Existence criterion for the solutions of fractional order p-Laplacian boundary value problems

Abstract

Abstract The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form:

D γ

(

ϕ p

(

D θ

z ( t ) ) ) + a ( t ) f ( z ( t ) ) = 0

$D^{\gamma}(\phi_{p}(D^{\theta}z(t)))+a(t)f(z(t)) =0$ ,

3 < θ

$3<{\theta}$ ,

γ ≤ 4

$\gamma\leq{4}$ ,

t ∈ [ 0 , 1 ]

$t\in[0,1]$ ,

z ( 0 ) =

z ‴

( 0 )

$z(0)=z'''(0)$ ,

η

D α

z ( t )

|

t = 1

=

z ′

( 0 )

$\eta D^{\alpha}z(t)|_{t=1}= z'(0)$ ,

ξ

z ″

( 1 ) −

z ″

( 0 ) = 0

$\xi z''(1)-z''(0)=0$ ,

0 < α < 1

$0<\alpha<1$ ,

ϕ p

(

D θ

z ( t ) )

|

t = 0

= 0 =

(

ϕ p

(

D θ

z ( t ) ) )

′

|

t = 0

$\phi_{p}(D^{\theta}z(t))|_{t=0}=0 =(\phi_{p}(D^{\theta}z(t)))'|_{t=0}$ ,

(

ϕ p

(

D θ

z ( t ) ) )

″

|

t = 1

=

1 2

(

ϕ p

(

D θ

z ( t ) ) )

″

|

t = 0

$(\phi_{p}(D^{\theta} z(t)))''|_{t=1} = \frac{1}{2}(\phi_{p}(D^{\theta} z(t)))''|_{t=0}$ ,

(

ϕ p

(

D θ

z ( t ) ) )

‴

|

t = 0

= 0

$(\phi_{p}(D^{\theta}z(t)))'''|_{t=0}=0$ , where

0 < ξ , η < 1

$0<\xi, \eta<{1}$ and

D θ

$D^{\theta}$ ,

D γ

$D^{\gamma}$ ,

D α

$D^{\alpha}$ are Caputo’s fractional derivatives of orders θ, γ, α, respectively. For this purpose, we apply Schauder’s fixed point theorem and the results are checked by illustrative examples.