Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

Abstract

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

1. Introduction

In the context of fixed-point theory, the metric fixed-point theory is the branch where metric conditions on the involved spaces and mappings play a crucial role in establishing theoretical results. In a certain sense, this theory is a far-reaching outgrowth of a well-known theorem of Banach [1] which states that if (𝑋,𝑑) is a complete metric space and if 𝑇∶𝑋→𝑋 is a mapping which satisfies the condition 𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦,∀𝑥,𝑦∈𝑋, (1) where 0≤𝑘<1 is a constant, then 𝑇 has a unique fixed point. Since most of the spaces studied in mathematical analysis share many algebraic and topological properties as well as metric properties, there is no clear line separating the metric fixed-point theory from the topological or the set-theoretic branch of the theory. Consequently, several authors considered the problem of existence (and uniqueness) of a fixed point for generalized contractions in a metric space (see, e.g., [2–9]). On the other hand, many definitions and theorems in the literature do not require that all of the properties of a metric hold true. For this reason, in the last decades, various concepts of generalized metrics were introduced (see, e.g., [10, 11]). Here, we are interested in the so-called gauge spaces that are characterized by the fact that the distance between two points of the space may be zero even if the two points are distinct. For instance, Frigon [12] and Chiş and Precup [13] gave generalizations of the Banach contraction principle on gauge spaces (see also [14–16]). Consistent with this line of research, the aim of this paper is to present some fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Then, to illustrate the usefulness of our theorems, we apply our results to the study of existence and uniqueness of solutions to a second-order nonlinear initial-value problem.

2. Preliminaries

In this section, we recall some preliminaries on gauge spaces and introduce some basic definitions.

Definition 1. Let 𝑋 be a nonempty set. A function 𝑑∶𝑋×𝑋→[0,+∞) is called a pseudometric in 𝑋 whenever the following hold:(i)𝑑(𝑥,𝑥)=0 for all 𝑥∈𝑋,(ii)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋,(iii)𝑑(𝑥,𝑦)≤𝑑(𝑥,𝑧)+𝑑(𝑧,𝑦) for all 𝑥,𝑦,𝑧∈𝑋.

Definition 2. Let 𝑋 be a nonempty set endowed with a pseudometric 𝑑. The 𝑑-ball of radius 𝜀>0 centered at 𝑥∈𝑋 is the set 𝐵(𝑥;𝑑,𝜀)=𝑦∈𝑋∣𝑑𝑥,𝑦<𝜀. (2)

Definition 3. A family ℱ={𝑑_𝜆∣𝜆∈𝒜} of pseudometrics is called separating if, for each pair (𝑥,𝑦) with 𝑥≠𝑦, there is a 𝑑_𝜆∈ℱ such that 𝑑_𝜆(𝑥,𝑦)≠0.

Definition 4. Let 𝑋 be a nonempty set, and let ℱ={𝑑_𝜆∣𝜆∈𝒜} be a separating family of pseudometrics on 𝑋. The topology 𝒯(ℱ) having for a subbasis the family ℬ(ℱ)=𝐵𝑥;𝑑_𝜆,𝜀∣𝑥∈𝑋,𝑑_𝜆∈ℱ,𝜀>0 (3) of balls is called the topology in 𝑋 induced by the family ℱ. The pair (𝑋,𝒯(ℱ)) is called a gauge space. Note that (𝑋,𝒯(ℱ)) is Hausdorff because we require ℱ to be separating.

Definition 5. Let (𝑋,𝒯(ℱ)) be a gauge space with respect to the family ℱ={𝑑_𝜆∣𝜆∈𝒜} of pseudometrics on 𝑋. Let {𝑥_𝑛} be a sequence in 𝑋, and 𝑥∈𝑋. Then, the following are considered.(a)The sequence {𝑥_𝑛} converges to 𝑥 if and only if ∀𝜆∈𝒜,∀𝜀>0,∃𝑁∈ℕsuchthat𝑑_𝜆𝑥_𝑛,𝑥<𝜀,∀𝑛≥𝑁. (4) In this case, we denote that 𝑥−−_𝑛^ℱ→𝑥.(b)The sequence {𝑥_𝑛} is Cauchy if and only if ∀𝜆∈𝒜,∀𝜀>0,∃𝑁∈ℕsuchthat𝑑_𝜆𝑥_𝑛+𝑝,𝑥_𝑛<𝜀,∀𝑛≥𝑁,𝑝∈ℕ. (5)(c)(𝑋,𝒯(ℱ)) is complete if and only if any Cauchy sequence in (𝑋,𝒯(ℱ)) is convergent to an element of 𝑋. (d)A subset of 𝑋 is said to be closed if it contains the limit of any convergent sequence of its elements.

For more details on gauge spaces, we refer the reader to [17]. Obviously, every metric space is automatically a pseudometric space. On the contrary, if a pseudometric 𝑑 is not a metric, it is because there are at least two points 𝑥≠𝑦 for which 𝑑(𝑥,𝑦)=0. In most situations, this does not happen; indeed, metrics come up in mathematics more often than pseudometrics. However, pseudometrics arise in a natural way in functional analysis and in the theory of hyperbolic complex manifolds [18].

3. Main Results

Let Ψ denote the set of all functions 𝜓∶[0,+∞)→[0,+∞) which satisfy the following conditions. (i)𝜓 is continuous and nondecreasing.(ii)𝜓(𝑡)=0 if and only if 𝑡=0.Similarly, let Φ denote the set of all functions 𝜑∶[0,+∞)→[0,+∞) which satisfy the following conditions. (i)𝜙 is lower semicontinuous. (ii)𝜙(𝑡)=0 if and only if 𝑡=0.

Definition 6. Let (𝑋,𝒯(ℱ)) be a gauge space, and let 𝑇∶𝑋→𝑋 be a mapping. For 𝑥∈𝑋, let 𝒪(𝑥) denote the orbit of 𝑥; that is, 𝒪(𝑥)={𝑇^𝑛𝑥,𝑛=0,1,…}.

Our first result is the following theorem.

Theorem 7. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for all 𝑥∈𝒪(𝑢), and for all 𝑦∈𝒪(𝑢),𝑑_𝜆(𝑥,𝑦)=0⇒𝑑_𝜆(𝑇𝑥,𝑇𝑦)=0.(ii)For all 𝜆∈𝒜 and for all 0<𝑎<𝑏<∞, there exist 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that ∀𝑥,𝑦∈𝑋,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (6)Then, 𝑇 has a unique fixed point.

Proof. First, we will remark that ∀𝜆∈𝒜,∀𝑥,𝑦∈𝑋,𝑑_𝜆𝑥,𝑦≠0⟹𝑑_𝜆𝑇𝑥,𝑇𝑦<𝑑_𝜆𝑥,𝑦. (7) In fact, let 𝜆∈𝒜, and let 𝑥,𝑦∈𝑋 such that 𝑑_𝜆(𝑥,𝑦)≠0; by taking 𝑎=𝑑_𝜆(𝑥,𝑦) and 𝑏=2𝑑_𝜆(𝑥,𝑦), we get, by applying assumption (ii) of the theorem, the existence of 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (8) Since 𝑑_𝜆(𝑥,𝑦)≠0, we have 𝜑_{𝜆,𝑎,𝑏}(𝑑_𝜆(𝑥,𝑦))>0, and so we get 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥,𝑇𝑦<𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (9) Using the monotony of 𝜓_{𝜆,𝑎,𝑏}, we obtain the desired result (7). Moreover, assumption (i) implies that ∀𝜆∈𝒜,∀𝑢∈𝑋,∀𝑥∈𝒪(𝑢),∀𝑦∈𝑑𝒪(𝑢),_𝜆𝑇𝑥,𝑇𝑦≤𝑑_𝜆𝑥,𝑦. (10)
Now, let us take 𝑥₀∈𝑋, and let {𝑥_𝑛} be the sequence in 𝑋 given by 𝑥_𝑛+1=𝑇𝑥_𝑛,∀𝑛≥0. (11) Thus, we complete the proof in the following four steps.
Step  1. We will prove that 𝑑_𝜆𝑥_𝑛+1,𝑥_𝑛⟶0as𝑛⟶+∞,∀𝜆∈𝒜. (12) Let 𝜆∈𝒜. Since 𝑥_𝑛,𝑥_𝑛+1∈𝒪(𝑥₀), by applying (10), we obtain 𝑑_𝜆𝑥_𝑛+1,𝑥_𝑛=𝑑_𝜆𝑇𝑥_𝑛,𝑇𝑥_𝑛−1≤𝑑_𝜆𝑥_𝑛,𝑥_𝑛−1,∀𝑛≥1. (13) It follows that {𝑑_𝜆(𝑥_𝑛+1,𝑥_𝑛)} is a decreasing sequence of nonnegative real numbers, and, hence, there is 𝑟=𝑟(𝜆)≥0 such that 𝑑_𝜆𝑥_𝑛+1,𝑥_𝑛⟶𝑟as𝑛⟶+∞. (14) Suppose that 𝑟>0; this implies that 𝑟≤𝑑_𝜆(𝑥_𝑛,𝑥_𝑛−1)≤𝑑_𝜆(𝑥₁,𝑥₀),forall𝑛≥1. Now, taking 𝑎=𝑟/2 and 𝑏=𝑑_𝜆(𝑥₁,𝑥₀), by applying (ii), we get the existence of 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that 𝜓_{𝜆,𝑎,𝑏}𝑑𝑥_𝑛+1,𝑥_𝑛≤𝜓_{𝜆,𝑎,𝑏}𝑑𝑥_𝑛,𝑥_𝑛−1−𝜑_{𝜆,𝑎,𝑏}𝑑𝑥_𝑛,𝑥_𝑛−1,∀𝑛≥1. (15) Letting 𝑛→+∞ in the above inequality, using the continuity of 𝜓_{𝜆,𝑎,𝑏} and the lower semicontinuity of 𝜑_{𝜆,𝑎,𝑏}, we get 𝜓_{𝜆,𝑎,𝑏}(𝑟)≤𝜓_{𝜆,𝑎,𝑏}(𝑟)−𝜑_{𝜆,𝑎,𝑏}(𝑟), (16) which implies that 𝑟=0. This is a contradiction, and, therefore, our claim (12) holds.
Step  2. We will prove that {𝑥_𝑛} is a Cauchy sequence in (𝑋,𝒯(ℱ)). Suppose that {𝑥_𝑛} is not a Cauchy sequence. Then, there exists (𝜆,𝜀)∈𝒜×(0,+∞) for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that, for all positive integers 𝑘, 𝑛(𝑘)>𝑚(𝑘)>𝑘,𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)𝑑≥𝜀,_𝜆𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)<𝜀. (17) Using (17) and the triangular inequality, we get 𝜀≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑_𝜆𝑥_{𝑛(𝑘)−1},𝑥_𝑚(𝑘)<𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}+𝜀. (18) Thus, we have 𝜀≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)<𝑑𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}+𝜀. (19) Letting 𝑘→+∞ in the above inequality and using (12), we obtain 𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)⟶𝜀as𝑘⟶+∞. (20) By the triangular inequality, we have 𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+1−𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+𝑑_𝜆𝑥_𝑚(𝑘),𝑥_𝑚(𝑘)+1. (21) Letting 𝑘→+∞ in the above inequality, using (12) and (20), we get 𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+1⟶𝜀as𝑘⟶+∞. (22) On the other hand, from (20), there exists a positive integer 𝑘₀ such that 𝜀2≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤2𝜀,∀𝑘≥𝑘₀. (23) Applying the hypothesis of the theorem by taking 𝑎=𝜀/2 and 𝑏=2𝜀, we get that there exist 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥_𝑛(𝑘),𝑇𝑥_𝑚(𝑘)≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘),∀𝑘≥𝑘₀; (24) that is, 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑚(𝑘)+1≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘),∀𝑘≥𝑘₀. (25) Letting 𝑘→+∞ in the above inequality, from (20), (22), the continuity of 𝜓_{𝜆,𝑎,𝑏}, and the lower semicontinuity of 𝜑_{𝜆,𝑎,𝑏}, we obtain 𝜓_{𝜆,𝑎,𝑏}(𝜀)≤𝜓_{𝜆,𝑎,𝑏}(𝜀)−𝜑_{𝜆,𝑎,𝑏}(𝜀), (26) which implies that 𝜑_{𝜆,𝑎,𝑏}(𝜀)=0, which leads to the contradiction 𝜀=0. Finally, we deduce that {𝑥_𝑛} is a Cauchy sequence.
Step  3 (existence of fixed point). As {𝑥_𝑛} is a Cauchy sequence in the complete gauge space (𝑋,𝒯(ℱ)), then there exists 𝑥^∗∈𝑋 such that 𝑥⟶_𝑛^ℱ𝑥^∗as𝑛⟶+∞. (27) This implies that 𝑥^∗∈𝒪(𝑥₀). On the other hand, by (10), for all 𝑥_𝑛∈𝒪(𝑥₀), we get 𝑑_𝜆𝑇𝑥_𝑛,𝑇𝑥^∗≤𝑑_𝜆𝑥_𝑛,𝑥^∗,∀𝜆∈𝒜,∀𝑛≥0. (28) Now, from 𝑑_𝜆𝑥^∗,𝑇𝑥^∗≤𝑑_𝜆𝑥^∗,𝑥_𝑛+1+𝑑_𝜆𝑇𝑥_𝑛,𝑇𝑥^∗≤𝑑_𝜆𝑥^∗,𝑥_𝑛+1+𝑑_𝜆𝑥_𝑛,𝑥^∗ (29) for all 𝜆∈𝒜, letting 𝑛→+∞, we obtain 𝑑_𝜆(𝑥^∗,𝑇𝑥^∗)≤0, for all 𝜆∈𝒜.
In the virtue of the separating structure of ℱ, we deduce that 𝑇𝑥^∗=𝑥^∗, and, hence, the existence of the fixed point is proved.
Step  4 (uniqueness). Suppose that there exist 𝑥^∗,𝑦^∗ such that 𝑥^∗≠𝑦^∗, 𝑇𝑥^∗=𝑥^∗, and 𝑇𝑦^∗=𝑦^∗. Then, there exists 𝜆∈𝒜 such that 𝑑_𝜆(𝑥^∗,𝑦^∗)≠0, and so, by using (7), we get a contradiction.

Remark 8. Theorem 7 is a generalization of the main result of [6].
By taking 𝜓_{𝜆,𝑎,𝑏}(𝑡)=𝑡 in Theorem 7, we get the following corollary.

Corollary 9. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for all 𝑥∈𝒪(𝑢), and for all 𝑦∈𝒪(𝑢),𝑑_𝜆(𝑥,𝑦)=0⇒𝑑_𝜆(𝑇𝑥,𝑇𝑦)=0.(ii)For all 𝜆∈𝒜 and for all 0<𝑎<𝑏<+∞, there exists 𝜑_{𝜆,𝑎,𝑏}∈Φ such that ∀𝑥,𝑦∈𝑋,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑑_𝜆𝑥,𝑦−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (30)Then, 𝑇 has a unique fixed point.

By taking 𝜑_{𝜆,𝑎,𝑏}(𝑡)=(1−𝑘)𝑡 in Corollary 9, we get the following result.

Corollary 10. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for all 𝑥∈𝒪(𝑢), and for all 𝑦∈𝒪(𝑢),𝑑_𝜆(𝑥,𝑦)=0⇒𝑑_𝜆(𝑇𝑥,𝑇𝑦)=0.(ii)For all 𝜆∈𝒜 and for all 0<𝑎<𝑏<∞, there exists 𝑘=𝑘(𝜆,𝑎,𝑏)∈(0,1) such that ∀𝑥,𝑦∈𝑋,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑘𝑑_𝜆𝑥,𝑦. (31)Then, 𝑇 has a unique fixed point.

Next, we give other consequences of Theorem 7.

Corollary 11. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following condition.
For all 𝜆∈𝒜, there exist 𝜓_𝜆∈Ψ and 𝜑_𝜆∈Φ such that 𝜓_𝜆𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝜓_𝜆𝑑_𝜆𝑥,𝑦−𝜑_𝜆𝑑_𝜆𝑥,𝑦∀𝑥,𝑦∈𝑋. (32) Then, 𝑇 has a unique fixed point.

Proof. Condition (i) of Theorem 7 is satisfied from inequality (32), and taking 𝜓_{𝜆,𝑎,𝑏}(𝑡)=𝜓_𝜆(𝑡) and 𝜑_{𝜆,𝑎,𝑏}(𝑡)=𝜑_𝜆(𝑡), we get Corollary 11.

By taking 𝜓_𝜆(𝑡)=𝑡 in Corollary 11, we get the following result.

Corollary 12. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following condition.
For all 𝜆∈𝒜, there exists 𝜑_𝜆∈Φ such that ∀𝑥,𝑦∈𝑋,𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑑_𝜆𝑥,𝑦−𝜑_𝜆𝑑_𝜆𝑥,𝑦. (33) Then, 𝑇 has a unique fixed point.

Also, by taking 𝜑_𝜆(𝑡)=(1−𝑘)𝑡 in Corollary 12, we get the following result.

Corollary 13. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following condition.
For all 𝜆∈𝒜, there exists 𝑘=𝑘(𝜆)∈(0,1) such that ∀𝑥,𝑦∈𝑋,𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑘𝑑_𝜆𝑥,𝑦. (34) Then, 𝑇 has a unique fixed point.

Finally, we give a result involving a condition of integral type. Let Γ denote the set of all functions 𝛼∶[0,+∞)→[0,+∞) which satisfy the following conditions.(c1)𝛼 is Lebesgue integrable on [0,+∞).(c2) For all 𝜀>0, we have ∫^𝜀₀𝛼(𝑡)𝑑𝑡>0.

Corollary 14. Let (𝑋,𝒯(ℱ)) be a complete gauge space, and let 𝑇∶𝑋→𝑋 be a mapping satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for all 𝑥∈𝒪(𝑢), and for all 𝑦∈𝒪(𝑢),𝑑_𝜆(𝑥,𝑦)=0⇒𝑑_𝜆(𝑇𝑥,𝑇𝑦)=0.(ii)For all 0<𝑎<𝑏<∞ and for all 𝜆∈𝒜, there exist 𝛼_{𝜆,𝑎,𝑏},𝛽_{𝜆,𝑎,𝑏}∈Γ such that ∀𝑥,𝑦∈𝑋,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹^𝑑𝜆₀^{(𝑇𝑥,𝑇𝑦)}𝛼_{𝜆,𝑎,𝑏}(𝑡)≤𝑑𝑡_{𝑑𝜆0(𝑥,𝑦)}𝛼_{𝜆,𝑎,𝑏}(𝑡)𝑑𝑡−^𝑑𝜆₀^(𝑥,𝑦)𝛽_{𝜆,𝑎,𝑏}(𝑡)𝑑𝑡. (35)Then, 𝑇 has a unique fixed point.

Proof. It follows from Theorem 7 that, by taking 𝜓_{𝜆,𝑎,𝑏}(𝑥)=^𝑥₀𝛼_{𝜆,𝑎,𝑏}(𝑡)𝜑𝑑𝑡,_{𝜆,𝑎,𝑏}(𝑥)=^𝑥₀𝛽_{𝜆,𝑎,𝑏}(𝑡)𝑑𝑡,𝑥≥0. (36)

Remark 15. Corollary 14 is a generalization of the main result of [19].

4. Results for Cyclic Mappings

In [20], Kirk et al. obtained extensions of the Banach contraction principle for cyclic mappings, by considering a cyclical contractive condition as given by the following theorem.

Definition 16. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty subsets of a nonempty set 𝑋. Then, 𝑇∶𝑋→𝑋 is called a cyclic mapping associated with (𝐴₁,…,𝐴_𝑝) if the following conditions hold:(i)𝑋=⋃^𝑝_𝑖=1𝐴_𝑖,(ii)𝑇(𝐴_𝑖)⊆𝐴_𝑖+1for1≤𝑖≤𝑝, where 𝐴_𝑝+1=𝐴₁.

Theorem 17. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a metric space (𝑋,𝑑), and suppose that 𝑇∶𝑋→𝑋 is a cyclic mapping associated with (𝐴₁,…,𝐴_𝑝) satisfying the following: ∃𝑘∈(0,1)𝑠𝑢𝑐ℎ𝑡ℎ𝑎𝑡𝑑𝑇𝑥,𝑇𝑦≤𝑘𝑑𝑥,𝑦,∀𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑓𝑜𝑟1≤𝑖≤𝑝. (37) Then, 𝑇 has a unique fixed point.

Inspired by this result, other fixed-point theorems with cyclical contractive conditions were obtained (see [21–30]). Our aim in this section is to extend Theorem 7 for cyclic mappings.

Theorem 18. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete gauge space (𝑋,𝒯(ℱ)), and suppose that 𝑇∶𝑋→𝑋 is a cyclic mapping associated with (𝐴₁,…,𝐴_𝑝) satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for 1≤𝑖≤𝑝, and for all 𝑥∈𝒪(𝑢)⋂𝐴_𝑖, 𝑦∈𝒪(𝑢)⋂𝐴_𝑖+1, 𝑑_𝜆𝑥,𝑦=0⟹𝑑_𝜆𝑇𝑥,𝑇𝑦=0. (38)(ii)For all 𝜆∈𝒜 and for all 0<𝑎<𝑏<∞, there exist 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that, for 1≤𝑖≤𝑝, ∀𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (39)Then, 𝑇 has a unique fixed point.

Proof. As in the proof of Theorem 7, one can prove that ∀𝜆∈𝒜,∀𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑑_𝜆𝑥,𝑦≠0⟹𝑑_𝜆𝑇𝑥,𝑇𝑦<𝑑_𝜆,(𝑢)𝑥,𝑦∀𝜆∈𝒜,∀𝑢∈𝑋,∀𝑥∈𝒪∩𝐴_𝑖,𝑦∈𝒪(𝑢)∩𝐴_𝑖+1,𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑑_𝜆𝑥,𝑦. (40)(41) Next, in establishing the existence of a fixed point, since 𝑇(⋂^𝑝_𝑖=1𝐴_𝑖)⊂⋂^𝑝_𝑖=1𝐴_𝑖, one only needs to prove that ⋂^𝑝_𝑖=1𝐴_𝑖≠∅. In fact, applying Theorem 7 to the restriction of 𝑇 on the complete gauge space (⋂^𝑝_𝑖=1𝐴_𝑖,𝒯(ℱ)), we obtain that 𝑇 has a fixed point in ⋂^𝑝_𝑖=1𝐴_𝑖.
Now, let 𝑥₀∈𝐴₁, and we construct a sequence {𝑥_𝑛} in 𝑋 such that 𝑥_𝑛+1=𝑇𝑥_𝑛,∀𝑛≥0. (42) Remark that, for all 𝑛≥0, there exists 𝑖=𝑖(𝑛)∈{1,…,𝑝} such that (𝑥_𝑛,𝑥_𝑛+1)∈𝐴_𝑖×𝐴_𝑖+1. Then, as in the proof of Theorem 7, we obtain that 𝑑_𝜆𝑥_𝑛−1,𝑥_𝑛⟶0as𝑛⟶+∞. (43) We will prove that {𝑥_𝑛} is a Cauchy sequence in the complete gauge space (𝑋,𝒯(ℱ)). Suppose that {𝑥_𝑛} is not a Cauchy sequence. Then, there exists (𝜆,𝜀)∈𝒜×(0,+∞) for which we can find two sequences of positive integers {𝑚(𝑘)} and {𝑛(𝑘)} such that, for all positive integers 𝑘, 𝑛(𝑘)>𝑚(𝑘)>𝑘,𝑑_𝜆𝑥_𝑚(𝑘),𝑥_𝑛(𝑘)𝑑≥𝜀,_𝜆𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}<𝜀. (44) Using (44) and the triangular inequality, we get 𝜀≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑_𝜆𝑥_𝑚(𝑘),𝑥_{𝑛(𝑘)−1}+𝑑_𝜆𝑥_{𝑛(𝑘)−1},𝑥_𝑛(𝑘)<𝜀+𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (45) Thus, we have 𝜀≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)<𝜀+𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑛(𝑘)−1}. (46) Letting 𝑘→+∞ in the above inequality and using (43), we obtain 𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)⟶𝜀as𝑘⟶+∞. (47) Let 𝑘 be a positive integer, and let 𝑗(𝑘)∈{1,…,𝑝} be the integer such that 𝑛(𝑘)−𝑚(𝑘)=𝑗(𝑘)mod𝑝. (48) Using the triangular inequality, we have 𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑛(𝑘)+1+𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)+𝑗(𝑘)}+^{𝑗(𝑘)−1}_𝑖=0𝑑_𝜆𝑥_{𝑚(𝑘)+1+𝑖},𝑥_{𝑚(𝑘)+𝑖},𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)+𝑗(𝑘)}≤𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)+^{𝑗(𝑘)−1}_𝑖=0𝑑_𝜆𝑥_{𝑚(𝑘)+1+𝑖},𝑥_{𝑚(𝑘)+𝑖}. (49) Thus, we have 𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)+𝑗(𝑘)}−𝑑_𝜆𝑥_𝑛(𝑘),𝑥_𝑚(𝑘)≤𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+^{𝑗(𝑘)−1}_𝑖=0𝑑_𝜆𝑥_{𝑚(𝑘)+1+𝑖},𝑥_{𝑚(𝑘)+𝑖}≤𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_𝑛(𝑘)+^𝑝−1_𝑖=0𝑑_𝜆𝑥_{𝑚(𝑘)+1+𝑖},𝑥_{𝑚(𝑘)+𝑖}. (50) Letting 𝑘→+∞ in the above inequality, using (43) and (47), we get that 𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)+𝑗(𝑘)}⟶𝜀as𝑘⟶+∞. (51) Using the same technique, we obtain that 𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}⟶𝜀as𝑘⟶+∞. (52) On the other hand, from (52), there exists a positive integer 𝑘₀ such that 𝜀2≤𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}≤2𝜀,∀𝑘≥𝑘₀. (53) Applying the hypothesis of the theorem by taking 𝑎=𝜀/2 and 𝑏=2𝜀, we get the existence of 𝜓_{𝜆,𝑎,𝑏}∈Ψ and 𝜑_{𝜆,𝑎,𝑏}∈Φ such that 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑇𝑥_𝑛(𝑘),𝑇𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}; (54) that is, 𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘)+1,𝑥_{𝑚(𝑘)+𝑗(𝑘)}≤𝜓_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥_𝑛(𝑘),𝑥_{𝑚(𝑘)+𝑗(𝑘)−1}. (55) Letting 𝑘→+∞ in the above inequality, using (51), (52), the continuity of 𝜓_{𝜆,𝑎,𝑏}, and the lower semicontinuity of 𝜑_{𝜆,𝑎,𝑏}, we obtain 𝜓_{𝜆,𝑎,𝑏}(𝜀)≤𝜓_{𝜆,𝑎,𝑏}(𝜀)−𝜑_{𝜆,𝑎,𝑏}(𝜀), (56) which implies that 𝜑_{𝜆,𝑎,𝑏}(𝜀)=0, which leads to the contradiction 𝜀=0. Finally, we deduce that {𝑥_𝑛} is a Cauchy sequence in the complete gauge space (𝑋,𝒯(ℱ)); therefore, {𝑥_𝑛} converges to some 𝑥^∗∈𝑋. In view of condition (ii) of Definition 16, an infinite number of elements of the sequence {𝑥_𝑛} lie in each 𝐴_𝑖, 𝑖∈{1,…,𝑝}. As 𝐴_𝑖 is a closed subset, we deduce that 𝑥^∗∈𝐴_𝑖, for 𝑖∈{1,…,𝑝}. Thus, 𝑥^∗∈⋂^𝑝_𝑖=1𝐴_𝑖, and so ⋂^𝑝_𝑖=1𝐴_𝑖≠∅.
In order to complete the proof, we have to show that uniqueness of the fixed point.
From condition (ii) of Definition 16, any fixed point of 𝑇 lies necessarily in ⋂^𝑝_𝑖=1𝐴_𝑖. Suppose that there exist 𝑥^∗,𝑦^∗∈𝑋 such that 𝑥^∗≠𝑦^∗, 𝑇𝑥^∗=𝑥^∗, and 𝑇𝑦^∗=𝑦^∗. Then, there exists 𝜆∈𝒜 such that 𝑑_𝜆(𝑥^∗,𝑦^∗)≠0, and so, by using (40), we get a contradiction.

Remark 19. Theorem 18 is a generalization of the main result of [24].

By taking 𝜓_{𝜆,𝑎,𝑏}(𝑡)=𝑡 in Theorem 18, we get the following corollary.

Corollary 20. Let {𝐴_𝑖}^𝑝_𝑖=1 be nonempty closed subsets of a complete gauge space (𝑋,𝒯(ℱ)), and suppose that 𝑇∶𝑋→𝑋 is a cyclic mapping associated with (𝐴₁,…,𝐴_𝑝) satisfying the following conditions.(i)For all 𝜆∈𝒜, for all 𝑢∈𝑋, for 1≤𝑖≤𝑝, and for all 𝑥∈𝒪(𝑢)⋂𝐴_𝑖,𝑦∈𝒪(𝑢)⋂𝐴_𝑖+1, 𝑑_𝜆𝑥,𝑦=0⟹𝑑_𝜆𝑇𝑥,𝑇𝑦=0. (57)(ii)For all 𝜆∈𝒜 and for all 0<𝑎<𝑏<∞, there exists 𝜑_{𝜆,𝑎,𝑏}∈Φ such that, for 1≤𝑖≤𝑝, ∀𝑥∈𝐴_𝑖,𝑦∈𝐴_𝑖+1,𝑎≤𝑑_𝜆𝑥,𝑦≤𝑏⟹𝑑_𝜆𝑇𝑥,𝑇𝑦≤𝑑_𝜆𝑥,𝑦−𝜑_{𝜆,𝑎,𝑏}𝑑_𝜆𝑥,𝑦. (58)Then, 𝑇 has a unique fixed point.

The analogous of Corollaries 10–14 for cyclic mappings can be obtained easily, and so, to avoid repetitions, we omit the details.

5. Application to Ordinary Differential Equation

In this section, we present an example where our obtained results can be applied. Precisely, we study the existence of solution for the following second-order nonlinear initial-value problem: 𝑥^′′(𝑡)((𝑡𝑥(0)𝑥=𝑘𝑡,𝑥)),𝑡>0,=𝛼,_′(0)=𝛽, (59) where 𝑘∶[0,+∞)×ℝ^𝑛→ℝ^𝑛 is a continuous function. This problem is equivalent to the integral equation 𝑥(𝑡)=^𝑡₀(𝑡−𝑠)𝑘(𝑠,𝑥(𝑠))𝑑𝑠+𝛽𝑡+𝛼,𝑡≥0. (60) Denote by 𝑋=𝐶([0,+∞),ℝ^𝑛) the set of continuous functions defined on [0,+∞). For each positive integer 𝑝≥1, we define the function ‖⋅‖_𝑝∶𝑋→[0,+∞) by ‖𝑥‖_𝑝=max_{0≤𝑡≤𝑝}|𝑥(𝑡)|,∀𝑥∈𝑋. (61) This function is a seminorm on 𝑋. Also, define 𝑑_𝑝𝑥,𝑦=𝑥−𝑦_𝑝,∀𝑝≥1,𝑥,𝑦∈𝑋. (62) Then, ℱ={𝑑_𝑝}_𝑝≥1 is a separating family of pseudometrics on 𝑋, and (𝑋,𝒯(ℱ)) is a complete gauge space.

We will prove the following result.

Theorem 21. Suppose that, for all 0<𝑎<𝑏<∞, for each 𝑠≥0, and for all 𝑥,𝑦∈𝑋, one has 𝑎<𝑑_𝑝𝑘((𝑠(𝑠)(𝑠)()𝑑𝑥,𝑦<𝑏⟹𝑠,𝑥))−𝑘𝑠,𝑦≤𝛾𝑏−𝑎_𝑝𝑥,𝑦²1+(𝑏−𝑎)𝑑_𝑝,𝑥,𝑦 (63) where 𝛾∶[0,+∞)→[0,+∞) is such that the function 𝑡↦∫^𝑡₀(𝑡−𝑠)𝛾(𝑠)𝑑𝑠 is bounded on [0,+∞) and sup_𝑡≥0^𝑡₀(𝑡−𝑠)𝛾(𝑠)𝑑𝑠≤1. (64)
Then, (59) has a unique solution 𝑥^∗∈𝐶([0,+∞),ℝ^𝑛).

Proof. Consider the operator 𝑇∶𝑋→𝑋 given by 𝑇𝑥(𝑡)=^𝑡₀(𝑡−𝑠)𝑘(𝑠,𝑥(𝑠))𝑑𝑠+𝛽𝑡+𝛼,𝑡≥0,𝑥∈𝑋. (65) Clearly, 𝑇 is well defined since 𝑘 is a continuous function.
Now, 𝑥^∗ is a solution of (59) if and only if 𝑥^∗ is a fixed point of 𝑇. Next, we will show that the two conditions of Corollary 9 hold true.
Condition (i). Let 𝑝 be a positive integer such that 𝑝≥1, and let 𝑥,𝑦∈𝑋 such that 𝑑_𝑝(𝑥,𝑦)=0. This means that 𝑦(𝑠)=𝑥(𝑠),∀𝑠∈0,𝑝. (66) Let us take 𝑡∈[0,𝑝]; then, we have (𝑡)(𝑡)=𝑇𝑦−𝑇𝑥_𝑡0(𝑡−𝑠)𝑘𝑠,𝑦(𝑠)−𝑘(𝑠,𝑥(𝑠))𝑑𝑠=0. (67) Hence, 𝑑_𝑝𝑇𝑥,𝑇𝑦=max_{0≤𝑡≤𝑝}𝑇𝑦(𝑡)−𝑇𝑥(𝑡)=0. (68) Thus, the condition is satisfied.
Condition (ii). Let 𝑎,𝑏∈ℝ such that 0<𝑎<𝑏, and let 𝑥,𝑦∈𝑋 such that 𝑎<𝑑_𝑝(𝑥,𝑦)<𝑏. For all 𝑡∈[0,𝑝], 𝑝≥1, we have 𝑇𝑥(𝑡)−𝑇𝑦(𝑡)≤^𝑡₀(𝑘(𝑠)((𝑠≤𝑡−𝑠)𝑠,𝑦−𝑘𝑠,𝑥))𝑑𝑠_𝑡0()𝛾(𝑠)()𝑑𝑡−𝑠𝑏−𝑎_𝑝𝑥,𝑦²1+(𝑏−𝑎)𝑑_𝑝≤()𝑑𝑥,𝑦𝑑𝑠𝑏−𝑎_𝑝𝑥,𝑦²1+(𝑏−𝑎)𝑑_𝑝𝑥,𝑦^𝑡₀()𝛾(𝑠)≤()𝑑𝑡−𝑠𝑑𝑠𝑏−𝑎_𝑝𝑥,𝑦²1+(𝑏−𝑎)𝑑_𝑝.𝑥,𝑦 (69) Then, for all 𝑝≥1, we have 𝑑_𝑝≤()𝑑𝑇𝑥,𝑇𝑦𝑏−𝑎_𝑝𝑥,𝑦²1+(𝑏−𝑎)𝑑_𝑝.𝑥,𝑦 (70) Hence, for all 𝑝≥1, for all 0<𝑎<𝑏<∞, and for all 𝑥,𝑦∈𝑋 such that 𝑎<𝑑_𝑝(𝑥,𝑦)<𝑏, we have 𝑑_𝑝𝑇𝑥,𝑇𝑦≤𝑑_𝑝𝑥,𝑦−𝜑_{𝑝,𝑎,𝑏}𝑑_𝑝𝑥,𝑦, (71) where 𝜑_{𝑝,𝑎,𝑏}(𝑡)=𝑡−(𝑏−𝑎)𝑡²/(1+(𝑏−𝑎)𝑡), and, obviously, 𝜑_{𝑝,𝑎,𝑏}∈Φ. Finally, applying Corollary 9, we obtain that 𝑇 has a unique fixed point 𝑥^∗∈𝑋; that is, 𝑥^∗∈𝐶([0,+∞),ℝ^𝑛) is the unique solution of (59).

Acknowledgment

The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for the funding of this research through the Research Group Project no. RGP-VPP-237.