A fixed point technique for solving boundary value problems in Branciari Suprametric Spaces

The technique to broaden the scope of fi xed point theory is to extend the class of spaces that have stronger conceptual frameworks than metric spaces. Therefore, this paper explores the introduction of novel metric spaces, namely, Branciari suprametric spaces

and s i ¹ .Furthermore, by employing the results obtained, the present study intends to provide an appropriate solution for the nonlinear fractional differential equations of the Riemann-Liouville type.

Introduction and Preliminaries
Fixed point approaches, especially are being applied in domains that include biology, engineering, chemistry, physics, game theory, and economics.In several disciplines of mathematics and computational science, the problems of the existence varied solutions to mathematical models relates to the existence of a fixed point problem for a particular map.Therefore, in order a variety of scientific fields across all areas rely substantially on the study of fixed points.
The development of fixed point theory was facilitated by the French mathematician Frechet's idea of metric spaces.In an attempt to improve the possibility of obtaining more generic fixed point results, the conception of a metric space has undergone numerous approaches to be extended.In analysis, the triangle inequality is among the most essential and effective inequalities.The triangle inequality has been extended to varying generalizations that are satisfied by different distance functions.These include the b-metric spaces by Czerwik [7], at which the triangle inequality equation is multiplied by the constant b on the right side; the extended b-metric spaces by Kamran [23], from which the triangle inequality equation is multiplied by the funtion q t i ( ) , on the right side; and the generalized metric spaces by Branciari [5], wherein the triangular inequality of metric spaces has been supplanted by a new inequality that is referred to as rectangular inequality.
Furthermore, a significant number of new distance functions were developed by combining, easing, or expanding some of the tenets of the already-existing distance functions with the objective to address the growing uncertainty of practical applications.A number of articles (see [1,11,[24][25][26][27]30] with the cited works therein) have addressed fixed point theory for single-valued mappings throughout various abstract spaces.
The idea of b-metric spaces was established by Czerwik [7] as follows: Definition 1.1.[7]   Rectangular metric spaces were initially developed by Branciari [5] with the following definition: Recently, a modified triangular inequality was employed in the new metric known as suprametric, originally presented by Maher Berzig [2], who additionally investigated several important aspects of its topology.Following that, the author demonstrated that specific contraction maps in suprametric spaces possess an unique fixed point if the space is complete or comprises a non-empty  -limit set.For more current works on suprametric spaces, researchers can refer to the articles [3,4,28,31].
The following gives the definition of suprametric space: 1 respectively.
On the other hand, Liouville and Riemann established the very first definition of fractional derivative at the culmination of the 19th century, however Leibniz and L'Hospital initially proposed the idea of non-integer derivative and integral in 1695 as an interpretation of the standard integer order differential and integral calculus.In practical terms, derivatives with fractional values provides an ideal approach to articulate the memory and inherited qualities associated with various processes and techniques.In recent years, the research on fractional differential equations has increased significantly.In order to determine the existence of and distinctiveness of or the multiplicity of solutions to nonlinear fractional differential equation boundary value problems, as well as to deal with other problems involving nonlinear fractional differential equations, nonlinear analysis techniques, which serve as the primary approach for accomplishing this, play a significant part in the investigations within this field (see [8-10, 13-22, 27, 29, 32-38] and the sources listed therein).
Incited by all of the works listed above, in the current study, we extend the works of Maher Berzig [2] in Section 2 by setting up the idea of Branciari (or rectangular) suprametric spaces by presenting an example for the given metric.In Section 3, we establish several intriguing fixed point results under varying contractive conditions.The boundary value problem of a class of fractional differential equations consisting of the Riemann-Liouville fractional derivative is then examined in Section 4 by employing the established fixed point result, with the goal of determining if such solutions exist and whether they are unique.

Main Results
The present section explores the conception of Branciari suprametric spaces which is an appropriate extension that encompasses suprametric spaces [2] and rectangular metric spaces [5].

Fixed Point Theorems on Branciari Suprametric Spaces
In this section, we establish two intriguing fixed point results depending on specific contractive conditions in the conceptual framework of Branciari suprametric spaces.
where g Î[ , ) 0 1 .Then the fixed point of G is unique.
Prof. For every t 0 ÎY , the iterative sequence { } t y is specified by t t Similarly to that, we observe Thereby, the sequences { ( , and { ( , are decreasing and with regard to all fixed integer k and for all y > k, it fulfils 1, where m ≥ 1, then the following is derived from inequality (3.4): y y y y y y m y ( , ) where where Therefore, we get This implies lim ( , ) y y y ®¥ + = d  t t 1 0 , subsequently there exists k Î  so that for all y ≥ k, we have ( , ) ( , )] ( , ) In order to illustrate { } t y is Cauchy, we take into in a pair of distinct cases.

Case 1:
We proceed by delving into an odd number represented as 1, where m ≥ 1, then the following is obtained from (3.12): y y y y m y i i y ) ( , where Inequality (3.14) subsequently follows   Case 2: Now, suppose by considering an even number represented by z m = 2 , where m ≥ 1, then the following is derived from inqualities (3.12) and (3.13):

Conclusion
The conceptual framework of Branciari suprametric spaces, which appears to be more effective than the ideas of rectangular metric spaces and suprametric spaces and constitutes an alternate perspective on the existence and uniqueness of the solutions to nonlinear fractional differential equations of the Riemann-Liouville type, was employed in this work.In the context of Branciari suprametric spaces, we presented an illustration and defined the terms convergence of sequences, Cauchy sequences, and completeness.A number of fixed point theorems, including the Banach fixed point theorem, were also proved in this space.

Declarations
Then d b is said to be a b-metric on Y and (Y, d b ) is called a b-metric space.

4 )Case 2 :
thus leads to the conclusion that d  ( , ) as y m, tend to infinity.Now, suppose an even number represented by z m = 2 , where m ≥ 1, then the following is obtained from inequalities (3.4) and (3.5):

Definition 2.1. Let Y be a nonempty set and
> 0 and t 0 ÎY is called an open ball of radius r 0 and center t 0 .
Definition 2.4.Let ( , ) Y d  be a Branciari suprametric space.A sequence { } t y in Y referred to as: t Furthermore, we establish that t is a fixed point of G. Consider Letting y ® ¥ in the previously given inequality, we find d G Through the use of inequality (3.1), we can easily show that t is a unique fixed point of G.
 ( , ) t t = 0 i.e., Gt t = .Thereby t is a fixed point of G. Theorem 3.2.Let ( , )Y d  be a complete Branciari suprametric space and G Y ]