Solvability for a system of Hadamard-type hybrid fractional differential inclusions

Definition 1

[4,5] The Hadamard-type fractional derivative of order q > 0 for an integrable function g : [ 1 , + ∞ ) → R is defined as:

where n = [ q ] + 1 , [ q ] is the smallest integer greater than or equal to q and log ( ⋅ ) = log e ( ⋅ ) .

Definition 2

[4,5] The Hadamard-type fractional integral of order q > 0 for an integrable function g is defined as:

provided that the integral exists.

Lemma 1

Let ϕ 1 , ϕ 2 ∈ L 1 ( [ 1 , e ] , R ) be continuous functions. Then, the integral solution of the Hadamard-type fractional differential system

is given by:

Proof

As argued in Chapter 9 in [28], the solution for (2) can be written as:

where c 1 , c 2 , d 1 , d 2 ∈ R . Note that x ( 1 ) = x ( e ) = y ( 1 ) = y ( e ) = 0 in (2), we have

Substituting these values into (4), we can obtain (3).□

Definition 3

A pair of function ( x , y ) is called the solution of system (1) if there exists a pair of function ( ϕ 1 , ϕ 2 ) ∈ L 1 ( [ 1 , e ] , R ) × L 1 ( [ 1 , e ] , R ) with ϕ 1 ∈ F 1 ( t , x ( t ) , y ( t ) ) and ϕ 2 ∈ F 2 ( t , x ( t ) , y ( t ) ) such that D α 1 x ( t ) f 1 ( t , x ( t ) , y ( t ) ) = ϕ 1 ( t ) , D α 2 x ( t ) f 2 ( t , x ( t ) , y ( t ) ) = ϕ 2 ( t ) almost every on [ 1 , e ] and x ( 1 ) = x ( e ) = y ( 1 ) = y ( e ) = 0 .

Next, we introduce some preliminary materials about norm spaces and multi-valued maps. Let X = C ( [ 1 , e ] , R ) denote a Banach space of continuous functions from [ 1 , e ] into R under the norm ‖ x ‖ = sup t ∈ [ 1 , e ] ∣ x ( t ) ∣ , which becomes a Banach algebra with respect to the multiplication “ ⋅ ” defined by ( x , y ) ( t ) = x ( t ) ⋅ y ( t ) for arbitrarily x , y ∈ X . The product space Ξ = X × X is also a Banach space under the norm ‖ ( x , y ) ‖ = ‖ x ‖ + ‖ y ‖ , which further becomes a Banach algebra with respect to the multiplication “ ⋅ ” defined by ( ( x , y ) ⋅ ( x ¯ , y ¯ ) ) ( t ) = ( x , y ) ( t ) ⋅ ( x ¯ , y ¯ ) ( t ) = ( x ( t ) x ¯ ( t ) , y ( t ) y ¯ ( t ) ) for ( x , y ) , ( x ¯ , y ¯ ) ∈ X × X . For more details about the results concerning algebraic structure of the product space Ξ = X × X , please see [41,42].

As aforementioned, ( Ξ , ‖ ( ⋅ , ⋅ ) ‖ ) is a Banach space. Now, we redeclare some basic definitions on multi-valued maps (see [43]):

Definition 4

A multi-valued map G : Ξ → T ( Ξ ) is called convex (closed) valued if G ( x , y ) is convex (closed) for all ( x , y ) ∈ Ξ .

Definition 5

The map G is called bounded on bounded sets if G ( B ) = ∪ ( x , y ) ∈ B G ( x , y ) is bounded in Ξ for any bounded set B of Ξ (i.e., sup ( x , y ) ∈ B { ‖ ( u , v ) ‖ : ( u , v ) ∈ G ( x , y ) } < ∞ ).

Definition 6

The map G is called upper semi-continuous (u.s.c.) on Ξ if for each ( x , y ) ∈ Ξ , the set G ( x , y ) is a nonempty closed subset of Ξ , and if for each open set B of Ξ containing G ( x , y ) , there exists an open neighborhood N of ( x , y ) such that G ( N ) ⊂ B .

Definition 7

The map G is called completely continuous if G ( B ) is relatively compact for every bounded subset B of Ξ .

Definition 8

A multi-valued map F : [ 1 , e ] × R 2 → T ( R ) is called L 1 -Carathéodory if

1. t → F ( t , x , y ) is measurable for each ( x , y ) ∈ R × R ,

2. ( x , y ) → F ( t , x , y ) is upper semicontinuous for almost all t ∈ [ 1 , e ] ,

3. there exists a function g r ∈ L 1 ( [ 1 , e ] , R + ) such that

   ‖ F ( t , x , y ) ‖ = sup { ∣ v ∣ : v ∈ F ( t , x , y ) ≤ g r ( t ) } ,

   for all x , y ∈ R with ‖ ( x , y ) ‖ ≤ r and for almost every t ∈ [ 1 , e ] .

Lemma 2

[40] Let X be a Banach algebra, A : X → X a single-valued operator, and B : X → T cp , cv ( X ) a multi-valued operator satisfying the following conditions:

1. A is a single-valued Lipschitz with Lipschitz constant k ,

2. B is compact and upper semicontinuous,

3. 2 M k < 1 , where M = ‖ B ( X ) ‖ .

1. the operator inclusion x ∈ A x B x has a solution,

1. the set Φ = { u ∈ X ∣ μ u ∈ A u B u , μ > 1 } is unbounded.

Lemma 3

[43] If G : X → T cl ( Y ) is upper semi-continuous, then G r ( G ) = { ( x , y ) ∈ X × Y , y ∈ G ( x ) } is a closed subset of X × Y ; i.e., for every sequence { x n } n ∈ N ⊂ X and { y n } n ∈ N ⊂ Y , if when n → ∞ , x n → x ∗ , y n → y ∗ and y n ∈ G ( x n ) , then y ∗ ∈ G ( x ∗ ) . Conversely, if G is completely continuous and has a closed graph, then it is upper semi-continuous.

Lemma 4

[44] Let X be a Banach space, F : [ 1 , e ] × R → T cp , cv ( X ) an L 1 -Carathéodory multi-valued map, and G a linear continuous mapping from L 1 ( [ 1 , e ] , X ) to C ( [ 1 , e ] , X ) . Then, the operator

is a closed graph operator in C ( [ 1 , e ] , X ) × C ( [ 1 , e ] , X ) .

Theorem 1

Suppose that

(H1) Functions f i : [ 1 , e ] × R × R → R \ { 0 } are continuous and there exist constants L i > 0 such that

for almost every t ∈ [ 1 , e ] , ∀ x , y , x ¯ , y ¯ ∈ R ;

(H2) operators F i : [ 1 , e ] × R × R → T ( R ) are L 1 -Carathéodory and have nonempty compact and convex values;

(H3) there exists a positive real number r such that

where 2 k ‖ g 1 r ‖ L 1 Γ ( α 1 ) + 2 k ‖ g 2 r ‖ L 1 Γ ( α 2 ) < 1 2 , k = L 1 + L 2 , F 10 = sup t ∈ [ 1 , e ] ∣ f 1 ( t , 0 , 0 ) ∣ , F 20 = sup t ∈ [ 1 , e ] ∣ f 2 ( t , 0 , 0 ) ∣ , here g 1 r ( t ) and g 2 r ( t ) have similar approach as Definition 8.

Proof

Applying Lemma 1, we transform syetem (1) into an equivalent fixed-point problem. Define the operator T : Ξ → T ( Ξ ) as T ( x , y ) ( t ) = ( T 1 ( x , y ) ( t ) , T 2 ( x , y ) ( t ) ) , where

and

Next, we consider two operators A = ( A 1 , A 2 ) and B = ( B 1 , B 2 ) , where A i : Ξ → Ξ are given by:

and B i : Ξ → T ( Ξ ) are given by:

Note that T i ( x , y ) = A i ( x , y ) B i ( x , y ) , i = 1 , 2 , then T ( x , y ) = ( A 1 ( x , y ) B 1 ( x , y ) , A 2 ( x , y ) B 2 ( x , y ) ) . We next show that the operators A and B satisfy all the conditions of Lemma 2. For clarity, we display the proof in several steps.

Step 1. We first show that Lemma 2(a) holds, i.e., the single-valued operator A is a Lipschitz on Ξ . Let ( x , y ) , ( x ¯ , y ¯ ) ∈ Ξ , by (H1), we have

Consequently,

So, the single-valued operator A is a Lipschitz on Ξ with Lipschitz constant k = L 1 + L 2 .

Step 2. We show that Lemma 2(b) holds, i.e., the multi-valued operator B is compact and upper semicontinuous on Ξ .

(i) We show that the operator B has convex values. Let u 11 , u 12 ∈ B 1 ( x , y ) , u 21 , u 22 ∈ B 2 ( x , y ) . Then, there are v 11 , v 12 ∈ F 1 , x y , v 21 , and v 22 ∈ F 2 , x y such that

For any constant λ ∈ [ 0 , 1 ] , we have

where v ¯ 1 ( t ) = λ v 11 ( t ) + ( 1 − λ ) v 12 ( t ) ∈ F 1 , x y , v ¯ 2 ( t ) = λ v 21 ( t ) + ( 1 − λ ) v 22 ( t ) ∈ F 2 , x y for all t ∈ [ 1 , e ] .

Therefore, we have

Then, we obtain the operator B ( x , y ) which is convex for each ( x , y ) ∈ Ξ . Then, operator B defines a multi-valued operator B : Ξ → T cv ( Ξ ) .

(ii) We display that the operator B maps bounded sets into bounded sets in Ξ . Let Ω = { ( x , y ) ∣ ‖ ( x , y ) ‖ < r , ( x , y ) ∈ Ξ } . Then, for each h i ∈ B i ( x , y ) , i = 1 , 2 , there are v i ∈ F i , x y ( i = 1 , 2 ) such that

From (H2), we have

and

This implies that