Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems

Definition 2.6

The non-negative number

assigned with a bounded chain C in a partially ordered normed linear space P is called P H -MNC.

Definition 2.7

[15] A P H -MNC satisfies the following axioms on a partially ordered Banach space P

1. β p ( C ) = 0 iff C ∈ P r c p , c n ( P ) ;

2. β p ( C ) = β p ( C ¯ ) , C ∈ P b d , c n ( P ) ;

3. C ⊂ D implies β p ( C ) ≤ β p ( D ) , C , D ∈ P b d , c n ( P ) ;

4. If lim n → ∞ β p ( C n ) = 0 for a sequence of closed chains { C n } in P b d , c n ( P ) such that C n + 1 ⊂ C n , then C ∞ = ⋂ n = 1 ∞ C n is partial compact and nonempty;

5. β p ( C ∪ D ) = max { β p ( C ) , β p ( D ) } , where C , D ∈ P b d , c n ( P ) ;

6. β p ( λ C ) = ∣ λ ∣ β ( C ) for any real number λ and C ∈ P b d , c n ( P ) ;

7. β p ( C + D ) ≤ β ( C ) + β ( D ) , C , D ∈ P b d , c n ( P ) .

Definition 2.8

[15] An operator T : P → P is said to be partial condensing if

for each C ∈ P b d , c n ( P ) such that β p ( C ) > 0 , where β p is P H -MNC.

Definition 2.9

[18] A coupled operator T : P × P → P is said to be coupled partial condensing if

for all C , D ∈ P b d , c n ( P ) such that β p ( C ) + β p ( D ) > 0 , where β p is P H -MNC.

Theorem 2.1

[15] Let T be a nondecreasing, partial bounded, partial continuous, and partial condensing operator on a ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete partial ordered normed linear space ( P , ⊑ , ∥ ⋅ ∥ ) . Then T admits at least one fixed point if P  contains ω 0 such that ω 0 ⊑ T ( ω 0 ) or T ( ω 0 ) ⊑ ω 0 .

Theorem 2.2

[18] Let T : P × P → P be a partial bounded, partial continuous, mixed monotone, and coupled partial condensing operator, where ( P , ⊑ , ∥ ⋅ ∥ ) be a ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete partial ordered normed linear space. Then T admits at least one coupled fixed point if P 2 contains ( ω 0 , υ 0 ) such that ω 0 ⊑ T ( ω 0 , υ 0 ) and T ( υ 0 , ω 0 ) ⊑ υ 0 .

Definition 2.10

[24] A mapping ζ : R + × R + → R satisfying

1. ζ ( t 1 , t 2 ) < t 2 − t 1 for all t 1 , t 2 > 0 ,

2. if there are two sequences { s j } and { t j } in R + \ { 0 } such that lim j → ∞ s j = lim j → ∞ t j > 0 and t j < s j , then lim j → ∞ sup ζ ( t j , s j ) < 0 ,

Example 2.1

A monotonic nondecreasing upper semicontinuous function η : R + → R + satisfying η ( 0 ) = 0 is called a D -function [15]. Let ζ D ( t , s ) = η ( s ) − t , for s , t ∈ R + , where η be a D -function with η ( r ) < r . Then ζ D ∈ Z .

Definition 2.11

[1,2] Let g ∈ ℒ 1 ( a , b ) . The integral

is the left side Riemann-Liouville fractional integral of order μ > 0 of the function g .

Definition 2.12

[1,2] The left side Riemann-Liouville fractional derivative of order μ of g is given by the expression

provided right-hand side exists.

Lemma 2.1

[4] For s > a , we have

Definition 2.13

[3] The right-hand side Hilfer fractional derivative operator of order 0 < μ < 1 and type 0 ≤ ν ≤ 1 is defined by

Alternatively, we can write

Remark 2.3

[3]

1. If ν = 0 , then D a + μ , ν = D a + μ which is Riemann-Liouville derivative.

2. If ν = 1 , then D a + μ , ν = I a + 1 − μ which is Caputo derivative.

1. Following relation holds for the parameter γ :

   0 < γ ≤ 1 , γ ≥ μ , γ > ν , 1 − γ < 1 − ν ( 1 − μ ) .

Lemma 2.2

[25] Let γ ∈ [ 0 , 1 ) , a < b < c , g ∈ C γ [ a , b ] , g ∈ C γ [ b , c ] , and g is continuous at b . Then g ∈ C γ [ a , c ] .

Lemma 2.3

[25] For μ > 0 , I a + μ maps C [ a , b ] into C [ a , b ] .

Lemma 2.4

[25] Let μ > 0 and γ ∈ [ 0 , 1 ) , then I a + μ maps C γ [ a , b ] into C γ [ a , b ] and is bounded.

Lemma 2.5

[25] Let μ > 0 and 0 ≤ γ < 1 , then I a + μ is bounded from C γ [ a , b ] into C [ a , b ] if γ ≤ μ .

Lemma 2.6

[25] Let γ ∈ [ 0 , 1 ) and g ∈ C γ [ a , b ] . Then I a + μ g ( a ) = lim s → a I a + μ g ( s ) = 0 , γ < μ .

Lemma 2.7

Let g ∈ L 1 ( a , b ) . Then lim t → b + ∫ a b ( t − s ) μ − 1 g ( s ) d s = ∫ a b ( b − s ) μ − 1 g ( s ) = Γ ( μ ) I a + μ g ( b ) , μ > 0 .

Lemma 2.8

[4,25] Let μ , ν ≥ 0 and g ∈ L 1 ( a , b ) . Then I a + μ I a + ν g ( t ) = I a + μ + ν g ( t ) a.e., t ∈ [ a , b ] . In particular, if g lies in C γ [ a , b ] or C [ a , b ] , then equality holds at every t in ( a , b ] or [ a , b ] .

Lemma 2.9

[4] Let μ > 0 , γ ∈ [ 0 , 1 ) , and g ∈ C γ [ a , b ] . Then D a + μ I a + μ g ( t ) = g ( t ) , for all t ∈ ( a , b ] .

Lemma 2.10

[4] Let μ ∈ ( 0 , 1 ) , γ ∈ [ 0 , 1 ) , g ∈ C γ [ a , b ] , and I a + 1 − μ g ∈ C γ 1 [ a , b ] . Then

for all t ∈ ( a , b ] .

Lemma 2.11

[4] Let μ ∈ ( 0 , 1 ) , ν ∈ [ 0 , 1 ] , and γ = μ + ν − μ ν . If g ∈ C 1 − γ γ [ a , b ] , then

and

Lemma 2.12

[4] Let g ∈ ℒ 1 ( a , b ) . If D a + ν ( 1 − μ ) g exists and lies in ℒ 1 ( a , b ) , then

Lemma 2.13

[4] Let μ ∈ ( 0 , 1 ) , ν ∈ [ 0 , 1 ] , and γ = μ + ν − μ ν . If g ∈ C 1 − γ γ [ a , b ] and I a + 1 − ν ( 1 − μ ) g ∈ C 1 − γ 1 [ a , b ] , then

Lemma 3.1

Let T be a nondecreasing and partial nonlinear Z -contraction on ( P , ⊑ , ∥ ⋅ ∥ ) , then T is partial condensing with respect to P H -MNC β p defined on P , that is,

for every C ∈ P b d , c n ( P ) with β p ( C ) > 0 .

Proof

The details of proof are similar to that in [26, Theorem 2.1] for metric space setting.□

Theorem 3.1

Assume that ( P , ⊑ , ∥ ⋅ ∥ ) is ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete. Let A ∈ ℳ n d , b d p a r ( P ) be a partial nonlinear Z -contraction and ℬ ∈ ℳ n d , c t p a r ( P ) be partial compact. Then ( A + ℬ ) admits at least one fixed point if P contains ω 0 such that ω 0 ⊑ A ω 0 + ℬ ω 0 or A ω 0 + ℬ ω 0 ⊑ ω 0 .

Proof

From the hypotheses assumed on operators A and ℬ , we can conclude that ( A + ℬ ) is nondecreasing partial bounded and continuous. We also have ω 0 ⊑ ( A + B ) ω 0 . Let C be any chain in P such that β p ( C ) > 0 . Then we have

As ℬ is partial compact and A is nonlinear partial Z -contraction, so by application of Lemma 3.1, we get from (3.1),

Thus, necessary requirements of Theorem 2.1 are fulfilled. Hence, ( A + ℬ ) admits at least one fixed point in P .□

Corollary 3.1

Assume that ( P , ⊑ , ∥ ⋅ ∥ ) is ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete. Let A ∈ ℳ n d , b d p a r ( P ) be a partial nonlinear D -contraction and ℬ ∈ ℳ n d , c t p a r ( P ) be partial compact. Then ( A + ℬ ) admits at least one fixed point if P contains ω 0 such that ω 0 ⊑ A ω 0 + ℬ ω 0 or A ω 0 + ℬ ω 0 ⊑ ω 0 .

Corollary 3.2

Assume that ( P , ⊑ , ∥ ⋅ ∥ ) is ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete. Let A ∈ ℳ n d , b d p a r ( P ) be a partial nonlinear contraction mapping and ℬ ∈ ℳ n d , c t p a r ( P ) be partial compact. Then ( A + ℬ ) admits at least one fixed point if P contains ω 0 such that ω 0 ⊑ A ω 0 + ℬ ω 0 or A ω 0 + ℬ ω 0 ⊑ ω 0 .

Theorem 3.2

Assume that ( P , ⊑ , ∥ ⋅ ∥ ) is ( ∥ ⋅ ∥ − ⊑ ) -compatible, regular, and complete. Let A ∈ ℳ m m , b d c , p a r ( P ) be a partial nonlinear coupled Z -contraction and ℬ ∈ ℳ m m , c t c , p a r ( P ) be partial compact. Then ( A + ℬ ) admits at least one coupled fixed point if P 2 contains ( ω 0 , υ 0 ) such that ω 0 ⊑ A ( ω 0 , υ 0 ) + ℬ ( ω 0 , υ 0 ) and A ( υ 0 , ω 0 ) + ℬ ( υ 0 , ω 0 ) ⊑ υ 0 .

Proof

Let us define T : P × P → P by

Also

From the definition of T , it can be observed that T is well defined and T is mixed monotone, partial bounded, and partial continuous coupled operator. We need to show first that T is coupled partial condensing operator.

Let C 1 , C 2 ∈ P b d , c n such that β p ( C 1 ) > 0 and β p ( C 2 ) > 0 , where β p is P H -MNC. Then clearly C = C 1 × C 2 is nonempty bounded chain in P 2 . Let us define β ˜ p ( C ) = max { β p ( C 1 ) , β p ( C 2 ) } . Then β ˜ p is a P H -MNC on P 2 and β ˜ p ( C ) > 0 .

Let us define two sequences { t n } and { s n } defined by t n = β ˜ p ( C ) − ε n > 0 and s n = β ˜ p ( C ) + ε n > 0 , where { ε n } is such that ε n → 0 as n → ∞ . Then lim n → ∞ t n = lim n → ∞ s n = β ˜ p ( C ) > 0 and t n < s n . Then by ( ζ 2 ) in Definition 2.10 for a ζ ∈ Z , we have

Choosing ε = sup ε n sufficiently small, we can deduce from inequality (3.2) that there exists Δ < 0 so that

where, t ∈ [ β ˜ p ( C ) − ε , β ˜ p ( C ) ] and s ∈ [ β ˜ p ( C ) , β ˜ p ( C ) + ε ] . Let ℛ = β ˜ p ( C ) + ε . Assume that C = C 1 × C 2 has a ℛ -net, that is,

where u j ∈ P × P , j = 1 , 2 , … , K .

Let ℛ † = β ˜ p ( C ) − ε . We show that A ( C ) has a ℛ † -net. Let y = ( y 1 , y 2 ) ∈ A ( C ) . Then for every y ∈ A ( C ) , there exists x = ( x 1 , x 2 ) ∈ C such that A x = A ( x 1 , x 2 ) = ( y 1 , y 2 ) = y . Then from (3.4), there exists u j = ( u j 1 , u j 2 ) such that

Now if A ( x ) = A ( x 1 , x 2 ) = A ( u j 1 , u j 2 ) = A ( u j ) , then ∥ A ( u j ) − A ( x ) ∥ < ℛ † . Suppose A ( x ) ≠ A ( u j ) , then we have two cases.

Case I: If 0 < ∥ u j − x ∥ < ℛ † , then Z -contractivity of A gives us

which implies

Case II: If ℛ † < ∥ u j − x ∥ < ℛ , then either ∥ A u j − A x ∥ < ℛ † or ∥ A u j − A x ∥ ≥ ℛ † . Suppose ∥ A u j − A x ∥ ≥ ℛ † , then (3.3) gives us