Some fixed point theorems for (φ, ψ, p)-contractive maps and (φ, k, p)-contractive maps on a complete metric space are proved. Presented fixed point theorems generalize many results existing in the literature.

1. Introduction and Preliminaries

Branciari [1] established a fixed point result for an integral type inequality, which is a generalization of Banach contraction principle. Kada et al. [2] introduced and studied the concept of w-distance on a metric space. They give examples of w-distances and improved Caristi’s fixed point theorem, Ekeland’s ϵ-variational’s principle, and the nonconvex minimization theorem according to Takahashi (see many useful examples and results on w-distance in [2–5] and in references therein). Kada et al. [2] defined the concept of w-distance in a metric space as follows.

Definition 1.1 (see [2].)Let X be a metric space endowed with a metric d. A function p : X × X → [0, ∞) is called a w-distance on X if it satisfies the following properties:

(1)
p(x, z) ≤ p(x, y) + p(y, z) for any x, y, z ∈ X,
(2)
p is lower semicontinuous in its second variable, that is, if x ∈ X and y_n → y in X then p(x, y) ≤ liminf _n→∞ p(x, y_n),
(3)
for each ϵ > 0, there exists δ > 0 such that p(z, x) ≤ δ and p(z, y) ≤ δ imply d(x, y) ≤ ϵ.

We denote by Φ the set of functions φ : [0, +∞)→[0, +∞) satisfying the following hypotheses:

(c1)
φ is continuous and nondecreasing,
(c2)
φ(t) = 0 if and only if t = 0.

We denote by Ψ the set of functions ψ : [0, +∞)→[0, +∞) satisfying the following hypotheses:

(h1)
ψ is right continuous and nondecreasing,
(h2)
ψ(t) < t for all t > 0.

Let p be a w-distance on metric space (X, d), φ ∈ Φ and ψ ∈ Ψ. A map T from X into itself is a (φ, ψ, p)-contractive map on X if for each x, y ∈ X, φp(Tx, Ty) ≤ ψφp(x, y).

The following lemmas are used in the next section.

Lemma 1.2 (see [3].)If ψ ∈ Ψ, then lim _n→∞ψⁿ(t) = 0 for each t > 0, and if φ ∈ Φ, {a_n}⊆[0, ∞) and lim _n→∞φ(a_n) = 0, then lim _n→∞a_n = 0.

Lemma 1.3 (see [2].)Let (X, d) be a metric space and let p be a w-distance on X.

(i)
If {x_n} is a sequence in X such that lim _np(x_n, x) = lim _np(x_n, y) = 0, then x = y. In particular, if p(z, x) = p(z, y) = 0, then x = y.
(ii)
If p(x_n, y_n) ≤ α_n p(x_n, y) ≤ β_n for any n ∈ ℕ, where {α_n} and {β_n} are sequences in [0, ∞) converging to 0, then {y_n} converges to y.
(iii)
Let p be a w-distance on metric space (X, d) and {x_n} a sequence in X such that for each ɛ > 0 there exist N_ɛ ∈ N such that m > n > N_ɛ implies p(x_n, x_m) < ɛ (or lim _m,n→∞p(x_n, x_m) = 0), then {x_n} is a Cauchy sequence.

Note that if p(a, b) = p(b, a) = 0 and p(a, a) ≤ p(a, b) + p(b, a) = 0, then p(a, a) = 0 and, by Lemma 1.3, a = b.

In [3], Razani et al. proved a fixed point theorem for (φ, ψ, p)-contractive mappings, which is a new version of the main theorem in [1], by considering the concept of the w-distance.

The main aim of this paper is to present some generalization fixed point Theorems by Kada et al. [2], Hicks and Rhoades [6] and several other results with respect to (φ, ψ, p)-contractive maps on a complete metric space.

2. (φ, ψ, p)-Contractive Maps

In the next theorem we state one of the main results of this paper generalizing Theorem 4 of [2]. In what follows, we use φp to denote the composition of φ with p.

Theorem 2.1. Let p be a w-distance on complete metric space (X, d), φ ∈ Φ and ψ ∈ Ψ. Suppose T : X → X is a map that satisfies

()

for each x ∈ X and that

()

for every y ∈ X with y ≠ Ty. Then there exists u ∈ X such that u = Tu. Moreover, if v = Tv, then p(v, v) = 0.

Proof. Fix x ∈ X. Set x_n+1 = Tx_n with x₀ = x. Then by (2.1)

()

thus lim _nφp(x_n, x_n+1) = 0 and Lemma 1.2 implies

()

and similarly

()

Now we proof that {x_n} is a Cauchy sequence. By triangle inequality, continuity of φ and (2.4), we have

()

as n → ∞ and so lim _n→∞φp(x_n, x_n+2) = 0 which concludes

()

By induction, for any k > 0 we have

()

So, by Lemma 1.3, {x_n} is a Cauchy sequence, and since X is complete, there exists u ∈ X such that x_n → u in X.

Now we prove that u is a fixed point of T.

From (2.8), for each ɛ > 0, there exists N_ɛ ∈ ℕ such that n > N_ɛ implies but x_n → u and p(x, ·) is lower semicontinuous, thus

()

Therefore,

. Set ɛ = 1/k, N_ɛ = n_k and we have

()

Now, assume that u ≠ Tu. Then by hypothesis, we have

()

as n → ∞ by (2.4) and (2.10). This is a contradiction. Hence u = Tu.

If v = Tv, we have

()

This is a contradiction. So φp(v, v) = 0, and by hypothesis p(v, v) = 0.

Here we give a simple example illustrating Theorem 2.1. In this example, we will show that Theorem 4 in [2] cannot be applied.

Example 2.2. Let X = {(1/n)∣n ∈ ℕ} ∪ {0}, which is a complete metric space with usual metric d of reals. Moreover, by defining p(x, y) = y, p is a w-distance on (X, d). Let T : X → X be a map as T(1/n) = 1/(n + 1), T0 = 0. Suppose φ(t) = t^1/t is a continuous and strictly nondecreasing map and ψ(t) = (1/3)t, for any t > 0. We have

()

and so there is not any r ∈ [0,1) such that p(Tx, T²x) ≤ rp(x, Tx), and hence Theorem 4 in [2] dose not work. But

()

because for any n ∈ ℕ we have ((n+1)/(n+2))ⁿ⁺¹1/(n + 2) ≤ 1/3. Also for any n ∈ ℕ we have 1/n ≠ T(1/n). So for arbitrary n ∈ ℕ, inf {p(1/m, 1/n) + p(1/m, 1/(m + 1)) : m ∈ ℕ} = 1/n > 0, hence T is satisfied in Theorem 2.1. We note that 0 is a fixed point for T.

The next examples show the role of the conditions (2.1) and (2.2).

Example 2.3. Let X = [−1,1], d(x, y) = |x − y|, and define p : X → X by p(x, y) = |3x − 3y|, where x, y ∈ X. Set ψ(t) = rt and φ(t) = t for all t ∈ [0, ∞). Let us define T : X → X by T0 = 1 and Tx = x/10 if x ≠ 0. We have

()

If x ≠ 0, then

()

and hence (2.1) holds.

Now, we remark that 0 ≠ T(0), and

()

Thus, the condition (2.2) is not satisfied, and there is no z ∈ X with Tz = z. In this case we observe that Theorem 2.1 is invalid without condition (2.2).

Example 2.4. Let X = [2, ∞) ∪ {0,1}, d(x, y) = |x − y|, x, y ∈ X, and set p = d. Let ψ, φ be as Example 2.3. Let us define T : X → X by T0 = 1 and Tx = 0 if x ≠ 0. Clearly, T has no fixed point in X. Now, for each x ∈ X and that

()

for every y ∈ X with y ≠ Ty, so condition (2.2) is satisfied. But, for x = 0, d(Tx, T²x) > rd(x, Tx) for any r ∈ [0,1). Hence, condition (2.1) dose not hold. We note that Theorem 2.1 dose not work without condition (2.1).

Suppose θ : ℝ⁺ → ℝ⁺ is Lebesgue-integrable mapping which is summable and , for each ɛ > 0. Now, in the next corollary, set and ψ(t) = ct, where c ∈ [0,1[. Then, φ ∈ Φ and ψ ∈ Ψ. Hence we can conclude the following corollary as a special case.

Corollary 2.5. Let T be a selfmap of a complete metric space (X, d) satisfying

()

for all x ∈ X. Suppose that

()

with y ≠ Ty. Then there exists a u ∈ X such that Tu = u.

Note that Corollary 2.5 is invalid without condition (2.20). For example, take X = {0}∪{1/2ⁿ : n ≥ 1}, which is a complete metric space with usual metric d of reals. Define T : X → X by T(0) = 1/2 and T(1/2∧n ) = 1/2ⁿ⁻¹ for n ≥ 1. Set φ(t) ≡ 1. It is easy to check that , for any x ∈ X; however, y ≠ Ty for any y ∈ X and inf {d(x, y) + d(x, Tx) : x ∈ X} = 0. Clearly, T has got no fixed point in X.

Remark 2.6. From Theorem 2.1, we can obtain Theorem 4 in [2] as a special case. For this, in the hypotheses of Theorem 2.1, set ψ(t) = rt and φ(t) = t for all t ∈ [0, ∞).

Corollary 2.7. Let p be a w-distance on complete metric space (X, d), φ ∈ Φ and ψ ∈ Ψ. Suppose T is a continuous mapping for X into itself such that (2.1), is satisfied. Then there exists u ∈ X such that u = Tu. Moreover, if v = Tv, then p(v, v) = 0.

Proof. Assume that there exists y ∈ X with y ≠ Ty and inf {p(x, y) + p(x, Tx) : x ∈ X} = 0. Then there exists a sequence {x_n} such that

()

as n → ∞. Hence p(x_n, y) → 0 and p(x_n, Tx_n) → 0 as n → ∞. Lemma 1.3 implies that Tx_n → y as n → ∞. Now by assumption

()

and so φp(Tx_n, T²x_n) → 0 as n → ∞. By Lemma 1.2, p(Tx_n, T²x_n) → 0 as n → ∞. We also have

()

hence p(x_n, T²x_n) → 0 as n → ∞. By Lemma 1.3, we conclude that {T²x_n} converges to y. Since T is continuous, we have

()

This is a contradiction. Therefore, if y ≠ Ty, then inf {p(x, y) + p(x, Tx) : x ∈ X} > 0. So, Theorem 2.1 gives desired result.

In Example 2.3, T is satisfied in condition (2.1), but it is not continuous. So, the hypotheses in Corollary 2.7are not satisfied. We note that T has no fixed point.

It is an obvious fact that, if f : X → X is a map which has a fixed point x ∈ X, then x is also a fixed point of fⁿ for every natural number n. However, the converse is false. If a map satisfies F(f) = F(fⁿ) for each n ∈ ℕ, where F(f) denotes a set of all fixed points of f, then it is said to have property P [7, 8]. The following theorem extends and improves Theorem 2 of [7].

Theorem 2.8. Let (X, d) be a complete metric space with w-distance p on X. Suppose T : X → X satisfies

(i)

()
or
(ii)
with strict inequality, ψ ≡ 1 and for all x ∈ X, x ≠ Tx. If F(T) ≠ ∅, then T has property P.

Proof. We shall always assume that n > 1, since the statement for n = 1 is trivial. Let u ∈ F(Tⁿ). Suppose that T satisfies (i). Then,

()

and so p(u, Tu) = 0. Now from

()

we have p(u, u) = 0. Hence, by Lemma 1.3, we have u = Tu, and u ∈ F(T). Suppose that T satisfies (ii). If Tu = u, then there is nothing to prove. Suppose, if possible, that Tu ≠ u. Then a repetition of the argument for case (i) leads to φp(u, Tu) < ψφp(u, Tu), that is a contradiction. Therefore, in all cases, u = Tu and F(Tⁿ) = F(T).

The following theorem extends Theorem 2.1 of [6]. A function G mapping X into the real is T-orbitally lower semicontinuous at z if {x_n} is a sequence in O(x, ∞) and x_n → z implies that G(p) ≤ liminf _n→∞G(x_n).

Theorem 2.9. Let (X, d) be a complete metric space with w-distance p on X. Suppose T : X → X and there exists an x such that

()

Then,

(i)
lim Tⁿx = z exists,
(ii)

()
(iii)
p(z, Tz) = 0 if and only if G(x) = p(x, Tx) is T-orbitally lower semicontinuous at z.

Proof. Observe that (i) and (ii) are immediate from the proof of Theorem 2.1. We prove (iii). It is clear that p(z, Tz) = 0 impling G(x) is T-orbitally lower semicontinuous at z.

x_n = Tⁿx → z and G is T-orbitally lower semicontinuous at x implies

()

So, p(z, Tz) = 0.

The mapping T is orbitally lower semicontinuous at u ∈ X if implies that . In the following, we improve Theorem 2 of [9] that it is correct form Theorem 1 of [7].

Theorem 2.10. Let p be a w-distance on complete metric space (X, d), φ ∈ Φ and ψ ∈ Ψ. Suppose T : X → X is orbitally lower semicontinuous map on X that satisfies

()

for each x ∈ X. Then there exists u ∈ X such that u ∈ F(T). Moreover, if v = Tv, then p(v, v) = 0.

Proof. Observe that the sequence {x_n} is a Cauchy sequence immediate from the proof of Theorem 2.1 and so there exists a point u in X such that x_n → u as n → ∞. Since T is orbitally lower semicontinuous at u, we have p(u, Tu) ≤ liminf _n→∞p(x_n, x_n+1) = 0. Now, we have

()

and so p(u, Tu) = 0. Similarly, p(Tu, u) = 0. Hence, u ∈ F(T). By Theorem 2.1 we can conclude that if v = Tv, then p(v, v) = 0.

The following example shows that Theorem 2 in [9] cannot be applicable. So our generalization is useful.

Example 2.11. Let = [0, ∞) be a metric space with metric d defined by d(x, y) = (40/3)|x − y|,x, y ∈ X, which is complete. We define p : X → X by p(x, y) = (1/3) | y|. Let φ be as defined before in Corollary 2.5 and ψ(t) = (1/10)t,t > 0. Assume that T : X → X by Tx = x/10 for any x ∈ X. We have, d(Tx, T²x) = (4/3)d(x, Tx), x ∈ X, and so Theorem 2 in [9] dose not work. But

()

for each x ∈ X. Hence by Theorem 2.10 there exists a fixed point for T. We note that 0 is fixed point for T.

3. (φ, k, p)-Contractive Maps

In this section we obtain fixed points for (φ, k, p)-contractive maps (i.e., (φ, ψ, p)-contractive maps that ψ(t) = k for all t ∈ [0, ∞), where k ∈ [0,1)).

In 1969, Kannan [10] proved the following fixed point theorem. Contractions are always continuous and Kannan maps are not necessarily continuous.

Theorem 3.1 (see [10].)Let (X, d) be a complete metric space. Let T be a Kannan mapping on X, that is, there exists k ∈ [0,1/2) such that

()

for all x, y ∈ X. Then, T has a unique fixed point in X. For each x ∈ X, the iterative sequence {Tⁿx} _n≥1 converges to the fixed point.

In the next theorem, we generalize this theorem as follows.

Theorem 3.2. Let (X, d) be a complete metric space. Let T be a (φ, k)-Kannan mapping on X, that is, there exists k ∈ [0,1/2) such that

()

for all x, y ∈ X. Then, T has a unique fixed point in X. For each x ∈ X, the iterative sequence {Tⁿx} _n≥1 converges to the fixed point.

Proof. Let x ∈ X and define x_n+1 = Tⁿx for any n ∈ N, and set r = k/(1 − k). Then, r ∈ [0,1),

()

and so

()

Then, from the proof of Theorem 2.1, lim Tⁿx = z exists. From (3.4), we have

()

Thus, lim Tⁿx = Tz, and so z = Tz. Clearly, z is unique. This completes the proof.

The set of all subadditive functions φ in Φ is denoted by Φ^′. In the following theorems, we generalize Theorems 3.4 and 3.5 due to Suzuki and Takahashi [4].

Theorem 3.3. Let p be a w-distance on complete metric space (X, d), φ ∈ Φ^′ and T be a selfmap. Suppose there exists k ∈ [0,1/2) such that

(i)
φp(Tx, T²x) ≤ kφp(x, T²x) for each x ∈ X,
(ii)
inf {p(x, z) + p(x, Tx) : x ∈ X} > 0 for every z ∈ X with z ≠ Tz.

Then T has a fixed point in X. Moreover, if v is a fixed point of T, then p(v, v) = 0.

Proof. Fix x ∈ X. Define x₀ = x and x_n = Tⁿx₀ for every n ∈ ℕ. Put r = k/(1 − k). Then, 0 ≤ r < 1. By hypothesis, since φ ∈ Φ^′, we have

()

for all n ∈ ℕ. It follows that

()

for all n ∈ ℕ. Using the similar argument as in the proof of Theorem 2.1, we can prove that the sequence {u_n} is Cauchy and so there exists u ∈ X such that x_n → u as n → ∞. Also, we have u ∈ F(T). Since

()

we have φp(v, v) = 0 and so p(v, v) = 0. The proof is completed.

Corollary 3.4. Let p be a w-distance on complete metric space (X, d), φ ∈ Φ^′ and let T be a continuous map. Suppose there exists k ∈ [0, 1/2) such that

()

for each x ∈ X.

Then T has a fixed point in X. Moreover, if v is a fixed point of T, then p(v, v) = 0.

Proof. It suffices to show that inf {p(x, z) + p(x, Tx) : x ∈ X} > 0 for every u ∈ X with u ≠ Tu. Assume that there exists u ∈ X with u ≠ Tu and inf {p(x, u) + p(x, Tx) : x ∈ X} = 0. Then there exists a sequence {x_n} in X such that lim _n→∞[p(x_n, u) + p(x_n, Tx_n)] = 0. It follows that p(x_n, u) → 0 and p(x_n, Tx_n) → 0 as n → ∞. Hence, Tx_n → u. On the other hand, since φ ∈ Φ^′ and (3.9), we have

()

and hence

()

for all n ∈ ℕ. Thus, p(x_n, T²x_n) → 0 as n → ∞. Therefore, T²x_n → u. Since T : X → X is continuous, we have

()

which is a contradiction. Therefore, using Theorem 3.3, p(v, v) = 0. This completes the proof.

Question 1. Can we generalize Theorems 3.2, 3.3, and Corollary 3.4 for (φ, ψ, p)-contractive maps?

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