On some fixed point results for (s, p, α)-contractive mappings in b-metric-like spaces and applications to integral equations

Fixed point theory has received much attention due to its applications in pure mathematics and applied sciences. Recently, a number of generalizations of metric spaces were introduced and extensively studied. In 1989, Bakhtin [1] (and also Czerwik [2]) introduced the concept of b-metric spaces and presented contraction mappings in such metric spaces thus obtaining a generalization of Banach contraction principle. For xed point theory in b-metric spaces, see [3] – [11] and the references therein. Amini-Harandi [12] introduced the notion of metric-like spaces, in which the self distance of a point need not be equal to zero. Such spaces play an important role in topology and logical programming. In 2013, Alghamdi et al. [13] generalized the notion of a b-metric by introduction of the concept of a b-metric-like and proved some related xed point results. Recently, many results on xed points, of mappings under certain contractive conditions in such spaces have been obtained (see [11] – [29]). Fixed point theory has been extended in various directions either by using generalized contractions, or by using more general spaces. Under these directions, in the rst part of this paper, we introduce the concept of (s, p,α)-contractions and quasi-contractions and prove some xed point results. In the second part, we generalize further this new class of contractions for self-mappings, introducing the class of (s, p)weak contractions. Considering such more general, and much wider classes of contractions, the obtained results greatly extend and improve some classical and recent xed point results in the existing literature.


Introduction
Fixed point theory has received much attention due to its applications in pure mathematics and applied sciences. Recently, a number of generalizations of metric spaces were introduced and extensively studied. In 1989, Bakhtin [1] (and also Czerwik [2]) introduced the concept of b-metric spaces and presented contraction mappings in such metric spaces thus obtaining a generalization of Banach contraction principle. For xed point theory in b-metric spaces, see [3] - [11] and the references therein.
Amini-Harandi [12] introduced the notion of metric-like spaces, in which the self distance of a point need not be equal to zero. Such spaces play an important role in topology and logical programming. In 2013, Alghamdi et al. [13] generalized the notion of a b-metric by introduction of the concept of a b-metric-like and proved some related xed point results. Recently, many results on xed points, of mappings under certain contractive conditions in such spaces have been obtained (see [11] - [29]).
Fixed point theory has been extended in various directions either by using generalized contractions, or by using more general spaces. Under these directions, in the rst part of this paper, we introduce the concept of (s, p, α)-contractions and quasi-contractions and prove some xed point results. In the second part, we generalize further this new class of contractions for self-mappings, introducing the class of (s, p)weak contractions. Considering such more general, and much wider classes of contractions, the obtained results greatly extend and improve some classical and recent xed point results in the existing literature. x ∈ X such that lim n,m→∞ The limit of a sequence in a b-metric-like space need not be unique.
Lemma 2.7 ([19]). Let (X, σ b ) be a b-metric-like space with parameter s, and f ∶ X → X be a mapping. Suppose that f is continuous at u ∈ X. Then for all sequences {x n } in X such that x n → u, we have fx n → fu that is Lemma 2.8 ([15]). Let (X, σ b ) be a b-metric-like space with parameter s ≥ , and suppose that {x n } and {y n } are σ b -convergent to x and y, respectively. Then we have In particular, if σ b (x, y) = , then we have lim Moreover, for each z ∈ X, we have The following result is useful.
Proof. The proof is obvious.
Proof. If lim n,m→∞ σ b (x n , x m ) ≠ , then there exist an ε > and sequences {m (k)} ∞ k= and {n (k)} ∞ k= of positive integers with n k > m k > k, such that n k is the smallest index for which This means that From (2) and property of De nition 2.2, we have Taking the upper limit as k → ∞ in (4), using the assumption (1) and relations (2) and (3) we get By the triangular inequality, we have so, taking the upper limit as k → ∞ and using (1), we get lim sup By (5) and (6) we have Also we have and, taking the upper limit as k → ∞, we get Again Taking the upper limit as k → ∞ and using (1), we get By (2) we have lim Consequently, Also Then from (7), (8) and (1) we have This completes the proof.

Main results
In this section, we introduce the concept of generalized (s, p, α)-contractions and obtain some xed point theorems for such class of contractions in the framework of b-metric-like spaces.
De nition 3.1. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ . If f ∶ X → X is a selfmapping that satis es: for some α ∈ [ , ) and all x, y ∈ X, then f is called an (s, α)-Banach contraction.

De nition 3.2.
Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ . If f ∶ X → X is a selfmapping that satis es: for some constants p ≥ and α ∈ [ , ) and for all x, y ∈ X, then f is called an (s, p, α)-Banach contraction.
We denote by Ψ , Φ the families of altering distance functions satisfying the following condition, respectively: is an increasing and continuous function and is a lower semicontinuous function and Φ(t) = , i t = .
Based on the de nition ofCiric's quasi-contractions, we introduce the following de nition in the setting of a b-metric-like space.

De nition 3.3.
Let (X, σ b ) be a b-metric-like space with parameter s ≥ . Let ψ ∈ Ψ , and let constants α, p be such that ≤ α < and p ≥ . A mapping f ∶ X → X is said to be a (ψ, s, p, α)-quasicontraction mapping, if for all x, y ∈ X Remark 3.4. 1. It is obvious that by taking ψ(t) = t (or the identity mapping ψ(t) = t) the above notion reduces to an (s, p, α)-quasicontraction.
2. Taking ψ(t) = t and the arbitrary constant p = we obtain the de nition of an (s, α)-quasi-contraction given in [30]. 3. If we take s = , it corresponds to the case of metric-like spaces.
Our rst main result is as follows: Proof. Let x be an arbitrary point in X. We construct a Picard iteration sequence {x n } with initial point x as usual: If we assume σ b (x n , x n + ) = for some n ∈ N, then we have Hence, x n is a xed point of f and the proof is completed. From now on, we assume that for all n ∈ N, By condition (13), we have If σ b (x n− , x n ) ≤ σ b (x n , x n+ ) for some n ∈ N, then we nd from inequality (14) that By the properties of ψ the above inequality gives σ b (x n , x n+ ) = , which is a contradiction, since we have that is, the sequence {σ b (x n , x n+ )} is decreasing and bounded below. Thus there exists r ≥ such that Let us prove that r = . If we suppose that r > , then applying the condition (14), we have Taking limit as n → ∞ in (16), using (15), since ≤ α < and by the properties of ψ, we get which is a contradiction. Hence lim In the next step, we claim that lim n,m→∞ Suppose, on the contrary that lim n,m→∞ Then by Lemma 2.10, there exist ε > and sequences From the contractive condition (13), we have Taking the upper limit as k → ∞ in (19) and using (17), (18), we obtain which is a contradiction due to the properties of ψ and the assumption ε > . Hence the sequence By continuity of f and Lemma 2.7, we have fx n → fu that is lim On the other hand lim n→∞ σ b (x n , u) = = σ b (u, u) and so by Lemma 2.8 This implies that In view of the properties of ψ, constant p ≥ , (20), (21) and using (13), we have From (22) and the properties of ψ, we get σ b (u, fu) = , which implies fu = u. Hence u is a xed point of f . If the self-map f is not continuous then, we consider By taking the upper limit as n → ∞, using Lemmas 2.8 and 2.10, and the relation (17), we obtain From above inequality and the properties of ψ, we get σ b (u, fu) = , which implies fu = u. Hence u is a xed point of f . Uniqueness: Let us suppose that u and v are two xed points of f , i.e. fu = u and fv = v. We will show that u = v. If not, by using condition (13), we have Since ≤ α < and p ≥ , the above inequality implies σ b (u, v) = which yields u = v.
The following example illustrates the theorem. For all x, y ∈ [ , ], and the function ψ(t) = t, and constant p = , we have

All conditions of Theorem 3.5 are satis ed and clearly x = is a unique xed point of f .
In particular, by taking ψ(t) = t in Theorem 3.5, we have the following result for a self-mapping (seen as a generalization ofCiric type quasi-contraction).
Corollary 3.7. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ . If f ∶ X → X is a self-mapping that satis es: for some constants α ∈ [ , ) and p ≥ all x, y ∈ X, then f has a unique xed point in X.
The following is a version of Hardy-Rogers result in [31].
Corollary 3.8. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ . If f ∶ X → X is a self-mapping and there exist p ≥ and constants a i ≥ , i = , . . . , with a + a + a + a + a < such that for all x, y ∈ X, then f has a unique xed point in X.
Proof. This result can be considered as a consequence of Corollary 3.7, since we have

Remark 3.9. Theorem 3.5 generalizes Theorem 1.2 in [32]. Theorem 3.2 in [28] is a special case of Corollary 3.7 (and so also of Theorem 3.5) for choice constant p = . Also, Theorems 3.1 and 3.4 in [6] are special cases of our Theorem 3.5. In Corollary 3.8, by choosing the constants a i in certain manner, we obtain certain classes of(s, p, α)-contractions.
The following corollaries are also consequences of Theorem 3.5, where self-maps satisfy contractive conditions given by rational expressions, and functions ψ ∈ Ψ , φ ∈ Φ are used. To proceed with them, we denote by M(x, y) the maximum of the set Corollary 3.10. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ and f ∶ X → X be a self-map. If there exist ψ ∈ Ψ , ≤ α < and p ≥ , such that the condition is satis ed for all x, y ∈ X, where M(x, y) is de ned as in (23), then f has a unique xed point in X.
Proof. Taking into account that for all x, y ∈ X and ≤ α < , where M(x, y) is de ned as in (23), we get that condition (24) implies condition (13). As a consequence, Theorem 3.5 guarantees the existence of a unique xed point of f .

Corollary 3.11. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ and f ∶ X → X be a self-map.
If there exist ψ ∈ Ψ , φ ∈ Φ, ≤ α < and p ≥ , such that the condition is satis ed for all x, y ∈ X, where M (x, y) is de ned as in (23), then f has a unique xed point in X.

Corollary 3.12.
Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ and f ∶ X → X a self-map. If there exist ψ ∈ Ψ , φ ∈ Φ, ≤ α < and p ≥ , such that the condition is satis ed for all x, y ∈ X , where M (x, y) is de ned as in (23), then f has a unique xed point in X.

Corollary 3.13. Let (X, σ b ) be a complete b-metric-like space with parameter s ≥ and f ∶ X → X a self-map.
If there exist ψ ∈ Ψ , φ ∈ Φ, ≤ α < and p ≥ , such that the condition is satis ed for all x, y ∈ X, where M(x, y) is de ned as in (23), then f has a unique xed point in X.
Proof. Taking into account that φ is a lower semi continuous function with φ(t) = ⇔ t = , we have for all x, y ∈ X and ≤ α < , where M(x, y) is de ned as in (23). Hence inequality (27) implies inequality (13). Hence the conclusion follows from Theorem 3.5.
The basic result, related to the notion of weakly contractive maps, is due to Rhoades [33]. Further, this result has been generalized and extended by many authors to the notion of (ψ − ϕ)-weakly contractive mappings. The aim of this part of the section is to extend and generalize the main classical result from [33] and other existing results in the literature on b-metric and metric-like spaces to the setup of b-metric-like spaces. Before presenting our results, we revise the weak contraction condition by introducing the notion of (s, p)-weak contraction.
Let (X, σ b ) be a b-metric-like space with parameter s ≥ . For a self-mapping f ∶ X → X we denote by N(x, y) the following: for all x, y ∈ X.
De nition 3.14. Let (X, σ b ) be ab-metric-like space with parameter s ≥ . A self-mapping f ∶ X → X is called a generalized (s, p)-weak contraction, if there exist ψ ∈ Ψ and a constant p ≥ , such that for all x, y ∈ X, where N(x, y) is de ned as in (28).

Remark 3.15. The above de nition reduces to the de nition of (s, p)-weak contraction if N(x, y) = σ b (x, y).
We now present the following result. If we assume that σ b (x n , x n+ ) = for some n ∈ N, then we have x n+ = x n that is x n = x n+ = f (x n ), so x n is a xed point of f and the proof is completed. From now on, we will assume that σ b (x n , x n+ ) > for all n ∈ N (that is x n+ ≠ x n ). Using De nition of N(x, y), we have If we assume that for some n ∈ N then from the inequality (30), we get By the condition (29), we have From (31) and (32), we have N(x n− , x n ) = σ b (x n , x n+ ).
From (29), and using (33), we obtain The above inequality gives a contradiction, since we have assumed σ b (x n , x n+ ) > . Hence, for all n ∈ N, σ b (x n , x n+ ) < σ b (x n− , x n ), and the sequence {σ b (x n , x n+ )} is decreasing and bounded below. So there exists l ≥ such that σ b (x n , x n+ ) → l. Also Since the function φ is lower semi continuous, we have Let us prove that l = . If we suppose that l > , taking the limit in (34) we have that is a contradiction since l > . Thus l = .
Hence lim Next, we show that lim From the de nition of N(x, y), we have Taking the upper limit as k → ∞ in (37) and using (35) and (36), we get Also, as in Lemma 2.10, we can show that From the (s, p)-weak contractive condition, we have Taking the upper limit in (40) and using (38) and (39), we obtain that is a contradiction since ε > . So lim n,m→∞ σ b (x n , x m ) = , and the sequence {x n } is a Cauchy sequence in If f is a continuous mapping, similarly as in Theorem 3.5 we get that u is a xed point of f . If the self-map f is not continuous then we consider Taking the upper limit in (41) and using Lemma 2.8 and the result (35), we obtain Now using the (s, p)-weak contractive condition, we have Taking the upper limit in (43), and using Lemma 2.8 and result (42), it follows that Hence, since p ≥ , the inequality (44) implies σ b (u, fu) = and so fu = u. Let us suppose that u and v, (u ≠ v) are two xed points of f where fu = u and fv = v. Firstly, since u is a xed point of f , we have σ b (u, u) = . From (s, p)-weak contractive condition, we have where N(u, u) = max σ b (u, u), σ b (u, u), σ b (u, u), σ b (u, u) + σ b (u, u) s = σ b (u, u).
Also, we have where N(u, v) = σ b (u, v). The inequality (46) implies σ b (u, v) = . Therefore u = v and the xed point is unique.
The following example illustrates the theorem.

Remark 3.19.
Since a b-metric-like space is a metric-like space when s = , so our results can be seen as a generalizations and extensions of several comparable results in metric-like spaces and b-metric spaces.

Application
In this section we will use Theorem 3.16 to show that there is a solution to the following integral equation: x(t) = ( x(t) + y(t) ) m for all x, y ∈ X, where m > . It is evident that (X, σ b ) is a complete b-metric-like space with parameter s = m− .
Consider the mapping f ∶ X → X given by fx(t) = ∫ T L(t, r, x(r))dr.