems. Boundary Value Problems, 2015, 138. http://dx.doi.org/10.1186/s13661-015-0403-8

6. 6. Caballero Mena, J., Harjani, J. and Sadarangani, K. (2009) Existence and Uni-queness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems. Boundary Value Problems, 2009, Article ID: 421310. http://dx.doi.org/10.1155/2009/421310

7. 7. Harjani, J. and Sadarangani, K. (2009) Fixed Point Theorems for Weakly Contractive Mappings in Partially Ordered Sets. Nonlinear Analysis: Theory, Methods & Applications, 71, 3403-3410. http://dx.doi.org/10.1016/j.na.2009.01.240

8. 8. Nieto, J.J. and Rodríguez-López, R. (2005) Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations. Order, 22, 223-239. http://dx.doi.org/10.1007/s11083-005-9018-5