On the existence of solution for fractional differential equations of order 3 < δ 1 ≤ 4 $3<\delta_{1}\leq4$

Abstract

Abstract In this paper, we deal with a fractional differential equation of order

δ 1

∈ ( 3 , 4 ]

$\delta_{1}\in(3,4]$ with initial and boundary conditions,

D

δ 1

ψ ( x ) = − H ( x , ψ ( x ) )

$\mathcal{D}^{\delta_{1}}\psi(x)=-\mathcal{H}(x,\psi(x))$ ,

D

α 1

ψ ( 1 ) = 0 =

I

3 −

δ 1

ψ ( 0 ) =

I

4 −

δ 1

ψ ( 0 )

$\mathcal{D}^{\alpha_{1}} \psi(1)=0=\mathcal{I}^{3-\delta_{1}}\psi(0)= \mathcal{I}^{4-\delta_{1}}\psi(0)$ ,

ψ ( 1 ) =

Γ (

δ 1

−

α 1

)

Γ (

ν 1

)

I

δ 1

−

α 1

H ( x , ψ ( x ) ) ( 1 )

$\psi(1) = \frac{\Gamma(\delta_{1}-\alpha_{1})}{\Gamma(\nu_{1})}\mathcal{I}^{\delta _{1}-\alpha_{1}} \mathcal{H}(x,\psi(x))(1)$ , where

x ∈ [ 0 , 1 ]

$x\in[0,1]$ ,

α 1

∈ ( 1 , 2 ]

$\alpha_{1} \in(1,2]$ , addressing the existence of a positive solution (EPS), where the fractional derivatives

D

δ 1

$\mathcal{D}^{\delta_{1}}$ ,

D

α 1

$\mathcal{D}^{\alpha_{1}}$ are in the Riemann-Liouville sense of the order

δ 1

$\delta_{1}$ ,

α 1

$\alpha_{1}$ , 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

$\mathcal{I}^{\delta_{1}-\alpha_{1}}\mathcal{H}(x,\psi(x))(1)=\frac {1}{\Gamma(\delta_{1}-\alpha_{1})} \int_{0}^{1}(1-z)^{\delta_{1}-\alpha_{1}-1}\mathcal{H}(z,\psi(z))\,dz$ . 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.