Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach

Definition 1.1

A mapping T : P ∪ Q → P ∪ Q is called

1. cyclic if T ( P ) ⊆ Q and T ( Q ) ⊆ P ;

2. noncyclic if T ( P ) ⊆ P and T ( Q ) ⊆ Q ;

3. relatively nonexpansive if it satisfies ‖ T a − T b ‖ ≤ ‖ a − b ‖ whenever a ∈ P and b ∈ Q ;

4. nonexpansive if ‖ T a − T b ‖ ≤ ‖ a − b ‖ whenever a ∈ P and b ∈ Q and P = Q ;

5. compact if ( T ( P ) ¯ , T ( Q ) ¯ ) is compact.

Theorem 1.2

[4] A relatively nonexpansive cyclic mapping T : P ∪ Q → P ∪ Q has a best proximity point if T is compact and P 0 is nonempty, where ( P , Q ) is a nonempty, bounded, closed and convex ( NℬCC ) pair in a Banach space X .

Theorem 1.3

[4] Let X be a strictly convex Banach space and ( P , Q ) is an NℬCC pair in X . Then, a relatively nonexpansive noncyclic mapping T : P ∪ Q → P ∪ Q admits a best proximity pair, provided it is compact and P 0 is nonempty.

Definition 1.4

[7–9] An MNC is a mapping μ : B ( X ) → R + satisfying the following axioms:

1. μ ( P ) = 0 if and only if P is relatively compact,

2. μ ( P ) = μ ( P ¯ ) , P ∈ B ( X ) ,

3. μ ( P ∪ Q ) = max { μ ( P ) , μ ( Q ) } , where P , Q ∈ B ( X ) .

Example 1.5

[8] The non-negative numbers

and

assigned with a bounded subset C of a metric space X are called Kuratowski MNC (K-MNC) and Hausdorff MNC (H-MNC), respectively.

Theorem 1.6

Let T be a continuous self-mapping on a NℬCC subset C of a Banach space X , for every M ⊂ C satisfying one of the followings:

1. There exists a 0 ≤ λ < 1 such that μ ( T ( M ) ) ≤ λ μ ( M ) ,

2. μ ( M ) > 0 , μ ( T ( M ) ) < μ ( M ) .

Theorem 2.1

Let ( P , Q ) be a nonempty and convex pair in a Banach space E with P 0 being nonempty and μ an MNC on E . A relatively nonexpansive cyclic mapping T : P ∪ Q → P ∪ Q has at least one best proximity point if for every NℬCC , proximinal and T invariant pair ( M 1 , M 2 ) ⊆ ( P , Q ) with dist ( M 1 , M 2 ) = dist ( P , Q ) and for continuous mappings Δ i : R + → R + , i ∈ { 1 , 2 , 3 } such that for t > 0 , Δ 1 ( t ) > Δ 2 ( t ) − Δ 3 ( t ) , T satisfies

where κ ∈ ℋ and φ : [ 0 , ∞ ) → [ 0 , ∞ ) is a nondecreasing and continuous function.

Proof

As P 0 is nonempty, ( P 0 , Q 0 ) is nonempty. Also one can show that ( P 0 , Q 0 ) is convex, closed, T -invariant and proximinal pair considering the conditions on T (for more details see [12]). For a ∈ P 0 , there is a b ∈ Q 0 such that ‖ a − b ‖ = dist ( P , Q ) . Since T is relatively nonexpansive cyclic mapping,

which gives T a ∈ Q 0 , that is, T ( P 0 ) ⊆ Q 0 . Similarly, T ( Q 0 ) ⊆ P 0 and so T is cyclic on P 0 ∪ Q 0 .

Let us define a pair ( G n , H n ) as G n = con ¯ ( T ( G n − 1 ) ) and H n = con ¯ ( T ( H n − 1 ) ) , n ≥ 1 with G 0 = P 0 and H 0 = Q 0 . We claim that G n + 1 ⊆ H n and H n ⊆ G n − 1 for all n ∈ N . We have H 1 = con ¯ ( T ( H 0 ) ) = con ¯ ( T Q 0 ) = con ¯ ( P 0 ) ⊆ P 0 = G 0 . Therefore, T ( H 1 ) ⊆ T ( G 0 ) . So H 2 = con ¯ ( T ( H 1 ) ) ⊆ con ¯ ( T ( G 0 ) ) = G 1 . Continuing this pattern, we obtain H n ⊆ G n − 1 by using induction. Similarly, we can see that G n + 1 ⊆ H n for all n ∈ N . Thus, G n + 2 ⊆ H n + 1 ⊆ G n ⊆ H n − 1 for all n ∈ N . Hence, we obtain a decreasing sequence { ( G 2 n , H 2 n ) } of nonempty, closed and convex pairs in P 0 × Q 0 . Moreover, T ( H 2 n ) ⊆ T ( G 2 n − 1 ) ⊆ con ¯ ( T ( G 2 n − 1 ) ) = G 2 n and T ( G 2 n ) ⊆ T ( H 2 n − 1 ) ⊆ con ¯ ( T ( H 2 n − 1 ) ) = H 2 n . Therefore, for all n ∈ N , the pair ( G 2 n , H 2 n ) is T -invariant.

Now if ( u , v ) ∈ P 0 × Q 0 is a proximinal pair, then

Next, we show that the pair ( G n , H n ) is proximinal using mathematical induction. Obviously for n = 0 , the pair ( G 0 , H 0 ) is proximinal. Suppose that ( G k , H k ) is proximinal, we show that ( G k + 1 , H k + 1 ) is also proximinal. Let x be an arbitrary member in G k + 1 = con ¯ ( T ( G k ) ) . Then, it is represented as x = ∑ l = 1 m λ l T ( x l ) with x l ∈ G k , m ∈ N , λ l ≥ 0 and ∑ l = 1 m λ l = 1 . Due to proximinality of the pair ( G k , H k ) , there exists y l ∈ H k for 1 ≤ l ≤ m such that ‖ x l − y l ‖ = dist ( G k , H k ) = dist ( P , Q ) . Take y = ∑ l = 1 m λ l T ( y l ) . Then, y ∈ con ¯ ( T ( H k ) ) = H k + 1 and

This means that the pair ( G k + 1 , H k + 1 ) is proximinal and induction does the rest to prove ( G n , H n ) is proximinal for all n ∈ N .

Now if possible, let us take κ ( μ ( G 2 n ∪ H 2 n ) , φ ( μ ( G 2 n ∪ H 2 n ) ) ) > 0 for all n ∈ N . As κ is nondecreasing and { ( G 2 n , H 2 n ) } is a decreasing sequence,

holds for all n ∈ N . Therefore, the sequence { κ ( μ ( G 2 n ∪ H 2 n ) , φ ( μ ( G 2 n ∪ H 2 n ) ) ) } is nonnegative, bounded below and decreasing. Thus, there exists a ≥ 0 such that

Now if possible, let a > 0 . From the assumed hypothesis on T , we have

As n → ∞ , we have Δ 1 ( a ) ≤ Δ 2 ( a ) − Δ 3 ( a ) , which is a contradiction to the assumption on mapping Δ 1 , Δ 2 and Δ 3 . This means that a = 0 . Therefore,

By definition of mapping κ , we deduce that

That is, lim n → ∞ μ ( G 2 n ∪ H 2 n ) = max { lim n → ∞ μ ( G 2 n ) , lim n → ∞ μ ( H 2 n ) } = 0 . Now let G ∞ = ∩ n = 0 ∞ G 2 n and H ∞ = ∩ n = 0 ∞ H 2 n . By property ( d ) of MNC, the pair ( G ∞ , H ∞ ) is nonempty, convex, compact and T -invariant with dist ( G ∞ , H ∞ ) = dist ( P , Q ) . Now we obtain, T : G ∞ ∪ H ∞ → G ∞ ∪ H ∞ , which is a cyclic, relatively nonexpansive mapping on a nonempty, convex, compact and T -invariant pair ( G ∞ , H ∞ ) with dist ( G ∞ , H ∞ ) = dist ( P , Q ) . Therefore, Theorem 1.2 ensures that T admits a best proximity point.□

Theorem 2.2

Let ( P , Q ) be a nonempty and convex pair in a strictly convex Banach space E with P 0 nonempty and μ an MNC on E . A relatively nonexpansive noncyclic mapping T : P ∪ Q → P ∪ Q has at least one best proximity pair if for every NℬCC , proximinal and T invariant pair ( M 1 , M 2 ) ⊆ ( P , Q ) with dist ( M 1 , M 2 ) = dist ( P , Q ) equation (1) is satisfied.

Proof

It is clear that ( P 0 , Q 0 ) is NℬCC pair which is proximinal and T -invariant (see [12] for more details on proof). Let ( a , b ) ∈ P 0 × Q 0 be such that ‖ a − b ‖ = dist ( P , Q ) . Since T is relatively nonexpansive noncyclic mapping,

which gives T a ∈ P 0 , that is, T ( P 0 ) ⊆ P 0 . Similarly, T ( Q 0 ) ⊆ Q 0 and so T is noncyclic on P 0 ∪ Q 0 .

Let us define a pair ( G n , H n ) as G n = con ¯ ( T ( G n − 1 ) ) and H n = con ¯ ( T ( H n − 1 ) ) , n ≥ 1 with G 0 = P 0 and H 0 = Q 0 . We have H 1 = con ¯ ( T ( H 0 ) ) = con ¯ ( T ( Q 0 ) ) ⊆ Q 0 = H 0 . Therefore, T ( H 1 ) ⊆ T ( H 0 ) . Thus, H 2 = con ¯ ( T ( H 1 ) ) ⊆ con ¯ ( T ( H 0 ) ) = H 1 . Continuing this pattern, we obtain H n ⊆ H n − 1 by using induction. Similarly, we can see that G n + 1 ⊆ G n for all n ∈ N . Hence, we obtain a decreasing sequence { ( G n , H n ) } of nonempty, closed and convex pairs in P 0 × Q 0 . Also, T ( H n ) ⊆ T ( H n − 1 ) ⊆ con ¯ ( T ( H n − 1 ) ) = H n and T ( G n ) ⊆ T ( G n − 1 ) ⊆ con ¯ ( T ( G n − 1 ) ) = G n . Therefore, for all n ∈ N , the pair ( G n , H n ) is T -invariant. From the proof of Theorem 2.1, we have ( G n , H n ) is a proximinal pair such that dist ( G n , H n ) = dist ( P , Q ) for all n ∈ N ∪ { 0 } .

Now since { μ ( G n ∪ H n ) } is a positive nonincreasing sequence and following the proof of Theorem 2.1, we can prove that { μ ( G n ∪ H n ) } converges to 0.

Therefore, μ ( G n ∪ H n ) → 0 as n → ∞ . That is, lim n → ∞ μ ( G n ∪ H n ) = max { lim n → ∞ μ ( G n ) , lim n → ∞ μ ( H n ) } = 0 . Now let G ∞ = ∩ n = 0 ∞ G n and H ∞ = ∩ n = 0 ∞ H n . By property ( d ) of MNC, ( G ∞ , H ∞ ) is nonempty, convex, compact and T -invariant pair with dist ( G ∞ , H ∞ ) = dist ( P , Q ) . Thus, application of Theorem 1.3 ensures that T admits a best proximity pair.□

Corollary 2.3

Let ( P , Q ) be a nonempty and convex pair in a (strictly convex) Banach space E with P 0 nonempty and μ an MNC on E . A relatively nonexpansive cyclic (noncyclic) mapping T : P ∪ Q → P ∪ Q has at least one best proximity point (pair) if for every NℬCC , proximinal and T -invariant pair ( M 1 , M 2 ) ⊆ ( P , Q ) with dist ( M 1 , M 2 ) = dist ( P , Q ) and for continuous mappings Δ i : R + → R + , i ∈ { 1 , 2 , 3 } such that for t > 0 , Δ 1 ( t ) > Δ 2 ( t ) − Δ 3 ( t ) , T satisfies

where φ : [ 0 , ∞ ) → [ 0 , ∞ ) is a nondecreasing and continuous function.

Proof

Taking κ ( a , b ) = a + b in Theorem 2.1 (Theorem 2.2), we obtain the desired result.□

Corollary 2.4

Let ( P , Q ) be a nonempty and convex pair in a (strictly convex) Banach space E with P 0 nonempty and μ an MNC on E . A relatively nonexpansive cyclic (noncyclic) mapping T : P ∪ Q → P ∪ Q has at least one best proximity point (pair) if for every NℬCC , proximinal and T -invariant pair ( M 1 , M 2 ) ⊆ ( P , Q ) with dist ( M 1 , M 2 ) = dist ( P , Q ) and for continuous mappings Δ i : R + → R + , i ∈ { 1 , 2 , 3 } such that for t > 0 , Δ 1 ( t ) > Δ 2 ( t ) − Δ 3 ( t ) , T satisfies

Proof

If we take φ as a constant zero function in Corollary 2.3, we obtain the desired result.□

Corollary 2.5

Let ( P , Q ) be a nonempty and convex pair in a (strictly convex) Banach space E with P 0 is nonempty and μ an MNC on E . A relatively nonexpansive cyclic (noncyclic) mapping T : P ∪ Q → P ∪ Q has at least one best proximity point (pair) if for every NℬCC , proximinal and T -invariant pair ( M 1 , M 2 ) ⊆ ( P , Q ) with dist ( M 1 , M 2 ) = dist ( P , Q ) , T satisfies

where λ ∈ [ 0 , 1 ) .

Proof

If we take Δ 1 ( t ) = t + 1 2 , Δ 2 ( t ) = 2 λ t + 1 , Δ 3 ( t ) = λ t + 1 2 ( λ ∈ [ 0 , 1 ) ) and φ as a constant zero function in Corollary 2.3, we obtain the required result.□

Definition 3.1

[25] For k > 0 ( k ∈ R ) , the k -gamma function of a complex number z ∈ C with Re ( z ) > 0 is defined by

Definition 3.2

[26] Let f ∈ L 1 ( a , b ) and k ∈ R + . The integral

is called a ( k , ψ ) -Riemann-Liouville fractional integral of function f on [ a , b ] of order η ∈ R + .

Definition 3.3

[27] Let η , k ∈ R + , ν ∈ [ 0 , 1 ] , and m ∈ N . Let f , ψ ∈ C m ( [ a , b ] , R ) be two functions such that ψ ( x ) is increasing and ψ ′ ( t ) ≠ 0 for all t ∈ [ a , b ] . The ( k , ψ ) -Hilfer fractional derivative of order η and type ν of function f is defined as the following expression:

provided the right-hand side exists.

Remark 3.4

The ( k , ψ ) -Hilfer fractional derivative is considered to be the most general and unified definition of fractional derivative. In fact, by choosing different values of ψ ( x ) , k , a and taking limits on parameters η , ν in definition of ( k , ψ ) -Hilfer fractional derivative, we obtain a wide variety of fractional derivatives in the literature. See [27–29] for more information related to this.

Lemma 3.5

[27] Let μ , k ∈ R + and let ξ ∈ R such that ξ k > − 1 . Then, we have