Solving a Nonlinear Fractional Differential Equation Using Fixed Point Results in Orthogonal Metric Spaces

: This research article aims to solve a nonlinear fractional differential equation by ﬁxed point theorems in orthogonal metric spaces. To achieve our goal, we deﬁne an orthogonal Θ -contraction and orthogonal ( α , Θ ) -contraction in the setting of complete orthogonal metric spaces and prove ﬁxed point theorems for such contractions. In this way, we consolidate and amend innumerable celebrated results in ﬁxed point theory. We provide a non-trivial example to show the legitimacy of the established results.


Introduction
In fixed point theory, the Banach contraction principle [1] is one of the most prominent and substantial results that was first introduced and established by Stefan Banach in 1922.Based on the intelligibility, adequacy , and applications of this result, it has become a very famous tool in solving existence problems in numerous branches of mathematical analysis.So several researchers have boosted, broadened, and elongated this theorem in various directions.In 2014, Jleli et al. [2] introduced a new variant of contractions in the setting of generalized metric spaces, which is known as Θ-contraction.As a consequence, they obtained a fixed point result in complete metric space, which is a generalization of Banach's fixed point theorem.Hussain et al. [3] introduced a different condition in the notion of Θ-contraction and proved a result that is an extension of the result of Jleli et al. [2].Ahmad et al. [4] changed the third postulate of Θ-contraction with an easy one.Later on, Imdad et al. [5] gave the notion of weak Θ-contraction by omitting some conditions of Θ-contraction and established some related theorems in the framework of complete metric spaces.Subsequently, Ameer et al. [6,7] presented Ćirić type α * -η * -Θ-contractions and Suzuki-type Θ-contractions and obtained a fixed point theorem for multivalued mappings.For further details in this field, we refer the researchers to [8][9][10][11].
Gordji et al. [12] innovated the concept of orthogonality in metric spaces and set up the fixed point result for self-mappings in the background of orthogonal metric spaces.Baghani et al. [13] improved the leading result of Gordji et al. [12] by proving some new fixed point theorems.They also investigated the existence and uniqueness of a solution to a Volterratype integral equation in L p space as application of their main theorem.Afterward, Baghani et al. [14,15] manifested fixed and coinciding point results for multivalued mappings.Hazarika et al. [16] discussed the general convergence methods in the setting of orthogonal metric spaces and studied the applications of fixed point results to obtain the existence of a solution of differential and integral equations.For more achievements in this direction, we refer researchers to [17][18][19][20].
On the other hand, abstract spaces like metric spaces, normed spaces, and inner product spaces are all examples of "topological spaces", which are more general spaces.
These spaces have been specified in order of increasing structure; that is, every inner product space is a normed space, and in turn, every normed space is a metric space.Two vectors are said to be orthogonal if and only if their inner product is zero, i.e., they make an angle of 90 • (π/2 radians), or one of the vectors is zero in the context of inner product spaces.The complete inner product space is called a Hilbert space.Some fixed point theorems for contractive and nonexpansive mappings in the setting of Hilbert spaces are given in the literature [21][22][23].However, no one has obtained fixed point theorems for Θ-contraction mappings in Hilbert spaces.
In this research, we introduce the notion of Θ-contraction mappings in orthogonally complete metric spaces and obtain some fixed point results for these mappings.Also, we give an example to illustrate the validity of our results.Moreover, we apply our results to investigate the solution to a differential equation.As a consequence of our leading result, we deduce the prime theorem of Jleli et al. [2] and several well-known results from the literature.

Preliminaries
In this article, we represent by N and R + the set of natural numbers and the set of positive real numbers, respectively.
Jleli et al. [2] initiated the notion of Θ-contraction along the following lines.
Gordji et al. [12] present the concept of the orthogonal set (O-set, for short) in this way.

Definition 2 ([12]
).Let X be a non-empty set and ⊥ ⊆ X × X be a binary relation.Then (X ,⊥) is said to be an O-set if there exists ξ 0 ∈ X such that ς ⊥ ξ 0 or ξ 0 ⊥ ς for all ς ∈ X .The element ξ 0 is said to be an orthogonal element.
for all  ∈ N.
Definition 4 ([12]).The triplet (X , ⊥, τ) is said to be an orthogonal metric space if the pair (X ,⊥) is an orthogonal set and the pair (X , τ) is a metric space.
Samet et al. [24] introduced the notion of α-admissible mapping as follows: Ramezani [25] presented the idea of orthogonal αadmissibility in the following way.
In this manuscript, we prove some fixed point results for orthogonal Θ-contraction and orthogonal (α, Θ)-contraction in the context of O-COMS.The established results will combine and modify many celebrated results from the literature.
In what follows, we shall present another result in which we replace (J 3 ) and (J 4 ) with the general condition (J / 3 ).
The following theorem is a direct outcome of Theorem 4.
Then, V has a unique fixed point ξ * ∈ X .
Proof.Define a binary relation on X by Fix ξ 0 ∈ R. Since V is a Θ-contraction, we have ξ 0 ⊥ς for all ς ∈ X .Hence, from Theorem 4, there exists a unique fixed point of V.

Applications
In this section, we will investigate the solution for the nonlinear fractional differential equation (0 < t < 1, 1 < η ≤ 2) via the integral boundary conditions where ξ ∈ C([0, 1], R) (family of all continuous functions).We symbolize and define the Caputo fractional derivative of order η as C D η and Recall that the Riemann-Liouville fractional integral of order η is given as for 0 < ϑ < 1.Then, the differential equation ( 19) has a unique solution.
Proof.For all t ∈ [0, 1], suppose that the orthogonality relation on X is given as The set X is orthogonal with this orthogonality relation because, for all ξ ∈ X , there exists ς(t) = 0 such that ξ(t)ς(t) = 0.

Conclusions
In this manuscript, we have proven some fixed point theorems in O-COMS for orthogonal Θ-contractions and orthogonal (α, Θ)-contraction.We have also explored the solution to a nonlinear fractional differential equation as the implementation of our foremost results.Furthermore, a significant example is also given to show the authenticity of the proved result.
In the context of O-COMS, establishing fixed points and common fixed points of fuzzy mappings and set-valued mapping for orthogonal Θ-contractions and orthogonal (α, Θ)contractions can be an interesting contribution in fixed point theory.Also, the solution to fractional differential inclusion can be investigated by applying these proposed outlines.