Existence results for ABC-fractional BVP via new fixed point results of F-Lipschitzian mappings

Theorem 1

[20] Let W be a nonempty set, ρ and δ two metrics on W , and ϒ : W → W . If

1. ∃ c > 0 : ρ ( ϒ ( υ ) , ϒ ( ω ) ) ≤ c δ ( υ , ω ) , for all υ , ω ∈ W ,

2. ( W , ρ ) is a complete metric space,

3. ϒ : ( W , ρ ) → ( W , ρ ) is continuous,

4. ∃ α ∈ ] 0 , 1 [ : δ ( ϒ ( υ ) , ϒ ( ω ) ) ≤ α δ ( υ , ω ) , for all υ , ω ∈ W ,

5. then ϒ has a unique fixed point υ ∗ , which can be calculated as lim n → ∞ ϒ n ( υ 0 ) for any υ 0 ∈ W .

Theorem 2

[15] Let W be a nonempty set, ρ and δ two metrics on W , and ϒ n , ϒ : W → W . If

1. ( W , ρ ) , ( W , δ ) , and ϒ satisfy the hypotheses of Theorem 1,

2. the sequence ( ϒ n ) n ∈ N converges uniformly on ( W , ρ ) to ϒ ,

3. F ϒ n ≠ ϕ ,

4. ∃ c 1 > 0 : δ ( υ , ω ) ≤ c 1 ρ ( υ , ω ) , for all υ , ω ∈ W ,

5. c c 1 α < 1 ,

6. then for every υ n ∈ F ϒ n , the sequence ( υ n ) n ∈ N converges in ( W , ρ ) to the unique fixed point, υ ∗ , of ϒ .

Definition 3

[8] Let ℱ denote the collection of all functions F : R + ⟶ R with the following properties:

1. F increases strictly, i.e. for s , t ∈ R + with s < t implies that F ( s ) < F ( t ) ;

2. For any sequence { α n } n ∈ N of real numbers with α n ≥ 0 , lim n ⟶ ∞ α n = 0 if and only if lim n ⟶ ∞ F ( α n ) = − ∞ ;

3. There exists k ∈ ( 0 , 1 ) satisfying lim α ⟶ 0 + α k F ( α ) = 0 .

The function F which satisfies ( F 1 ) – ( F 3 ) is called the Wardowski function. In [14], it has been given that ( F 2 ) can be replaced by ( F 2 ) ′ for any sequence ( ℓ n ) ⊂ ( 0 , ∞ ) , if lim t n ⟶ 0 + F ( ℓ n ) = − ∞ , then lim n ⟶ ∞ ℓ n = 0 .

Any function F that satisfies ( F 1 ) and ( F 2 ) ′ is called the semi-Wardowski function.

Theorem 4

[21] Let ( W , ρ ) be a complete metric space and let ϒ : W ⟶ W be a mapping. If there exist F ∈ ℱ and τ > 0 such that

where

then ϒ has a unique fixed point in W, provided that ϒ or F is continuous.

Theorem 5

[13] Let a complete metric space ( W , ρ ′ ) be given and ρ be another metric on W . Let ϒ : W ⟶ W be a mapping and F ∈ ℱ be continuous. If there exists τ > 0 with

where

Moreover, if ρ ≱ ρ ′ and ϒ is a uniformly continuous function from ( W , ρ ) into ( W , ρ ′ ) , and if ρ ≠ ρ ′ , with ϒ is a continuous function from ( W , ρ ′ ) into ( W , ρ ′ ) , then there is a point z ∈ W with z = ϒ z .

Theorem 6

[22] Let a complete metric space ( W , ρ ′ ) be given and ρ be another metric on W . Let ϒ : W ⟶ W be a mapping. Suppose there exists q ∈ ( 0 , 1 ) such that for υ , ω ∈ W we have

Moreover, if ρ ≱ ρ ′ , and ϒ is a uniformly continuous function from ( W , ρ ) into ( W , ρ ′ ) , and if ρ ≠ ρ ′ , with ϒ is a continuous function from ( W , ρ ′ ) into ( W , ρ ′ ) , then there is a point z ∈ W with z = ϒ z .

Definition 7

[23] Let W be a nonempty set and ϒ : W ⟶ W . The orbit of an element υ ∈ W is defined by O ( υ ) = { υ , ϒ υ , ϒ 2 υ , … } .

Definition 8

[23] Let ( W , ρ ) be a metric space and ϒ : W ⟶ W be a mapping, ϒ is called orbitally continuous if

implies

for each υ ∈ W .

A space ( W , ρ ) is ϒ -orbitally complete if every Cauchy sequence from orbit { ϒ n k υ } k = 1 ∞ , υ ∈ W converges in W .

Definition 9

[24] Let ϒ be a self-mapping on a metric space ( W , ρ ) . We say that ϒ is called a Lipschitz mapping with constant k ≥ 0 if

Clearly every Lipschitz mapping is uniformly continuous on W . A Lipschitz mapping with k < 1 is called strict (or Banach) contractions with k = 1 called nonexpansive. Lipschitz mappings are generally also known as Lipschitzian.

Lemma 10

[25] For a given semi-Wardowski function F, there exists a countable subset Ω ⊂ ( 0 , ∞ ) satisfying

for each t ∈ ( 0 , ∞ ) \ Ω .

Lemma 11

[25] Let F be a given semi-Wardowski function. Then

for all t , s ∈ ( 0 , ∞ ) .

Theorem 12

Let ρ 1 and ρ 2 be two metrics on a nonempty set W , let ϒ : W → W be an orbitally ρ 2 -continuous and ( W , ρ 2 ) be an orbitally complete metric space, if there exists τ such that

where the function F is a semi-Wardowski function and

provided ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) ≠ 0 and ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) ≠ 0 , for all υ , ω ∈ W .

If ρ 1 2 ( υ , ϒ ω ) + ρ 1 2 ( ϒ υ , ω ) = 0 or ρ 1 ( υ , ϒ ω ) + ρ 1 ( ϒ υ , ω ) = 0 , then M ( υ , ω ) = 0 .

In addition,

and

Then ϒ has a fixed point.

Proof

For υ 0 ∈ W , consider the iterative sequence

and substituting υ = υ n − 1 and ω = υ n in ( 3 ) , we have

where

So we have

If

then, we have

if and only if

which is a contradiction. So

Then, we have

if and only if

and

So we have

where

So

Then, by ( F 2 ) ′

From (7), we have { ρ 1 ( υ n − 1 , υ n ) } is nonincreasing and bounded from below so converges to some α ≥ 0 . We claim α = 0 , if not then there exists some k ∈ N such that ρ 1 ( υ n − 1 , υ n ) > 0 for all n ≥ k . Using (7), we have α < α , which is a contradiction, therefore α = 0 . Hence, in both cases ( τ > 0 and τ ∈ R ) we have

Now to prove { υ n } is a Cauchy sequence, we use Lemma ( 10 ) , since F is strictly increasing. Let Ω be the set of all points of discontinuities of F , clearly Ω can be at most countable. Now suppose contrary that { υ n } is not Cauchy, and there exists ε > 0 , ε ∉ Ω such that for every k ≥ 0 , we can choose natural numbers n k and m k such that

Denote by m ¯ k and n ¯ k the least of m k and n k , respectively, such that

for all k ≥ 0 . Choose k 0 ∈ N such that ε k < ε for all k ≥ k 0 , we have

for all k ≥ k 0 . Taking lim as k → ∞ , we have

Now for all k ≥ 0 , we have

Also as

From (A) and (B), we have

Taking lim as k → ∞ , we have

Now from ( 6 ) , we have

Taking again lim as k → ∞ and using the fact that F is continuous at ε , we have

which is a contradiction. Also from ( 8 ) , we have

Using the fact that F is continuous at ε , we have

which is a contradiction.

Hence, { υ n } is a Cauchy sequence in ( W , ρ 1 ) .

Now we claim that { υ n } is a Cauchy sequence in ( W , ρ 2 ) . The case ρ 1 ≥ ρ 2 is trivial. Suppose ρ 1 ≱ ρ 2 and by using the orbitally uniform continuity of ϒ , there exists δ ( ε ) > 0 , such that ρ 2 ( ϒ υ , ϒ ω ) < ε whenever ρ 1 ( υ , ω ) < δ ( ε ) . This implies ρ 2 ( υ m , υ n ) < ε for all m , n ≥ k , whenever ρ 1 ( υ m − 1 , υ n − 1 ) < δ . Hence, { υ n } is a Cauchy sequence in ( W , ρ 2 ) . As ( W , ρ 2 ) is ϒ orbitally complete, so there exists υ ∈ W such that υ n → υ . Also ϒ is ρ 2 -orbitally continuous, so υ n → υ implies ϒ υ n → ϒ υ . Then for n → ∞ , we have

If ρ 1 ≠ ρ 2 , then

For n → ∞ and using ( 5 ) , we have

Now if ρ 1 = ρ 2 and υ ≠ ϒ υ , then there exists a natural number k 1 and a subsequence { υ n k } of { υ n } such that ρ 1 ( υ n k , ϒ υ ) > 0 for all k ≥ k 1 . (If not then clearly υ n k → ϒ υ , and we are done.) From ( 3 ) , we have

where

So (9) becomes

if and only if

For k → ∞ , we have

If

then we have

which is a contradiction. If

then we have

which is a contradiction. Hence,

and

So (9) becomes