武競力, 楊喜陶

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帶積分邊界條件的非線性高階分數(shù)階微分方程解的存在性

武競力, 楊喜陶

(湖南科技大學數(shù)學與計算科學學院, 湖南湘潭, 411201)

首先通過拉普拉斯變換得出一類帶積分邊界條件的非線性高階分數(shù)階微分方程滿足邊界條件的解, 再利用壓縮映射原理和Krasnosel’skii不動點理論, 討論了這類方程解的存在性和惟一性。

分數(shù)階微分方程; 邊界值問題; 壓縮映射原理; 不動點理論

分數(shù)階微積分涉及到將整數(shù)階導數(shù)和積分推廣到任意階, 在最近40年引起了極大關注。隨著分數(shù)階微積分理論的發(fā)展, 分數(shù)階微分方程被廣泛應用于科學和工程領域[1-7]。近年來, 帶積分邊界條件的分數(shù)階微分方程的解問題被普遍研究, 而且在實際應用中有很大價值。

考慮帶一個積分邊界條件的分數(shù)階微分方程:

1 幾個引理

考慮下面一個邊界值問題:

運用另一個邊界條件, 有

。 (4)

對式(3)兩邊同時從0到1積分, 得到

。 (5)

將式(6)代入式(3), 得到問題(2)的解

引理2 (壓縮映射原理[10]) 設是一個Banach空間,是的閉子集,是一個嚴格壓縮映射, 即對任意不等式成立, 其中。那么有惟一一個不動點。

引理3 (Arzela-Ascoli[10]) 如果的一個緊子集中的點列是一致有界和等度連續(xù)的, 則它有一個一致收斂的子列。

引理4(Krasnosel’skii[11]) 設是一個Banach空間,是的一個凸的非空閉子集,和是2個算子, 使得: (1) 對任意(2)是緊的和連續(xù)的; (3)是壓縮映射。那么存在一個, 使得。

2 主要結論

應用上面的引理, 可以得到2個定理。

。

舉例:

例1 考慮邊界值問題

例2 考慮邊界值問題

參考文獻：

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(責任編校:劉曉霞)

Existence of solutions for nonlinear high-order fractional boundary value problem with integral boundary condition

Wu Jingli, Yang Xitao

(School of Mathematics, Hunan University of Science and Technology, Xiangtan411201, China)

The existence of solutions of a class of nonlinear high-order fractional differential equations with integral boundary conditions is studied. By using Laplace transform, the contraction mapping principle and Krasnosel’skii fixed point theorem, the existence and uniqueness of solution are obtained, which enriches the theory for the solution of fractional differential equations.

fractional differential equations; boundary value problem; contraction mapping principle; fixed point theorem

10.3969/j.issn.1672–6146.2015.03.001

O 175.6

1672–6146(2015)03–0001–05

武競力, beckhsm7777777@sina.com。

2015–03–13

湖南省自然科學基金項目(2015JJ2063); 湖南科技大學研究生創(chuàng)新基金項目(S140035)。