Existence of solutions for nonlinear problems involving mixed fractional derivatives with p(x)-Laplacian operator

Definition 1.1

We say ψ ∈ X is a solution to functional BVP (1.1), which means that ψ satisfies the equation and boundary conditions in (1.1).

Definition 2.1

Let g be a real function defined on [ 0 , 1 ] and γ > 0 . Then, the left and right Riemann-Liouville fractional integrals of order γ > 0 of a function g are defined, respectively, by

The left Riemann-Liouville fractional derivative and the right Caputo fractional derivative of order γ > 0 of a function g ∈ A C n ( [ 0 , 1 ] , R ) are defined, respectively, by

where n = [ γ ] + 1 , [ γ ] denotes the integer part of number γ .

Lemma 2.2

Assume f ∈ L [ 0 , 1 ] , p > 0 , and q > 0 ; then, the following relations hold almost everywhere on [ 0 , 1 ] :

Lemma 2.3

Let n − 1 < γ < n , f ∈ L [ 0 , 1 ] , and I 0 + n − α f ( t ) ∈ A C n [ a , b ] , then

1. I 0 + γ D 0 + γ f ( t ) = f ( t ) − ∑ k = 1 n ( I 0 + n − γ f ( t ) ) ( n − k ) ( 0 ) Γ ( γ − k + 1 ) t γ − k ;

1. I 1 − γ D 1 − γ C f ( t ) = f ( t ) − ∑ k = 0 n − 1 ( − 1 ) k f ( k ) ( 1 ) k ! ( 1 − t ) k .

Lemma 2.4

If the fractional derivatives D 0 + γ g ( t ) and D 0 + γ + m g ( t ) exist, then

( D 0 + γ g ( t ) ) ( m ) = D 0 + γ + m g ( t ) , where γ > 0 , m is a positive integer.

Lemma 2.5

[6] For any ( x , u ) ∈ [ 0 , 1 ] × R , φ p ( x ) ( u ) = ∣ u ∣ p ( x ) − 2 u , is a homeomorphism from R to R and strictly monotone increasing for any fixed t. Moreover, its inverse operator φ p ( x ) − 1 ( ⋅ ) is defined by

which is continuous and sends bounded sets to bounded sets.

Definition 2.6

(see [34–37]) Let X and Y be two Banach spaces. A continuous operator M : X ∩ dom M → Y is said to be quasi-linear if

1. Im M ≔ M ( X ∩ dom M ) is a closed subset of Y ;

2. Ker M ≔ { ψ ∈ X ∩ dom M : M ψ = 0 } is linearly homeomorphic to R n , n < ∞ ,

Definition 2.7

(See [37]) Suppose that N λ : Ω ¯ → Y , λ ∈ [ 0 , 1 ] is a continuous and bounded operator. Denote N 1 by N . Let Σ λ = { ψ ∈ Ω ¯ : M ψ = N λ ψ } . N λ is said to be M -quasi-compact in Ω ¯ if there exists a vector subspace Y 1 of Y satisfying dim Y 1 = dim X 1 and two operators Q and R such that for λ ∈ [ 0 , 1 ] ,

1. Ker Q = Im M ;

2. Q N λ ψ = θ , λ ∈ ( 0 , 1 ) ⇔ Q N ψ = θ ;

3. ( R ⋅ , 0 ) is the zero operator and ( R ⋅ , λ ) ∣ Σ λ = ( I − P ) ∣ Σ λ ;

4. M [ P + R ( ⋅ , λ ) ] = ( I − Q ) N λ ,

Theorem 2.8

(See [34– 37]) Let X and Y be two Banach spaces and Ω ⊂ X be an open and bounded nonempty set. Suppose

is a quasi-linear operator and that N λ : Ω ¯ → Y , λ ∈ [ 0 , 1 ] is M-quasi-compact. In addition, if the following conditions hold:

1. M ψ ≠ N λ ψ , ∀ ψ ∈ dom M ∩ ∂ Ω , λ ∈ ( 0 , 1 ) ;

2. deg ( J Q N , Ω ∩ Ker M , 0 ) ≠ 0 ,