A Common Fixed Point Theorem for Nonlinear Quasi-Contractions on b -Metric Spaces with Application in Integral Equations

In this paper


Introduction
In 1974, Ćirić presented the first fixed point result for quasicontractive mappings.This Ćirić's theorem is one of the most general results with linear comparison function in classical metrical fixed point theory (see [1,2]).The existence and uniqueness of fixed point for mappings defined on metric spaces, which satisfies a quasi-contractive inequality with a nonlinear comparison function, were considered by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6].Alshehri et al. [7] proved a fixed point theorem for quasicontractive mappings, defined by linear quasi-contractive conditions on b-metric spaces.Common fixed point generalizations of Ćirić result was obtained by Das and Naik [8], with linear comparison functions and by Di Bari and Vetro [9], with a nonlinear comparison function.
The notion of symmetric spaces, which is the oldest and one of the most important generalizations of metric spaces, was introduced by Fréchet [10].He used the name E-space for a symmetric space.In the last 50 years, many authors (see [11][12][13][14][15][16]) called them semimetric (in German halb-metrisher) spaces.Now, the term symmetric space is usual.After 1955, the term semimetric space is widely used to denote a symmetric space in which the closure operator is idempotent, which started the papers of Heath, Brown, Mc Auley, Jones, and Burke (see [17,18]).Fixed point investigation was started by Cicchese [19] and Jachymski et al. [18] on semimetric spaces and by Hicks and Rhoades [20] on symmetric spaces.
In [10], Fréchet also considered the class of E-spaces with regular écart which include the class of b-metric spaces.Important examples of b-metric spaces are quasi-normed spaces introduced by Bourgin [21] and Hyers [22] and spaces of homogeneous type which have many applications in the theory of analytic functions (see Coifman and Weiss [23]).First, fixed point results on b-metric spaces were presented by Bakhtin [24] and Czerwik [25].
In this paper, we present a common fixed point result for a pair of mappings defined on a b-metric space, which satisfies a quasi-contractive inequality with a nonlinear comparison function.

Symmetric Spaces and b-Metric Spaces
The ordered pair ðΔ, μÞ, where Δ is a nonempty set and μ : Δ 2 → ½0,∞Þ, is a symmetric space, if and only if it satisfies: (W1) μðι, κÞ = 0 if and only if ι = κ (W2) μðι, κÞ = μðκ, ιÞ for any ι, κ ∈ Δ The difference between symmetric spaces and more convenient metric spaces is in the absence of triangle inequality, but many notions in symmetric spaces are defined similar to those in metric spaces.For instance, in symmetric space ðΔ, μÞ, the limit point of a sequence ðι n Þ is defined by Also, we say that a sequence fι n g ⊆ Δ is a Cauchy sequence, if for any given ε > 0, there exists a positive integer n 0 such that μðι m , ι n Þ < ε for every m, n ≥ n 0 .If each Cauchy sequence in symmetric space ðΔ, μÞ is convergent, then we say that ðΔ, μÞ is a complete symmetric space. By we indicate the diameter of the set A.
Let ðΔ, μÞ be a symmetric space.We can introduce the topology τ d by defining the family of all closed sets as follows: a set A ⊆ Δ is closed if and only if for each ι ∈ Δ, μðι, AÞ = 0 implies ι ∈ A, where The convergence of a sequence ðι n Þ in the topology τ d need not imply μðι n , ιÞ → 0, but the converse is true.
Any s ∈ ½0,∞Þ which satisfies inequality (3) of Definition 1 for all ι, κ, z ∈ Δ, where ðΔ, μÞ is a b-metric space, is said to be the b constant of space ðΔ, μÞ.It is clear that if s = 1, then ðΔ, μÞ is a metric space.Lemma 2. Let ðΔ, μÞ be a b-metric space with b constant s.Then, s ≥ 1.
In [17], the following result was proved.

Main Results
First, recall some standard terminology and notations from the fixed point theory.
Let Δ be a nonempty set, and let Y : Δ → Δ be an arbitrary mapping.
Let Δ and Λ be nonempty sets, Y, Γ : Δ → Λ, and YðΔÞ ⊆ ΓðΔÞ.Choose a point is called a Jungck sequence with an initial point ι 0 .Note that a Jungck sequence might not be determined by its initial point ι 0 .
Let Δ be a nonempty set and Y, Γ : Δ → Δ. Y and Γ are called weakly compatible if they commute at their coincidence points.
Lemma 9 (see [28]).Let Δ be a nonempty set and let Y, Γ : Δ → Δ be weakly compatible self mappings.If Y and Γ have a unique point of coincidence κ = YðιÞ = ΓðιÞ, then κ is the unique common fixed point of Y and Γ.Now, we present our main result.Before stating the result, we make a convention to abbreviate YðιÞ and ΓðιÞ in order to avoid too much parenthesis.
Theorem 10.Let ðΛ, μÞ be a b -metric space with b constant s and let Y, Γ : Δ → Λ be two mappings.Suppose that the range of Γ contains the range of Y and that ΓðΔÞ is a complete subspace of Λ.If there exist χ 1 , χ 2 , χ 3 , χ 4 , χ 5 : ½0,∞Þ → ½0,∞Þ such that for any ι, κ ∈ Δ, then there exists z ∈ Λ which is the limit of every Jungck sequence defined by Y and Γ.Further, z is the unique point of coincidence of Y and Γ.Moreover, if Δ = Λ and Y, Γ are weakly compatible, then z is the unique common fixed point for Y and Γ.
Proof.We shall, first, reduce the statement to the case Indeed, from Lemma 7, it follows that there exist functions χ * k : ½0,∞Þ → ½0,∞Þ for each ι > 0 and for all 1 ≤ k ≤ 5, whereas from Lemma 8, it follows that there exists a real function χ : ½0,∞Þ → ½0,∞Þ such that s Thus, we can assume that χ j = χ for all 1 ≤ j ≤ 5 and s Let ι 0 ∈ Δ be arbitrary and let ðι n Þ be an arbitrary sequence such that Yðι n Þ is a Jungck sequence with an initial point ι 0 .
Let d 0 = μðΓι 0 , Yι 0 Þ.We will prove that there exists a real number r 0 > 0 such that: Consider the set D = fr | t − s • χðtÞ > d 0 ∀t > rg which is nonempty, since r − χðrÞ → ∞ as r → ∞.Also, if q ∈ D and p > q imply p ∈ D, and hence, D is an unbounded interval.Set r 0 = inf D. For each positive integer n, there is r n ∉ D such that r 0 − 1/n < r n , and therefore, there is Next, we prove that for all positive integer k, n.Since χ is nondecreasing, it commutes with max, and for all k ≤ i, j ≤ k + n, we have By induction, from (13), we obtain that For 1 ≤ i, j ≤ n, we have Yι i , Yι j ∈ O n−1 ðι 1 Þ, and hence, by (13) Hence, we get which implies that diamðO n ðι 0 ÞÞ ≤ r 0 , and hence Hence, all Jungck sequences defined by Y and Γ are bounded.Now, we shall prove that our Jungck sequence is a Cauchy sequence.Let m > n be positive integers.Then, Yι n , Yι m ∈ O m−n+1 ðι n Þ.Using (15) (with l = n) and ( 19), we get as m, n → ∞.Since YðΔÞ ⊆ ΓðΔÞ, and ΓðΔÞ is complete, it follows that Yι n is convergent.Let κ ∈ Δ be its limit.
Clearly, κ ∈ ΓðΔÞ.So, there is z ∈ Δ such that ΓðzÞ = κ.Let us prove that YðzÞ is also equal to κ.By (8), we have If n → ∞, then the left-hand side in the previous inequality tends to μðκ, YzÞ, and the first, the second, and the fifth argument of max tend to μðκ, κÞ = 0, whereas the third and the fourth tend to μðκ, YzÞ.Thus, we have which is impossible, unless μðκ, YzÞ = 0. Finally, we prove that the point of coincidence is unique.Suppose that there is two points of coincidence κ and κ′ obtained by z and z ′ , i.e., Yz = Γz = κ and Yz ′ = Γz ′ = κ ′ .Then, by (8) we have unless μðκ, κ ′ Þ = 0. Since every Jungck sequence converges to some point of coincidence, and the point of coincidence is unique, it follows that all Jungck sequences converge to the same limit.

Advances in Mathematical Physics
Let Δ = Λ and let Y, Γ be weakly compatible.By Lemma 3, we get that κ = z which is the unique common fixed point of Y and Γ.
The previous theorem extended earlier results for nonlinear contractions on metric spaces obtained by Danes [3], Ivanov [4], Aranđelović et al. [5], and Bessenyei [6] and common fixed point results of Das and Naik [8] and Di Bari and Vetro [9].It also generalizes the fixed point theorem of Aleksić et al. [7] which proved the fixed point theorems for quasi-contractive mappings on b-metric spaces, defined by linear quasi-contractive conditions.

Application
The existence of the solution for the following integral equation is the main purpose in this section.
Proof.Let us consider the operator Y : BC½0,∞Þ → BC½0,∞Þ defined by In view of the given assumptions, we infer that the function YðσÞ is continuous for arbitrarily σ ∈ BC½0,∞Þ.Now, we show that YðσÞ is bounded in BC½0, ∞Þ.As 5

Advances in Mathematical Physics
we have f ι, Thus, f ι, From the above calculations, we have Due to the above inequality, the function Y is bounded.Now, we show that Y satisfies all the conditions of Theorem 10.Let σ 1 , σ 2 be some elements of BC½0, ∞Þ.Then, we have where Mðσ 1 , σ 2 Þ is defined by Thus, we obtain that Using Theorem 10, we obtain that the operator Y admits a fixed point.Thus, the functional integral equation ( 28) admits at least one solution in BC½0, ∞Þ.
Some further inclusion between different classes of comparison functions will be presented in the next statements.If χ ∈ Ξ 3 ∩ Ξ 8 , then χ ∈ Ξ 4 .