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

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 farreaching 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


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 farreaching outgrowth of a well-known theorem of Banach [1] which states that if (, ) is a complete metric space and if  :  →  is a mapping which satisfies the condition  (, ) ≤  (, ) , ∀,  ∈ , where 0 ≤  < 1 is a constant, then  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 settheoretic 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][3][4][5][6][7][8][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 Chis ¸and Precup [13] gave generalizations of the Banach contraction principle on gauge spaces (see also [14][15][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 initialvalue problem.

Preliminaries
In this section, we recall some preliminaries on gauge spaces and introduce some basic definitions.
of balls is called the topology in  induced by the family F. The pair (, T(F)) is called a gauge space.Note that (, T(F)) is Hausdorff because we require F to be separating.
Definition 5. Let (, T(F)) be a gauge space with respect to the family F = {  |  ∈ A} of pseudometrics on .Let {  } be a sequence in , and  ∈ .Then, the following are considered.
(a) The sequence {  } converges to  if and only if In this case, we denote that   F  → .
(b) The sequence {  } is Cauchy if and only if (c) (, T(F)) is complete if and only if any Cauchy sequence in (, T(F)) is convergent to an element of .
(d) A subset of  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  is not a metric, it is because there are at least two points  ̸ =  for which (, ) = 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].
(i)  is continuous and nondecreasing.
Our first result is the following theorem.
Then,  has a unique fixed point.
Step 2. We will prove that {  } is a Cauchy sequence in (, T(F)).Suppose that {  } is not a Cauchy sequence.Then, there exists (, ) ∈ A × (0, +∞) for which we can find two sequences of positive integers {()} and {()} such that, for all positive integers , Using ( 17) and the triangular inequality, we get Thus, we have Letting  → +∞ in the above inequality and using (12), we obtain By the triangular inequality, we have Letting  → +∞ in the above inequality, using ( 12) and ( 20), we get On the other hand, from (20), there exists a positive integer  0 such that Letting  → +∞ in the above inequality, from (20), (22), the continuity of  ,, , and the lower semicontinuity of  ,, , we obtain which implies that  ,, () = 0, which leads to the contradiction  = 0. Finally, we deduce that {  } is a Cauchy sequence.
In the virtue of the separating structure of F, we deduce that  * =  * , and, hence, the existence of the fixed point is proved.
Remark 8. Theorem 7 is a generalization of the main result of [6].
By taking  ,, () =  in Theorem 7, we get the following corollary.
Corollary 9. Let (, T(F)) be a complete gauge space, and let  :  →  be a mapping satisfying the following conditions.
Then,  has a unique fixed point.
Then,  has a unique fixed point.
Next, we give other consequences of Theorem 7.
Corollary 11.Let (, T(F)) be a complete gauge space, and let  :  →  be a mapping satisfying the following condition.For all  ∈ A, there exist   ∈ Ψ and   ∈ Φ such that ∀,  ∈ . (32) Then,  has a unique fixed point.
By taking   () =  in Corollary 11, we get the following result.
Corollary 12. Let (, T(F)) be a complete gauge space, and let  :  →  be a mapping satisfying the following condition.
Corollary 13.Let (, T(F)) be a complete gauge space, and let  :  →  be a mapping satisfying the following condition.
Finally, we give a result involving a condition of integral type.Let Γ denote the set of all functions  : [0, +∞) → [0, +∞) which satisfy the following conditions.
Corollary 14.Let (, T(F)) be a complete gauge space, and let  :  →  be a mapping satisfying the following conditions.

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.
Letting  → +∞ in the above inequality and using (43), we obtain Let  be a positive integer, and let () ∈ {1, . . ., } be the integer such that Using the triangular inequality, we have Then,  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.

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 initialvalue problem: where  : [0, +∞) × R  → R  is a continuous function.This problem is equivalent to the integral equation Denote by  = ([0,+∞),R  ) the set of continuous functions defined on [0, +∞).For each positive integer  ≥ 1, we define the function ‖ ⋅ ‖  :  → [0, +∞) by This function is a seminorm on .Also, define Then, F = {  } ≥1 is a separating family of pseudometrics on , and (, T(F)) is a complete gauge space.
We will prove the following result.
Theorem 21.Suppose that, for all 0 <  <  < ∞, for each  ≥ 0, and for all ,  ∈ , one has (65) Clearly,  is well defined since  is a continuous function.Now,  * is a solution of (59) if and only if  * is a fixed point of .Next, we will show that the two conditions of Corollary 9 hold true.