We extend the concept of α-ψ-contractive mappings introduced recently by Samet et al. (2012) to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

1. Introduction and Preliminaries

Fixed point theory plays an important role in nonlinear analysis. This is because many practical problems in applied science, economics, physics, and engineering can be reformulated as a problem of finding fixed points of nonlinear mappings. The Banach contraction principle [1] is one of the fundamental results in fixed point theory. It guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces and provides a constructive method to approximate those fixed points.

Theorem 1 (see [1].)Let (X, d) be a complete metric space. Let T : X → X be a contraction self-mapping on X; that is, there exists a constant k ∈ (0,1) such that

()

for all (x, y) ∈ X × X. Then the map T admits a unique fixed point. Moreover, for any x₀ ∈ X, the sequence {Tⁿx₀} converges to this fixed point.

During the last few decades, several extensions of this famous principle have been established. In 1961, Edelstein [2] established the following result.

Theorem 2 (see [2].)Let (X, d) be complete and ε-chainable for some ε > 0. Let T : X → X be such that

()

where k ∈ (0,1) is a constant. Then T has a unique fixed point.

Kirk et al. [3] introduced the concept of cyclic mappings and proved the following fixed point theorem.

Theorem 3 (see [3].)Let A and B be two nonempty closed subsets of a complete metric space (X, d). Let T : X → X be a self-mapping such that

()

where k ∈ (0,1) is a constant. Suppose also that T(A)⊆B and T(B) ⊂ A. Then T has a unique fixed point in A∩B.

Ran and Reurings [4] extended the Banach contraction principle to a metric space endowed with a partial order. They established the following result.

Theorem 4 (see [4].)Let (X, d) be a complete metric space endowed with a partial order ⪯. Let T : X → X be a continuous mapping such that

()

where k ∈ (0,1) is a constant. Suppose also that there exists x₀ ∈ X such that x₀⪯Tx₀. Then T has a fixed point.

Many extensions of the previous result exist in the literature; for more details, we refer the reader to [5–11] and the references therein.

Observe that all the contractive conditions (2), (3), and (4) can be written as

()

where H is a subset of X × X. In Theorem 2, we have

()

In Theorem 3, we have

()

In Theorem 4, we have

()

The contractive condition (5) is said to be a partial contraction, that is, a contraction satisfied only on a subset of H⊆X × X.

Very recently, Samet et al. [12] observed that a partial contraction can be considered as a total contraction, that is, a contraction satisfied for every pair (x, y) ∈ X × X. More precisely, if we define the function α : X × X → [0, ∞) by

()

we show that (5) is equivalent to

()

In [12], the authors considered a more general inequality; that is,

()

where ψ : [0, ∞)→[0, ∞) is a function satisfying some conditions. The above inequality is called an α-ψ-contraction. In [12], some fixed point results were established under this contractive condition. For other works in this direction, we refer the reader to [13–16].

The aim of this work is to extend, generalize, and improve the obtained results in [12]. More precisely, the concept of α-ψ-contractive mappings is extended to the setting of gauge spaces. New fixed point results are established on such spaces, and some applications to nonlinear integral equations on the half-line are presented.

Through this paper, 𝔼 will denote a gauge space endowed with a separating gauge structure 𝒟 = {d_λ} _λ∈Λ, where Λ is a directed set.

A sequence {x_n} ⊂ 𝔼 is said to be convergent if there exists an x ∈ 𝔼 such that for every ε > 0 and λ ∈ Λ, there is an N ∈ ℕ with d_λ(x_n, x) < ε, for all n ≥ N.

A sequence {x_n} ⊂ 𝔼 is said to be Cauchy, if for every ε > 0 and λ ∈ Λ, there is an N ∈ ℕ with d_λ(x_n, x_n+p) < ε, for all n ≥ N and p ∈ ℕ.

A gauge space is called complete if any Cauchy sequence is convergent.

A subset of 𝔼 is said to be closed if it contains the limit of any convergent sequence of its elements.

For more details on guage spaces, we refer the reader to Dugundji [17].

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

(C1)
ψ is nondecreasing;
(C2)
, for all t > 0, where ψⁿ is the nth iterate of ψ;
(C3)
ψ(a) + ψ(b) ≤ ψ(a + b), for all a, b ≥ 0.

It is easy to show that under the conditions (C1) and (C2), we have ψ(t) < t, for all t > 0. Moreover, under the conditions (C1) and (C3), we have

()

for all n ∈ ℕ and a, b ≥ 0.

Example 5. Let ψ : [0, ∞)→[0, ∞) be the function defined by

()

Then ψ ∈ Ψ.

Definition 6. Let α : 𝔼 × 𝔼 → [0, ∞) be a given function. Let N ∈ ℕ and x, y ∈ 𝔼. We say that is an N-α-path from x to y if

()

We denote

()

Let N ∈ ℕ and x ∈ 𝔼. For λ ∈ Λ and y ∈ x[N, α], and let

()

2. Fixed Point Results for α-ψ_λ-Contractions

Definition 7. Let T : 𝔼 → 𝔼 be a given self-mapping. Let α : 𝔼 × 𝔼 → [0, ∞) be a given function, and let {ψ_λ} _λ∈Λ ⊂ Ψ. We say that T is an α-ψ_λ-contraction if

()

for all λ ∈ Λ and x, y ∈ 𝔼.

Definition 8. Let T : 𝔼 → 𝔼 be a given self-mapping. Let α : 𝔼 × 𝔼 → [0, ∞) be a given function. We say that T is α-admissible if

()

The following lemma will be useful to establish our fixed point results.

Lemma 9. Let T : 𝔼 → 𝔼 be a self-mapping. Suppose that there exist α : 𝔼 × 𝔼 → [0, ∞) and {ψ_λ} _λ∈Λ ⊂ Ψ such that the following conditions hold:

(i)
T is an α-ψ_λ-contraction;
(ii)
T is α-admissible.

Let ε > 0 and N ∈ ℕ. Then, for every λ ∈ Λ, x ∈ X, and y ∈ x[N, α], one has

(I)
Ty ∈ Tx[N, α] with
()
(II)
for all k ∈ ℕ ∪ {0}, one has T^ky ∈ T^kx[N, α] with
()

Proof. Let λ ∈ Λ, x ∈ X, and y ∈ x[N, α]. Let be an N-α-path from x to y such that

()

Since α(x, x¹) ≥ 1, we have

()

Since α(x¹, x²) ≥ 1, we have

()

Recursively from i = 3 to N, since α(xⁱ⁻¹, xⁱ) ≥ 1, we have

()

On the other hand, since T is α-admissible,

is an N-α-path from Tx to Ty. So, we have

()

Thus, we proved (I).

Again, since α(Tx, Tx¹) ≥ 1, we have

()

Since α(Tx¹, Tx²) ≥ 1, we have

()

Recursively from i = 3 to N, since α(Txⁱ⁻¹, Txⁱ) ≥ 1, we have

()

On the other hand, since T is α-admissible,

is an N-α-path from T²x to T²y. So, we have

()

Continuing this process, by induction, we get (II).

Definition 10. Let T : 𝔼 → 𝔼 be a self-mapping, and let α : 𝔼 × 𝔼 → [0, ∞) be a given function. For N ∈ ℕ, we say that a sequence {x_n} ⊂ 𝔼 is an N-α-Picard trajectory from x₀ if x_n = Tx_n−1 ∈ x_n−1[N, α] for all n ∈ ℕ. We denote by 𝒯_N(T, α, x₀), the set of all N-α-Picard trajectories from x₀.

Definition 11. Let T : 𝔼 → 𝔼 be a self-mapping, and let α : 𝔼 × 𝔼 → [0, ∞) be a given function. For N ∈ ℕ, we say that T is N-α-Picard continuous from x₀ ∈ 𝔼 if the limit of any convergent sequence {x_n} ∈ 𝒯_N(T, α, x₀) is a fixed point of T.

We have the following fixed point result.

Theorem 12. Let T : 𝔼 → 𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψ_λ} _λ∈Λ ⊂ Ψ and α : 𝔼 × 𝔼 → [0, ∞) be a given function. Suppose that the following conditions hold:

(i)
T is an α-ψ_λ-contraction;
(ii)
T is α-admissible;
(iii)
there exist N ∈ ℕ and x₀ ∈ 𝔼 such that Tx₀ ∈ x₀[N, α];
(iv)
T is N-α-Picard continuous from x₀.

Then T has a fixed point.

Proof. Let λ ∈ Λ and ε > 0. From condition (iii) and Lemma 9, we have T²x₀ ∈ Tx₀[N, α] and

()

Again, from Lemma 9, we have T³x₀ ∈ T²x₀[N, α] and

()

Continuing this process, by induction, for n ≥ 2, we have Tⁿ⁺¹x₀ ∈ Tⁿx₀[N, α] and

()

Thus, {Tⁿx₀} ∈ 𝒯_N(T, α, x₀) and, for m ≥ 1,

()

From condition (C2), we have

()

which implies that {Tⁿx₀} is a Cauchy sequence in the complete gauge space 𝔼. Since T is N-α-Picard continuous from x₀, the limit of {Tⁿx₀} is a fixed point of T.

Corollary 13. Let T : 𝔼 → 𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψ_λ} _λ∈Λ ⊂ Ψ and α : 𝔼 × 𝔼 → [0, ∞) be a given function. Suppose that the following conditions hold:

(i)
T is an α-ψ_λ-contraction;
(ii)
T is α-admissible;
(iii)
there exist N ∈ ℕ and x₀ ∈ 𝔼 such that Tx₀ ∈ x₀[N, α];
(iv)
T is continuous.

Then T has a fixed point.

Proof. Let {x_n} ∈ 𝒯_N(T, α, x₀) be such that x_n → x ∈ 𝔼. Since T is continuous, we have x_n+1 = Tx_n → Tx. Since 𝔼 is endowed with a separating gauge structure, we have x = Tx. The conclusion follows from Theorem 12.

Corollary 14. Let T : 𝔼 → 𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψ_λ} _λ∈Λ ⊂ Ψ and α : 𝔼 × 𝔼 → [0, ∞) be a given function. Suppose that the following conditions hold:

(i)
T is an α-ψ_λ-contraction;
(ii)
T is α-admissible;
(iii)
there exist N ∈ ℕ and x₀ ∈ 𝔼 such that Tx₀ ∈ x₀[N, α];
(iv)
for every {x_n} ∈ 𝒯_N(T, α, x₀) such that x_n → x ∈ 𝔼, there exist a subsequence {x_n(k)} of {x_n} and k₀ ∈ ℕ such that α(x_n(k), x) ≥ 1 for k ≥ k₀.

Then T has a fixed point.

Proof. Let {x_n} ∈ 𝒯_N(T, α, x₀) be such that x_n → x ∈ 𝔼. From condition (iv), there exist a subsequence {x_n(k)} of {x_n} and k₀ ∈ ℕ such that α(x_n(k), x) ≥ 1 for k ≥ k₀. Since T is an α-ψ_λ-contraction, for all λ ∈ Λ and k ≥ k₀, we have

()

Letting k → ∞ in the above inequality, we obtain that

()

Since 𝔼 is endowed with a separating gauge structure, we have x = Tx. The conclusion follows from Theorem 12.

For T : 𝔼 → 𝔼, we denote by Fix (T) the set of fixed points of T; that is,

()

The next result gives us a sufficient condition that ensures the uniqueness of the fixed point.

Theorem 15. Suppose that all the conditions of Theorem 12 are satisfied. Moreover, suppose that

(v)
for every (x, y) ∈ Fix (T) × Fix (T) with x ≠ y, there exists N(x, y) ∈ ℕ such that y ∈ x[N(x, y), α].

Then T has a unique fixed point.

Proof. From Theorem 12, The mapping T has at least one fixed point. Suppose that u, v ∈ 𝔼 are two fixed points of T with u ≠ v. From the condition (v), there exists N(u, v) ∈ ℕ such that v ∈ u[N(u, v), α]. Let λ ∈ Λ and ε > 0. From Lemma 9, we have

()

for all n ∈ ℕ. Letting n → ∞ in the above inequality, we obtain that d_λ(u, v) = 0 for all λ ∈ Λ, which is a contradiction with u ≠ v (since we have a separating gauge structure). We deduce that u = v.

The following result follows immediately from Theorems 12 and 15 with N = 1 and α(x, y) = 1 for every x, y ∈ 𝔼.

Corollary 16. Let T : 𝔼 → 𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψ_λ} _λ∈Λ ⊂ Ψ. Suppose that for all λ ∈ Λ, for all x, y ∈ 𝔼, one has

()

Then T has a unique fixed point.

Corollary 17. Let T : 𝔼 → 𝔼 be a self-mapping on the complete gauge space 𝔼. Let {ψ_λ} _λ∈Λ ⊂ Ψ and α : 𝔼 × 𝔼 → [0, ∞) be a given function. Suppose that the following conditions hold:

(i)
T is an α-ψ_λ-contraction;
(ii)
T is α-admissible;
(iii)
there exists x₀ ∈ 𝔼 such that α(x₀, Tx₀) ≥ 1;
(iv)
T is continuous

(or) for any sequence {x_n} ⊂ 𝔼 such that x_n = Tx_n−1, x_n → x ∈ 𝔼 and α(x_n−1, Tx_n−1) ≥ 1 for n ∈ ℕ, there exist a subsequence {x_n(k)} of {x_n} and k₀ ∈ ℕ such that α(x_n(k), x) ≥ 1 for k ≥ k₀.

Then T has a fixed point. Moreover, if

(v)
for every (x, y) ∈ Fix (T) × Fix (T) with x ≠ y, there exists z ∈ 𝔼 such that α(x, z) ≥ 1 and α(z, y) ≥ 1,

one has uniqueness of the fixed point.

Proof. The existence follows from Theorem 12 with N = 1. The uniqueness follows from Theorem 15 with N(x, y) = 2.

Corollary 18. Let ⪯ be a partial order on the complete gauge space 𝔼. Let T : 𝔼 → 𝔼 be a self-mapping and {ψ_λ} _λ∈Λ ⊂ Ψ. Suppose that the following conditions hold:

(i)
for all λ ∈ Λ, for all x, y ∈ 𝔼 such that x and y are comparable, one has
()
(ii)
x, y ∈ 𝔼, x and y are comparable ⇒Tx and Ty are comparable;
(iii)
there exists x₀ ∈ 𝔼 such that x₀ and Tx₀ are comparable;
(iv)
T is continuous,

(or) for any sequence {x_n} ⊂ 𝔼 such that x_n = Tx_n−1, x_n → x ∈ 𝔼, x_n−1 and x_n = Tx_n−1 are comparable for n ∈ ℕ, there exist a subsequence {x_n(k)} of {x_n} and k₀ ∈ ℕ such that x_n(k) and x are comparable for k ≥ k₀.

Then T has a fixed point. Moreover, if

(v)
for every (x, y) ∈ Fix (T) × Fix (T) with x ≠ y, there exists z ∈ 𝔼 such that x and z are comparable, z and y are comparable,

one has uniqueness of the fixed point.

Proof. It follows from Corollary 17 with

()

3. Applications

In this section, we are interested in the study of the existence of solutions to the nonlinear integral equation on the real axis

()

where h ∈ C([0, ∞), ℰ) and F ∈ C([0, ∞)×[0, ∞) × ℰ, ℰ). Here, ℰ is a Banach space with respect to a given norm ∥·∥_ℰ.

Let 𝔼 : = C([0, ∞), ℰ) and the family of pseudonorms {∥·∥_n} _n∈ℕ defined by

()

For every n ∈ ℕ, define now

()

Then 𝒟≔{d_n} _n∈ℕ is a separating gauge structure on 𝔼.

We have the following existence result.

Theorem 19. Suppose that the following conditions hold:

(i)
there exist a nonempty set Γ⊆ℰ × ℰ and a constant k < τ such that
()

for all t, s ≥ 0, (u, v) ∈ Γ;

(ii)
there exists a nonempty set Γ_𝔼⊆𝔼 × 𝔼 such that
()
(iii)
for all (x, y) ∈ Γ_𝔼, one has
()

for all t ≥ 0;

(iv)
there exists x₀ ∈ 𝔼 such that
()

for all t ≥ 0;

(v)
if {x_p} ⊂ 𝔼 is a sequence such that (x_p, x_p+1) ∈ Γ_𝔼 for p ∈ ℕ and x_p → x ∈ 𝔼 (with respect to 𝒟), then there exist a subsequence {x_p(k)} of {x_p} and k₀ ∈ ℕ such that
()

for all k ≥ k₀, t ≥ 0.

Then (42) has at least one solution in 𝔼.

Proof. Consider the mapping T : 𝔼 → 𝔼 defined by

()

for all x ∈ 𝔼. We have to prove that T has at least a fixed point.

Define the function α : 𝔼 × 𝔼 → [0, ∞) by

()

We claim that for all n ∈ ℕ, for all x, y ∈ 𝔼,

()

where ψ_n(t) = (k/τ)t for all n ∈ ℕ, for all t ≥ 0. Clearly, since k < τ, {ψ_n} _n∈ℕ ⊂ Ψ. If α(x, y) = 0, (52) holds immediately. So, suppose that α(x, y) ≠ 0; that is, (x, y) ∈ Γ_𝔼. Let n ∈ ℕ. From conditions (i) and (ii), for all t ∈ [0, n], we have

()

which implies that

()

Thus, we proved (52).

We will prove that T is α-admissible. Let (x, y) ∈ 𝔼 × 𝔼 such that α(x, y) ≥ 1; that is, (x, y) ∈ Γ_𝔼. From condition (iii), we have (Tx(t), Ty(t)) ∈ Γ for all t ≥ 0, which implies from condition (ii) that (Tx, Ty) ∈ Γ_E; that is, α(Tx, Ty) ≥ 1. So, T is α-admissible.

From conditions (iv) and (ii), we have (x₀, Tx₀) ∈ Γ_𝔼, which is equivalent to say that α(x₀, Tx₀) ≥ 1.

Finally, condition (v) implies that for every {x_p} ∈ 𝒯₁(T, α, x₀) such that x_p → x ∈ 𝔼, there exist a subsequence {x_p(k)} of {x_p} and k₀ ∈ ℕ such that α(x_p(k), x) ≥ 1 for k ≥ k₀.

Now, All the hypotheses of Corollary 14 are satisfied; we deduce that T has at least a fixed point, which is a solution to (52).

Theorem 20. In addition to the assumptions of Theorem 19, suppose that

(vi)
for all (x, y) ∈ 𝔼 × 𝔼, there exists z ∈ 𝔼 such that (x, z) ∈ Γ_𝔼 and (z, y) ∈ Γ_𝔼.

Then (42) has one and only one solution in 𝔼.

Proof. It follows immediately from Theorem 15.

Conflict of Interests

The authors declare that there is no competing/conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgment

This work is supported by the Research Center, College of Science, King Saud University.

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Fixed Point Results for α-ψ_λ-Contractions on Gauge Spaces and Applications

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1. Introduction and Preliminaries

2. Fixed Point Results for α-ψ_λ-Contractions

3. Applications

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Fixed Point Results for α-ψλ-Contractions on Gauge Spaces and Applications

Abstract

1. Introduction and Preliminaries

2. Fixed Point Results for α-ψλ-Contractions

3. Applications

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Fixed Point Results for α-ψ_λ-Contractions on Gauge Spaces and Applications

2. Fixed Point Results for α-ψ_λ-Contractions