文章基本信息

标题：On the existence of solution for fractional differential equations of order 3 < δ 1 ≤ 4 $3<\delta_{1}\leq4$
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作者：Dumitru Baleanu ; Ravi P Agarwal ; Hasib Khan 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2015
卷号：2015
期号：1
页码：362
DOI：10.1186/s13662-015-0686-1
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this paper, we deal with a fractional differential equation of order δ 1 ∈ ( 3 , 4 ] $\delta_)\in(3,4]$ with initial and boundary conditions, D δ 1 ψ ( x ) = − H ( x , ψ ( x ) ) $\mathcal{D}^{\delta_)}\psi(x)=-\mathcal{H}(x,\psi(x))$ , D α 1 ψ ( 1 ) = 0 = I 3 − δ 1 ψ ( 0 ) = I 4 − δ 1 ψ ( 0 ) $\mathcal{D}^{\alpha_)} \psi(1)=0=\mathcal{I}^{3-\delta_)}\psi(0)= \mathcal{I}^{4-\delta_)}\psi(0)$ , ψ ( 1 ) = Γ ( δ 1 − α 1 ) Γ ( ν 1 ) I δ 1 − α 1 H ( x , ψ ( x ) ) ( 1 ) $\psi(1) = \frac{\Gamma(\delta_)-\alpha_))}{\Gamma(\nu_))}\mathcal{I}^{\delta _)-\alpha_)} \mathcal{H}(x,\psi(x))(1)$ , where x ∈ [ 0 , 1 ] $x\in[0,1]$ , α 1 ∈ ( 1 , 2 ] $\alpha_) \in(1,2]$ , addressing the existence of a positive solution (EPS), where the fractional derivatives D δ 1 $\mathcal{D}^{\delta_)}$ , D α 1 $\mathcal{D}^{\alpha_)}$ are in the Riemann-Liouville sense of the order δ 1 $\delta_)$ , α 1 $\alpha_)$ , respectively. The function H ∈ C ( [ 0 , 1 ] × R , R ) $\mathcal{H}\in C([0,1]\times{R} , {R})$ and I δ 1 − α 1 H ( x , ψ ( x ) ) ( 1 ) = 1 Γ ( δ 1 − α 1 ) ∫ 0 1 ( 1 − z ) δ 1 − α 1 − 1 H ( z , ψ ( z ) ) d z $\mathcal{I}^{\delta_)-\alpha_)}\mathcal{H}(x,\psi(x))(1)=\frac ){\Gamma(\delta_)-\alpha_))} \int_(^)(1-z)^{\delta_)-\alpha_)-1}\mathcal{H}(z,\psi(z))\,dz$ . To this aim, we establish an equivalent integral form of the problem with the help of a Green’s function. We also investigate the properties of the Green’s function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results.
关键词：existence of positive solutions ; Green’s function ; Krasnosel’skiĭ theorem ; Arzela-Ascoli theorem