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

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Peer-Reviewed Research
  • SDG 17
  • SDG 7
  • 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.