Common fixed point results for two families of multivalued A–dominated contractive mappings on closed ball with applications

Definition 1.1

[23] Let M be a nonempty set and d_b : M × M → [0, ∞) be a function. If, for any x, y, z ∈ M, the following conditions hold:

1. d_b(x, y) ≤ b[d_b(x, z) + d_b(z, y)], (where b ≥ 1);

2. d_b(x, y) = 0 implies x = y;

3. d_b(x, y) = d_b(y, x).

Then d_b is called a dislocated b-metric with coefficient b (or simply d_b-metric) and the pair (M, d_b) is called a dislocated b-metric space (or simply DBM space). It should be noted that every dislocated metric is a dislocated b-metric with b = 1. Also, if x = y, then d_b(x, y) may not be 0. For x ∈ M and ε > 0, B(x, ε) = {y ∈ M : d_b(x, y) ≤ ε} is a closed ball in M.

Definition 1.2

[23] Let (M, d_b) be a D.B.M space.

1. A sequence {x_n} in (M, d_b) is called Cauchy sequence if given ε > 0, there corresponds n₀ ∈ N such that for all n, m ≥ n₀ we have d_b(x_m, x_n) < ε or limn,m→∞ d_b(x_n, x_m) = 0.

2. A sequence {x_n} dislocated b-converges (for short d_b-converges) to x if limn→∞ d_b(x_n, x) = 0. In this case x is called a d_b-limit of {x_n}.

3. (M, d_b) is called complete if every Cauchy sequence in M converges to a point x ∈ M such that d_b(x, x) = 0.

Definition 1.3

Let K be a nonempty subset of D.B.M space of M and let x ∈ M. An element y₀ ∈ K is called a best approximation in K if

We denote P(M) be the set of all closed proximinal subsets of M.

Definition 1.4

[12] The function H_db : P(M) × P(M) → R⁺, defined by

is called dislocated Hausdorff b–metric on P(M).

Definition 1.5

Let (M, d_b) be a D.B.M space. Let S : M → P(M) be multivalued mapping, α : M × M → [0, +∞) and α_∗(i, Si) = ∈ f {α(i, l) : l ∈ Si}. Let H ⊆ M, then S is said to be α_∗-dominated on H, whenever α_∗(i, Si) ≥ 1 for all i ∈ H. If H = M, then we say that the S is α_∗-dominated. If S : M → M is a self mapping, then S is α-dominated on H, whenever α (i, Si) ≥ 1 for all i ∈ H.

Lemma 1.6

[13] Let (M, d_b) be a D.B.M space and (P(M), H_db) be a dislocated Hausdorff b-metric space. For all G, H in P(M) and for any g ∈ G such that d_b(g, H) = d_b(g, h_g), where h_g ∈ H. Then H_db(G, H) ≥ d_b(g, h_g) holds.

Theorem 2.1

Let (M, d_b) be a complete D.B.M space with constant b ≥ 1. Let r > 0, c₀ ∈ B_db(c₀, r) ⊆ M, α : M × M → [0, ∞) and {S_σ : σ ∈ Ω}, {T_β : β ∈ Φ} be two families of α_∗-dominated multivalued mappings from M to P(M) on B_db(c₀, r). Suppose that the following are satisfied:

1. There exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and a strictly increasing mapping A such that

   τ+A(Hdb(Sσe,Tβy))≤Aμ1db(e,y)+μ2db(e,Sσe)+μ3db(e,Tβy)+μ4db(e,Sσe).db(y,Tβy)1+db(e,y), (2.1)

   whenever e, y ∈ B_db(c₀, r) ∩ \{T_β S_σ(c_n)}, α (e, y) ≥ 1, σ ∈ Ω, β ∈ Φ and H_db(S_σ e, T_β y) > 0.

2. If η=μ1+μ2+bμ31−bμ3−μ4, then

   db(c0,Sac0)≤η(1−bη)r. (2.2)

   Then {T_β S_σ(c_n)} is a sequence in B_db(c₀, r), α (c_n, c_n+1) ≥ 1 for all n ∈ ℕ ∪ {0} and {T_β S_σ(c_n)} → u ∈ B_db(c₀, r). Also, if u satisfies (2.1) and either α(c_n, u) ≥ 1 or α (u, c_n) ≥ 1 for all n ∈ ℕ ∪ {0}, then S_σ and T_β have common fixed point u in B_db(c₀, r) for all σ ∈ Ω and β ∈ Φ.

Proof

Consider a sequence {T_β S_σ(c_n)}. From (2.2), we get

It follows that,

Let c₂, ⋯, c_j ∈ B_db(c₀, r) for some j ∈ ℕ. If j is odd, then j = 2ì + 1 for some ì ∈ ℕ. Since {S_σ : σ ∈ Ω} and {T_β : β ∈ Φ} are two families of α_∗-dominated multivalued mappings on B_db(c₀, r), so α_∗(c_2ì, S_σ c_2ì) ≥ 1 and α_∗(c_2ì+1, T_βc_2ì+1) ≥ 1 for all σ ∈ Ω and β ∈ Φ. As α_∗(c_2ì, S_σ c_2ì) ≥ 1, this implies ∈ f {α (c_2ì, b) : b ∈ S_σ c_2ì} ≥ 1. Also c_2ì+1 ∈ S_f c_2ì for some f ∈ Ω, so α (c_2ì, c_2ì+1) ≥ 1. Also c_2ì+2 ∈ T_g c_2ì+1 for some g ∈ Φ. Now by using Lemma 1.6, we have

This implies

As A is strictly increasing, we obtain

Which implies

By assumptions η=μ1+μ2+bμ31−bμ3−μ4<1. Hence

Similarly, if j is even, we have

Summing up, we have

It follows,

As μ₁, μ₂, μ₃, μ₄ > 0, b ≥ 1 and bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1, so |bη| < 1. Then, we have

the last inequality following by bη < 1, that is the assumption (i). So c_j+1 ∈ B_db(c₀, r). Hence, by induction c_n ∈ B_db(c₀, r) for all n ∈ N. Also α(c_n, c_n+1) ≥ 1 for all n ∈ ℕ ∪ {0}. Now,

Hence, for any positive integers m, n(n > m), we have

Hence {T_β S_σ(c_n)} is a Cauchy sequence in B_db(c₀, r). Since (B_db(c₀, r), d_b) is a complete metric space, so there exists u ∈ B_db(c₀, r) such that {T_β S_σ(c_n)} → u as n → ∞, then

by assumption, α(c_n, u) ≥ 1. Suppose that d_b(u, T_βu) > 0, then there exists a positive integer k such that d_b(c_n, T_βu) > 0 for all n ≥ k. For n ≥ k, we have

Now, there exists some e ∈ Ω such that c_2n+1 ∈ S_e c_2n and d_b(c_2n, S_e c_2n) = d_b(c_2n, c_2n+1). By using Lemma 1.6 and inequality (2.1), we have

Letting n → ∞, and by using (2.5) we get

which is a contradiction. So our supposition is wrong. Hence d_b(u, T_βu) = 0 or u ∈ T_βu for all β ∈ Φ. Similarly, by using Lemma 1.6 and inequality (2.1), we can show that d_b(u, S_σu) = 0 or u ∈ S_σu for all σ ∈ Ω. Hence the S_σ and T_β have a common fixed point u in B_db(c₀, r) for all σ ∈ Ω and β ∈ Φ. Now,

This implies that d_b(u, u) = 0.

Example 2.2

Let M = Q⁺ ∪ {0} and let d_b : M × M → M be the complete D.B.M space defined by

with b = 2. Define, two families of multivalued mappings S_σ, T_β : M × M → P(M) by

and

Suppose that, x₀ = 1, r = 225, then B_db(x₀, r) = [0, 14] ∩ M. Now, d_b(x₀, S₁x₀) = d_b(1, S₁1) = d_b(1, 13 ). So x₁ = 13 . Now, db(x1,T1x1)=db(13,T113)=db(13,112). , So x2=112. Now, db(x2,S2x2)=db(112,S2112)=db(112,172). So x3=172. Continuing in this way, we have {TnSm(xn)}={1,13,112,172....}. Take μ1=110,μ2=120,μ3=160,μ4=130, then bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and η=1156. Now

Consider the mapping α : M × M → [0, ∞) by

Now, if x, y ∈ B_db(x₀, r) ∩ {T_β S_σ(x_n)} with α (x, y) ≥ 1, we have

Thus,

which implies that, for any τ∈(0,1295] and for a strictly increasing mapping A(s) = ln s, we have

Note that, for 16, 15 ∈ M, then α(16, 15) ≥ 1. But, we have

So condition (2.1) does not holds on all M but holds only on B_db(1, 225). Thus all the conditions of Theorem 2.1 are satisfied. Hence S_σ and T_β have a common fixed point for all σ ∈ Ω and β ∈ Φ.

If, we take {S_σ : σ ∈ Ω} = {T_β : β ∈ Φ} in Theorem 2.1, then we have the following result.

Corollary 2.3

Let (M, d_b) be a complete D.B.M space with constant b ≥ 1. Let r > 0, c₀ ∈ B_db(c₀, r) ⊆ M, α : M × M → [0, ∞) and {S_σ : σ ∈ Ω} be a family of α_∗-dominated multivalued mappings from M to P(M) on B_db(c₀, r). Suppose that the following satisfy:

1. There exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and a strictly increasing mapping A such that

   τ+A(Hdb(Sσe,Sβy))≤Aμ1db(e,y)+μ2db(e,Sσe)+μ3db(e,Sβy)+μ4db(e,Sσe).db(y,Sβy)1+db(x,y), (2.6)

   whenever e, y ∈ B_db(c₀, r) ∩ {S_σ(c_n)}, α(e, y) ≥ 1, σ, β ∈ Ω and H_db(S_σe, S_βy) > 0.

2. If η=μ1+μ2+bμ31−bμ3−μ4, then

   db(c0,Sσc0)≤η(1−bη)r.

   Then {MS_σ(c_n)} is a sequence in B_db(c₀, r), α(c_n, c_n+1) ≥ 1 for all n ∈ ℕ ∪ {0} and {S_σ(c_n)} → u ∈ B_db(c₀, r). Also, if u satisfies (2.6) and either α(c_n, u) ≥ 1 or α (u, c_n) ≥ 1 for all n ∈ ℕ ∪ {0}, then {S_σ : σ ∈ Ω} have common fixed point u in B_db(c₀, r).

Definition 3.1

Let X be a nonempty set and G = (V(G), E(G)) be a graph such that V(G) = X, A ⊆ X. A mapping F : X → P(X) is said to be multi graph dominated on A if (x, y) ∈ E(G), for all y ∈ Fx and x ∈ A.

Theorem 3.2

Let (M, d_b) be a complete D.B.M space endowed with a graph G with constant b ≥ 1. Let r > 0, c₀ ∈ B_db(c₀, r) and {S_σ : σ ∈ Ω}, {T_β : β ∈ Φ} be two families of multivalued mappings from M to P(M). Suppose that the following are satisfied:

1. {S_σ : σ ∈ Ω}, {T_β : β ∈ Φ} are two families of multi graph dominated on B_db(c₀, r) ∩ {T_β S_σ(c_n)}.

2. There exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and a strictly increasing mapping A such that

   τ+A(Hdb(Sσe,Tβy))≤Aμ1db(e,y)+μ2db(e,Sσe)+μ3db(e,Tβy)+μ4db(e,Sσe).db(y,Tβy)1+db(e,y), (3.1)

   whenever e, y ∈ B_db(c₀, r) ∩ {T_β S_σ(c_n)}, (e, y) ∈ E(G), σ ∈ Ω, β ∈ Φ and H_db(S_σe, T_βy) > 0.

3. d_b(c₀, S_σ c₀) ≤ η (1 – bη)r, where η=μ1+μ2+bμ31−bμ3−μ4.

Then, {T_β S_σ(c_n)} is a sequence in B_db(c₀, r), (c_n, c_n+1) ∈ E(G) and {T_β S_σ(c_n)} → m^∗. Also, if m^∗ satisfies (3.1) and (c_n, m^∗) ∈ E(G) or (m^∗, c_n) ∈ E(G) for all n ∈ N ∪ {0}, then S_σ and T_β have common fixed point m^∗ in B_db(c₀, r) for all σ ∈ Ω and β ∈ Φ.

Proof

Define α : M × M → [0, ∞) by

As S_σ and T_β are two families of graph dominated on B_db(c₀, r), then for e ∈ ≤, (e, y) ∈ E(G) for all y ∈ S_σe and (e, y) ∈ E(G) for all y ∈ T_βe. So, α (e, y) = 1 for all y ∈ S_σe and α (e, y) = 1 for all y ∈ T_βe. This implies that ∈ f {α (e, y) : y ∈ S_σe} = 1 and ∈ f {α (e, y) : y ∈ T_βe} = 1. Hence α_∗(e, S_σe) = 1, α_∗(e, T_βe) = 1 for all e ∈ B_db(c₀, r). So, S_σ, T_β : M → P(M) are two families of α_∗-dominated mappings on B_db(c₀, r). Moreover, inequality (3.1) can be written as

whenever e, y ∈ B_db(c₀, r) ∩ {T_β S_σ(c_n)}, α (e, y) ≥ 1 and H_db(S_σe, T_βy) > 0. Also, (iii) holds. Then, by Theorem 2.1, we have {T_β S_σ(c_n)} is a sequence in B_db(c₀, r) and {T_β S_σ(c_n)} → m^∗ ∈ B_db(c₀, r). Now, c_n, m^∗ ∈ B_db(c₀, r) and either (c_n, m^∗) ∈ E(G) or (m^∗, c_n) ∈ E(G) implies that either α (c_n, m^∗) ≥ 1 or α (m^∗, c_n) ≥ 1. So, all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, S_σ and T_β have a common fixed point m^∗ in B_db(c₀, r) and d_b(m^∗, m^∗) = 0.

Theorem 4.1

Let (M, d_b) be a complete D.B.M space with constant b ≥ 1. Let r > 0, c₀ ∈ B_db(c₀, r) ⊆ M, α : M × M → [0, ∞) and {S_σ : σ ∈ Ω}, {T_β : β ∈ Φ} be two families of α-dominated mappings from M to M on B_db(c₀, r). Suppose that the following are satisfied:

1. There exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and a strictly increasing mapping A such that

   τ+A(Hdb(Sσe,Tβy))≤Aμ1db(e,y)+μ2db(e,Sσe)+μ3db(e,Tβy)+μ4db(e,Sσe).db(y,Tβy)1+db(e,y), (4.1)

   whenever e, y ∈ B_db(c₀, r) ∩ {T_β S_σ(c_n)}, α (e, y) ≥ 1, σ ∈ Ω, β ∈ Φ and d_b(S_σe, T_βy) > 0.

2. If η=μ1+μ2+bμ31−bμ3−μ4, then

   db(c0,Sσc0)≤η(1−bη)r.

   Then {T_β S_σ(c_n)} is a sequence in B_db(c₀, r), α (c_n, c_n+1) ≥ 1 for all n ∈ ℕ ∪ {0} and {T_β S_σ(c_n)} → u ∈ B_db(c₀, r). Also, if u satisfies (4.1) and either α(c_n, u) ≥ 1 or α (u, c_n) ≥ 1 for all n ∈ ℕ ∪ {0}, then S_σ and T_β have common fixed point u in B_db(c₀, r) for all σ ∈ Ω and β ∈ Φ.

Proof

The proof of the above Theorem is similar to Theorem 2.1.

If, we take {S_σ : σ ∈ Ω} = {T_β : β ∈ Φ} in Theorem 4.1, then we have the following result.

Corollary 4.2

Let (M, d_b) be a complete D.B.M space with constant b ≥ 1. Let r > 0, c₀ ∈ B_db(c₀, r) ⊆ M, α : M × M → [0, ∞) and {S_σ : σ ∈ Ω} be a family of α-dominated mappings from M to M on B_db(c₀, r). Suppose that the following satisfy:

1. There exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and a strictly increasing mapping A such that

   τ+A(Hdb(Sσe,Sβy))≤Aμ1db(e,y)+μ2db(e,Sσe)+μ3db(e,Sβy)+μ4db(e,Sσe).db(y,Sβy)1+db(x,y), (4.2)

   whenever e, y ∈ B_db(c₀, r) ∩ {MS_σ(c_n)}, α (e, y) ≥ 1, σ, β ∈ Ω, and d_b(S_σe, S_σy) > 0.

2. If η=μ1+μ2+bμ31−bμ3−μ4, then

   db(c0,Sσc0)≤η(1−bη)r.

   Then {MS_σ(c_n)} is a sequence in B_db(c₀, r), α (c_n, c_n+1) ≥ 1 for all n ∈ ℕ ∪ {0} and {MS_σ(c_n)} → u ∈ B_db(c₀, r). Also, if u satisfies (4.2) and either α(c_n, u) ≥ 1 or α (u, c_n) ≥ 1 for all n ∈ ℕ ∪ {0}, then S_σ has a fixed point u in B_db(c₀, r) for all σ ∈ Ω.

Theorem 5.1

Let (M, d_b) be a complete D.B.M space with coefficient b ≥ 1. Let c₀ ∈ M and {S_σ : σ ∈ Ω}, {T_β : β ∈ Φ} be two families of mappings from M to M. Assume that there exist τ, μ₁, μ₂, μ₃, μ₄ > 0 satisfying bμ₁ + bμ₂ + (1 + b)bμ₃ + μ₄ < 1 and A : ℝ₊ → ℝ is a strictly increasing mapping such that the following holds:

whenever e, y ∈ {T_β S_σ(c_n)}, σ ∈ Ω, β ∈ Φ and d_b(S_σe, T_β y) > 0. Then {T_β S_σ(c_n)} → u ∈ M. Also, if inequality (5.1) holds for e, y ∈ {u}, then S_σ and T_β have unique common fixed point u in M for all σ ∈ Ω and β ∈ Φ.

Proof

The proof of this theorem is similar to Theorem 2.1. We have to prove the uniqueness only. Let v be another common fixed point of S_σ and T_β. Suppose d_b(S_σ u, T_β v) > 0. Then, we have