文章基本信息

标题：On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions
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作者：Dumitru Baleanu ; Hasib Khan ; Hossein Jafari 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：318
DOI：10.1186/s13662-015-0651-z
语种：English
出版社：Hindawi Publishing Corporation
摘要：We investigate sufficient conditions for existence and uniqueness of solutions for a coupled system of fractional order hybrid differential equations (HDEs) with multi-point hybrid boundary conditions given by D ω ( x ( t ) H ( t , x ( t ) , z ( t ) ) ) = − K 1 ( t , x ( t ) , z ( t ) ) , ω ∈ ( 2 , 3 ] , D ϵ ( z ( t ) G ( t , x ( t ) , z ( t ) ) ) = − K 2 ( t , x ( t ) , z ( t ) ) , ϵ ∈ ( 2 , 3 ] , x ( t ) H ( t , x ( t ) , z ( t ) ) t = 1 = 0 , D μ ( x ( t ) H ( t , x ( t ) , z ( t ) ) ) t = δ 1 = 0 , x ( 2 ) ( 0 ) = 0 , z ( t ) G ( t , x ( t ) , z ( t ) ) t = 1 = 0 , D ν ( z ( t ) G ( t , x ( t ) , z ( t ) ) ) t = δ 2 = 0 , z ( 2 ) ( 0 ) = 0 , $$\begin{aligned}& \mathcal{D}^{\omega}\biggl(\frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)=-\mathcal {K}_)\bigl(t,x(t),z(t)\bigr), \quad\omega\in(2,3], \\& \mathcal{D}^{\epsilon}\biggl(\frac{z(t)}{\mathcal {G}(t,x(t),z(t))}\biggr)=-\mathcal{K}_, \bigl(t,x(t),z(t)\bigr), \quad\epsilon\in (2,3],\\& \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\bigg _{t=1}=0,\qquad \mathcal{D}^{\mu}\biggl( \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)\bigg _{t= \delta_) }=0,\qquad x^{(2)}(0)=0, \\& \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\bigg _{t=1}=0, \qquad\mathcal{D}^{\nu}\biggl( \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\biggr)\bigg _{t= \delta_,}=0,\qquad z^{(2)}(0)=0, \end{aligned}$$ where t ∈ [ 0 , 1 ] $t\in[0,1]$ , δ 1 , δ 2 , μ , ν ∈ ( 0 , 1 ) $\delta_), \delta_,, \mu, \nu\in(0,1)$ , and D ω $\mathcal{D}^{\omega}$ , D ϵ $\mathcal{D}^{\epsilon}$ , D μ $\mathcal{D}^{\mu}$ and D ν $\mathcal{D}^{\nu}$ are Caputo’s fractional derivatives of order ω, ϵ, μ and ν, respectively, K 1 , K 2 ∈ C ( [ 0 , 1 ] × R × R , R ) $\mathcal{K}_), \mathcal{K}_,\in C([0,1]\times\mathcal{R}\times \mathcal{R},\mathcal{R} )$ and G , H ∈ C ( [ 0 , 1 ] × R × R , R − { 0 } ) $\mathcal{G},\mathcal{H}\in C([0,1]\times\mathcal{R}\times\mathcal {R},\mathcal{R} - \{0\} )$ . We use classical results due to Dhage and Banach’s contraction principle (BCP) for the existence and uniqueness of solutions. For applications of our results, we include examples.
关键词：coupled system of hybrid fractional differential equations ; existence of solutions ; uniqueness of solutions