A Best Proximity Point Result in Modular Spaces with the Fatou Property

Definition 1. Let X_ρ be a modular space.

• (1)

  The sequence {x_n} ⊂ X_ρ is said to be ρ-convergent to x ∈ X_ρ if ρ(x_n − x) → 0, as n → ∞.

• (2)

  The sequence {x_n} ⊂ X_ρ is said to be ρ-Cauchy if ρ(x_n − x_m) → 0, as n, m → ∞.

• (3)

  A subset C of X_ρ is called ρ-closed if the ρ-limit of a ρ-convergent sequence of C always belongs to C.

• (4)

  A subset C of X_ρ is called ρ-complete if any ρ-Cauchy sequence in C is ρ-convergent and its ρ-limit belongs to C.

Definition 2. The modular ρ has the Fatou property if ρ(x) ≤ liminf _n→∞ρ(x_n) whenever {x_n}  ρ-converges to x.

Definition 3. Let T : A → B be a given nonself-mapping. We say that z ∈ A₀ is a best proximity point of T if

Definition 4. A nonself-mapping T : A → B is said to be a proximal quasicontraction if there exists a number q ∈ (0,1) such that

where x, y, u, v ∈ A.

Lemma 5. Let T : A → B be a nonself-mapping. Suppose that

• (i)

  A₀ ≠ ∅;

• (ii)

  T(A₀)⊆B₀.

Then, for any a ∈ A₀, there exists a sequence {x_n} ⊂ A₀ such that

Proof. Let a ∈ A₀. From (ii), we have Ta ∈ B₀. By definition of the set B₀, there exists x₁ ∈ A₀ such that ρ(x₁ − Ta) = γ(A, B). Again, we have Tx₁ ∈ B₀, which implies that there exists x₂ ∈ A₀ such that ρ(x₂ − Tx₁) = γ(A, B). Continuing this process, by induction, we obtain a sequence {x_n} ⊂ A₀ satisfying (6).

Definition 6. Under the assumptions of Lemma 5, any sequence {x_n} ⊂ A₀ satisfying (6) is called a proximal Picard sequence associated to a ∈ A₀. We denote by PP(a) the set of all proximal sequences associated to a ∈ A₀.

Definition 7. Under the assumptions of Lemma 5, we say that A₀ is proximal T-orbitally ρ-complete if every ρ-Cauchy sequence {x_n} ∈ PP(a) for some a ∈ A₀ρ-converges to an element in A₀.

Lemma 8. Let X_ρ be a modular space. Suppose that a nonself-mapping T : A → B, where (A, B) is a pair of subsets of X, satisfies the following conditions:

• (i)

  ∃a ∈ A₀   |   δ_p(a) < ∞;

• (ii)

  T(A₀)⊆B₀;

• (iii)

  T is proximal quasi-contraction.

Then, for any {x_n} ∈ PP(a), one has for any n ≥ 1 and m ∈ ℕ.

Proof. Let {x_n} ∈ PP(a) and (s, r) ∈ ℕ². From the definition of PP(a), for all n ≥ 1, we have

which implies, since T is a proximal quasi-contraction, that This implies immediately that for all n ≥ 1. Hence, for any n ∈ ℕ, we have Using the above inequality, for all n ≥ 1 and m ∈ ℕ, we have

Lemma 9. Let (A, B) be a pair of subsets of a modular space X_ρ. Let T : A → B be a given nonself-mapping. Suppose that

• (i)

  A₀ is proximal T-orbitally ρ-complete;

• (ii)

  T(A₀)⊆B₀;

• (iii)

  ∃a ∈ A₀ such that δ_p(a) < ∞;

• (iv)

  T is proximal quasi-contraction;

• (v)

  ρ satisfies the Fatou property.

Then, any sequence {x_n} ∈ PP(a)  ρ-converges to some z ∈ A₀ such that for all n ≥ 1. Moreover, there exists w ∈ A₀ such that

Proof. Let {x_n} ∈ PP(a). From Lemma 8, we know that {x_n} is ρ-Cauchy. Since A₀ is proximal T-orbitally ρ-complete, then there exists z ∈ A₀ such that {x_n}  ρ-converges to z. Again, by Lemma 8, we have

for any n ≥ 1 and m ∈ ℕ. Letting m → ∞ in the above inequality and using the Fatou property, we obtain for all n ≥ 1. Now, since Tz ∈ B₀, by the definition of B₀, there exists some w ∈ A₀ such that ρ(w − Tz) = γ(A, B).

Theorem 10. Suppose that the assumptions of the previous lemma are satisfied. Assume ρ(z − w) < ∞ and ρ(a − w) < ∞. Then, the ρ-limit z ∈ A₀ of {x_n} ∈ PP(a) is a best proximity point of T. Moreover, if u ∈ A₀ is any best proximity point of T such that ρ(z − u) < ∞, then one has z = u.

Proof. By Lemma 9, we have

On the other hand, from the definition of {x_n}, we have Since T is proximal quasi-contraction, we get that Using Lemmas 8 and 9, we obtain that Again, from the definition of {x_n}, we have Since T is proximal quasi-contraction, we get that Thus, we proved that Continuing this process, by induction, we get that for all n ≥ 1. Therefore, we have Using the Fatou property, we get which implies, since q < 1, that ρ(z − w) = 0; that is, z = w. Thus, from (19), we get that Hence, z is a best proximity point of T.

Suppose now that u ∈ A₀ is a best proximity point of T such that ρ(z − u) < ∞. Since T is proximal quasi-contraction, we obtain that

Since q < 1, we have ρ(z − u) = 0, which implies that u = z.

Definition 11. We say that A is T-orbitally ρ-complete if {T^na} is a ρ-Cauchy for every a ∈ A, then it is ρ-convergent to an element of A.

Definition 12. The self-mapping T : A → A is said to be a quasicontraction if there exists a constant q ∈ (0,1) such that

for all x, y ∈ A.

Corollary 13. Consider a self-mapping T : A → A, where A is a nonempty subset of X_ρ. Suppose that the following conditions hold:

• (i)

  A is T-orbitally ρ-complete;

• (ii)

  ∃a ∈ A such that sup {ρ(T^sa − T^ra) : s, r ∈ ℕ} < ∞;

• (iii)

  ρ satisfies the Fatou property;

• (iv)

  T is quasi-contraction.

Then, the sequence {T^na}  ρ-converges to some z ∈ A. Moreover, if ρ(z − Tz) < ∞ and ρ(a − Tz) < ∞, then z is a fixed point of  T. If u ∈ A is a fixed point of  T with ρ(z − u) < ∞, then u = z.