SOME FIXED POINT RESULTS ON E -METRIC SPACES USING CONTRACTION OPERATORS

: In this paper, we prove the generalizations of the Bianchini contraction and the Ciri ´ c-Reich-Rus contraction mappings ﬁxed point results using the concept of convergence criteria for semi-interior points in E -metric spaces with non-solid and non-normal set of positive elements E + of real normed space E (positive cone). Additionally, many examples are included to illustrate the existence of semi-interior points of E + with empty interior. In addition, we present some applications in the ﬁeld of applied mathematics that support our main ﬁndings.


INTRODUCTION
Fixed point theory (FPT ) in ordered normed spaces has played an important role in computer science, economics, optimization theory, astronomy, dynamical systems, decision theory, parameter estimation, and many other subjects over a period of several decades.Large scale problems requiring fixed point theory are highly esteemed for their lightning-fast solutions.
As a result, in recent years, many scholars have focused on developing FPT approaches and have provided various useful techniques for discovering FP in complex issues.Poincare [20], Recently in 2017, the concept of semi-interior points was defined by Polyrakis et al. in [5].
According to Proposition 3.2 of [5], any semi-interior point of E + is also an interior point of E + with respect to the norm ||.|| of E which coincides with the initial norm ||.|| of E in E + .So E + is ||.||-normal if and only if E + is ||.||-normal.So all the main results of this paper, can followed automatically by using the new norm ||.|| of E, but a systematic way to prove the fixed point results is defined in the main results.The class of cones with semi-interior point and empty interior is a class of cones larger than the one with nonempty interior as the examples of [5].It is worth noting that fixed points results for ordered normed spaces can also proved for this larger class of cones with semi-interior points.For more relevant results, see also [2], [3], [7], [9], [10], [11], and [12].
On the other hand, the first researchers to investigate a generalization of the Banach fixed point theorem while simultaneously using a contraction condition of the rational type were Dass and Gupta [28].Jaggi [30], used a contraction condition of the rational type to prove a fixed point theorems in complete metric spaces.Moreover, rational contraction conditions have been heavily employed in both the FP and common FP locations.
The remaining parts of this manuscript are displayed as follows: In Section 2, we recall the notations, basic notions, and essential definitions needed throughout the paper.In Section 3, we prove the main concept related to fixed point results using the B-contraction, the Bianchini contraction, and some rational type contraction mappings in the setting of E metric spaces with non-solid and possibly non-normal cones.In Section 4, we present some applications in the field of applied mathematics related to the main findings of this paper.Finally, in Section 5, we reach a conclusion.

PRELIMINARIES
In this section, some notations, basic notions, essential definitions and lemmas from earlier works are recalled.Throughout in this article, let E be an ordered normed space with norm ., which is ordered by its positive cone E + , such that for all p, q ∈ E, p ≤ q iff q − p ∈ E + .Let E * be the dual space of E. The following are the basics of ordered normed spaces and E-metric spaces.
Definition 2.1.[24] An ordered space E is a vector space over the real numbers, with a partial order relation such that (N1) for all p, q and r ∈ E, p q implies p + r q + r; (N2) for all a ∈ R + and p ∈ E with p 0 E , ap 0 E .
Moreover if E is equipped with norm ., then E is called normed ordered space.Definition 2.2.[24] [7]The positive cone E + of a normed ordered space X is called; (D1) normal, if there exist a constant M 0 such that for all p, q ∈ E where 0 p q implies p M q .
(D2) solid, if E + has non-empty interior, (D3) reflexive, iff E + ∩U is weakly compact, where U is the unit ball in X, (D4) strongly reflexive, iff E + ∩U is compact.
Definition 2.3.[24] Let T be a non-empty set and let E be an ordered space, over the real scalars.An ordered E-metric on T is an E-valued mapping d : T × T → E such that for all p, q, r ∈ T , the following hold: (CM1) d(p, q) 0 and d(p, q) = 0 if and only if p = q; (CM2) d(p, q) = d(q, p); Then the pair (T, d) is called E-metric space.
Let us assume that int(E + ) is non-empty.Now, we recall the the definitions of convergent and Cauchy sequence in an E-metric space.Next, we present the the notion of semi-interior points of the positive cone E + of an ordered space E and we define new convergence criteria.Further, we provide non-trivial examples from literature to ensure the applications of our presented notions.
Remark 2.5.Let E be an ordered normed space ordered by the positive cone E + , we shall denote by 0 E the zero of E, be the closed unit ball of E, and by U + mean the positive part of unit ball defined by the set Definition 2.6.[24] The point p 0 ∈ E + is a semi-interior point of E + if there exists a real number µ > 0 such that Clearly any interior point of E + is semi-interior point.
Note that the set of all semi-interior points of E + is denoted by (E + ) , and for p, q ∈ E + , p ≪ q if and only if q − p ∈ (E + ) .Moreover, the following example from [5] insure the existence of semi-interior points of cones having empty interior, these spaces are useful for linear and nonlinear optimization problems and operator analysis.
Example 2.7.[24] This example clearly shown that a strong reflexive cone ) and E + 1 be a set of positive elements of L 1 ([0, 1]) generated by the set ω = 3q + V , where q = ∑ ∞ k=1 β k−1 e k for β ∈ (0, 1), where {e i } is the set of standard normalized basis.It has been shown in [5], that the cone E + has the empty interior but has semi-interior points.
For more examples related to the positive cone with semi-interior points but empty interior point, we refer the readers to [5].Also, it is clear that, if int(E + ) is nonempty then every interior point of the cone E + is the semi-interior point.The following lemma is the converse in complete, ordered normed spaces.
Lemma 2.8.[24] If E is the complete ordered normed space with generating and closed cone E + , then any semi-interior point of E + is an interior point of E + .Now we define the e-convergence and e-Cauchy convergence criteria in the ordered normed space E, with non-solid cone E + .Definition 2.9.[24] Let E be a ordered normed space with assumption that (E + ) is nonempty and (T, d) be an E-metric space.Let {p n } be a sequence in T and p ∈ T .Then; (i) A sequence {p n } is e-convergence to some p whenever for every e ≫ 0, there exists a natural number k such that d(p n , p) ≪ e for all n ≥ k.We denote this by lim n→∞ p n = p or p n → p.
(ii) A sequence {p n } is e-Cauchy, if for every e ∈ E with 0 ≪ e there is natural number N such that for all m, n > N, d(p n , p m ) ≪ e.
(iii) (T, d) is said to be e-complete if every e-Cauchy sequence is e-convergent.

MAIN RESULTS
In this section, we prove some FP by using various contraction mappings such as the Bianchini contraction and the Cirić-Reich-Rus contraction and their related consequences on Emetric spaces.For that, assume (K, d) be a e-complete e-metric space.Firstly, we consider the Bianchini contraction mapping to prove our main FP theorem.Theorem 3.1.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the Bianchini contraction condition, i.e., d(T p, T q) ≤ ed(p, q), where d(p, q) = max{d(p, T p), d(q, T q)}, for all p, q ∈ K and e ∈ [0, 1).Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Proof.For any p 0 ∈ K, let us take the iterative sequence p n+1 = T p n = T n p 0 with p n = p n+1 for some n ∈ N. Case 2. Consider d(p, q) = d(q, T q).Then, we have which is impossible because e ∈ [0, 1).Therefore, Case 2 does not exist.Now, by Case 1 and n m, we get Let e ≫ 0 be given, choose ν > 0 such that e − νU + ⊆ E + and a natural number Let q ∈ k be such that p = q = T q, then consider which implies d(p, q) = 0 E .This proves the theorem.
Corollary 3.2.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the contraction condition, i.e., d(T p, T q) ≤ ed(p, T p), for all p, q ∈ K and e ∈ (0, 1).Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Proof.Substituting d(p, q) = d(p, T p) in Theorem 3.1 completes this corollary.
Secondly, we consider another contraction mapping known as the Cirić-Reich-Rus contraction to prove our next main FP theorem.
Theorem 3.3.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the condition d(T p, T q) ≤ e 1 d(p, q) + e 2 (d(p, T p) + d(q, T q)), for all p, q ∈ K and some e 1 , e 2 ∈ [0, 1) with e 1 + 2e 2 < 1.Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Proof.For any p 0 ∈ K, let us take the iterative sequence p n+1 = T p n = T n p 0 with p n = p n+1 for some n ∈ N. Consider, . . .
Now for n m, consider, Let e ≫ 0 be given, choose ν > 0 such that e − νU + ⊆ E + and a natural number Let q ∈ k be such that p = q = T q, then consider which implies d(p, q) = 0 E .This proves the theorem.
Corollary 3.4.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the condition d(T p, T q) ≤ e 1 d(p, q), for all p, q ∈ K and some e 1 ∈ (0, 1).Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Corollary 3.5.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the condition d(T p, T q) ≤ e 2 (d(p, T p) + d(q, T q)), for all p, q ∈ K and some e 2 ∈ (0, 1).Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Next, we prove the same kind of result by using rational-type contraction mapping to find the existence and uniqueness of the FP.
Theorem 3.6.Let (K, d) be an e-complete e-metric space with closed positive cone E + such that (E + ) = φ .Let T : K → K be a mapping satisfying the condition d(T p, T q) ≤ e 1 d(q, T q)[1 + d(p, T p)] 1 + d(p, q) + e 2 d(p, q), for all p, q ∈ K and some e 1 , e 2 ∈ [0, 1) with 2e 1 + e 2 < 1.Then T has a unique FP in K and for each p ∈ K, the iterative sequence {T n p} n≥0 converges to the FP p of T .
Proof.For any p 0 ∈ K, let us take the iterative sequence p n+1 = T p n = T n p 0 with p n = p n+1 for some n ∈ N. Consider, Let e ≫ 0 be given, choose ν > 0 such that e − νU + ⊆ E + and a natural number Let q ∈ k be such that p = q = T q, then consider which implies d(p, q) = 0 E .This proves the theorem.

APPLICATIONS
The FP covers a wide range of applications in the field of mathematics, particularly differential geometry, numerical analysis, and so on.By reading [35]    Then, we have d(T p, T q) ≤ e 1 d(p, q) + e 2 (d(p, T p) + d(q, T q)).

CONCLUSION
This paper has introduced some new FP theorems that are applicable to both contraction and rational contraction operators on E-metric spaces.In particular, going in the same direction as [24], we provide the results in the setting of contraction mappings, namely the Bianchini contraction type, Ciric-Reich-Rus contraction type, and their consequences.Additionally, we provide the FP theorems by using the rational contraction mapping, which were discussed mostly in [28] and [30].In order to confirm the presence of the FP theorems, alternative discoveries presented in the later can be demonstrated in a lower environment.

ACKNOWLEDGEMENT
All the authors thank the anonymous referee(s) of the paper for their valuable recommendations.
Once again, we thank the editor for giving us the opportunity to reset the manuscript in a nice way.

Case 1 .
Consider d(p, q) = d(p, T p).Then, we have d(p n , p n+1 ) = d(T p n−1 , T p n ) ed(p n−1 , T p n−1 ) ed(p n−1 , p n ) ed(T p n−2 , T p n−1 ) e 2 d(p n−2 , T p n−2 ) e 2 d(p n−2 , p n−1 ) . . .e n d(p 0 , p 1 ) for all m, n ≥ k 1 which implies {p n } is an e-cauchy sequence, since K is e-complete so there exists some p ∈ K such thatp n → p.For a given e ≫ 0 E , choose k 2 ∈ N, such that d(p, p n ) ≪ e 2 ,for all n ≥ k 2 .Consider for all n ≥ k 2 , d(p, T p) ≤ d(p, p n ) + d(p n , T p) ≤ d(p, p n ) + νd(p, p n−1 ) and m ∈ N, therefore e m − d(p, T p) ∈ E + , for all m ∈ N, which implies −d(p, T p) ∈ E + , but d(p, T p) ∈ E + , therefore d(p, T p) = 0 E .Hence p = T p.