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
