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

Kastriot Zoto; Stojan Radenović; Arslan H. Ansari

doi:10.1515/math-2018-0024

Article Open Access

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

Kastriot Zoto , Stojan Radenović and Arslan H. Ansari

Published/Copyright: March 20, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

In this work, we introduce the notions of (s, p, α)-quasi-contractions and (s, p)-weak contractions and deduce some fixed point results concerning such contractions, in the setting of b-metric-like spaces. Our results extend and generalize some recent known results in literature to more general metric spaces. Moreover, some examples and applications support the results.

Keywords: (s, p, α)-quasi-contraction; (s, p)-weak contraction; b-metric-like space; Fixed point

MSC 2010: 47H10; 54H25

1 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 fixed point theory in b-metric spaces, see [3, 4,5,6,7,8,9,10, 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 fixed point results. Recently, many results on fixed points, of mappings under certain contractive conditions in such spaces have been obtained (see [11, 12, 13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,[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 first part of this paper, we introduce the concept of (s, p, α)-contractions and quasi-contractions and prove some fixed 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 fixed point results in the existing literature.

2 Preliminaries

Definition 2.1

([12]). LetXbe a nonempty set. A mappingσ : X × X → [0, ∞) is called metric-like if the following conditions hold for allx, y, z ∈ X:

σ (x, y) = 0 impliesx = y,
σ(x, y) = σ(y, x)
σ(x, y) ≤ σ(x, z) + σ(z, y).

The pair (X, σ) is called a metric-like space.

Definition 2.2

([13]). LetXbe a nonempty set. A mappingσ_b :X × X → [0, ∞) is calledb-metric-like if the following conditions hold for somes ≥ 1 and for allx, y, z ∈ X:

σ_b (x, y) = 0 impliesx = y,
σ_b(x, y) = σ_b(y, x)
σ_b(x, y) ≤ s[σ_b(x, z) + σ_b(z, y)].

The pair (X, σ_b) is called ab-metric-like space.

In a b-metric-like space (X, σ_b), if x, y ∈ X and σ_b (x, y) = 0, then x = y, but the converse need not be true, and σ_b (x, x) may be positive for some x ∈ X.

Example 2.3

IfX = ℝ, thenσ_b(x, y) = |x| + |y| defines a metric-like onX.

Example 2.4

LetX = ℝ⁺ ∪ {0} andα > 0 be any constant. Define the distance functionσ : X × X → [0, ∞) byσ (x, y) = α (x + y). Then, the pair (X, σ) is a metric-like space.

Example 2.5

IfX = ℝ⁺ ∪ {0}, thenσ_b (x, y) = (x + y)²defines ab-metric-like onXwith parameters = 2.

Definition 2.6

([13]). Let (X, σ_b) be ab-metric-like space with parameters, and let {x_n} be any sequence inXandx ∈ X. Then

The sequence {x_n} is said to be convergent toxif $limn→∞$ σ_b(x_n, x) = σ_b(x, x);
The sequence {x_n} is said to be a Cauchy sequence in (X, σ_b) if $limn,m→∞$ σ_b(x_n, x_m) exists and is finite;
(X, σ_b) is said to be a completeb-metric-like space if, for every Cauchy sequence {x_n} inX, there exists anx ∈ Xsuch that $limn,m→∞$ σ_b(x_n, x_m) = $limn→∞$ σ_b(x_n, x) = σ_b(x, x).

The limit of a sequence in a b-metric-like space need not be unique.

Lemma 2.7

([19]). Let (X, σ_b) be ab-metric-like space with parameters, and f : X → Xbe a mapping. Suppose thatfis continuous atu ∈ X. Then for all sequences {x_n} inXsuch thatx_n → u, we havefx_n → futhat is

$limn→∞σb(fxn,fu)=σb(fu,fu).$

Lemma 2.8

([15]). Let (X, σ_b) be ab-metric-like space with parameters ≥ 1, and suppose that {x_n} and {y_n} areσ_b-convergent toxandy, respectively. Then we have

$1s2σb(x,y)−1sσb(x,x)−σb(y,y)≤lim infn→∞σb(xn,yn)≤lim supn→∞σb(xn,yn)≤sσb(x,x)+s2σb(y,y)+s2σb(x,y).$

In particular, ifσ_b (x, y) = 0, then we have $limn→∞$ σ_b (x_n, y_n) = 0.

Moreover, for eachz ∈ X, we have

$1sσb(x,z)−σb(x,x)≤lim infn→∞σb(xn,z)≤lim supn→∞σb(xn,z)≤sσb(x,z)+sσb(x,x).$

In particular, ifσ_b (x, x) = 0, then

$1sσb(x,z)≤lim infn→∞σb(xn,z)≤lim supn→∞σb(xn,z)≤sσb(x,z).$

The following result is useful.

Lemma 2.9

Let (X, σ_b) be ab-metric-like space with parameters ≥ 1. Then

Ifσ_b (x, y) = 0, thenσ_b (x, x) = σ_b (y, y) = 0;
If (x_n) is a sequence such that $limn→∞$ σ_b (x_n, x_n+1) = 0, then we have
$limn→∞σb(xn,xn)=limn→∞σb(xn+1,xn+1)=0;$
Ifx ≠ y, thenσ_b (x, y) > 0;

□

Proof

The proof is obvious. □

Lemma 2.10

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1 and let {x_n} be a sequence such that

$limn→∞σb(xn,xn+1)=0$ (1)

If for the sequence {x_n}, $limn,m→∞$ σ_b (x_n, x_m) ≠ 0, then there existε > 0 and sequences ${mk}k=1∞and{nk}k=1∞$ of positive integers withn_k > m_k > k, such thatσ_b (x_{m_k}, x_{n_k}) ≥ ε, σ_b (x_{m_k}, x_{n_k−1}) < ε, ε/s² ≤ $lim supk→∞$ σ_b (x_{m_k−1}, x_{n_k−1}) ≤ εs, andε/s ≤ $lim supk→∞$ σ_b (x_{n_k−1}, x_{m_k}) ≤ ε, ε/s ≤ $lim supk→∞$ σ_b (x_{m_k−1}, x_{n_k}) ≤ εs².

Proof

If $limn,m→∞$ σ_b (x_n, x_m) ≠ 0, then there exist an ε > 0 and sequences ${mk}k=1∞and{nk}k=1∞$ of positive integers with n_k >m_k > k, such that n_k is the smallest index for which

$nk>mk>k,σb(xmk,xnk)≥ε.$ (2)

This means that

$σb(xmk,xnk−1)<ε$ (3)

From (2) and property of Definition 2.2, we have

$ε≤σb(xmk,xnk)≤sσb(xmk,xmk−1)+sσb(xmk−1,xnk)≤sσb(xmk,xmk−1)+s2σb(xmk−1,xnk−1)+s2σb(xnk−1,xnk).$ (4)

Taking the upper limit as k → ∞ in (4), using the assumption (1) and relations (2) and (3) we get

$εs2≤lim supk→∞σb(xmk−1,xnk−1).$ (5)

By the triangular inequality, we have

$σb(xmk−1,xnk−1)≤sσb(xmk−1,xmk)+sσb(xmk,xnk−1),$

so, taking the upper limit as k → ∞ and using (1), we get

$lim supk→∞σb(xmk−1,xnk−1)≤εs.$ (6)

By (5) and (6) we have

$εs2≤lim supk→∞σb(xmk−1,xnk−1)≤εs.$ (7)

Also we have

$ε≤σb(xmk,xnk)≤sσb(xmk,xmk−1)+sσb(xmk−1,xnk),$

and, taking the upper limit as k → ∞, we get

$εs≤lim supk→∞σb(xmk−1,xnk).$ (8)

Again

$ε≤σb(xmk,xnk)≤sσb(xmk,xnk−1)+sσb(xnk−1,xnk).$

Taking the upper limit as k → ∞ and using (1), we get

$εs≤lim supk→∞σb(xnk−1,xmk).$ (9)

By (2) we have

$limk→∞σb(xnk−1,xmk)≤ε.$ (10)

Consequently,

$εs≤lim supk→∞σb(xnk−1,xmk)≤ε.$ (11)

Also

$σbxmk−1,xnk≤sσbxmk−1,xnk−1+sσbxnk−1,xnk.$

Then from (7), (8) and (1) we have

$lim supk→∞σb(xmk−1,xnk)≤slim supk→∞σb(xmk−1,xnk−1)≤εs2.$

Consequently,

$εs≤lim supk→∞σb(xmk−1,xnk)≤εs2.$ (12)

This completes the proof. □

3 Main results

In this section, we introduce the concept of generalized (s, p, α)-contractions and obtain some fixed point theorems for such class of contractions in the framework of b-metric-like spaces.

Definition 3.1

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1. Iff : X→ Xis a self-mapping that satisfies:

$sσbfx,fy≤ασbx,y$

for someα ∈ [0, 1) and allx, y ∈ X, thenfis called an (s, α)-Banach contraction.

Definition 3.2

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1. Iff : X→ Xis a self-mapping that satisfies:

$spσbfx,fy≤ασbx,y$

for some constantsp ≥ 1 andα ∈ [0, 1) and for allx, y ∈ X, thenfis called an (s, p, α)-Banach contraction.

We denote by Ψ, Φ the families of altering distance functions satisfying the following condition, respectively:

Ψ : [0, ∞)→ [0, ∞) is an increasing and continuous function and Ψ(t) = 0, iff t = 0,
Φ :[0, ∞)→ [0, ∞) is a lower semicontinuous function and Φ(t) = 0, iff t = 0.

Based on the definition of C̀iric̀’s quasi-contractions, we introduce the following definition in the setting of a b-metric-like space.

Definition 3.3

Let (X, σ_b) be ab-metric-like space with parameters ≥ 1. Letψ ∈ Ψ, and let constantsα, pbe such that 0 ≤ α < 1 andp ≥ 2. A mappingf : X→ Xis said to be a (ψ, s, p, α)-quasicontraction mapping, if for allx, y ∈ X

$ψ(2spσb(fx,fy))≤αψ(maxσbx,y,σbx,fx,σby,fy,σbx,fy,σby,fx).$ (13)

Remark 3.4

It is obvious that by takingψ(t) = $12$ t(or the identity mappingψ(t) = t) the above notion reduces to an (s, p, α)-quasicontraction.
Takingψ(t) = $12$ tand the arbitrary constantp = 2 we obtain the definition of an (s, α)-quasi-contraction given in [30].
If we takes = 1, it corresponds to the case of metric-like spaces.

Our first main result is as follows:

Theorem 3.5

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1, f : X→ Xbe a given self-mapping. Iffis an (ψ, s, p, α)-quasicontraction, thenfhas a unique fixed point.

Proof

Let x₀ be an arbitrary point in X. We construct a Picard iteration sequence {x_n} with initial point x₀ as usual:

$x1=f(x0),x2=f(x1),…,xn+1=f(xn),…for n∈N.$

If we assume σ_b (x_n₀, x_n₀+1) = 0 for some n₀ ∈ ℕ, then we have x_n₀+1 = x_n₀ that is x_n₀ = x_n₀+1 = f(x_n₀). Hence, x_n₀ is a fixed point of f and the proof is completed. From now on, we assume that for all n ∈ ℕ, σ_b (x_n, x_n+1) > 0 (that is x_n+1 ≠ x_n).

By condition (13), we have

$ψ2sσbxn,xn+1=ψ2spσbxn,xn+1=ψ2spσbfxn−1,fxn≤αψmaxσbxn−1,xn,σbxn−1,fxn−1,σbxn,fxn,σbxn−1,fxn,σbxn,fxn−1=αψmaxσbxn−1,xn,σbxn−1,xn,σbxn,xn+1,σbxn−1,xn+1,σbxn,xn≤αψmaxσbxn−1,xn,σbxn−1,xn,σbxn,xn+1,sσbxn−1,xn+σbxn,xn+1,2sσbxn−1,xn.$ (14)

If σ_b (x_n−1, x_n) ≤ σ_b (x_n, x_n+1) for some n ∈ ℕ, then we find from inequality (14) that

$ψ(2sσb(xn,xn+1))≤αψ(2sσb(xn,xn+1))<ψ(2sσb(xn,xn+1)).$

By the properties of ψ the above inequality gives σ_b (x_n, x_n+1) = 0, which is a contradiction, since we have supposed σ_b (x_n, x_n+1) > 0. Hence, for all n ∈ ℕ

$σb(xn,xn+1)<σb(xn−1,xn),$

that is, the sequence {σ_b (x_n, x_n+1)} is decreasing and bounded below. Thus there exists r ≥ 0 such that

$limn→∞σbxn,xn+1=r.$ (15)

Let us prove that r = 0. If we suppose that r > 0, then applying the condition (14), we have

$ψ(2sσb(xn,xn+1))≤ψ(2spσb(xn,xn+1))≤αψ(2sσb(xn−1,xn)).$ (16)

Taking limit as n → ∞ in (16), using (15), since 0 ≤ α < 1 and by the properties of ψ, we get

$ψ(2sr)≤αψ(2sr),$

which is a contradiction. Hence

$limn→∞σb(xn,xn+1)=0.$ (17)

In the next step, we claim that

$limn,m→∞σb(xn,xm)=0.$

Suppose, on the contrary that $limn,m→∞$ σ_b (x_n, x_m) ≠ 0. Then by Lemma 2.10, there exist ε > 0 and sequences {m(k)} and {n(k)} of positive integers with n_k >m_k > k, such that σ_b (x_{m_k},x_{n_k}) ≥ ε, σ_b (x_{m_k}, x_{n_k−1}) < ε and

$εs2≤lim supk→∞σb(xmk−1,xnk−1)≤εs,εs≤lim supk→∞σb(xnk−1,xmk)≤ε,εs≤lim supk→∞σb(xmk−1,xnk)≤εs2.$ (18)

From the contractive condition (13), we have

$ψ2s2σb(xmk,xnk)≤ψ2spσb(xmk,xnk)=ψ2spσb(fxmk−1,fxnk−1)≤αψmax{σb(xmk−1,xnk−1),σb(xmk−1,fxmk−1),σb(xnk−1,fxnk−1),σb(xmk−1,fxnk−1),σb(xnk−1,fxmk−1)}=αψmax{σb(xmk−1,xnk−1),σb(xmk−1,xmk),σb(xnk−1,xnk),σb(xmk−1,xnk),σb(xnk−1,xmk)}.$ (19)

Taking the upper limit as k → ∞ in (19) and using (17), (18), we obtain

$ψ2s2ε≤αψmaxεs,0,0,es2,ε≤αψ2εs2,$

which is a contradiction due to the properties of ψ and the assumption ε > 0. Hence the sequence {x_n} is a Cauchy sequence in the complete b-metric-like space (X, σ_b). So there is some u ∈ X such that

$limn→∞σb(xn,u)=σb(u,u)=limn,m→∞σb(xn,xm)=0.$ (20)

By continuity of f and Lemma 2.7, we have fx_n → fu that is $limn→∞$ σ_b (x_n, fu) = σ_b (fu, fu).

On the other hand $limn→∞$ σ_b (x_n, u) = 0 = σ_b (u, u) and so by Lemma 2.8

$1sσb(u,fu)≤limn→∞σb(xn,fu)≤sσb(u,fu).$

This implies that

$1sσb(u,fu)≤σb(fu,fu)≤sσb(u,fu).$ (21)

In view of the properties of ψ, constant p ≥ 2, (20), (21) and using (13), we have

$ψσbu,fu≤ψsσbfu,fu≤ψ2spσbfu,fu≤αψmaxσbu,u,σbu,fu,σbu,fu,σbu,fu,σbu,fu=αψσbu,fu.$ (22)

From (22) and the properties of ψ, we get σ_b (u, fu) = 0, which implies fu = u. Hence u is a fixed point of f.

If the self-map f is not continuous then, we consider

$ψ2s2σbxn+1,fu≤ψ2spσbxn+1,fu=ψ2spσbfxn,fu≤αψmaxσb(xn,u),σb(xn,fxn),σb(u,fu),σbxn,fu,σb(u,fxn)=αψmaxσbxn,u,σbxn,xn+1,σbu,fu,σbxn,fu,σb(u,xn+1).$

By taking the upper limit as n → ∞, using Lemmas 2.8 and 2.10, and the relation (17), we obtain

$ψ2sσbu,fu=ψ2s21sσbu,fu≤ψlim supn→∞2s2σbxn+1,fu≤αψ2sσbu,fu.$

From above inequality and the properties of ψ, we get σ_b (u, fu) = 0, which implies fu = u. Hence u is a fixed point of f.

Uniqueness: Let us suppose that u and v are two fixed points of f, i.e. fu = u and fv = v. We will show that u = v. If not, by using condition (13), we have

$ψ2spσbu,v=ψ2spσbfu,fv≤αψmaxσbu,v,σbu,fu,σbv,fv,σbu,fv,σbv,fu=αψmaxσbu,v,σbu,u,σbv,v,σbu,v,σbv,u≤αψ2sσbu,v.$

Since 0 ≤ α < 1 and p ≥ 2, the above inequality implies σ_b (u, v) = 0 which yields u = v. □

The following example illustrates the theorem.

Example 3.6

LetX = [0, 1] andσ_b (x, y) = (x + y)²for allx, y ∈ X. It is clear thatσ_bis ab-metric-like onXwith parameters = 2 and (X, σ_b) is complete. Also, σ_byis not a metric-like or ab-metric onX. Define a self-mappingf : X→ Xbyfx = $x6.$

For allx, y ∈ [0, 1], and the functionψ(t) = 2t, and constantp = 2, we have

$ψ2s2σbfx,fy=ψ8x6+y62=ψ8x+y236=1636x+y2=8362x+y2=8362σbx,y=836ψσbx,y≤αψσbx,y≤αψmaxσbx,y,σbx,fx,σby,fy,σbx,fy,σby,fx.$

All conditions ofTheorem3.5are satisfied and clearlyx = 0 is a unique fixed point off.

In particular, by taking ψ(t) = $12$ t in Theorem 3.5, we have the following result for a self-mapping (seen as a generalization of C̀iric̀ type quasi-contraction).

Corollary 3.7

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1. Iff : X→ Xis a self-mapping that satisfies:

$spσb(fx,fy)≤αmaxσb(x,y),σb(x,fx),σb(y,fy),σb(x,fy),σb(y,fx)$

for some constantsα ∈ [0, 1 /2 .) andp ≥ 2 allx, y ∈ X, thenfhas a unique fixed point inX.

The following is a version of Hardy-Rogers result in [31].

Corollary 3.8

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1. Iff : X→ Xis a self-mapping and there existp ≥ 2 and constantsa_i ≥ 0, i = 1, …, 5 witha₁ + a₂ + a₃ + a₄ + a₅ < 1 such that

$spσb(fx,fy)≤α1σb(x,y)+α2σb(x,fx)+α3σb(y,fy)+α4σb(x,fy)+α5σb(y,fx),$

for allx, y ∈ X, thenfhas a unique fixed point inX.

Proof

This result can be considered as a consequence of Corollary 3.7, since we have

$α1σb(x,y)+α2σb(x,fx)+α3σb(y,fy)+α4σb(x,fy)+α5σb(y,fx)≤(α1+α2+α3+α4+α5)maxσb(x,y),σb(x,fx),σb(y,fy),σb(x,fy),σb(y,fx)=αmaxσb(x,y),σb(x,fx),σb(y,fy),σb(x,fy),σb(y,fx).$

□

Remark 3.9

Theorem3.5generalizes Theorem 1.2in[32]. Theorem 3.2 in[28]is a special case ofCorollary3.7(and so also ofTheorem 3.5) for choice constantp = 2. Also, Theorems 3.1 and 3.4 in[6] are special cases of ourTheorem 3.5. InCorollary3.8, by choosing the constantsa_iin 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

$σbx,y,σbx,fx,σby,fy,σbx,fy,σby,fx.$ (23)

Corollary 3.10

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1 andf : X→ Xbe a self-map. If there existψ ∈ Ψ, 0 ≤ α < $12$ andp ≥ 2, such that the condition

$ψ2spσbfx,fy≤αψMx,y1+ψMx,y$ (24)

is satisfied for allx, y ∈ X, whereM(x, y) is defined as in(23), thenfhas a unique fixed point inX.

Proof

Taking into account that

$αψMx,y1+ψMx,y=α11+ψMx,yψMx,y≤αψMx,y$

for all x, y ∈ X and 0 ≤ α < $12$ , where M(x, y) is defined as in (23), we get that condition (24) implies condition (13). As a consequence, Theorem 3.5 guarantees the existence of a unique fixed point of f. □

Corollary 3.11

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1 andf : X → Xbe a self-map. If there existψ ∈ Ψ, ϕ ∈ Φ, 0 ≤ α < $12$ andp ≥ 2, such that the condition

$ψ(2spσb(fx,fy))≤αψ(M(x,y))1+ϕ(M(x,y))$ (25)

is satisfied for allx, y ∈ X, where M(x, y) is defined as in(23), thenfhas a unique fixed point inX.

Proof

The conclusion follows from Theorem 3.5, since the inequality (25) implies the inequality (13). □

Corollary 3.12

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1 andf : X→ Xa self-map. If there existψ ∈ Ψ, ϕ ∈ Φ, 0 ≤ α < $12$ andp ≥ 2, such that the condition

$ψ(2spσb(fx,fy))≤αψ(M(x,y))ϕ(M(x,y))1+ϕ(M(x,y))$ (26)

is satisfied for allx, y ∈ X, whereM(x, y) is defined as in(23), thenfhas a unique fixed point inX.

Proof

The inequality (26) implies the inequality (13). Hence the conclusion follows from Theorem 3.5. □

Corollary 3.13

$ψ(2spσb(fx,fy))≤αψ(M(x,y))−ϕ(M(x,y))1+ϕ(M(x,y))$ (27)

is satisfied for allx, y ∈ X, where M(x, y) is defined as in (23), thenfhas a unique fixed point inX.

Proof

Taking into account that ϕ is a lower semi continuous function with ϕ(t) = 0 ⇔ t = 0, we have

$αψM(x,y)−ϕM(x,y)1+ϕM(x,y)≤αψM(x,y)1+ϕM(x,y)≤α11+ϕM(x,y)ψM(x,y)≤αψM(x,y)$

for all x, y ∈ X and 0 ≤ α < $12$ , where M(x, y) is defined 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 ≥ 1. For a self-mapping f : X→ X we denote by N(x, y) the following:

$N(x,y)=max{σb(x,y),σb(x,fx),σb(y,fy),σb(x,fy)+σb(y,fx)4s}$ (28)

for all x, y ∈ X.

Definition 3.14

Let (X, σ_b) beab-metric-like space with parameters ≥ 1. A self-mappingf : X→ Xis called a generalized (s, p)-weak contraction, if there existψ ∈ Ψand a constantp ≥ 1, such that

$spσb(fx,fy)≤N(x,y)−ϕ(N(x,y))$ (29)

for allx, y ∈ X, whereN(x, y) is defined as in(28).

Remark 3.15

The above definition reduces to the definition of (s, p)-weak contraction ifN(x, y) = σ_b (x, y).

We now present the following result.

Theorem 3.16

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1. Iff : X→ Xis a self-mapping that is a generalized (s, p)-weak contraction, thenfhas a unique fixed point inX.

Proof

Let x₀ be an arbitrary point in X. Define the iterative sequence {x_n} as: x₁ = f(x₀), x₂ = f(x₁), …,x_n+1 = f(x_n), … for n ∈ ℕ.

If we assume that σ_b (x_n, x_n+1) = 0 for some n ∈ ℕ, then we have x_n+1 = x_n that is x_n = x_n+1 = f(x_n), so x_n is a fixed point of f and the proof is completed. From now on, we will assume that σ_b (x_n, x_n+1) > 0 for all n ∈ ℕ (that is x_n+1 ≠ x_n). Using Definition of N(x, y), we have

$N(xn−1,xn)=maxσb(xn−1,xn),σb(xn−1,fxn−1),σb(xn,fxn),σbxn−1,fxn+σbxn,fxn−14s=maxσbxn−1,xn,σbxn−1,xn,σbxn,xn+1,σbxn−1,xn+1+σbxn,xn4s≤maxσb(xn−1,xn),σb(xn−1,xn),σb(xn,xn+1),sσb(xn−1,xn)+σb(xn,xn+1)+2sσb(xn−1,xn)4s.$ (30)

If we assume that for some n ∈ ℕ

$σbxn−1,xn≤σbxn,xn+1,$

then from the inequality (30), we get

$Nxn−1,xn≤σb(xn,xn+1).$ (31)

By the condition (29), we have

$σbxn,xn+1≤spσb(xn,xn+1)=spσb(fxn−1,fxn)≤Nxn−1,xn−ϕN(xn−1,xn)≤N(xn−1,xn).$ (32)

From (31) and (32), we have

$N(xn−1,xn)=σb(xn,xn+1).$ (33)

From (29), and using (33), we obtain

$σb(xn,xn+1)≤spσb(xn,xn+1)=spσb(fxn−1,fxn)≤N(xn−1,xn)−ϕN(xn−1,xn)=σb(xn,xn+1)−ϕσb(xn,xn+1).$ (34)

The above inequality gives a contradiction, since we have assumed σ_b (x_n, x_n+1) > 0.

Hence, for all n ∈ ℕ, σ_b (x_n, x_n+1) < σ_b (x_n−1, x_n), and the sequence {σ_b (x_n, x_n+1)} is decreasing and bounded below. So there exists l ≥ 0 such that σ_b (x_n, x_n+1)→ l. Also

$limn→∞σb(xn,xn+1)=limn→∞N(xn−1,xn)=l.$

Since the function ϕ is lower semi continuous, we have

$ϕ(l)≤limn→∞infϕN(xn−1,xn).$

Let us prove that l = 0. If we suppose that l > 0, taking the limit in (34) we have

$l≤l−ϕ(l),$

that is a contradiction since l > 0. Thus l = 0.

Hence

$limn→∞σb(xn,xn+1)=limn→∞N(xn−1,xn)=0.$ (35)

Next, we show that $limn,m→∞$ σ_b (x_n, x_m) = 0. Suppose the contrary, that is, $limn,m→∞$ σ_b (x_n, x_m) ≠ 0. Then by Lemma 2.10, there exist ε > 0 and sequences {m_k} and {n_k} of positive integers with n_k >m_k > k, such that

$σb(xmk,xnk)≥ε,σb(xmk,xnk−1)<ε$

and

From the definition of N(x, y), we have

$N(xmk−1,xnk−1)=maxσb(xmk−1,xnk−1),σb(xmk−1,fxmk−1),σb(xnk−1,fxnk−1),σb(xmk−1,fxnk−1)+σb(xnk−1,fxmk−1)4s=maxσb(xmk−1,xnk−1),σb(xmk−1,xmk),σb(xnk−1,xnk),σb(xmk−1,xnk)+σb(xnk−1,xmk)4s.$ (37)

Taking the upper limit as k → ∞ in (37) and using (35) and (36), we get

$limk→∞supN(xmk−1,xnk−1)=limk→∞supmaxσb(xmk−1,xnk−1),σb(xmk−1,xmk),σb(xnk−1,xnk),σb(xmk−1,xnk)+σb(xnk−1,xmk)4s≤maxεs,0,0,εs2+ε4s≤εs.$ (38)

Also, as in Lemma 2.10, we can show that

$limk→∞infσb(xmk−1,xnk−1)≥εs2,limk→∞infσb(xmk−1,xnk)≥εs,limk→∞infσb(xnk−1,xmk)≥εs,$

and

$limk→∞infMxmk−1,xnk−1≥ε2s2.$ (39)

From the (s, p)-weak contractive condition, we have

$sσb(xmk,xnk)≤spσb(fxmk−1,fxnk−1)≤N(xmk−1,xnk−1)−ϕ(N(xmk−1,xnk−1)).$ (40)

Taking the upper limit in (40) and using (38) and (39), we obtain

$εs≤εs−ϕε2s2,$

that is a contradiction since ε > 0. So $limn,m→∞$ σ_b (x_n, x_m) = 0, and the sequence {x_n} is a Cauchy sequence in the complete b-metric-like space (X, σ_b). Thus, there is some u ∈ X, such that

$limn→∞σb(xn,u)=σb(u,u)=limn,m→∞σb(xn,xm)=0.$

If f is a continuous mapping, similarly as in Theorem 3.5 we get that u is a fixed point of f.

If the self-map f is not continuous then we consider

$N(xn,u)=maxσb(xn,u),σb(xn,fxn),σb(u,fu),σb(xn,fu)+σb(u,fxn)4s=maxσb(xn,u),σb(xn,xn+1),σb(u,fu),σb(xn,fu)+σb(u,xn+1)4s.$ (41)

Taking the upper limit in (41) and using Lemma 2.8 and the result (35), we obtain

$limn→∞supN(xn,u)≤max0,0,bd(u,fu),sσb(u,fu)4s=σb(u,fu).$ (42)

Now using the (s, p)-weak contractive condition, we have

$spσb(xn+1,fu)=spσb(fxn,fu)≤N(xn,u)−ϕ(N(xn,u)).$ (43)

Taking the upper limit in (43), and using Lemma 2.8 and result (42), it follows that

$sp−1σbu,fu=sp⋅1sσbu,fu≤σb(u,fu)−ϕσb(u,fu).$ (44)

Hence, since p ≥ 1, the inequality (44) implies σ_b (u, fu) = 0 and so fu = u.

Let us suppose that u and v, (u ≠ v) are two fixed points of f where fu = u and fv = v.

Firstly, since u is a fixed point of f, we have σ_b (u, u) = 0. From (s, p)-weak contractive condition, we have

$spσbu,u≤sσb(fu,fu)≤N(u,u)−ϕ(N(u,u))≤bd(u,u)−ϕ(σb(u,u)),$ (45)

where

$N(u,u)=maxσb(u,u),σb(u,u),σb(u,u),σb(u,u)+σb(u,u)4s=σb(u,u).$

From the inequality (45) it follows that σ_b (u, u) = 0 (also σ_b (v, v) = 0).

Also, we have

$spσb(u,v)≤sσb(fu,fv)≤N(u,v)−ϕN(u,v)≤σb(u,v)−ϕσb(u,v),$ (46)

where N(u, v) = σ_b (u, v). The inequality (46) implies σ_b (u, v) = 0. Therefore u = v and the fixed point is unique. □

The following example illustrates the theorem.

Example 3.17

LetX = [0, ∞) andσ_b (x, y) = x² + y² + |x − y|²for allx, y ∈ X. It is clear thatσ_bis ab-metric-like onX, with parameters = 2 and (X, σ_b) is complete. Also, σ_bis not a metric-like nor ab-metric (and nor a metric onX). Define the self-mappingf : X→ Xbyfx = $ln1+x4.$ For allx, y ∈ X, and the functionϕ(t) = $34$ tand constantp = 2, we have

$s2σb(fx,fy)=4f2x+f2y+fx−fy2=4lnx+142+lny+142+lnx+14−lny+142≤4x216+y216+x4−y42=14x2+y2+x−y2=14σb(x,y)≤14N(x,y)=N(x,y)−34N(x,y)=N(x,y)−ϕ(N(x,y)).$

All of the conditions ofTheorem3.16are satisfied and clearlyx = 0 is a unique fixed point off.

Corollary 3.18

Let (X, σ_b) be a completeb-metric-like space with parameters ≥ 1 andf : X→ Xbe a self-mapping such that for some coefficientp ≥ 2 and for allx, y ∈ Xit satisfies

$spσb(fx,fy)≤αmaxσb(x,y),σb(x,fx),σb(y,fy),σb(x,fy)+σb(y,fx)4s,$ (47)

whereα ∈ (0, 1). Thenfhas a unique fixed point.

Proof

In Theorem 3.16, taking ϕ(t) = (1 − α) t for all t ∈ [0, ∞), we get Corollary 3.18. □

Remark 3.17

Since ab-metric-like space is a metric-like space whens = 1, so our results can be seen as a generalizations and extensions of several comparable results in metric-like spaces andb-metric spaces.

4 Application

In this section we will use Theorem 3.16 to show that there is a solution to the following integral equation:

$x(t)=∫0TL(t,r,x(r))dr$ (48)

Let X = C([0, T]) be the set of real continuous functions defined on [0, T] for T > 0.

We endow X with

$σb(x,y)=maxt∈0,1x(t)+y(t)mfor all x,y∈X,$

where m > 1. It is evident that (X, σ_b) is a complete b-metric-like space with parameter s = 2^m−1.

Consider the mapping f : X→ X given by fx(t) = $∫0T$ L(t, r, x(r))dr.

Theorem 4.1

Consider equation(48) and suppose that

L : [0, T] × [0, T] × ℝ → ℝ⁺, (that isL(t, r, x(r)) ≥ 0) is continuous;
there exists a continuousγ : [0, T] × [0, T] → ℝ;
$supt∈0,T∫0T$ γ (t, r) dr ≤ 1;
there exists a constant λ ∈ (0, 1) such that for all (t, r) ∈ [0, T]²andx, y ∈ R,

$Lt,r,x(r)+Lt,r,y(r)≤λs31mγ(t,r)x(r)+y(r).$

Then the integral equation (48) has a unique solutionx ∈ X.

Proof

For x, y ∈ X, from conditions (3) and (4), for all t, we have

$s2σb(fx(t),fy(t))=s2fx(t)+fy(t)m=s2∫0TL(t,r,x(r))dr+∫0TL(t,r,y(r))drm≤s2∫0TL(t,r,x(r))dr+∫0TLt,r,y(r)drm≤s2∫0Tλs31mγ(t,r)x(r)+y(r)m1mdrm≤s2∫0Tλs31mγ(t,r)σb1mx(r),y(r)drm≤s2⋅λs3σb(x(r),y(r))∫0Tγ(t,r)drm=λsσb(x(r),y(r))∫0Tγ(t,r)drm≤λsσb(x(r),y(r))≤λsN(x,y)=N(x,y)−1−λsN(x,y)=N(x,y)−ϕ(N(x,y)),$

Therefore, taking the coefficient p = 2, and function ϕ (x) = (1 − λ/s.)x, where λ /s. ∈ (0, 1), all of the conditions of Theorem 3.16 are satisfied, and as a result, the mapping f has a unique fixed point in X, which is a solution of the integral equation in (48). □

5 Conclusions

Contractive conditions (13) and (29) are much wider than some previously used, and theorems related to these conditions are more general, since parameter s and the coefficient p ≥ 1 are optional. Theorems 3.5 and 3.16 extend and generalize some existing results to a wider domain such as b-metric-like-spaces. Also, the generalized (s, p, α)-contractions and (s, p)-weak contractions unify a large class of existing contractions in the literature. Theoretical results are supported by applications.

Competing interests: The authors declare that they have no competing interests.
Authors’ contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final version of manuscript.

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Received: 2017-11-15

Accepted: 2018-01-31

Published Online: 2018-03-20

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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DOI: doi.org/10.1515/math-2018-0024

Keywords for this article

(s, p, α)-quasi-contraction; (s, p)-weak contraction; b-metric-like space; Fixed point

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