System Requirements Document for lime-shadow

1. Introduction

The project name is lime-shadow.

1.1 Source overview

New Fixed Point Results for F-Contractions in

ordered Dislocated b-Metric Spaces with

Applications

Djelloul Chennoufi1, Zahia Mostefaoui2, Said Beloul3*†

1Laboratory of Mathematical Analysis and Applications, Department of

Mathematics, Benyoucef Benkhedda University, Algiers 1, 02 Didouche

Mourad Street, Algiers, Algeria.

2Department of Mathematics, Higher Normal School of Kouba, Algeria.

3Department of Mathematics, El Oued University, Algeria.

*Corresponding author(s). E-mail(s): beloulsaid@gmail.com;

Contributing authors: chennoufi.dj@gmail.com;

zahia.moustefaoui@g.ens-kouba.dz;

†

Abstract

In this paper, we establish several existence and uniqueness results for com-

mon fixed points of two pairs of self-mappings in complete ordered dislocated

b-metric spaces by using the framework of F -contractions. Moreover, we apply

the obtained results to prove the existence of solutions for a boundary value

problem and provide illustrative example to demonstrate the effectiveness and

applicability of our theoretical findings.

MSC Classification: 47H10 , 54H25

1 Introduction

Fixed point theory has become one of the most active and influential areas of non-

linear analysis due to its wide range of applications in differential equations, integral

1

equations, optimization, approximation theory, game theory, and mathematical eco-

nomics. Over the years, many generalizations of metric spaces have been introduced in

order to accommodate situations where the classical metric axioms are too restrictive.

Among these generalizations, b-metric spaces which introduced by Bakhtin [5] and

developed by Czerwik [8] as a natural extension of a metric space, replaces the usual

triangle inequality by a relaxed form involving a constant s ≥1, later, some fixed

point results were obtained in such spaces, for single valued or set valued mappings,

for instance see [9, 10, 12, 19]. On other hand, the concept of a dislocated metric [3]

introduced to weaken the self-distance condition of a metric, allows the possibility

that the self-distance of a point may be nonzero. This feature has proved useful in

computer science, domain theory, and approximation processes, as well as in several

branches of nonlinear analysis. By combining the ideas of a dislocated metric and a

b-metric, one obtains the notion of a dislocated b-metric space [2], which extends

both structures simultaneously. Such spaces constitute an appropriate framework for

investigating fixed point problems under generalized contractive conditions, especially

when the geometry of the underlying space does not permit the use of standard

metric methods. Consequently, the study of fixed point theory in dislocated b-metric

spaces has become an important and growing topic. Another major development in

modern fixed point theory is the introduction of new contractive conditions that go

beyond the classical linear contraction. Among them, the notion of an F-contraction,

introduced by Wardowski [21], has generated substantial interest and has led to many

important extensions and applications and many authors investigate in this way

[4, 11, 13, 15, 18].

Motivated by these developments, the aim of the present paper is to investigate

fixed point results for F-contractions and related contractive mappings in the setting of

dislocated b-metric spaces. We establish new fixed point theorems for F-contractions

and related mappings in ordered dislocated b-metric spaces. We also discuss some

consequences and special cases that follow from our results. In the final section, we

provide an application to the existence of solution for a boundary value problem to

illustrating the effectiveness of the obtained theorems.

2 Preliminaries

In this section, we recall some basic definitions and auxiliary results that will be used

throughout the paper.

3 Main results

2. System Overview

lime-shadow is a research project whose aim is to investigate fixed point results for F-contractions and related contractive mappings in the setting of dislocated b-metric spaces. The work establishes new fixed point theorems for F-contractions and related mappings in ordered dislocated b-metric spaces, and it also discusses some applications of these results. The project is motivated by prior work on extensions and applications of this kind, cited in the source as [4, 11, 13, 15, 18].

2.1 Actors

The evidence identifies no human roles, so no personas are derived here; this batch does not establish a persona catalog. The only named actor is the project itself, lime-shadow, which carries the stated research aim. Supporting actors are the mathematical objects the project works with: F-contractions, related contractive mappings, dislocated b-metric spaces, and ordered dislocated b-metric spaces. The authors cited as [4, 11, 13, 15, 18] are referenced as background, not as system actors.

2.2 Capabilities and Outcomes

The project's capabilities are its stated mathematical contributions:

1. New fixed point theorems for F-contractions in dislocated b-metric spaces.
2. New fixed point theorems for related mappings in the same setting.
3. Fixed point results in ordered dislocated b-metric spaces for F-contractions and related mappings.
4. Discussion of applications arising from these results.

The outcome of each capability is a fixed point theorem or a related result within the named space setting; the applications discussion is an accompanying outcome, not a separate product feature.

2.3 Boundaries

The evidence defines the scope narrowly: the project concerns F-contractions and related contractive mappings in dislocated b-metric spaces and their ordered variant, together with a discussion of applications. No user-facing pages, components, workflows, or interface behaviors are supported by this evidence, so none are described. No global page inventory is claimed from this partial material.

2.4 Defaults

No defaults are required for this chapter; every statement above is drawn from the supplied source facts. The project name lime-shadow is taken from the user's explicit choice.

3. Functional Requirements

No additional details specified.

4. User Personas

No additional details specified.

5. Core User Flows

No additional details specified.

6. Visuals Colors and Theme

[Design proposal; unspecified choices are defaults.]

Muse and headline. The muse is the fixed point itself: the single value a mapping returns unchanged. The headline is “Every contraction lands somewhere.” The product is lime-shadow, a reading and reference surface for the paper New Fixed Point Results for F-Contractions in ordered Dislocated b-Metric Spaces with Applications (Chennoufi, Mostefaoui, Beloul; MSC 47H10, 54H25).

Mode. Light-first, paper-toned, with a dark “shadow” mode for long reading sessions. No indigo/blue-on-white.

Colour tokens by role.

• --bg-paper #F4F6EC (page ground, faint green cast)
• --bg-shadow #14180F (dark mode ground)
• --ink #1B2114 (body text on paper)
• --ink-muted #5A6350 (captions, metadata, MSC codes)
• --lime #A8D42B (primary accent: links, active states, the fixed point)
• --lime-deep #6E8F14 (hover/pressed, accessible on paper)
• --shadow #2E3A22 (structural rules, theorem frames, the “shadow” of each lime mark)
• --surface #FFFFFF (cards, theorem blocks)
• --border #D6DCC4

Type. Headings: Fraunces (serif, optical-size axis) for theorem titles and the hero headline. Body: Source Serif 4 for prose; JetBrains Mono for formulas, MSC codes, and author emails. Scale: 12 / 14 / 16 / 20 / 26 / 34 / 48 / 64 px, 1.25 ratio; body line-height 1.65, headings 1.15.

Shape and spacing. Radius: 2 px on cards, 999 px on pills (MSC tags, author chips). Spacing rhythm on a 4 px base: 4 / 8 / 12 / 16 / 24 / 32 / 48 / 64. Hairline 1 px --border rules; theorem blocks get a 3 px left bar in --lime with a 1 px --shadow offset shadow.

Imagery. No stock photography. Visual language is mathematical: contour lines of a contraction converging to a fixed point, rendered as thin --shadow strokes with one --lime dot at the limit. Diagrams are line-art, monochrome plus lime.

7. Signature Design Concept

[Design proposal; unspecified choices are defaults.]

Signature concept: “The Shadow Lands.” Every interactive element casts a short, offset lime-tinted shadow that collapses to zero when the element reaches its resting state — a literal reading of a contraction mapping onto its fixed point. The public entry is the paper's front matter recomposed as a single landing surface: title, the three authors with their affiliations (Algiers 1, ENS Kouba, El Oued), the abstract, MSC 47H10 / 54H25 as pills, and one primary control, Read the paper, which scrolls to the Introduction.

Hero composition. Left column: the headline in Fraunces at 64 px, the author line in Source Serif 4 with superscript affiliation markers, and the MSC pills. Right column: a line-art contour diagram — nested curves in --shadow converging inward, with a single --lime dot at the centre labelled fixed point. A thin --border rule separates hero from the abstract block, which sits on --surface with the 3 px lime left bar.

Section rhythm. Introduction → Preliminaries → Main results → Application (boundary value problem) → illustrative example, each as a numbered anchor with a mono section numeral in --ink-muted. Theorem and definition blocks reuse the lime-bar card; the boundary value problem application gets a distinct --shadow-framed panel to mark the shift from theory to application.

Defaults (labelled). Dark mode, the 4 px spacing base, and the 1.25 type ratio are coherent defaults chosen here, not source requirements.

8. Interaction Model & Motion Direction

[Design proposal; unspecified choices are defaults.]

Interaction Model: Animated Motion Tempo: restrained Hero Dimensionality: layered_2d

Landing Hero Motion Brief. Focal subject: the contour diagram converging on the lime fixed point. Thesis: on load, the nested contour curves draw inward from their outer radius toward the centre (input: page load; transformation: stroke-dashoffset reveal plus a 6% scale settle; outcome: the lime dot resolves last and holds) — using only the accepted hero content, no new behaviour. Motion vocabulary: short eased draws (cubic-bezier 0.2, 0.7, 0.2, 1), 600–900 ms, staggered 80 ms per curve; the lime dot fades in over 200 ms after the final curve. Composed first frame: curves at 40% opacity, dot absent, headline and author line already legible — nothing blocks reading. Reduced-motion state: all curves and the dot render at final position immediately; no stagger, no scale.

Supporting motion. The shadow-collapse hover on cards and pills (shadow offset 4 px → 0 over 160 ms) and a 120 ms underline draw on links. Section anchors use native smooth scroll, disabled under prefers-reduced-motion. No parallax, no scroll-linked depth, no WebGL — the tempo is restrained because the audience is reading mathematical prose and motion must never compete with formulas.

9. Non-Functional Requirements

No additional details specified.

10. Tech Stack

The evidence for lime-shadow is a mathematical research paper, not a software specification. It establishes no application, service, or delivery platform and no technology choices at any layer; no stack is required or suggested by the source, and none is inferred here.

Domain and subject matter. The work concerns fixed point theory in ordered dislocated b-metric spaces using F-contractions (Wardowski). It establishes existence and uniqueness results for common fixed points of two pairs of self-mappings in complete ordered dislocated b-metric spaces, derives consequences and special cases, and applies the results to a boundary value problem, supported by an illustrative example. MSC classification: 47H10, 54H25.

Mathematical foundations referenced. b-metric spaces (Bakhtin; developed by Czerwik), replacing the triangle inequality with a relaxed form involving a constant s ≥ 1; dislocated metrics, which weaken the self-distance condition so a point's self-distance may be nonzero; dislocated b-metric spaces, combining both; and F-contractions.

Defaults. Because the evidence specifies no software stack, no default technology is chosen; selecting one would add product behavior the source does not support. If a software deliverable is later intended, its layers remain unspecified and must be decided explicitly rather than assumed.

Future work. Any tooling for formal verification, symbolic computation, or numerical illustration of the boundary value problem is not established by this evidence and is deferred.

11. Assumptions and Constraints

11.1 Source overview

Theorem 1 Let (X, ≤, σb) be a partially ordered complete dislocated b-metric space, and let

S, T, A, B →X be self-mappings. Assume that the following conditions are satisfied:

• [H1] : SX ⊆BX, TX ⊆AX.

• [H2] : S, T are dominating, A, B are dominated.

• [H3] : (S, T) is an ordered F-contraction with respected to (A, B).

• [H4] : {xn} is nondecreasing , xn ≤yn for all n ∈N and yn →z, then xn ≤z.

• [H5] :(i) S, or A is continuous, (S, A) is compatible and (T, B) is weakly compatible.

(ii) T, or B is continuous, (T, B) is compatible and (S, A) is weakly compatible.

Then S, T, A and B has a common fixed point.

Proof Let x0 ∈X. Since SX ⊂BX there exists x1 ∈X such that y1 = Bx1 = Sx0. Let

y2 = Tx1, since TX ⊂AX, there exists x2 ∈X such that y2 = Ax2 = Tx1. Continuing this

process, we can construct sequence {yn} in X as follows :

y2n+1 = Sx2n = Bx2n+1,

y2n+2 = Tx2n+1 = Ax2n+2,

n ≥0.

(2)

4

If there exists n0 ∈N such that σb(y2n0, y2n0+1) = 0. Then y2n0 = y2n0+1, using (1) with

x = x2n0 and y = x2n0+1, we get

τ + F(sεσb(Sx2n0, Tx2n0+1)) ≤F(M(x2n0, x2n0+1)),

which implies that

τ + F(sεσb(y2n0+1, y2n0+2)) = τ + F(sεσb(Sx2n0, Tx2n0+1)) ≤F(M(x2n0, x2n0+1)),

so we have

M(x2n, x2n+1) = max{σb(Ax2n0, Bx2n0+1), σb(Ax2n0, Sx2n0), σb(Bx2n0+1, Tx2n0+1),

σb(Ax2n0, Tx2n0+1) + σb(Bx2n0+1, Sx2n0)

4s

}

= max{σb(y2n0, y2n0+1), σb(y2n0, y2n0+1), σb(y2n0+1, y2n0+2),

σb(y2n0, y2n0+2) + σb(y2n0+1, y2n0+1)

4s

}

= max{0, 0, σb(y2n0+1, y2n0+2), σb(y2n0, y2n0+2)

4s

}

≤max{σb(y2n0+1, y2n0+2), s[σb(y2n0, y2n0+1) + σb(y2n0+1, y2n0+2)]

4s

}

= σb(y2n0+1, y2n0+2),

since

σb(y2n0, y2n0+2) + σb(y2n0+1, y2n0+1)

4s

≤3sσb(y2n0, y2n0+1) + sσb(y2n0+1, y2n0+2)

4s

≤σb(y2n0+1, y2n0+2)

4s

.

So we get

τ + F(σb(y2n0+1, y2n0+2) ≤τ + F(sεσb(y2n0+1, y2n0+2) ≤F(σb(y2n0+1, y2n0+2),

which implies that y2n0+1 = y2n0+2 and this yields that y2n0 = y2n0+1 = y2n0+2, so for all

n ∈N the sequence (yn) is constant then it is a Cauchy one.

Now, suppose that y2n ̸= y2n+1 for all n ∈N.By(1), we have

τ + F(sεσb(y2n+1, y2n+2)) ≤F(M(x2n, x2n+1)),

(3)

Therefore, since F is strictly increasing, we get

σb(y2n+1, y2n+2) ≤M(x2n, x2n+1),

where

M(x2n, x2n+1) = max



σb(Ax2n, Bx2n+1), σb(Ax2n, Sx2n), σb(Bx2n+1, Tx2n+1),

σb(Ax2n, Tx2n+1) + σb(Bx2n+1, Sx2n)

4s

= max



σb(y2n, y2n+1), σb(y2n, y2n+1), σb(y2n+1, y2n+2),

σb(y2n, y2n+2) + σb(y2n+1, y2n+1)

4s

= max {σb(y2n, y2n+1), σb(y2n+1, y2n+2)} ,

since

σb(y2n, y2n+2) + σb(y2n+1, y2n+1)

4s

≤3sσb(y2n, y2n+1) + sσb(y2n+1, y2n+2)

4s

.

5

Therefore

M(x2n, x2n+1) = max



σb(y2n, y2n+1), σb(y2n+1, y2n+2)

.

If σb(y2n, y2n+1) ≤σb(y2n+1, y2n+2), then (3) gives

τ + F

�sε σb(y2n+1, y2n+2)

≤F

�σb(y2n+1, y2n+2)

,

which is contradiction since τ > 0, then we have

τ + F

�sε σb(y2n+1, y2n+2)

≤F

�σb(y2n, y2n+1)

∀n ≥0.

(4)

Applying the same argument to the comparable pair (y2n+1, y2n+2),

τ + F

�sε σb(y2n+2, y2n+3)

≤F

�σb(y2n+1, y2n+2)

,

∀n ≥0.

(5)

By (4) and (5) , we have

F

�sε σb(yn+1, yn+2)

≤F

�σb(yn, yn+1)

−τ.

∀n ≥0.

(6)

Continuing in this manner and using (F4), for all n ∈N we get

F(snεσb(yn+1, yn+2)) ≤F(σb(y0, y1)) −nτ.

Passing to the limit in the last inequality we get

lim

n→∞F(snεσb(y2n, y2n+1)) = −∞,

from (F2) we get

lim

n→∞snεσb(yn, yn+1) = 0.

By the definition of the limit there exits n0 ∈N such that for all n ≥n0 we have

σb(yn, yn+1 < s−nε,

For n, m ∈N with m > n ≥1, we have

σb(xn, xm) ≤

m−1

X

i=n

si−n+1σb(yi, yi+1) ≤

m−1

X

i=n

siσb(yi, yi+1)

≤

m−1

X

i=n

si(1−ε) ≤

∞

X

i=1

si(1−ε) < ∞.

Then σb(yn, ym) →0 as n, m →∞, which implies that (yn) is a Cauchy sequence.

Then {xn} is a Cauchy sequence in X. Since (X, σb) is complete, there exists z ∈X such that

yn →z

as n →∞.

Since {xn} is nondecreasing and by hypothesis (H4), we have yn ≤z,

∀n ∈N.

Also we have

lim

n→∞Sx2n = lim

n→∞Bx2n+1 = lim

n→∞Tx2n+1 = lim

n→∞Ax2n+2 = z.

(7)

We show that z is a common fixed point of S, T, A, and B. Since A is continuous, therefore,

lim

n→∞ASx2n = Az = lim

n→∞A2x2n+2.

(8)

Since the pair {S, A} is compatible, so,

lim

n→∞σb(SAx2n, ASx2n) = 0,

using Lemma 1, we get

1

s2 σb(Az, lim SAx2n) ≤lim

n→∞σb(SAx2n, ASx2n) = 0 ≤s2σb(Az, lim SAx2n),

6

which implies that

lim

n→∞SAx2n = Az.

(9)

Now put x = Ax2n and y = x2n+1 in (1) and suppose on the contrary that σb(Az, z) > 0;

we obtain

τ + F(sεσb(SAx2n, Tx2n+1)) ≤F(M(Ax2n, x2n+1))

(10)

where

M(Ax2n, x2n+1) = max



σb(A2x2n, Bx2n+1), σb(A2x2n, SAx2n), σb(Bx2n+1, Tx2n+1),

σb(A2x2n, Tx2n+1) + σb(Bx2n+1, SAx2n)

4s

.

Taking n →∞and using (7) and Lemma 1, we get

lim

n→∞M(Ax2n, x2n+1) = max



σb(Az, z), σb(Az, Az), σb(z, z), σb(Az, z) + σb(z, Az)

4s

= σb(Az, z),

Because

0 ≤σb(Az, Az) ≤s



σb(Az, Ax2n+1) + σb(Ax2n+1, Az)



= 2s σb(Ax2n+1, Az) n→∞

−−−−→0.

passing to the limit in (10) and using the continuity of F, we get

τ + F

�sεσb(Az, z)

≤F

�σb(Az, z)

.

(11)

This is a contradiction. Therefore, we get σb(Az, z) = 0, that is, Az = z. Again now put x = z

and y = x2n+1 in (1)and (2) and suppose on the contrary that (σb(Sz, z) > 0 , we obtain

τ + F(sε(σb(Sz, Tx2n+1)) ≤F(M(z, x2n+1))

(12)

where

M(z, x2n+1) = max



σb(Az, Bx2n+1), σb(Az, Sz), σb(Bx2n+1, Tx2n+1),

σb(Az, Tx2n+1) + σb(Bx2n+1, Sz)

4s

.

Letting n →∞on both sides in the above inequality, using Az = z, and (7) we obtain

M(Sz, z) = max{(σb(z, z), (σb(z, Sz), (σb(z, z),

σb(z, z) + σb(z, Sz)

4s

} = (σb(Sz, z),

passing to the limit in (10) and using the continuity of F and Lemma1, we get

τ + F(sε(σb(Sz, z)) ≤F((σb(Sz, z)).

(13)

This is a contradiction. Therefore, we get σb(Sz, z) = 0, that is, Sz = z.

On the other hand, since z = Sz ∈SX ⊂BX, there exists u ∈X such that z = Sz = Bu

. By It will need to prove Bu = Tu. Via z = Sz = Bu ≤u, it implies z ≤u. By utilizing (1),

we obtain

τ + F(sε(σb(z, Tu)) = τ + F(sε(σb(Sz, Tu)) ≤F(M(z, u))

where

M(z, u) = max{(σb(Az, Bu), (σb(Az, Sz), (σb(Bu, Tu), (σb(Az, Tu) + (σb(Bu, Sz)

4s

}.

= max{(σb(z, Bu), (σb(z, z), (σb(Bu, Tu), (σb(z, Tu) + (σb(Bu, z)

4s

} = σb(z, Tu),

Letting n →∞,we have

τ + F(sε(σb(z, Tu)) ≤(σb(z, Tu))

7

a contradiction, hence, (σb(z, Tu) = 0 implies Tu = z, and so Tu = z = Bu. Since the pair

(T, B) is weakly compatible, we have Tz = TBu = BTu = Bz. Now we prove that Tz = z.

From (1) , we have

τ + F(sε(σb(z, Tz)) = τ + F(sε(σb(Sz, Tz)) ≤F(M(z, z))

where

M(z, z) = max{(σb(Az, Bz), (σb(Az, Sz), (σb(Bz, Tz),

(σb(Az, Tz) + (σb(Bz, Sz)

4s

}

= max{(σb(z, z), (σb(z, z), (σb(z, Tz),

(σb(z, Tz) + (σb(z, z)

4s

} = (σb(z, Tz).

This implies that (σb(z, Tz) = 0, so Tz = z. Therefore, we obtain Sz = Az = Tz = Bz = z,

so z is a common fixed point of S, T, A, and B.

We take a similar argument when S is continuous.

Similarly, condition (ii) holds ,we can also fin the common fixed point of of S, T, A and B.

Next, we prove that S, T, A, and B have a unique common fixed point. Suppose that ω ∈X

is also a common fixed point of S, T, A and B, then then from (1), we have

τ + F((σb(z, ω)) = τ + F((σb(Sz, Tω)) ≤F(M(z, ω)),

where

4



,

Tx = ln



1 + x

5



.

(14)

Clearly, S, T, A, and B are continuous self-mappings complying with SX = TX = AX =

BX = [0, ∞). We note that the pair {A, S} is compatible. Indeed, let {xn} be a sequence in

X satisfying limn→∞Axn = limn→∞Sxn = t for some t ∈X. Then

lim

n→∞|Sxn −t| = lim

n→∞|Axn −t| = 0,

(15)

and equivalently

lim

n→∞

ln



1 + xn

4



−t

= lim

n→∞|e5xn −1 −t| = 0,

(16)

implies

lim

n→∞||xn −(4et −4)| = lim

n→∞

xn −ln(t + 1)

5

= 0.

(17)

The uniqueness of limit gives 4et −4 = ln(t + 1)

5

; thus, t = 0 is the only possible solution.

Due to the continuity of A, S we have

lim

n→∞σb(ASxn, SAxn) = lim

n→∞|ASxn + SAxn|2 = |A(0) + S(0)|2 = |0 + 0|2 = 0.

(18)

Similarly, the pair {B, T} is compatible. Now for each x, y ∈X, consider

σb(Sx, Ty) =

ln



1 + x

4



• ln



1 + y

5



2

≤

x

4 + y

5

2

=

1

20(5x + 4y)

2

≤

1

202

(e5x −1) + (e4y −1)

2

=

1

202 σb(Ax, By)

≤

1

202 M(x, y).

Hence

σb(Sx, Ty) ≤

1

202 M(x, y),

Hence

202 σb(Sx, Ty) ≤M(x, y),

(ε = 2, s = 2)

Then

102(4 σb(Sx, Ty)) ≤M(x, y),

2 ln 10 + ln

�4σb(Sx, Ty)

≤ln

�M(x, y)

.

(19)

Define the function F : R+ →R by F(t) = ln(t), for all t ∈R+. Hence, for all x, y ∈X, such

that σb(Sx, Ty) > 0, with τ = 2 ln 10 we obtain

τ + F

�sεσb(Sx, Ty)

≤F

�M(x, y)

.

(20)

Thus, the contractive condition is satisfied for all x, y ∈X. Hence, all the hypotheses of the

theorem are satisfied. Note that S, T, A, and B have a unique common fixed point x = 0.

9

4 Application

In this section, we will give some applications in the existence and uniqueness of

solutions to several equations.

First of all, we consider the following equation:

d2x

dt2 = f(t, x(t)),

t ∈[a, b],

x(a) = x(b) = 0,

(21)

Theorem 3 For Equation (21), if the following condition holds:

|f(t, x(t)) + f(t, y(t))| ≤λ|x(t) + y(t)|,

(22)

where λ > 0 is a constant, f(t, 0) = 0, and λ(b −a)2 ≤21−ε

2 with ε > 1, then Equation (21)

has a unique solution in [a, b] provided that 1

8λ(b −a)2 ≤1.

Proof Equation (21) is equivalent to the equation below:

x(t) =

Z b

a

Ω(t, r)f(r, x(r))ds,

(23)

where

Ω(t, r) =

( r−a

a−b (b −t), a ≤r ≤t,

t−a

a−b(b −r), t ≤r ≤b.

(24)

We use reverse thinking to prove (23). That is to say, we utilize (23) to show (21). Indeed,

by using (23) and (24), we have

x(t) =

Z b

a

Ω(t, r)f(r, x(r)) dr

=

Z r

a

Ω(t, r)f(r, x(r)) dr +

Z b

t

Ω(t, r)f(r, x(r)) dr

=

Z r

In order to find the solution of (23), we need to look for the fixed point of the following

mapping:

Sx(t) =

Z b

a

Ω(t, r)f(r, x(r)) dr.

∀t ∈[a, b].

(26)

Since f is continuous, then S is a self-mapping on X. Via (22) it is not hard to verify that

|Sx(t) + Ty(t)|2 =

Z b

a

Ω(t, s)f(s, x(s)) ds +

Z b

a

Ω(t, s)f(s, y(s)) ds

2

≤

Z b

a

|Ω(t, r)||f(r, x(r)) + f(r, y(r))| dr

!2

≤

Z b

a

|Ω(t, s)λ|x(r) + y(r)| dr

!2

≤λ2

Z b

a

|Ω(t, r)| dr

!2

max

r∈[a,b] |x(r) + y(r)|2

= λ2

1

2(t −a)(b −t)

2

σb(Ax, By)

≤1

64λ2(b −a)4σb(Ax, By)

≤1

26



22−ε

σb(Ax, By)

≤

1

24+ε σb(Ax, By)

≤

1

24+ε M(x, y)

Then

42 �2εσb(Sx, Ty)

≤M(x, y)

2 ln 4 + ln

�2εσb(Sx, Ty)

≤ln

�M(x, y)

.

(27)

Define the function F : R+ →R by F(t) = ln(t), for all t ∈R+. Hence, for all x, y ∈X, such

that σb(Sx, Ty) > 0, with τ = 2 ln 4 we obtain

τ + F

�sεσb(Sx, Ty)

≤F

�M(x, y)

.

(28)

11

Where S = T. Since f(t, 0) = 0, then by ((22)), it is easy to see that

|Sx| =

Z b

a

Ω(t, r)f(r, x(r)) dr

≤

Z b

a

|Ω(t, r)||f(r, x(r)) + 0| dr

=

Z b

a

|Ω(t, r)||λ(r, x(r)) + f(r, 0)| dr

≤

Z b

a

|Ω(t, r)|k|x(r) + 0| dr

≤λ

Z b

a

|Ω(t, r)| max

r∈[a,b] |x(r)| dr

= λ

Z b

a

|Ω(t, r)| dr max

r∈[a,b] |x(r)|

= 1

2λ(t −a)(b −t) max

r∈[a,b] |x(r)|

≤1

8λ(b −a)2 max

r∈[a,b] |x(r)|

≤max

r∈[a,b] |x(r)|,

so x ≤Sx, that is, S, T are dominating. Let A = B = I be an identity mapping on X.

Accordingly, A, B are dominated. Consequently, all the conditions of Theorem are satisfied.

Hence, by Theorem 1, f has a unique fixed point in X. Thus, (23) has a unique solution in

[a, b]. In other words, (21) has a unique solution in [a, b].

□

Conclusion

In this paper, we discussed the existence and uniqueness of common fixed point for

two pairs of self mappings under Wordowski contractive condition and suitable oth-

ers in dislocated ordered b-metric spaces. The obtained results are formulated in a

general framework and are designed to cover different types of contractive conditions

associated with F-contractions. In particular, our results improve and generalize a

number of earlier fixed point theorems proved in metric, b-metric, dislocated metric,

and dislocated b-metric spaces.

Acknowledgements.

The authors thank the referees for their careful reading of

the manuscript and insightful comments.

Author contributions.

All authors contributed equally in the writing of this paper.

All authors read and approved the final version of this manuscript.

Declarations

Conflict of interest.

All the authors declare that there is no conflict of interests.

Funding.

All the authors declare that there is no funding.

12. Glossary

12.1 Source overview

Keywords: common fixed point, F-contraction, weakly compatible, ordered

dislocated b-metric spaces

Definition 1 [8] Let X be a nonempty set and let s ≥1 be a real number. A function

bd : X × X →[0, ∞) is called a b-metric on X if, for all x, y, z ∈X, the following conditions

are satisfied:

1. bd(x, y) = 0 if and only if x = y;
2. bd(x, y) = bd(y, x);
3. bd(x, z) ≤s



bd(x, y) + bd(y, z)



.

Then the triple (X, bd, s) is called a b-metric space.

2

It is well known that every metric space is a b-metric space, but the converse is

not necessarily true.

Lemma 1 [1] Let (X, bd) be a b-metric space with s ≥1, and suppose that {xn} and {yn}

are b-convergent to x, y respectively. Then the following inequality holds.

1

s2 bd(x, y) ≤lim

n→∞inf bd(xn, yn) ≤lim

n→∞sup bd(xn, yn) ≤s2bd(x, y).

If x = y, then we have limnto∞bd(xn, yn) = 0 and for each z ∈X we have

1

sbd(x, z) ≤lim

n→∞inf bd(xn, z) ≤lim

n→∞sup bd(xn, z) ≤sd(x, z).

Definition 2 [2, 10] Let X be a nonempty set and let (s ≥1 be a constant. A function

σb : X × X →R+ is called a dislocated b-metric on X if, for all x, y, z ∈X, the following

conditions hold:

1. σb(x, y) = 0 implies x = y;
2. σb(x, y) = σb(y, x);
3. σb(x, z) ≤s



σb(x, y) + σb(y, z)



.

In this case, the triple (X, σb, s) is called a dislocated b-metric space with coefficient s.

In a dislocated b-metric space (X, σb, s)), if σb(x, y) = 0, then x = y. However, the

converse need not be true, since it may happen that σb(x, x) > 0 for some x ∈X.

Example 1 [2] Let X = R+ and let p ≥1 be a constant. Define σb : X × X →R+ by either

σb(x, y) = (x + y)p, or [ σb(x, y) =

�max x, y

p. Then X, σb, s) is a b-metric-like space with

coefficient s = 2p−1.

Clearly, (X, σb, s) is neither a b-metric space nor a metric-like space.

Definition 3 [1, 17] Let (X, ≤, σb) be a partially ordered dislocated b-metric space.

1. Two elements x, y ∈Xare said to be comparable if x ≤y or y ≤x.
2. A self-mapping T is said to be dominating if x ≤Tx, for all x ∈X.
3. A self-mapping T is said to be dominated if Tx ≤x, for all x ∈X.

Definition 4 [10] Let (X, ⪯, σb) be a partially ordered dislocated b-metric space. Two self-

mappings S, T are said to be compatible if

lim

n→∞σb

�TSyn, STyn

= 0, whenever {yn} is a

sequence in X) such that lim

n→∞Tyn = lim

n→∞Syn = t for some t ∈X.

Definition 5 Let (X, ⪯, σb) be a partially ordered dislocated b-metric space. Two self-

mappings S, T are said to be weakly compatible if they commute at their coincidence points,

that is, if for any z ∈X satisfying Tz = Sz, one has STz = TSz.

3

Definition 6 [21] Let F : (0, +∞) →R be a function satisfying the following conditions:

(F1) F is strictly increasing;

(F2) for every sequence {un} ⊂(0, +∞), limn→∞un = 0

⇐⇒

limn→∞F(un) =

−∞;

(F3) There exists k ∈(0, 1) such that limα→0+ αkF(α) = 0.

In [16], the authors have replaced the property (F3) by the following property:

(F ′

3. : F is continuous on [0, ∞).

Cosentino et al. [7] introduced the following condition:

(F4) : if (un) is a sequence such that τ + F(sun) ≤F(un−1), for all n ∈N, s ≥1 and

for some τ > 0, then τ + F(snun) ≤F(sn−1un−1), for all n ∈N.

We denote by F the class of all functions F : (0, +∞) →R) satisfying conditions

(F1)), (F2), (F ′

3. and (F4) :.

Example 2 Let Fi : (0, +∞) →R), (i ∈{1, 2, 3}), be defined by

F1(t) = ln t, F2(t) = t + ln t, F3(t) = −1

√

t.

Then Fi ∈F for each (i ∈1, 2, 3).

Definition 7 Let (X, ⪯, σb) be a partially ordered complete dislocated b-metric space, a pair

(S, T) of self mappings is said to be an ordered F-contractive pair with respected to (A, B)

if and only if for any comparable elements x, y ∈X, there exists τ > 0 and F ∈F such that

τ + F(sεσ(Sx, Ty)) ≤F(M(x, y)),

(1)

where s, ε > 1 and

M(x, y) = max{σb(Ax, By), σb(Ax, Sx), σb(By, Ty), 1

4s(σb(Ax, Ty) + σb(By, Sx))}.

M(z, ω) = max{(σb(Az, Bω), (σb(Az, Sz), (σb(Bω, Tω),

(σb(Az, Tω) + (σb(Bω, Sz)

4s

}

= max{(σb(z, ω), (σb(z, z), (σb(ω, ω),

(σb(z, ω) + (σb(ω, z)

4s

}.

Thus, we have

σb(z, ω) < (σb(z, ω),

which is a contradiction. Hence, z = ω and z is a unique common fixed point of the four

maps S, T, A, and B.

□

If S = T and A = B, we get the following corollary.

Corollary 2 Let (X, ≤, σb) be a partially ordered complete dislocated b-metric space, and

let S and A be self-mappings. Suppose that S is an ordered F-contraction with respect to A.

Assume, moreover, that the following conditions are satisfied:

• [H1] : SX ⊆AX.

• [H2] : S is dominating and A is dominated.

• [H3] : {xn} is nondecreasing, xn ≤yn for all n ∈N and yn →z, then xn ≤z.

• [H4] : S, or A is continuous, (S, A) is compatible.

Then S, T, A and B has a common fixed point.

Example 3 Let X = [0, ∞) with the partial order as

x ⪯y ⇐⇒x ≥y.

8

and define σb : X × X →R+ by σb(x, y) = |x + y|2, so (X, σb, 2) is a complete b-metric like

space. Define the mappings S, T, A, B : X →X, for all x ∈X, by

Ax = e5x −1,

Bx = e4x −1,

Sx = ln



1 + x

a

r −a

a −b (b −t)f(r, x(r)) dr +

Z b

t

t −a

a −b(b −r)f(r, x(r)) dr

= b −t

a −b

Z r

a

(r −a)f(r, x(r)) dr + t −a

a −b

Z b

t

(b −r)f(r, x(r)) dr,

(25)

then

dx

dt = −

1

a −b

Z r

a

(r −a)f(r, x(r)) dr + b −t

a −b(t −a)f(t, x(t))

1

a −b

Z b

t

(b −r)f(r, x(r)) dr −t −a

a −b(b −t)f(t, x(t))

= −

1

a −b

Z t

a

(r −a)f(r, x(r)) dr +

1

a −b

Z b

t

(b −r)f(r, x(r)) dr,

which infers that

d2x

dt2 = −

1

a −b(t −a)f(t, x(t)) −

1

a −b(b −t)f(t, x(t)) = f(t, x(t)).

By (3.5), x(a) = x(b) = 0. As a consequence, (3.1) holds.

10

Let X = C[a, b] with partial order as :

x ⪯y ⇐⇒

max

t∈[a,b] |x(t)| ≥max

t∈[a,b] |y(t)|.

Define a mapping σb : X × X →[0, +∞) by

σb(x, y) = max

t∈[a,b] |x(t) + y(t)|2.

It is evident that (X, σb) is a complete dislocated b-metric space with s = 2.

We introduce the self-mappings S, T : X →X defined by:

Sx(t) = Tx(t) =

Z b

a

x(t) =

Z b

a

Ω(t, r)f(r, x(r)) dr

and let A, B : X →X be the identity mappings Ax = Bx = x.

12

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14