On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions

Abstract

Abstract 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 ,

$$\begin{aligned}& \mathcal{D}^{\omega}\biggl(\frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)=-\mathcal {K}_{1}\bigl(t,x(t),z(t)\bigr), \quad\omega\in(2,3], \\& \mathcal{D}^{\epsilon}\biggl(\frac{z(t)}{\mathcal {G}(t,x(t),z(t))}\biggr)=-\mathcal{K}_{2} \bigl(t,x(t),z(t)\bigr), \quad\epsilon\in (2,3],\\& \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\bigg|_{t=1}=0,\qquad \mathcal{D}^{\mu}\biggl( \frac{x(t)}{\mathcal{H}(t,x(t),z(t))}\biggr)\bigg|_{t= \delta_{1} }=0,\qquad x^{(2)}(0)=0, \\& \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\bigg|_{t=1}=0, \qquad\mathcal{D}^{\nu}\biggl( \frac{z(t)}{\mathcal{G}(t,x(t),z(t))}\biggr)\bigg|_{t= \delta_{2}}=0,\qquad z^{(2)}(0)=0, \end{aligned}$$ where

t ∈ [ 0 , 1 ]

$t\in[0,1]$ ,

δ 1

,

δ 2

, μ , ν ∈ ( 0 , 1 )

$\delta_{1}, \delta_{2}, \mu, \nu\in(0,1)$ , 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 )

$\mathcal{K}_{1}, \mathcal{K}_{2}\in C([0,1]\times\mathcal{R}\times \mathcal{R},\mathcal{R} )$ 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.