Fixed point theorems via Generalized WF-Contractions with applications

The aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

They recall the following uniqueness result from [5], where a weak contractivity condition is considered by using functions in the set .
Theorem 1 [5] Let (X , d) be a complete metric space and f : X → X be a weak contractive mapping, that is, a mapping satisfying (x, y)), for all x, y ∈ X , where ψ, ϕ ∈ . Then f has a unique fixed point.
Some other interesting concepts to deal with weak contractions are, for instance, the notion of implicit contractive function given in 1997 by Popa [12], which was later discussed and considered by other authors, as we can see, for instance, in the research works [2,3,7,8,13]. Recently, Tiwari and Gupta [16] proved some new common fixed point theorems in metric spaces for weakly compatible mappings satisfying an implicit relation involving quadratic terms. Another interesting work is that by Wardowski [17], who gave in 2012 an extension of Contraction Mapping Principle for the case of the new concept of F -contractions, notion that fits adequately in the sense that the corresponding mappings exhibit uniqueness of fixed point in the context of complete metric spaces.

Preliminaries
We begin by introducing some concepts needed, where we denote R + = [0, ∞). The notion of F -contraction introduced by Wardowski is presented in Definition 1. For this definition, the author considered the class of functions F consisting of all functions F : R + → R satisfying the conditions (F1)-(F3) specified below: (F3): There exists k ∈ (0, 1) such that lim s→0 + s k F(s) = 0.

Definition 1 ([17])
We say that a self-mapping f on a metric space (X , d) is an F-contraction if there exists τ ≥ 0 such that, for all x, y ∈ X with d( f x, f y) > 0, we have where F : R + → R is a mapping satisfying conditions (F1)-(F3).
More recently, in [6], Gubran et al. introduced a new class of contractions, called WFcontractions, presented as a mixed type of weak and F-contractions, but different from both concepts, and defined as explained in Definition 2.
For this concept, they considered two families of functions: Definition 2 [6] A self-mapping f on a metric space (X , d) is said to be a WF -contraction if there exist two functions G, δ : where G ∈ G and δ ∈ .
The authors of [6] proved some fixed point results for this class of mappings, as we recall below.
for all n, where G ∈ G and δ ∈ . Then, the sequence {t n } is strictly decreasing and Theorem 2 [6, Theorem 2.1] Let (X , d) be a complete metric space and consider f : X → X a WF-contraction for some G ∈ G and δ ∈ . Then, f has a unique fixed point.
The authors in [6] also justified that the concept of WF -contraction is different from the notion of weak contraction, and that each WF -contraction f satisfies d( f x, f y) < d(x, y), for x, y different, so that the continuity of the mapping is guaranteed.
However, in the proof of [6, Theorem 2.1], some confusing points have been detected in the following sense. To prove that a particular sequence {x n }, which is understood to converge to a fixed point of the mapping, is a Cauchy sequence, it is considered M := min 0≤i≤n δ(t i ) and then deduced that where t n = d(x n , x n+1 ) → 0 as n → 0, and hence argued that it is possible to find N ∈ N such that n t k n ≤ 1 for all n ≥ N . However, M should be a sequence instead, that is, satisfying lim n→∞ (n + 1)t k n+1 M n = 0, and M n could tend to zero, so it is not possible to deduce that n t k n ≤ 1 for n large enough. The purpose of this paper is to revise some of these aspects and to introduce a new class of mixed contractions in order to generalize this result by Gubran et al. proved in [6], giving a more general class of mappings which allow to deduce the existence of a unique fixed point. We also show an example of a mapping in this class. Further, we show an application to the solvability of a two-point boundary value problem for second-order differential equations.

Main result
In this section, we introduce the concept of a generalized WF -contraction, and establish a fixed point theorem valid for complete metric spaces.
We consider two families of functions:

Definition 3
We say that a self-mapping f on a metric space where G satisfies condition (G1), and δ satisfies the conditions (G2) and (G3), that is, G ∈ G and δ ∈ .
Next, we give an example of mapping satisfying condition (4), which is similar to the one proposed in [6].

Example 1
We consider the base space of nonnegative real numbers X = [0, ∞) and define a self mapping f on X by Then the mapping f satisfies (4), taking the functions G(s) = s + 1 3(s+1) and δ(t) = t 9 , for which it is obvious the validity of conditions (G1)-(G3).
Indeed, we distinguish three cases: Case I. If 3 ≥ y ≥ x, then the inequality (4) is written as: .
Similar to the Example in [6], we denote a := 3 − x and b := y − 3. Thus, the previous inequality is written as: or, equivalently, which is obviously true.
Case III. If 3 ≤ x ≤ y, then nothing has to be checked since In any case, the inequality in (4) would be valid since it reduces to: which can be expressed as: .

Remark 3
Since it will be used later, we explain the hypotheses required on functions G and δ for the validity of Lemma 2 [6, Lemma 2.2]. By revising the proof in [6], we find that the conclusion holds just by assuming that G is strictly increasing and δ(t) > 0 for t ∈ (0, ∞).
Next, we provide the result on the existence of a unique fixed point for the mapping f . We use the notation N 0 := N ∪ {0}.

Theorem 4 Consider a complete metric space (X , d) and
suppose that the mapping f : X → X is continuous and satisfies the Generalized WF-contraction property for some G ∈ G and δ ∈ . Then, there exists a unique fixed point for the mapping f .

Proof
We start by selecting an arbitrary element x 0 ∈ X , and defining the sequence {x n } in X by recurrence, as follows: x n+1 := f x n , for all n ∈ N 0 .
In case that x n = x n+1 for some n ∈ N 0 , then we have proved the existence of a fixed point for f . Therefore, we assume that d(x n , x n+1 ) > 0 for every n ∈ N 0 . Similar to [6], we denote by t n := d(x n , x n+1 ), n ∈ N 0 , so that {t n } is a sequence of positive real numbers. Now, choosing x = x n and y = x n+1 in the inequality (4), we get By the positiveness of δ on (0, +∞) and the strictly increasing character of G, we get that and (7) is reduced to for every n ∈ N 0 . Hence, similar to Theorem 2.1 [6], by applying Lemma 2 (see Remark 3), we deduce the strictly decreasing character of the sequence {t n } and the convergence of the series ∞ i=0 δ(t i ) < ∞, thus, the general term is convergent to zero, that is, lim n→∞ δ(t n ) = 0, which implies that by virtue of condition (G2). By (G3), and the convergence of ∞ i=0 δ(t i ) < ∞, we deduce the convergence of Now, we prove that {x n } is a Cauchy sequence. Let m, n ∈ N with m > n, then we obtain due to the convergence of ∞ i=0 t i < ∞. This proves that {x n } is a Cauchy sequence. By the completeness of X , there exists x ∈ X which is its limit, and using the continuity of f , we have Since the uniqueness was justified in Lemma 3, the proof is finished. Proof We prove that ⊆ by checking that conditions (G3)-(G4) imply ( G3). Indeed, let {s n } be a strictly decreasing sequence of positive real numbers, and assume that: Consider the convergent sequence of partial sums {y n }, where y n = n j=0 δ(s j ), and the strictly increasing sequence {z n } of positive real numbers given by z n = n j=0 s j . Then, by (G3), δ(s j ) = y n , for all n ∈ N.
Since {y n } is convergent, then it is bounded, so {δ(z n )} is a bounded sequence. By (G4), {z n } is also bounded. Moreover, it is an increasing sequence, so {z n } is convergent, but this is the sequence of partial sums of the series ∞ n=0 s n , whose sum is finite. Remark 6 It is clear that a WF -contraction is continuous and also satisfies the inequality required to be a Generalized WF -contraction (by the nondecreasing character of G), so Theorem 4 provides a generalized inequality in comparison to Theorem 2.1 [6]. However, we have modified the families G (extending it) and (adding or modifying the conditions), in order to clarify the procedure.
In the following Corollary, we consider the particular case of G(t) = t, and δ(t) = μt, where μ > 0, which belong to the families G and ∩ , respectively.

Corollary 7
Consider a complete metric space (X , d) and suppose that the mapping f : X → X is continuous and that, for all x, y ∈ X with d( f x, f y) > 0, we have Then, there exists a unique fixed point for the mapping f .
Last result is an extension of Banach Contraction Principle since a function satisfying with c ∈ [0, 1), is continuous and satisfies for all x, y ∈ X , where μ := 1 − c ∈ (0, 1]. Also, by taking G(t) = ln(t + r ), where r is fixed with r ∈ (0, 1] and δ ∈ ∪ , we have the following Corollary.

Corollary 8
Suppose that r is fixed with r ∈ (0, 1] and that δ ∈ ∪ . Consider a complete metric space (X , d) and suppose that the mapping f : X → X is continuous and that, for all x, y ∈ X with d( f x, f y) > 0, we have Then, there exists a unique fixed point for the mapping f .

Application
Similar to the application shown in [6], we present an example of a two-point boundary value problem for differential equations of second order, and we apply the new results in order to deduce the existence of a unique solution. The problem considered is similar to equation (4.1) in [6]: where α, β ∈ R, and g : J × R → R is a continuous function. For the above-mentioned boundary value problem (13), the Green's function is defined as: Considering the space C(J ) consisting of all continuous real functions defined on J, it is well-known that (C(J ), d θ ) is a complete metric space if we consider the weighted supremum distance where θ ≥ 0 is fixed. In the following, we use d := d 0 the supremum norm. By applying the results proved, we can deduce the existence of a unique solution for problem (13).

Theorem 9
Suppose that there exists δ ∈ ∪ such that, for all t, s ∈ J and non-identical x, y ∈ C(J ), we have Then, problem (13) has a unique solution x * ∈ C 2 .
Proof It is well known that x ∈ C 2 is a solution to (13) if and only if x ∈ C is a solution to This definition of f clearly allows to affirm that the fixed points of f in C(J ) are the continuous solutions to (15), and, therefore, the solutions to the boundary value problem (13). Next, we check the validity of the conditions in Theorem 4. It is obvious that f is a continuous mapping. Let x, y ∈ C(J ) with d( f x, f y) > 0, that is, x not coincident with y, then we obtain, for t ∈ J , that | f x(t) − f y(t)| =   (d(x, y)).
This proves that condition (4) is satisfied for G chosen as the identity mapping. By applying Theorem 4, there exists a unique solution to problem (13).
We can extend Theorem 9 in the following way.

Theorem 10
Suppose that there exist G ∈ G, and δ ∈ ∪ such that, for all t, s ∈ J and non-identical x, y ∈ C(J ), we have Then, problem (13) has a unique solution x * ∈ C 2 .
Proof It is obvious since both G and G −1 are strictly increasing functions.