INNOVATIONS IN FIXED POINT THEORY ON STRONG CONTROLLED-BRANCIARI- b-DISTANCE WITH AN APPLICATION TO NONLINEAR BOUNDARY PROBLEM (http://doi.org/10.63386/620587)

DANIA SANTINA, WAN AINUN MIOR OTHMAN, KOK BIN WONG, NABIL MLAIKI, AND

SUHAD SUBHI AIADY

Abstract. This paper presents the strong-controlled Branciari b-distance, a novel distance proposed to integrate and unify various fixed point results previously documented in the relevant literature. We provide some examples to highlight our findings. In addition, we offer the application within the framework of strong-controlled Branciari b-distance to demonstrate the practicality of the acquired outcomes.

Keywords: Metric space (M_sp), Branciari distance (B_d), Strong-Controlled Branciari b-distance (SCB_bd) , θ-Branciari contraction (θ_BC), and Ciric-Reich-Rus type θ-Branciari contraction (CRRT − θ_BC).

Introduction

An essential method for demonstrating fixed point findings in M_sp is the contraction principle, which Banach first proposed in 1922 [1]. This has resulted in significant growth and enthusiasm in the discipline of fixed point theories, which has broad implications in numerous branches of mathematics. Bakhtin [2] developed b − M_sp which expanded the idea

of standard M_sp, this was further developed by Kamran and Samreen’s notion of extended b − M_sp[3]. Abdelawad then

established the concept of controlled b − M_sp [4]. This concept was elaborated upon by Santina et al. who progressed it into strong-controlled b M_sp [5]. Nevertheless, Branciari introduced the Branciari distance as a generalization of M_sp

[6]. It is often referred to as the Branciari metric or the Branciari distance function. Those generalizations open up new avenues and possibilities, forming a vibrant and developing field marked by ongoing research projects [7] [8] [9] [10]

[11] [12]. The resulting spaces offer fresh and fascinating interpretations on M_sp ideas, indicating potential for a wide range of uses. Moreover, existence and uniqueness issues are fundamentally resolved by fixed point theory, especially when dealing with differential and integral equations. It provides a basic structure for coping with a number of issues, such as integrodifferential equations. Under diverse contraction conditions, such as the θ-contraction presented by Jleli et

Samet [13] in the framework of Branciari M_sp, CRRT − θ_BC and interpolative- θ_BC [14] [15] [16] [17] many researchers have investigated a variety of b − M_sp. When determining the presence and uniqueness of fixed points for mappings in M_sp, the Banach contraction principle-a foundational finding in nonlinear analysis-is essential. Drawing inspiration from

Braciari’s concepts of B_d and Branciari b-distance, the idea of strong-controlled Branciari b-metric spaces (SCB_bd) is introduced in the first section of this article. Moreover, we provide the associated definitions and topological properties in the second section. In the third section, three intriguing principles are introduced: interpolative- θ_BC, CRRT − θ_BC, and θ_BC. Utilizing these novel contractions, we develop and demonstrate a few fixed point theorems within the context of SCB_bd. Additionally, an illustrating example that makes use of different sequences is provided. We offer a solution to a boundary value problem involving a fourth-order differential equation in the last section as an application.

Preliminary ASSERTIONS

We start by defining the Branciari distance (B_d) [6], which is one of the ideas put out to broaden and generalize the scope of metric.

Definition 2.1. Let X be a nonempty set and let Br : X × X −→ [0, ∞) such that for all g, h ∈ X and all k ̸= l ∈ X\{g, h}

(Br1) Br(g, h) = 0 if and only if g = h(selfdistance / indistancy)

(Br2) Br(g, h) = Br(h, g)(symmetry)

(Br3) Br(g, h) ≤ Br(g, k) + Br(k, l) + Br(l, h) (triangle Theory).

Then,Br is said to be B_d. Then, the pair (X, Br) is referred to as B_d space.

Now we revisit the notion of θ-contraction that is established by Jleli and Samet [13].

(0.1)

∞ → ∞

Consider the set θ that contains all the functions that are continuous and non-decreasing ϑ : (0, ) (1, ) such that it fulfills the conditions below:

(□) for each sequence {g_n} ⊂ (0, ∞), lim_n_→∞ ϑ (g_n) = 1 ⇔ lim_n_→∞ g_n = 0⁺;

ϑ(g)−1

(△) there exist e ∈ (0, 1) and L ∈ (0, ∞] such that lim_g_→0+ = L.

Several fixed-point results have been improved by using this notion. (see e.g. [19] [20] [21]) Recall the notion of controlled strong b − M_sp introduced by Santina et al. [5],

Definition 2.2. Let X be a non-empty set and let ω : X × X → [1, ∞). The following function SC : X × X → [0, ∞) is called a strong-controlled b − M_sp if

SC(g, h) = 0 iff g = h;
SC(g, h) = SC(h, g);
SC(g, h) ≤ SC(g, l) + ω(l, h)SC(l, h),

for all g, h, l ∈ X. The pair (X, SC ) is called as a strong-controlled b − M_sp.

—

We will merge these two concepts, strong-controlled b M_sp and B_d under the designation of a SCB_bd space according to

the following:

Definition 2.3. Consider the set X that contains at least one element and ω : X × X → [1, ∞) be a mapping. Hence, the function SCB_bd : X × X → [0, ∞) is said to be a Branciari strong controlled b-measure if it fulfills

SCB_bd(g, h) = 0 if and only if g = h;
SCB_bd(g, h) = SCB_bd(h, g);
SCB_bd(g, h) ≤ SCB_bd(g, l) + SCB_bd(l, k) + ω(k, h)SCB_bd(k, h),

for all g, h ∈ X and all distinct k, l ∈ X\{g, h}. The couple of the symbols (X, SCB_bd) denotes SCB_bd − M_sp. Example 2.1. Let X = {10, 11, 12, 13}. Define d_b : X × X → [0, ∞) as follows:

d_b(t, t) = 0, ∀t ∈ X, d_b(10, t) = d_b(t, 10) = 40, ∀t ∈ X − {10}

d_b(11, 12) = d_b(12, 11) = d_b(11, 13) = d_b(13, 11) = 211

d_b(13, 12) = d_b(12, 13) = 999

Consider the symmetric function ω : X × X → [1, ∞) with the following characteristics:

ω(t, t) = 10, ∀t ∈ X

ω(10, 11) = 3, ω(10, 12) = 4, ω(10, 13) = ω(11, 12) = 2, ω(11, 13) = 9, ω(12, 13) = 3

Therefore, (X, d_b) is a SCB_bd − M_sp. In spite of that, we shall observe that

d_b(12, 13) = 999 > d_b(12, 10) + ω(10, 13)d_b(10, 13) =
d_g(12, 13) = 999 > ω(12, 11)d_b(12, 11) + ω(11, 10)d_b(11, 10) + ω(10, 13)d_b(10, 13) =

Thus (X, d_b) is neither a strong controlled metric type space nor a controlled Branciari b-distance space.

{ } × →

Example 2.2. Let X = 1, 2, 3, 4 . Define SCB_bd : X X R by ω(g, h) = 10g + 10h + 100 then (X, SCB_bd) is a strong controlled Branciari b-distance space.

Once all other conditions are met, we shall demonstrate the amended quadrilateral inequality.

SCB_bd(g, h) = |g − h|2

= |g − l + l − k + k − h|2 where g ̸= h ̸= l ̸= k

= |g − l|2 + |l − k|2 + |k − h|2 + 2|g − l||l − k|

+ 2|l − k||k − h| + 2|k − h||g − l|

≤ |g − l| + |l − k| + (10g + 10h + 100)|k − h|

= SCB_bd(g, l) + SCB_bd(l, k) + ω(g, h)SCB_bd(k, h)

Therefore, SCB_bd(g, h) ≤ SCB_bd(g, l) + SCB_bd(l, k) + ω(g, h)SCB_bd(k, h)

Remark 2.1. If ω(g, h) = s = 1, then it is the standard B_d . Since it’s widely recognized, b-metric doesn’t require continuity.

As a result, SCB_bd is also not always continuous. We assume that every SCB_bd is continuous in the present article. Now, we present the topological properties of strong controlled Branciari b-distance ( SCB_bd ).

{ }

Definition 2.4. Let X be a set that includes at least one element and endowed with SCB_bd and a sequence g_n in X is said to be

Convergent to g if for every ϵ > 0 there exists N = N (ϵ) ∈ N such that SCB_bd (g_n, g) < ϵ, for all n ≥ N . Particularly, for this instance, we define lim_n_→∞ g_n = g.
Cauchy if for every ϵ > 0 there exists N = N (ϵ) ∈ N such that SCB_bd (g_m, g_n) < ϵ, for all m, n ≥ N .
A SCB_bd-metric space (X, SCB_bd) is said to be complete if every Cauchy sequence in X is
1. Primary Findings

We shall commence this portion by providing an introduction to the notion θ-Branciari contraction.

Definition 3.1. Let (X, SCB_bd) be a (SCB_bd) − M_sp and consider the self-mapping L : X → X where X is a non-empty set.

Then, L is called an θ_BC if ∃ ϑ ∈ θ satisfying

ϑ (SCB_bd(Lg, Lh)) ≤ [ϑ (SCB_bd(g, h))]^t if SCB_bd(Lg, Lh) ̸= 0 for g, h ∈ X.

Where t ∈ (0, 1) and g_p = L g₀ for g₀ ∈ X.

Theorem 3.1. Let (X, SCB_bd) be a complete SCB_bd − M_sp and L : X → X be a θ_BC. If

sups≥1 lim_i_→+_∞ w (g_2i+2, g_s) < 1 and for each g ∈ X, lim_p_→+_∞ w (g, g_p) exists and is finite, then L contains only one fixed point in X according to the following proof.

For any point g₀ ∈ X we generate the following iterative sequence {g_p} where g_p = L^pg₀ for all p ∈ N

L^pg, L^p⁺¹g

Assume L^p∗ g = L^p∗+1g for some p_∗ ∈ N, then L^p∗ g is certainl y a fixed po int of L.

ϑ (SCB_bd (g_p, g_p₊₁)) = ϑ (SCB_bd (Lg_p₋₁, Lg_p))

Recursively, we find that

≤ [ϑ (SCB_bd (g_p₋₁, g_p))]

≤ [ϑ (SCB_bd (g_p₋₂, g_p₋₁))]

ϑ (SCB (g , g

)) ≤ [ϑ (SCB (g , g ))] (3.1)

bd p

Accordingly, we obtain that

p+1

1 < ϑ (SCB (g , g

p+1 bd 0 1

)) ≤ [ϑ (SCB (g , g ))] for all p ∈ N (3.2)

Letting p → ∞ in (3.2), we get ϑ (SCB_bd (g_p, g_p₊₁)) → 1 as p → ∞.

From ( □ ), we have

Similarly, we can easily deduce that

lim SCB_bd (g_p, g_p₊₁) = 0 (3.3)

p→∞

lim SCB_bd (g_p, g_p₊₂) = 0 (3.4)

p→∞

From (△), there exist e ∈ (0, 1) and L ∈ (0, ∞] such that

lim

p→∞

Suppose that F < ∞. In this case, let C = ^F

ϑ (SCB_bd (g_p, g_p₊₁)) − 1 = F

[SCB_bd (g_p, g_p₊₁)]^e

> 0. Utilizing the definition of a limit, choose p₀ ∈ N such that

[SCB_bd (g_p, g_p₊₁)]^e

ϑ (SCB_bd (g_p, g_p₊₁)) − 1 − F ≤ C

for all p ≥ p₀.

[SCB_bd(g_p,g_p₊₁)]^e

This implies that ^ϑ^(SCBbd (gp,gp+1))−1 ≥ F − C = C for all p ≥ p₀.

Then, we derive that:

p [SCB_bd

(g_p, g

p+1

)]e ≤ p ϑ (SCB_bd (g_p, g_p₊₁)) − 1 for all p ≥ p

Suppose that F = ∞. Let C > 0 be an arbitrary positive number.

[SCB_bd(g_p,g_p₊₁)]^e

Using the limit definition, we find p₀ ∈ N such that ^ϑ^(SCBbd (gp,gp+1))−1 ≥ C for all p ≥ p₀. This implies that

p [SCB_bd (g_p, g_p₊₁)]^e ≤ p h ϑ(SCBbd (gp,gp+1))−1 i for all p ≥ p₀.

Thus, in all cases, there exist ¹ > 0 and p₀ ∈ N such that p [SCB_bd (g_p, g_p₊₁)]^e ≤ p h ϑ(SCBbd (gp,gp+1))−1 i for all

p ≥ p₀.

Using equation (3.2), we obtain:

p [SCB_bd Letting p → ∞, then it yields to:

(g_p, g_p₊₁

)]^e ≤ [ϑ (SCB

(g , g ))]^tp − 1 for all p ≥ p

bd 0 1 0

lim p [SCB_bd (g_p, g_p₊₁)]^e = 0

p→∞

Thus, there exists p₁ ∈ N such that

SCB_bd (g_p, g_p₊₁) ≤ p 1

for all p ≥ p₁ (3.5)

Let N = max {p₀, p₁}. Due to the modified triangle inequality, we have two cases:

L^ksh, L^ks⁺¹h for all

Case 1: Let g_p = g_q where p ̸= q. In case q > p, we have L^q⁻^p (g_p) = g_p. C hoose h = g p and s = q − p. Then L^sh = h.

∈

k N. Therefore, it is clear from the above reasoning that SCB_bd(h, Lh) = 0, so h = Lh, that is, h is a fixed point of L.

̸ ̸ { }

Case 2: Suppose that L^pg = L^qg for all integers p = q. Let p < q be two natural numbers; to show that g_p is a Cauchy sequence, we need to consider two subcases:

— → ∞

Subcase 1: We claim that if p q is odd, then SCB_bd (g_p, g_q) converges to 0 as p, q To prove this, we may assume that q = p + 2s + 1. Thus,

SCBbd (gp, gp+2s+1) ≤ SCBbd (gp, gp+1) + SCBbd (gp+1, gp+2) + w (gp+2, gp+2s+1) SCBbd (gp+2, gp+2s+1)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s+1) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+4, gp+2s+1) SCBbd (gp+4, gp+2s+1)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s+1) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) SCBbd (gp+4, gp+2s+1)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s+1) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) [SCBbd (gp+4, gp+5) + SCBbd (gp+5, gp+6)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) w (gp+6, gp+2s+1) SCBbd (gp+6, gp+2s+1)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s+1) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) [SCBbd (gp+4, gp+5) + SCBbd (gp+5, gp+6)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) w (gp+6, gp+2s+1) [SCBbd (gp+6, gp+7) + SCBbd (gp+7, gp+8)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) w (gp+6, gp+2s+1) w (gp+8, gp+2s+1) SCBbd (gp+8, gp+2s+1) .

≤ · · ·

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

+ w (g_p₊₂, g_p_+2s+1) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) · · · w (gp+2s−2, gp+2s+1) [SCBbd (gp+2s−2, gp+2s−1) + SCBbd (gp+2s−1, gp+2s)]

+ w (gp+2, gp+2s+1) w (gp+4, gp+2s+1) w (gp+6, gp+2s+1)

· · w (gp+2s, gp+2s+1) SCBbd (gp+2s, gp+2s+1) This can be written as,

SCB_bd (g_p, g_p_+2s+1) ≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s−2

i= ^p +1 p+2s

i= ^p +1

(SCB_bd (g_2i, g_2i+1) + SCB_bd (g_2i+1, g_2i+2))

w (g2i, gp+2s+1) SCBbd (gp+2s, gp+2s+1)

w (g2j, gp+2s+1)

≤SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s−2

i= ^p +1 p+2s

SCBbd (g2i, g2i+1)

w (g2j, gp+2s+1)

i= ^p +1

p+2s−2 2

w (g2i, gp+2s+1) SCBbd (gp+2s, gp+2s+1)

i= ^p +1

SCBbd (g2i+1, g2i+2)

j=Yp +1

w (g2j, gp+2s+1)

Here, we can combine the above inequality as follows:

SCB_bd (g_p, g_p_+2s+1) ≤SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s

i= ^p +1

p+2s−2 2

SCBbd (g2i, g2i+1)

w (g2j, gp+2s+1)

i= ^p +1

SCBbd (g2i+1, g2i+2)

j=Yp +1

w (g2j, gp+2s+1)

Applying (3.5) to the above inequality yields the following:

1 1



SCB_bd (g_p, g_p_+2s+1) ≤ p1/e + (p + 1) 1 .

p+2s



w (g2j, gp+2s+1) (2i)1/e

Σ2 Y 1



p+2s−2



w (g2j, gp+2s+1) (2i + 1)1/e

Σ2 Y 1

Now, let

L_c =

j=1

w (g2j, gp+2s+1) (2i)1/e .

Σ ,Yi  1

N_d =

and

Σ ,Yi  1

j=1

w (g2j, gp+2s+1) (2i + 1)1/e .

Then, we obtain:

SCB_bd (g_p, g_p_+2s+1) ≤ p1/2 + (p + 1)1/e +

Lp+2s − L 2

1 1 h

p i h p i

Np+2s−2 − N 2

By using the fact w (g_p, g_q) < 1 and applying the ratio test for convergence of series, we can deduce that

lim [Lp+2s − Lp ] = 0 and lim [Np+2s−2 − Np ] = 0

p,s→∞ 2 2 p,s→∞ 2 2

Moreover,

lim [

1 ] = 0

p→∞ pe

and

lim [

1 ] = 0, where e ∈ (0, 1).

p→∞ (p + 1) e

Therefore,

lim

p,s→∞

SCB_bd(g_p, g_p_+2s+1) = 0

— → ∞

Subcase 2: We claim that if p q is even, then SCB_bd (g_p, g_q) converges to a positive number as p, q To prove this we may assume that q = p + 2s. Thus,

SCB_bd (g_p, g_p_+2s) ≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s) SCB_bd (g_p₊₂, g_p_+2s)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+3) w (gp+4, gp+2s) SCBbd (gp+4, gp+2s)

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂) + w (g_p₊₂, g_p_+2s) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (g_p₊₂, g_p₊₃) w (g_p₊₄, g_p_+2s) [SCB_bd (g_p₊₄, g_p₊₅) + SCB_bd (g_p₊₅, g_p₊₆)]

+ w (gp+2, gp+3) w (gp+4, gp+2s) w (gp+6, gp+2s) SCBbd (gp+6, gp+2s)

≤ · · ·

≤ SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

+ w (g_p₊₂, g_p_+2s) [SCB_bd (g_p₊₂, g_p₊₃) + SCB_bd (g_p₊₃, g_p₊₄)]

+ w (gp+2, gp+3) w (gp+4, gp+2s) · · · w (gp+2s−3, gp+2s) [SCBbd (gp+2s−3, gp+2s−2) + SCBbd (gp+2s−2, gp+2s−1)]

+ w (gp+3, gp+2s) w (gp+5, gp+2s) w (gp+7, gp+2s)

· · w (gp+2s−1, gp+2s) SCBbd (gp+2s−1, gp+2s) This can be written as,

SCB_bd (g_p, g_q) ≤ SCB_bd (g_p₊₂, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s−3

i= ^p +1

p+2s−1 2

i= ^p +1

(SCB_bd (g_2i, g_2i+1) + SCB_bd (g_2i+1, g_2i+2))

w (g2i, gp+2s) SCBbd (gp+2s−1, gp+2s)

w (g2j, gp+2s)

SCB_bd (g_p, g_p_+2s) ≤SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s−3

i= ^p +1 p+2s−1

SCBbd (g2i, g2i+1)

w (g2j, gp+2s)

i= ^p +1

p+2s−3 2

w (g2i, gp+2s) SCBbd (gp+2s−1, gp+2s)

i= ^p +1

SCBbd (g2i+1, g2i+2)

j=Yp +1

w (g2j, gp+2s)

≤SCB_bd (g_p, g_p₊₁) + SCB_bd (g_p₊₁, g_p₊₂)

j=Y+1

p+2s−1

i= ^p +1

p+2s−3 2

SCBbd (g2i, g2i+1)

w (g2j, gp+2s)

Again, employing (3.5), we reach the following:

i=Σp +1

SCBbd (g2i+1, g2i+2)

j=Yp +1

w (g2j, gp+2s)

SCB

(g , g

1 1

p+2s

p1/e

(p + 1)1/e

) ≤ + +

p+2s−1

Σ2 1

Y w (g , g )

p+2s

p+2s−3

2 1

+ 1

Y w (g_2j

, gp+2s)

i= ^p +1 (2i + 1) e j= ^p +1

2 2

Now, let us consider the following substitutions to simplify the above inequality.

i=1

Let L_c = Σc

i j=1

(2i)1/e

w (g2j, gp+2s) 1

and N_d

d i=1

i j=1

w (g_2j

, gp+2s

)

(2i+1) e

Then, the simplified inequality will be as follows:

= Σ

p1/e

(p+1) e

SCB_bd (g_p, g_p_+2s) ≤ 1 + ¹ 1 + hLp+2s − Lp i + hNp+2s−2 − Np i

Using a similar procedure as in the previous part in correlation with the ratio test, we prove the following:

Therefore, {g_p} is Cauchy sequence in X.

lim

p,s→∞

SCB_bd(g_p, g_p_+2s) = 0

Given that (X, SCB_bd) is a complete SCB_bd, it implies that the sequence {g_p} converges to a point µ in X.

Following this, we shall show the continuity of L. Assume that if Lg ̸= Lh and by employing (3.1), we obtain

ln [ϑSCB_bd(Lg, Lh)] ≤ t ln [ϑSCB_bd(g, h)]

≤ ln [ϑSCB_bd(g, h)]

Given that ϑ is non-decreasing, the aforementioned observance leads to the conclusion that SCB_bd(Lg, Lh) ≤ SCB_bd(g, h) for all distinct g, h ∈ X.

From this inspection, we can get, SCB_bd (g_p₊₁, Lµ) = SCB_bd (Lg_p, Lµ) ≤ SCB_bd (g_p, µ)∀ p ∈ N.

→ ∞ →

If we take p in the preceding inequality, we obtain g_p₊₁ Lµ.

Utilizing axiom 3 in definition 2.3, we get

→ ∞

SCB_bd(µ, Lµ) ≤ SCB_bd (µ, g_p) + SCB_bd (g_p, g_p₊₁) + ω (g_p₊₁, Lµ) SCB_bd (g_p₊₁, Lµ) Letting p , applying (3.5) and the definition 1.1 of (ii), we get SCB_bd(µ, Lµ) = 0, this means Lµ = µ,

That opposes the argument that L lacks a periodic point. Then, consider period e and take µ to be a periodic point of L.

Assume that L has a phi set of fixed points.

∀ ∈

Hence, SCB_bd(z, Lz) > 0 z X and SCB_bd (z, L^ez) = 0 for e > 1 Using Definition 2.1, we get

ϑ (SCB_bd(z, Lz)) = ϑ SCB_bd L^ez, L^e⁺¹z

≤ [ϑ (SCB_bd(z, Lz))]

∈ ̸

< ϑ (SCB_bd(z, Lz)) Therefore, this results in a contradiction. Hence, ∃ a point µ ∈ X where Lµ = µ.

Assume that ξ L is a fixed point different than µ. Accordingly, SCB_bd(µ, ξ) = SCB_bd(Lµ, Lξ) = 0.

Now using the condition (3.1), we get,

ϑ (SCB_bd(µ, ξ)) = ϑ (SCB_bd(Lµ, Lξ)) = ϑ (SCB_bd (L^eµ, L^eξ))

≤ [ϑ (SCB_bd(µ, ξ))]

< ϑ (SCB_bd(µ, ξ)) , a contradiction.

Therefore, µ = ξ. Consequently, L is asserted to possess one particular fixed point in X. Now, let us consider the following example validate our findings. Example 3.1.

Construct the following sequence:

σ₁ = 1 × 2

σ₂ = 1 × 2 + 2 × 5

σ₃ = 1 × 2 + 2 × 5 + 3 × 10

σ₄ = 1 × 2 + 2 × 5 + 3 × 10 + 4 × 17 (3.6)

σ_p = 1 × 2 + 2 × 5 + 3 × 10 + 4 × 17 +………. + p p² + 1 = p³ + p

p(p + 1) 2

p(p + 1)

p⁴ + 2p³ + 3p² + 2p 4

Let g = {σ_p : p ∈ N}. Define SCB_bd : X × X → [0, ∞) as SCB_bd(g, h) = |g − h|2. Considerw : X × X → [1, ∞) as

ω(g, h) = 5g + 3h + 20. Subsequently, (X, SCB_bd) is a complete SCB_bd space.

→ ∀ ≥

Let L : X X be a self-mapping such that L (σ₁) = σ₁, L (σ_p) = σ_p₋₁ p 2. We will now demonstrate that L is a θ_BC

where ϑ(x) = e^x.

Since θ (SCB_bd we get,

(Lg, Lh)) ≤ [ϑ (SCB_bd

(g, h))]^t which yields e^(SCBbd (Lg,Lh)) ≤ e^(SCBbd (g,h)) t. Applying log on both sides

SCB_bd(Lg, Lh) ≤ tSCB_bd(g, h) Therefore, proving the preceding equation is sufficient to demonstrate that L is a θ_BC.

Case-1: If p = 1 and q > 2, then

SCB_bd (Lσ₁, Lσ_q) = SCB_bd (σ₁, σ_q₋₁)

(q − 1) + 2(q − 1) + 3(q − 1) + 2(q − 1) − 8

(q − 1) (q − 1)³ + 2(q − 1)² + 3(q − 1) + 2 − 8 2

(q − 1) q³ − 3q² + 3q − 1 + 2q² − 4q + 2 + 3q − 3 + 2 − 8 2

(q − 1) q³ − q² + 2q − 8 2

q − 2q + 3q − 2q − 8

and

q⁴ + 2q³ + 3q² + 2q − 8

Now consider,

SCB_bd (σ₁, σ_q) = 4

4 3 2 2

SCB (Lσ , Lσ ) q − 2q + 3q − 2q − 8

SCB (σ , σ ) q4 + 2q3 + 3q2 + 2q − 8

Case-2: For q > p > 1, we have

− − − −

SCB_bd (Lσ_p, Lσ_q) = SCB_bd (σ_p₋₁, σ_q₋₁)

< t given that t ∈ (0, 1).

= SCB_bd

(p 1)⁴ + 2(p 1)³ + 3(p 1)² + 2(p 1)

(q − 1)⁴ + 2(q − 1)³ + 3(q − 1)² + 2(q − 1) 4

(p − 1)⁴ + 2(p − 1)³ + 3(p − 1)² + 2(p − 1) (q − 1)⁴ + 2(q − 1)³ + 3(q − 1)² + 2(q − 1)

4 − 4

(p − 1)⁴ − (q − 1)⁴ + 2 (p − 1)³ − (q − 1)³ 3 (p − 1)² − (q − 1)² + 2(p − q) 2

and

SCB_bd (σ_p, σ_q) = SCB_bd

p⁴ + 2p³ + 3p² + 2p

q⁴ + 2q³ + 3q² + 2q

4 3 2 2

p⁴ + 2p³ + 3p² + 2p q + 2q + 3q + 2q

p⁴ − q⁴ + 2 p³ − q³ + 3 p² − q² + 2(p − q) 2

4 − 4

Now consider,

SCB (Lσ , Lσ ) (p − 1)⁴ − (q − 1)⁴ + 2 (p − 1)³ − (q − 1)³ + 3 (p − 1)² − (q − 1)² + 2(p − q) 2

SCB_bd (σ_p, σ_q)

(p − q ) + 2 (p − q ) + 3 (p − q ) + 2(p − q)

< t given that t ∈ (0, 1).

Hence, L meets θ_BC with ϑ(x) = e^x. Then, from Theorem 2.1, L has one particular fixed point σ₁. Letting w(g, h) = 1 in the preceding theorem, the following corollary is obtained.

Corollary 3.1.Consider L as a self mapping on a complete B_d space. If ∃ Θ ∈ θ and t ∈ (0, 1) satisfying

Θ(d(Lg, Lh)) ≤ [Θ(d(g, h))]^t when d(Lg, Lh) ̸= 0 for g, h ∈ X

then L possesses one particular fixed point in X.

Definition 3.2. Consider the SCB_bd − M_sp (X, SCB_bd). The self-mapping L : X → X is said to be C´iri´c-Reich-Rus type θ_BC, briefly, CRRT − θ_BC, if there exists a function ϑ ∈ θ and non-negative real number t < 1 such that

ϑ (SCB_bd(Lg, Lh)) ≤ [M_L_,ϑ(g, h)]^t (3.7)

for all g, h ∈ X, where

M_L_,ϑ(g, h) := max {ϑ (SCB_bd(g, h)) , ϑ (SCB_bd(h, Lh)) ϑ (SCB_bd(g, Lg))}

where lim supp,q→∞ ω (g_p, g_q) < 1, here g_p = f^pg₀ for g₀ ∈ X and t ∈ (0, 1).

→ −

Theorem 3.2. Consider (X, SCB_bd) a complete SCB_bd space and L : X X be a CRRT θ_BC. Then L has one particular fixed point.

As in Theorem 3.1, we establish an iterative sequence {g_p}₀∞ . Let g₀ ∈ X and define:

p p+1 p p +1

g_p = L^pg₀∀ p ∈ N

Keeping generality intact, we presume that SCB_bd L g, L g > 0 for all p ∈ N. Certainly, if L ∗ g = L ∗ g for some

p_∗ ∈ N, then L^p∗ g will be a fixed point of L. We show that lim_p_→∞ SCB_bd (g_p, g_p₊₁) = 0.

Applying the condition of contraction (3.7), we obtain,

ϑ (SCB_bd (g_p₊₁, g_p)) ≤ [M_L_,ϑ (g_p, g_p₋₁)]^t (3.8)

in which

M_L_,ϑ (g_p, g_p₋₁) = max {ϑ (SCB_bd (g_p, g_p₋₁)) , ϑ (SCB_bd (g_p, Lg_p)) , ϑ (SCB_bd (g_p₋₁, Lg_p₋₁))}

= max {ϑ (SCB_bd (g_p, g_p₋₁)) , ϑ (SCB_bd (g_p, g_p₊₁)) , ϑ (SCB_bd (g_p₋₁, g_p))}

≤ max {ϑ (SCB_bd (g_p, g_p₋₁)) , ϑ (SCB_bd (g_p, g_p₊₁))}

If M_L_,θ (g_p, g_p₋₁) = ϑ (SCB_bd (g_p, g_p₊₁)), afterwards the inequality (3.8) turns into

θ (SCB_bd (g_p₊₁, g_p)) ≤ ϑ (SCB_bd (g_p, g_p₊₁))^t ⇔ ln (ϑ (SCB_bd (g_p₊₁, g_p))) ≤ t ln (ϑ (SCB_bd (g_p₊₁, g_p))), which is a contra- diction (because t < 1). Hence, we have M_L_,ϑ (g_p, g_p₋₁) = ϑ (SCB_bd (g_p₋₁, g_p)). From (3.8), it follows that

ϑ (SCB_bd (g_p, g_p₊₁)) ≤ [ϑ (SCB_bd (g_p₋₁, g_p))]^t

Repeatedly, we discover that

ϑ (SCB

p+1

(g , g

)) ≤ [ϑ (SCB

(g , g ))]^tp

{ }

Following this insight, we deduce that {g_p} in X is Cauchy sequence by tracing the relevant lines in the Theorem 3.2 proof. In conclusion, the sequence {g_p} in X is Cauchy. Because, (X, SCB_bd) is a complete SCB_bd, there is a certain point µ in X in a way that g_p converges to µ.

Keeping generality intact, we presume that L^pg = µ for all p (or p tends to infinity). Assume that d(µ, Lµ) > 0. Using (3.7), we obtain

for all g, h ∈ X, in which

ϑ (SCB_bd (Lg_p, Lµ)) ≤ [M_L_,ϑ (g_p, µ)]^t (3.9)

M_L_,ϑ (g_p, µ) := max {ϑ (SCB_bd (g_p, µ)) , ϑ (SCB_bd (g_p, Lg_p)) , ϑ (SCB_bd(µ, Lµ))}

Taking p → ∞ in the preceding inequality, we get

ϑ (SCB_bd(µ, Lµ)) ≤ [ϑ (SCB_bd(µ, Lµ))]^t < ϑ (SCB_bd(µ, Lµ)) that is a contradiction. Therefore, Lµ = µ.

Hence, L contains a fixed point in X.

̸ ̸

Assume that µ = ξ are distinct fixed points of L. Afterwards, obviously SCB_bd(µ, ξ) = SCB_bd(Lµ, Lξ) = 0. Applying condition (3.10) now, we obtain,

1 < ϑ (SCB_bd(µ, ξ)) = ϑ (SCB_bd(Lµ, Lξ))

≤ [max {ϑ (SCB_bd(µ, ξ)) , ϑ (SCB_bd(µ, Lµ)) , ϑ (SCB_bd(ξ, Lξ))}]

< ϑ (SCB_bd(µ, ξ))

that is clearly a contradiction. Consequently, we obtain ξ = µ.

As a result, L has just one fixed point in X.

Definition 3.3. Consider the SCB_bd space, (X, SCB_bd) and a self-mapping L : X → X. Hence, L is considered an interpolative- θ_BC when ∃ a function ϑ ∈ θ such that t₁ + t₂ + t₃ < 1 where t₁, t₂, t₃ are positive real numbers satisfying

ϑ (SCB_bd(Lg, Lh)) ≤ [ϑ (SCB_bd(g, h))]^t1 [ϑ (SCB_bd(g, Lg))]^t2

[ϑ (SCB_bd(h, Lh))]^t3 , (3.10)

for all g, h ∈ X, where lim supp,q→∞ ω (g_p, g_q) < 1, here g_p = L^pg₀ for g₀ ∈ X and t ∈ (0, 1).

→

Theorem 3.3. Let (X, SCB_bd) be a complete and continuous function SCB_bd. If L : X X is an interpolative- θ_BC, then in X, L has a single fixed point.

We omit this proof because

[ϑ (SCB_bd(g, h))]^t1 [θ (SCB_bd(g, Lg))]^t2 [ϑ (SCB_bd(h, Lh))]^t3

t1+t2+t3

≤ [M_ϑ,_L(g, h)]

Then, selecting t := t₁ + t₂ + t₃ < 1 is sufficient in Theorem 3.2 to sum up the preceding Theorem.

Application to Differential Equations

Consider the following system of differential equations:

(4.1)

ψ⁴(χ) = g (χ, ψ(χ), ψ^′, ψ^′′, ψ^′′′)

ψ(0) = ψ^′(0) = ψ^′′(1) = ψ^′′′(1) = 0; χ ∈ [0, 1]

so that g is a continuous function defined as g : [0, 1] × R³ × R → R .

The focus of the study is on the boundary value problem (BVP), a fundamental concept in mathematical analysis, par- ticularly when applied to the modeling of complex physical phenomena. In this case, the problem is situated within the context of elastic beam deformations, with an emphasis on the equilibrium configuration. Specifically, the problem models scenarios where one end of the beam is free to move, while the other is fixed in place. This setup is commonly referred to as the cantilever beam problem in the field of mechanics, underlining its wide applicability and importance in both theoretical and applied mathematics.

The solutions to this type of BVP play a crucial role in numerous applications, ranging from structural engineering to mathematical physics. As such, determining the existence and uniqueness of solutions is of central importance. To achieve this, we apply the fixed-point technique, a well-established method in mathematical analysis that is particularly effective for solving BVPs of this nature.

Here we examine if a boundary value problem with a differential equation of fourth order has a solution. On the interval [0, 1], X = F[0, 1] is the notation for the space of continuous bounded functions. Within this space, we introduce the SCB_bd as a means to measure distances between functions. The metric is denoted by:

| − |

SCB_bd(L(v), g(v)) = max L(v) g(v) ²

v∈X

which provides the necessary structure for analyzing the convergence of sequences of functions in X.

→ ∞

Furthermore, we define a mapping ω : X⁴ [1, ), where ω (u₁, u₂, u₃, u₄) = 5u₁ + 5u₂ + 3. This mapping introduces an additional layer of complexity to our exploration, enabling us to further study the behavior of solutions within the context of the BVP.

Through this approach, we aim to demonstrate the existence of a solution to the boundary value problem, using the fixed-point theorem and the introduced metrics to rigorously establish the

conditions under which solutions can be found. Having established the foundational framework, we now proceed to reformulate the fourth-order ordinary differential equation (BVP) as an integral equation. The integral form in which the BVP is presented as below:

∫

ψ(t) = G(χ, v)g (v, ψ(v), ψ^′(v)) dv, ψ ∈ F[0, 1]

In this expression, G(χ, v) represents Green’s function associated with the homogeneous linear problem:

ψ⁽⁴⁾(χ) = 0, ψ(0) = ψ^′(0) = ψ^′′(1) = ψ⁽³⁾(1) = 0

This function provides a key component for solving the boundary value problem and offers valuable insight into the underlying structure of the equation. The use of Green’s function allows for a detailed understanding of the interactions between the boundary conditions and the solution, facilitating the analysis of the problem’s characteristics.

G(χ, v) =

( 1 χ²(3v − χ), 0 ≤ χ ≤ v ≤ 1

1 v²(3χ − v), 0 ≤ v ≤ χ ≤ 1

(4.2)

The subsequently properties of G(χ, v) could be simply verified from (3.2).

1 χ²v² ≤ G(χ, v) ≤ 1 χ² or 1 v² , χ, v ∈ [0, 1]

Theorem 4.1. Assume that the subsequent constraints are satisfied.

The mapping g : [0, 1] × R³ × R → R is
For every ψ, y ∈ X, there is a σ ∈ [1, ∞) such that the below condition is

|g (v, ψ, ψ^′) − g (v, y, y^′)| ≤ √20e −σ |ψ(v) − y(v)|, v ∈ [0, 1]

There exists ψ₀ ∈ X where for every χ ∈ [0, 1], we deduce

∫ ₁

Hence, the BVP problem contains a solution in X. Proof: Assume the self-mapping L : X → X is defined as

∫ ₁

Then ψ = Lψ, which imply that the BVP has only one solution.

Consider,

∫

|L(ψ)(χ) − L(y)(χ)| =

2 G(χ, v)g (v, ψ(v), ψ^′(v)) dv − G(χ, v)g (v, y(v), y^′(v)) dv

∫

≤ (G(χ, v))² | g (v, ψ(v), ψ^′(v)) − g

v, y(v), y^′(v)| dv

1 1

≤ v⁴20e⁻^σ|ψ(v) − y(v)|2dv

₀ 4

≤ 20e⁻^σSCB_bd(ψ, y)

v⁴dv

∫ ₁ 1

≤ 20e⁻^σSCB_bd

(ψ, y)

where this deduce to,

= e⁻^σSCB_bd(ψ, y)

SCB_bd(L(ψ), L(y)) ≤ e⁻^σSCB_bd(ψ, y)

√SCB_bd(L(ψ), L(y)) ≤ √e−σ SCB_bd(ψ, y)

√

√ ^√_e−σ

e SCB_bd(L(ψ),L(y)) ≤ e

√

SCB_bd(ψ,y)

; where e⁻^σ < 1 as σ ≥ 1

Thus, e√SCBbd

(L(ψ),L(y)) ≤ e√SCB_bd

(ψ,y)

^r with r = √e−σ which gives,

θ (SCB_bd(Lψ, Ly)) ≤ [θ (SCB_bd(ψ, y))]^r where θ(t) = e√χ

L has a fixed point as all conditions of Theorem 3.1 are met. Therefore, the solution to BVP is found in X.

Conclusion

As an extension of strong controlled metric type space and Branciari space, we introduced the notion of strong controlled b-Branciari metric type space in this work. Next, in the framework of strong controlled-b-Branciari metric type space, we proved several fixed point theorems concerning Theta-contraction, Ciric-Reich-Rus type Theta-Brainciari contraction, and interpolative-Theta-Branciari contraction. Additionally, we provide numerous examples to highlight our conclusions.

Acknowledgments: D. Santina and N. Mlaiki would like to express their gratitude to Prince Sultan University for covering the publication fees associated with this work through TAS LAB.

Competing Interest: No conflicting interests are disclosed by the authors.

Authors Contributions: All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Dania Santina

Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586. Institute of Mathematical Sciences, Faculty of Science, University Malaya, Kuala Lumpur, Malaysia

Email address: dsantina@psu.edu.sa

Wan Ainun Mior Othman

Institute of Mathematical Sciences, Faculty of Science, University Malaya, Kuala Lumpur, Malaysia

Email address: wanainun@um.edu.my

Kok Bin Wong

Institute of Mathematical Sciences, Faculty of Science, University Malaya, Kuala Lumpur, Malaysia

Email address: kbwong@um.edu.my

Nabil Mlaiki^∗

Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586

Email address: nmlaiki@psu.edu.sa, nmlaiki2012@gmail.com

Suhad Subhi Aiady^∗

Department of Mathematics and Sciences, Prince Sultan University Riyadh, Saudi Arabia 11586

Institute of Mathematical Sciences, Faculty of Science, University Malaya, Kuala Lumpur, Malaysia

Email address: ssubhi@psu.edu.sa

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