一类非线性分数阶微分方程的正解
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摘要
针对具有积分边值条件的分数阶微分方程正解的问题,利用算子不动点理论,结合迭代逼近的思想,给出一类非线性项带参数且具有积分边值条件的分数阶微分方程正解的存在唯一性,并通过构造迭代序列来逼近方程的正解;利用一类特殊算子方程正解的性质,结合所讨论方程格林函数的性质,给出方程正解依赖于参数的一些性质。结果表明,利用算子不动点理论讨论非线性项带参数的分数阶微分方程边值问题正解的存在唯一性是可行的。
In view of the positive solutions of fractional differential equations with integral boundary conditions, the existence and uniqueness of the positive solutions of fractional differential equations with integral boundary conditions were proved by using the fixed point theory of the operator and the idea of iterative approximation. An iterative sequence was constructed to approximate the solution. The properties of the positive solutions of a special operator equation and the properties of the Green's function were given. Some properties of positive solutions to the boundary value problem dependent on the parameter were presented. The results show that it is feasible to use the fixed point theory of the operator to discuss the existence and uniqueness of the positive solution of a boundary value problem for a fractional differential equation with a nonlinear term.
In view of the positive solutions of fractional differential equations with integral boundary conditions, the existence and uniqueness of the positive solutions of fractional differential equations with integral boundary conditions were proved by using the fixed point theory of the operator and the idea of iterative approximation. An iterative sequence was constructed to approximate the solution. The properties of the positive solutions of a special operator equation and the properties of the Green's function were given. Some properties of positive solutions to the boundary value problem dependent on the parameter were presented. The results show that it is feasible to use the fixed point theory of the operator to discuss the existence and uniqueness of the positive solution of a boundary value problem for a fractional differential equation with a nonlinear term.
引文
[1] ZHAI C B,HAO M R.Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems[J].Nonlinear Analysis,2012,75:2542-2551.
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[3] ZHAI C B,YANG C,ZHANG X Q.Positive solutions for non-linear operator equations and several classes of applications[J].Mathematische Zeitschrift,2010,266:43-63.
[4] WANG J R,ZHANG Y R.On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives[J].Applied Mathematics Letters,2015,39:85-90.
[5] SUN Y P,ZHAO M.Positive solutions for a class of fractional differential equations with integral boundary conditions[J].Applied Mathematics Letters,2014,34:17-21.
[6] ZHAO X K,CHAI C W,GE W G.Existence and nonexistence results for a class of fractional boundary value problems[J].Journal of Applied Mathematics and Computing,2013,41:17-31.
[7] ZHANG X Q,WANG L,SUN Q.Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter[J].Journal of Applied Mathematics and Computing,2014,226:708-718.
[8] JIANG M,ZHONG S M.Successively iterative method for fractional differential equations with integral boundary conditions[J].Applied Mathematics Letters,2014,38:94-99.
[9] ZHAI C B,WANG F.Properties of positive solutions for the operator equation Ax=λx and applications to fractional differential equations with integral boundary conditions[J].Advances in Difference Equations,2015,366:1-10.
[10] 郭大钧.非线性泛函分析[M].2版.济南:山东科技出版社,2001.
[11] OLDHAM K B,SPANIER J.The fractional calculus[M].San Diego:Academic Press,2006.
[2] ZHAI C B,YAN W P,YANG C.A sum operator method for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems[J].Communications in Nonlinear Science and Numerical Simulation,2013,18:858-866.
[3] ZHAI C B,YANG C,ZHANG X Q.Positive solutions for non-linear operator equations and several classes of applications[J].Mathematische Zeitschrift,2010,266:43-63.
[4] WANG J R,ZHANG Y R.On the concept and existence of solutions for fractional impulsive systems with Hadamard derivatives[J].Applied Mathematics Letters,2015,39:85-90.
[5] SUN Y P,ZHAO M.Positive solutions for a class of fractional differential equations with integral boundary conditions[J].Applied Mathematics Letters,2014,34:17-21.
[6] ZHAO X K,CHAI C W,GE W G.Existence and nonexistence results for a class of fractional boundary value problems[J].Journal of Applied Mathematics and Computing,2013,41:17-31.
[7] ZHANG X Q,WANG L,SUN Q.Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter[J].Journal of Applied Mathematics and Computing,2014,226:708-718.
[8] JIANG M,ZHONG S M.Successively iterative method for fractional differential equations with integral boundary conditions[J].Applied Mathematics Letters,2014,38:94-99.
[9] ZHAI C B,WANG F.Properties of positive solutions for the operator equation Ax=λx and applications to fractional differential equations with integral boundary conditions[J].Advances in Difference Equations,2015,366:1-10.
[10] 郭大钧.非线性泛函分析[M].2版.济南:山东科技出版社,2001.
[11] OLDHAM K B,SPANIER J.The fractional calculus[M].San Diego:Academic Press,2006.