In this section, we introduce the concept of a weakly (α, ψ, ξ)-contractive mapping and give fixed point result for such mapping.
Let (X, d) be a metric space.
(1)
A multi-valued mapping T :X → CL(X) is called a weakly (α, ψ, ξ)-contractive mapping if there exist three functions ψ ∈ Φ, ξ ∈ Ξ and α :X × X → [0, ∞)such that the following condition holds:
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where .
(2)
If ψ is strictly increasing, then the weakly (α, ψ, ξ)-contractive mapping is called a strictly weakly (α, ψ, ξ)-contractive mapping.
Next, we give first main result in this paper.
Let (X, d) be a complete metric space and T :X → CL(X) be a strictly weakly (α, ψ, ξ)-contractive mapping satisfying the following conditions:
(S1) T is an α-admissible multi-valued mapping;
(S2) there exist x0 ∊ X and x1 ∊ Tx0 such that α(x0,x1) ≥ 1;
(S3) T is a continuous multi-valued mapping.
Then T has a fixed point in X.
For x0, x1 in (S2), if x0 = x1 or x1 ∊ T x1, then x1 is a fixed point of T. We have nothing to prove.
So, we assume that x0 ≠ x1 and x1 ∉ Tx1. Since x1 ∊ Tx0 and α(x0,x1) ≥ 1, by the strictly weakly (α, ψ, ξ)-contractive condition of T, we get
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By the property of ψ, the above relation is impossible if max{d(x0,x1),d(x1,Tx1)} = d(x1,Tx1). Hence, from (3), it follows that
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Using Lemma 1.6, there exists x2 ∈ Tx1 such that
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where q is a fixed real number such that q > 1. If x1 = x2 or x2 ∈ Tx2, then x2 is a fixed point of T and hence we have noting to prove. Now, we may assume that x1 ≠ x2 and x2 ∉ Tx2. From (4) and (5), we obtain that
(6)
Applying ψ in the above inequality, we get
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and so
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Since T is an α-admissible multi-valued mapping, we get α(x1, x2) ≥ 1. From the weakly (α, ψ, ξ)-contractive condition of T, we have
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From the above inequality, it follows that and thus
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and thus
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For q1 > 1, by using Lemma 1.6, there exists x3 ∈ Tx2 such that
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If x2 = x3 or x3 ∈ Tx3, then x3 is a fixed point of T and hence we have noting to prove. Now, we may assume that x2 ≠ x3 and x3 ∉ Tx3. From (11) and (12), we get
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Since ψ is a strictly increasing function, we obtain
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By continuing this process, we can construct a sequence {xn} in X such that xn ≠ xn+1 ∈ Txn,
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and
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for all n ∈ ℕ∪{0}.
Next, we prove that {xn} is a Cauchy sequence in X. Let m, n ∈ ℕ. Without loss of generality, we may assume that m > n. From the triangle inequality and (16), we obtain
Using the property of ψ, we get . By the continuity of ξ and (ξ3), we have
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This shows that {xn} is a Cauchy sequence in (X, d). Since (X, d) is a complete metric space, there exists x* ∈ X such that xn → x* as n → ∞, that is, . From the continuity of T, we obtain
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Further, we have
By the closedness of Tx*, we get x* ∈ Tx*. Therefore, x* is a fixed point of T. This completes the proof. □
Next, we give second main result in this paper.
Let (X, d) be a complete metric space and T :X → CL(X) be a strictly weakly (α, ψ, ξ)-contractive mapping satisfying the following conditions:
(S1) T is an α-admissible multi-valued mapping;
(S2) there exist x0 and x1 ∈ Tx0 such that α(x0, x1) ≥ 1;
if {xn} is a sequence in X with xn+1 ∈ Txn, xn → x ∈ X as n → ∞ and α(xn, xn+1) ≥ 1 for all n ∈ ℕ, then we have ξ(d(xn+1,Tx)) ≥ ψ(ξ(M(xn,x))) for all n ∈ ℕ.
Then T has a fixed point in X.
Following the proof of Theorem 2.2, we can construct a Cauchy sequence {xn} in X such that xn → x* as n → ∞ and
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for all n ∈ ℕ. From the condition , we get
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for all n ∈ ℕ. Suppose that d(x*, Tx*) > 0 and let . Since xn → x* as n → ∞, we can find N1 ∈ ℕ such that
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for all n ≥ N1. Furthermore, we obtain
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for all n ≥ N1. Also, since {xn} is a Cauchy sequence, there exists N2 ∈ ℕ such that
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for all n ≥ N2. Since d(xn, Tx*) → d(x*, Tx*) as n → ∞, it follows that there exists N3 ∈ ℕ such that
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for all n ≥ N3. Using (21)-(24), it follows that
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for all n ≥ N := max{N1, N2, N3}. For each n ≥ N, from (20) and the triangle inequality, it follows that
Letting n → ∞ in the above inequality, we get
This implies that ξ(d(x*,Tx*)) = 0, which is a contradiction. Therefore, d(x*,Tx*)) = 0, that is, x* ∈ Tx*. This completes the proof. □
Next, we give an example to show that our result is more general than the results of Ali et al. [6] and many known results in literature.
Let X = [0, 100] and the metric d : X × X → ℝ defined by d(x, y) = |x – y| for all x, y ∈ X. Define T :X → CL(X) and α :X × X → [0,∞) by
and
Here we show that the contractive condition (1) does not hold for all functions ψ ∈ Ψand ξ ∈ Ξ. Let x = 0 and y = 2. We observe that
but
Therefore, the results of Ali et al. [6] are not applicable here.
Next, we show that Theorem 2.3 can be used to guarantee the existence of fixed point of T. Define the functions ψ, ξ : [0, ∞) → [0, ∞) by and for all t ∈ [0, ∞). It is easy to see that ψ ∈ Φ and ξ ∈ Ξ. First, we show that T is a strictly weakly (α, ψ, ξ)-contractive mapping. Suppose that x ∈ X, y ∈ Tx and α(x, y) > 1 and hence x ∈ [0,1] and y ∈ [0,0.01]. Also, we obtain
It is easy to observe that ψ is a strictly increasing function. Therefore, T is a strictly weakly (α, ψ, ξ)-contractive mapping. It is easy to see that T is an α-admissible multi-valued mapping. Moreover, there exists x0 = 1 ∈ X and x1 = 0.0001 ∈ Tx0 such that
Finally, for each sequence {xn} in X with xn+1 ∈ Txn, xn → x ∈ X as n → ∞ and α(xn, xn+1 > 1 for all n ∈ ℕ, we get xn, x ∈ [0,1]for all n ∈ ℕ and so
Therefore, the condition in Theorem 2.3 holds. By using Theorem 2.3, we can conclude that T has a fixed point in X. In this case, T has infinitely fixed points such as 0, 6 and 7.
From Remark 1.8, Theorems 2.2 and 2.3 apply particularly in the case when T is an α*-admissible multi-valued mapping.
It is easy to see that the contractive condition (1) implies the contractive condition (2). Therefore, we give the following results without the proof:
(Theorem 2.5 in [6]). Let (X, d) be a complete metric space and T :X → CL(X) be a strictly (α, ψ, ξ)-contractive mapping satisfying the following conditions:
(S1) T is an α-admissible (or a* -admissible) multi-valued mapping;
(S2) there exist x0 ∈ X and x1 ∈ TX0 suchthat α(x0, x1) > 1;
(S3) T is a continuous multi-valued mapping.
Then T has a fixed point in X.
(Theorem 2.6 in [6]). Let (X, d) be a complete metric space and T :X → CL(X) be a strictly (α, ψ, ξ)-contractive mapping satisfying the following conditions:
(S1) T is an a-admissible (or α* -admissible) multi-valued mapping;
(S2) there exist x0 and x1 ∈ Tx0 such that α(x0, x1) ≥ 1;
if {xn} is a sequence in X with xn → x ∈ X as n → ∞ and α(xn, xn + 1) > 1 for all n ∈ ℕ, then we have α(xn, x) > 1 for all n ∈ ℕ.
Then T has a fixed point in X.
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