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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

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

Kastriot Zoto
  • Corresponding author
  • Department of Mathematics and Computer Sciences, Faculty of Natural Sciences, University of Gjirokastra, Gjirokastra, Albania
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Stojan Radenović
  • Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120 Beograd, Serbia
  • Serbia State University of Novi Pazar, Novi Pazar, Serbia
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Arslan H. Ansari
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/math-2018-0024

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]). Let X be a nonempty set. A mapping σ : X × X → [0, ∞) is called metric-like if the following conditions hold for all x, y, zX:

  • σ (x, y) = 0 implies x = 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]). Let X be a nonempty set. A mapping σb :X × X → [0, ∞) is called b-metric-like if the following conditions hold for some s ≥ 1 and for all x, y, zX:

  • σb (x, y) = 0 implies x = y,

  • σb(x, y) = σb(y, x)

  • σb(x, y) ≤ s[σb(x, z) + σb(z, y)].

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

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

Example 2.3

If X = ℝ, then σb(x, y) = |x| + |y| defines a metric-like on X.

Example 2.4

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

Example 2.5

If X = ℝ+ ∪ {0}, then σb (x, y) = (x + y)2 defines a b-metric-like on X with parameter s = 2.

Definition 2.6

([13]). Let (X, σb) be a b-metric-like space with parameter s, and let {xn} be any sequence in X and xX. Then

  1. The sequence {xn} is said to be convergent to x if limn σb(xn, x) = σb(x, x);

  2. The sequence {xn} is said to be a Cauchy sequence in (X, σb) if limn,m σb(xn, xm) exists and is finite;

  3. (X, σb) is said to be a complete b-metric-like space if, for every Cauchy sequence {xn} in X, there exists an xX such that limn,m σb(xn, xm) = limn σb(xn, x) = σb(x, x).

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

Lemma 2.7

([19]). Let (X, σb) be a b-metric-like space with parameter s, and f : XX be a mapping. Suppose that f is continuous at uX. Then for all sequences {xn} in X such that xnu, we have fxnfu that is

limnσb(fxn,fu)=σb(fu,fu).

Lemma 2.8

([15]). Let (X, σb) be a b-metric-like space with parameter s ≥ 1, and suppose that {xn} and {yn} are σb-convergent to x and y, respectively. Then we have

1s2σb(x,y)1sσb(x,x)σb(y,y)lim infnσb(xn,yn)lim supnσb(xn,yn)sσb(x,x)+s2σb(y,y)+s2σb(x,y).

In particular, if σb (x, y) = 0, then we have limn σb (xn, yn) = 0.

Moreover, for each zX, we have

1sσb(x,z)σb(x,x)lim infnσb(xn,z)lim supnσb(xn,z)sσb(x,z)+sσb(x,x).

In particular, if σb (x, x) = 0, then

1sσb(x,z)lim infnσb(xn,z)lim supnσb(xn,z)sσb(x,z).

The following result is useful.

Lemma 2.9

Let (X, σb) be a b-metric-like space with parameter s ≥ 1. Then

  1. If σb (x, y) = 0, then σb (x, x) = σb (y, y) = 0;

  2. If (xn) is a sequence such that limn σb (xn, xn+1) = 0, then we have

    limnσb(xn,xn)=limnσb(xn+1,xn+1)=0;

  3. If xy, then σb (x, y) > 0;

 □

Proof

The proof is obvious. □

Lemma 2.10

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1 and let {xn} be a sequence such that

limnσb(xn,xn+1)=0(1)

If for the sequence {xn}, limn,m σb (xn, xm) ≠ 0, then there exist ε > 0 and sequences {mk}k=1and{nk}k=1 of positive integers with nk > mk > k, such that σb (xmk, xnk) ≥ ε, σb (xmk, xnk−1) < ε, ε/s2lim supk σb (xmk−1, xnk−1) ≤ ε s, and ε/slim supk σb (xnk−1, xmk) ≤ ε, ε/slim supk σb (xmk−1, xnk) ≤ ε s2.

Proof

If limn,m σb (xn, xm) ≠ 0, then there exist an ε > 0 and sequences {mk}k=1and{nk}k=1 of positive integers with nk >mk > k, such that nk is the smallest index for which

nk>mk>k,σb(xmk,xnk)ε.(2)

This means that

σb(xmk,xnk1)<ε(3)

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

εσb(xmk,xnk)sσb(xmk,xmk1)+sσb(xmk1,xnk)sσb(xmk,xmk1)+s2σb(xmk1,xnk1)+s2σb(xnk1,xnk).(4)

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

εs2lim supkσb(xmk1,xnk1).(5)

By the triangular inequality, we have

σb(xmk1,xnk1)sσb(xmk1,xmk)+sσb(xmk,xnk1),

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

lim supkσb(xmk1,xnk1)εs.(6)

By (5) and (6) we have

εs2lim supkσb(xmk1,xnk1)εs.(7)

Also we have

εσb(xmk,xnk)sσb(xmk,xmk1)+sσb(xmk1,xnk),

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

εslim supkσb(xmk1,xnk).(8)

Again

εσb(xmk,xnk)sσb(xmk,xnk1)+sσb(xnk1,xnk).

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

εslim supkσb(xnk1,xmk).(9)

By (2) we have

limkσb(xnk1,xmk)ε.(10)

Consequently,

εslim supkσb(xnk1,xmk)ε.(11)

Also

σbxmk1,xnksσbxmk1,xnk1+sσbxnk1,xnk.

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

lim supkσb(xmk1,xnk)slim supkσb(xmk1,xnk1)εs2.

Consequently,

εslim supkσb(xmk1,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, σb) be a complete b-metric-like space with parameter s ≥ 1. If f : XX is a self-mapping that satisfies:

sσbfx,fyασbx,y

for some α ∈ [0, 1) and all x, yX, then f is called an (s, α)-Banach contraction.

Definition 3.2

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1. If f : XX is a self-mapping that satisfies:

spσbfx,fyασbx,y

for some constants p ≥ 1 and α ∈ [0, 1) and for all x, yX, 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:

  • 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, σb) be a b-metric-like space with parameter s ≥ 1. Let ψΨ, and let constants α, p be such that 0 ≤ α < 1 and p ≥ 2. A mapping f : XX is said to be a (ψ, s, p, α)-quasicontraction mapping, if for all x, yX

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

Remark 3.4

  1. It is obvious that by taking ψ(t) = 12 t (or the identity mapping ψ(t) = t) the above notion reduces to an (s, p, α)-quasicontraction.

  2. Taking ψ(t) = 12 t and the arbitrary constant p = 2 we obtain the definition of an (s, α)-quasi-contraction given in [30].

  3. If we take s = 1, it corresponds to the case of metric-like spaces.

Our first main result is as follows:

Theorem 3.5

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1, f : XX be a given self-mapping. If f is an (ψ, s, p, α)-quasicontraction, then f has a unique fixed point.

Proof

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

x1=f(x0),x2=f(x1),,xn+1=f(xn),for nN.

If we assume σb (xn0, xn0+1) = 0 for some n0 ∈ ℕ, then we have xn0+1 = xn0 that is xn0 = xn0+1 = f(xn0). Hence, xn0 is a fixed point of f and the proof is completed. From now on, we assume that for all n ∈ ℕ, σb (xn, xn+1) > 0 (that is xn+1xn).

By condition (13), we have

ψ2sσbxn,xn+1=ψ2spσbxn,xn+1=ψ2spσbfxn1,fxnαψmaxσbxn1,xn,σbxn1,fxn1,σbxn,fxn,σbxn1,fxn,σbxn,fxn1=αψmaxσbxn1,xn,σbxn1,xn,σbxn,xn+1,σbxn1,xn+1,σbxn,xnαψmaxσbxn1,xn,σbxn1,xn,σbxn,xn+1,sσbxn1,xn+σbxn,xn+1,2sσbxn1,xn.(14)

If σb (xn−1, xn) ≤ σb (xn, xn+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 σb (xn, xn+1) = 0, which is a contradiction, since we have supposed σb (xn, xn+1) > 0. Hence, for all n ∈ ℕ

σb(xn,xn+1)<σb(xn1,xn),

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

limnσbxn,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(xn1,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

limnσb(xn,xn+1)=0.(17)

In the next step, we claim that

limn,mσb(xn,xm)=0.

Suppose, on the contrary that limn,m σb (xn, xm) ≠ 0. Then by Lemma 2.10, there exist ε > 0 and sequences {m(k)} and {n(k)} of positive integers with nk >mk > k, such that σb (xmk,xnk) ≥ ε, σb (xmk, xnk−1) < ε and

εs2lim supkσb(xmk1,xnk1)εs,εslim supkσb(xnk1,xmk)ε,εslim supkσb(xmk1,xnk)εs2.(18)

From the contractive condition (13), we have

ψ2s2σb(xmk,xnk)ψ2spσb(xmk,xnk)=ψ2spσb(fxmk1,fxnk1)αψmax{σb(xmk1,xnk1),σb(xmk1,fxmk1),σb(xnk1,fxnk1),σb(xmk1,fxnk1),σb(xnk1,fxmk1)}=αψmax{σb(xmk1,xnk1),σb(xmk1,xmk),σb(xnk1,xnk),σb(xmk1,xnk),σb(xnk1,xmk)}.(19)

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

ψ2s2εαψmaxεs,0,0,es2,εαψ2εs2,

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

limnσb(xn,u)=σb(u,u)=limn,mσb(xn,xm)=0.(20)

By continuity of f and Lemma 2.7, we have fxnfu that is limn σb (xn, fu) = σb (fu, fu).

On the other hand limn σb (xn, u) = 0 = σb (u, u) and so by Lemma 2.8

1sσb(u,fu)limnσb(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,fuψsσbfu,fuψ2spσbfu,fuαψmaxσbu,u,σbu,fu,σbu,fu,σbu,fu,σbu,fu=αψσbu,fu.(22)

From (22) and the properties of ψ, we get σb (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,fuψ2spσbxn+1,fu=ψ2spσbfxn,fuαψmaxσb(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,fuψlim supn2s2σbxn+1,fuαψ2sσbu,fu.

From above inequality and the properties of ψ, we get σb (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,fvαψmaxσbu,v,σbu,fu,σbv,fv,σbu,fv,σbv,fu=αψmaxσbu,v,σbu,u,σbv,v,σbu,v,σbv,uαψ2sσbu,v.

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

The following example illustrates the theorem.

Example 3.6

Let X = [0, 1] and σb (x, y) = (x + y)2 for all x, yX. It is clear that σb is a b-metric-like on X with parameter s = 2 and (X, σb) is complete. Also, σb yis not a metric-like or a b-metric on X. Define a self-mapping f : XX by fx = x6.

For all x, y ∈ [0, 1], and the function ψ(t) = 2t, and constant p = 2, we have

ψ2s2σbfx,fy=ψ8x6+y62=ψ8x+y236=1636x+y2=8362x+y2=8362σbx,y=836ψσbx,yαψσbx,yαψmaxσbx,y,σbx,fx,σby,fy,σbx,fy,σby,fx.

All conditions of Theorem3.5 are satisfied and clearly x = 0 is a unique fixed point of f.

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, σb) be a complete b-metric-like space with parameter s ≥ 1. If f : XX is 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 constants α ∈ [0, 1 /2 .) and p ≥ 2 all x, yX, then f has a unique fixed point in X.

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

Corollary 3.8

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1. If f : XX is a self-mapping and there exist p ≥ 2 and constants ai ≥ 0, i = 1, …, 5 with a1 + 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 all x, yX, then f has a unique fixed point in X.

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

Theorem3.5 generalizes Theorem 1.2 in [32]. Theorem 3.2 in [28] is a special case of Corollary3.7 (and so also of Theorem 3.5) for choice constant p = 2. Also, Theorems 3.1 and 3.4 in [6] are special cases of our Theorem 3.5. In Corollary3.8, by choosing the constants ai 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 ψΨ, ϕΦ 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, σb) be a complete b-metric-like space with parameter s ≥ 1 and f : XX be a self-map. If there exist ψΨ, 0 ≤ α < 12 and p ≥ 2, such that the condition

ψ2spσbfx,fyαψMx,y1+ψMx,y(24)

is satisfied for all x, yX, where M(x, y) is defined as in (23), then f has a unique fixed point in X.

Proof

Taking into account that

αψMx,y1+ψMx,y=α11+ψMx,yψMx,yαψMx,y

for all x, yX 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, σb) be a complete b-metric-like space with parameter s ≥ 1 and f : XX be a self-map. If there exist ψΨ, ϕΦ, 0 ≤ α < 12 and p ≥ 2, such that the condition

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

is satisfied for all x, yX, where M(x, y) is defined as in (23), then f has a unique fixed point in X.

Proof

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

Corollary 3.12

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1 and f : XX a self-map. If there exist ψΨ, ϕΦ, 0 ≤ α < 12 and p ≥ 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 all x, yX, where M(x, y) is defined as in (23), then f has a unique fixed point in X.

Proof

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

Corollary 3.13

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1 and f : XX a self-map. If there exist ψΨ, ϕΦ, 0 ≤ α < 12 and p ≥ 2, such that the condition

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

is satisfied for all x, yX, where M(x, y) is defined as in (23), then f has a unique fixed point in X.

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, yX 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 (ψφ)-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, σb) be a b-metric-like space with parameter s ≥ 1. For a self-mapping f : XX 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, yX.

Definition 3.14

Let (X, σb) be ab-metric-like space with parameter s ≥ 1. A self-mapping f : XX is called a generalized (s, p)-weak contraction, if there exist ψΨ and a constant p ≥ 1, such that

spσb(fx,fy)N(x,y)ϕ(N(x,y))(29)

for all x, yX, where N(x, y) is defined as in (28).

Remark 3.15

The above definition reduces to the definition of (s, p)-weak contraction if N(x, y) = σb (x, y).

We now present the following result.

Theorem 3.16

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1. If f : XX is a self-mapping that is a generalized (s, p)-weak contraction, then f has a unique fixed point in X.

Proof

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

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

N(xn1,xn)=maxσb(xn1,xn),σb(xn1,fxn1),σb(xn,fxn),σbxn1,fxn+σbxn,fxn14s=maxσbxn1,xn,σbxn1,xn,σbxn,xn+1,σbxn1,xn+1+σbxn,xn4smaxσb(xn1,xn),σb(xn1,xn),σb(xn,xn+1),sσb(xn1,xn)+σb(xn,xn+1)+2sσb(xn1,xn)4s.(30)

If we assume that for some n ∈ ℕ

σbxn1,xnσbxn,xn+1,

then from the inequality (30), we get

Nxn1,xnσb(xn,xn+1).(31)

By the condition (29), we have

σbxn,xn+1spσb(xn,xn+1)=spσb(fxn1,fxn)Nxn1,xnϕN(xn1,xn)N(xn1,xn).(32)

From (31) and (32), we have

N(xn1,xn)=σb(xn,xn+1).(33)

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

σb(xn,xn+1)spσb(xn,xn+1)=spσb(fxn1,fxn)N(xn1,xn)ϕN(xn1,xn)=σb(xn,xn+1)ϕσb(xn,xn+1).(34)

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

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

limnσb(xn,xn+1)=limnN(xn1,xn)=l.

Since the function ϕ is lower semi continuous, we have

ϕ(l)limninfϕN(xn1,xn).

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

llϕ(l),

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

Hence

limnσb(xn,xn+1)=limnN(xn1,xn)=0.(35)

Next, we show that limn,m σb (xn, xm) = 0. Suppose the contrary, that is, limn,m σb (xn, xm) ≠ 0. Then by Lemma 2.10, there exist ε > 0 and sequences {mk} and {nk} of positive integers with nk >mk > k, such that

σb(xmk,xnk)ε,σb(xmk,xnk1)<ε

and

εs2lim supkσb(xmk1,xnk1)εs,εslim supkσb(xnk1,xmk)ε,εslim supkσb(xmk1,xnk)εs2(36)

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

N(xmk1,xnk1)=maxσb(xmk1,xnk1),σb(xmk1,fxmk1),σb(xnk1,fxnk1),σb(xmk1,fxnk1)+σb(xnk1,fxmk1)4s=maxσb(xmk1,xnk1),σb(xmk1,xmk),σb(xnk1,xnk),σb(xmk1,xnk)+σb(xnk1,xmk)4s.(37)

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

limksupN(xmk1,xnk1)=limksupmaxσb(xmk1,xnk1),σb(xmk1,xmk),σb(xnk1,xnk),σb(xmk1,xnk)+σb(xnk1,xmk)4smaxεs,0,0,εs2+ε4sεs.(38)

Also, as in Lemma 2.10, we can show that

limkinfσb(xmk1,xnk1)εs2,limkinfσb(xmk1,xnk)εs,limkinfσb(xnk1,xmk)εs,

and

limkinfMxmk1,xnk1ε2s2.(39)

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

sσb(xmk,xnk)spσb(fxmk1,fxnk1)N(xmk1,xnk1)ϕ(N(xmk1,xnk1)).(40)

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

εsεsϕε2s2,

that is a contradiction since ε > 0. So limn,m σb (xn, xm) = 0, and the sequence {xn} is a Cauchy sequence in the complete b-metric-like space (X, σb). Thus, there is some uX, such that

limnσb(xn,u)=σb(u,u)=limn,mσb(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

limnsupN(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

sp1σbu,fu=sp1sσbu,fuσb(u,fu)ϕσb(u,fu).(44)

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

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

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

spσbu,usσb(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 σb (u, u) = 0 (also σb (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) = σb (u, v). The inequality (46) implies σb (u, v) = 0. Therefore u = v and the fixed point is unique. □

The following example illustrates the theorem.

Example 3.17

Let X = [0, ∞) and σb (x, y) = x2 + y2 + |xy|2 for all x, yX. It is clear that σb is a b-metric-like on X, with parameter s = 2 and (X, σb) is complete. Also, σb is not a metric-like nor a b-metric (and nor a metric on X). Define the self-mapping f : XX by fx = ln1+x4. For all x, yX, and the function ϕ(t) = 34 t and constant p = 2, we have

s2σb(fx,fy)=4f2x+f2y+fxfy2=4lnx+142+lny+142+lnx+14lny+1424x216+y216+x4y42=14x2+y2+xy2=14σb(x,y)14N(x,y)=N(x,y)34N(x,y)=N(x,y)ϕ(N(x,y)).

All of the conditions of Theorem3.16 are satisfied and clearly x = 0 is a unique fixed point of f.

Corollary 3.18

Let (X, σb) be a complete b-metric-like space with parameter s ≥ 1 and f : XX be a self-mapping such that for some coefficient p ≥ 2 and for all x, yX it satisfies

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

where α ∈ (0, 1). Then f has 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 a b-metric-like space is a metric-like space when s = 1, so our results can be seen as a generalizations and extensions of several comparable results in metric-like spaces and b-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)=maxt0,1x(t)+y(t)mfor all x,yX,

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

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

Theorem 4.1

Consider equation (48) and suppose that

  1. L : [0, T] × [0, T] × ℝ → ℝ+, (that is L(t, r, x(r)) ≥ 0) is continuous;

  2. there exists a continuous γ : [0, T] × [0, T] → ℝ;

  3. supt0,T0T γ (t, r) dr ≤ 1;

  4. there exists a constant λ ∈ (0, 1) such that for all (t, r) ∈ [0, T]2 and x, yR,

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

Then the integral equation (48) has a unique solution xX.

Proof

For x, yX, from conditions (3) and (4), for all t, we have

s2σb(fx(t),fy(t))=s2fx(t)+fy(t)m=s20TL(t,r,x(r))dr+0TL(t,r,y(r))drms20TL(t,r,x(r))dr+0TLt,r,y(r)drms20Tλs31mγ(t,r)x(r)+y(r)m1mdrms20Tλs31mγ(t,r)σb1mx(r),y(r)drms2λs3σb(x(r),y(r))0Tγ(t,r)drm=λsσb(x(r),y(r))0Tγ(t,r)drmλsσb(x(r),y(r))λsN(x,y)=N(x,y)1λsN(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.

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About the article

Received: 2017-11-15

Accepted: 2018-01-31

Published Online: 2018-03-20


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.


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 235–249, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0024.

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