Relation-theoretic ﬁxed point theorems under a new implicit function with applications to ordinary di ﬀ erential equations

: In this paper, we introduce a new implicit function without any continuity requirement and utilize the same to prove uniﬁed relation-theoretic ﬁxed point results. We adopt some examples to exhibit the utility of our implicit function. Furthermore, we use our results to derive some multidimensional ﬁxed point results. Finally, as applications of our results, we study the existence and uniqueness of solution for a ﬁrst-order ordinary di ﬀ erential equations.


Introduction
The celebrated Banach contraction principle [7] is an indispensable result of metric fixed point theory. This fundamental result has been extended and generalized in various directions. This principle is utilized in diverse applications in the domain of mathematics and outside it as well. In recent years, several researchers attempted to unify the existing extensions and generalizations of Banach contraction principle employing varies methods. A very simple and effective method of carrying out such unifications is essentially due to Popa [26] wherein the author initiated the idea of implicit functions.
In Section 3 of this manuscript, we have a new class of implicit functions which is general enough to deduce several known fixed point theorems in one go besides being general enough to yield new but unknown contractions. Some examples are also given to support this view point.
The branch of related metric fixed point theory is a relatively new branch was initially studied by Turinici [35]. Now a days, this direction of research becomes very active especially after the existence of the fantastic articles due to Ran and Reurings [29] and Nieto and Rodriguez-lopez [24,25] which also contain fruitful applications. Recently, this direction of research is undertaken by several researchers such as: Bhaskar and Lakshmikantham [11], Samet and Turinici [33], Ben-El-Mechaiekh [8], Imdad et al. [14,17], Mursaleen et al. [23] and some others.
In Section 4, we prove some relation-theoretic fixed point theorems utilizing our newly introduced implicit function. Some corollaries are deduced which cover several known as well as unknown fixed point results.
On the other hand, the extensions of coupled fixed point up to higher dimensional product set carried out by several authors are not unique (c. f [3]). The first attempt to unify the multi-tupled fixed point notions was due to Berzig and Samet [10], wherein they defined a unified notion of N-tupled fixed point. Thereafter, the notion of N-tupled fixed point was extended by Roldán et al. [31] by introducing Υ-fixed point. Soon, Alam et al. [3] modified the notion of Υ-fixed point by introducing * -fixed point.
In Section 5, we apply Theorems 4.1 and 4.2 to deduce some multidimensional fixed point results utilizing the notion of * -fixed point. The proved results unify numerous multidimensional fixed point results of the existing literature especially those contained in [3,[9][10][11].
The existing literature contains numerous results on the existence of solutions for ordinary differential equations (in short ODE) in the presence of lower as well as upper solutions of the ODE problems under consideration. In Section 6, inspired by [24,25], we establish the existence and uniqueness of the solution of the problem described by (6.1).
From now on, N, N 0 , R + and R, respectively, refer to the set of: natural numbers, whole numbers, non-negative real numbers and real numbers. Also, M is a nonempty set, f : for all n, is called a Picard sequence based on x 0 .

Relation theoretic notions and auxiliary results
A binary relation S on M is a subset of M × M. M × M is always a binary relation on M known as universal relation. We write xSy whenever (x, y) ∈ S and xS y whenever xSy and x y. Observe that S is also a binary relation on M such that S ⊆ S. The points x and y are said to be S-comparable if xSy or ySx which is often denoted by [x, y] ∈ S. Throughout this work, S stands for a binary relation defined on M, S M stands for the universal relation on M and M( f, S) = {x ∈ M : xS f x}. [1,12,20,21]) A binary relation S is said to be: (iii) transitive if xSz whenever xSy and ySz, ∀x, y, z ∈ M; (iv) antisymmetric if xSy and ySx imply x = y, ∀x, y ∈ M; (v) partial order if it satisfies (ii), (iii) and (iv); (vi) complete or connected if [x, y] ∈ S, ∀x, y ∈ M; (vii) f -closed if xSy implies f xS f y, ∀x, y ∈ M.
Definition 2.5. [16] Let B ⊆ M. If each S-preserving Cauchy sequence {x n } ⊆ B converges to some x ∈ M, then B is said to be S-precomplete.
Remark 2.1. Every precomplete subset of M is S-precomplete, for an arbitrary binary relation S.

An implicit function
A natural and simple way to present unified fixed point results was possible via implicit function when Popa [26] initiated the idea of implicit functions wherein he proved some common fixed point theorems using continuous implicit function which is general enough to deduce several known fixed point theorems in one go besides being general enough to deduce several fixed point results under new contractions. Thereafter, several authors used the idea of implicit functions assuming several suitable assumptions (e.g., [4-6, 15, 18, 22, 27, 28, 30] and references therein). We are not familiar with any article dealing with implicit functions without continuity assumption deducing contraction mappings in complete metric spaces but in this paper we endeavor to do so. With this idea in mind, we introduce a new implicit function without continuity with merely two requirements.
Definition 3.1. Let E be the class of all functions E : R 6 + → R satisfying: (E 1 ) E is non-increasing in the 4 th , 5 th and 6 th variables; where λ ∈ [0, 1) and p ≥ 1, then E ∈ E.

Fixed point results
Here, we provide unified relation-theoretic fixed point results via our newly introduced implicit function beginning with the following one. (c) f M is S -precomplete; (d) f is S -continuous; (e) ∃E ∈ E such that (∀x, y ∈ M with xS y), Then f has a fixed point.
, then we are done as x n 0 = f x n 0 . Now, suppose x n x n+1 , ∀n ∈ N 0 . As x 0 S f x 0 and x n x n+1 (∀ n ∈ N 0 ), we have x 0 S x 1 and in general x n S x n+1 (∀n ∈ N 0 ) due to f -closedness of S . Now, using (e), we get (∀n ∈ N 0 ) which together with triangle inequality and (E 1 ) give rise Using induction on n in (4.1), we have Letting n → ∞ in (4.2), we get lim Let n, m ∈ N with n < m. Now, on using triangular inequality and (4.2), we have As f is S -continuous and {x n } is S -preserving sequence converges to x, we have {x n+1 = f x n } → f x. Therefore, we have f x = x (as the limit is unique). The end.
Next, we present an analogous of Theorem 4.1 utilizing the d-self-closedness.
Proof. As in the proof of Theorem 4.1, one can see that {x n } is S -preserving Cauchy sequence converging to x. In view (d ), ∃{x n k } ⊆ {x n } such that [x, x n k ] ∈ S . This implies that either xS x n k or x n k S x. Assume that xS x n k . On using condition (e), ∃E ∈ E satisfying which together with triangle inequality and (E 1 ) give rise . So, f x = x (as the limit is unique). The proof of the case x n k S x is similar. The end.
The following condition is useful in the next result: (U) for each x, y ∈ Fix( f ) ∃z ∈ M such that z is S-comparable to both x and y. Due to the condition (U), there is z 0 ∈ M such that [x, z 0 ] ∈ S and [y, z 0 ] ∈ S. Let {z n } be the sequence given by z n+1 = f z n , ∀n ∈ N 0 . Now, we show x = y by proving {z n } → x and {z n } → y.
As [x, z 0 ] ∈ S, either xSz 0 or z 0 Sx. Suppose that xSz 0 . If x = z n 0 , for n 0 ∈ N 0 , then x = z n , for all n ≥ n 0 . Thus, {z n } → x. If x z n (∀n ∈ N 0 ), then xS z 0 . As S is f -closed, we have xS z n , for all n ∈ N 0 . Using condition (e), we have which on using triangle inequality and (E 1 ), gives rise The proof of the case z 0 Sx is similar. Also, by the same argument one can show that {z n } → y. This accomplishes the proof.

Corresponding multidimensional results
As consequences of Theorems 4.1-4.3, we provide here some existence and uniqueness multidimensional fixed point results.
Let B be a non-empty set. On the lines of [3], recall that a binary operation * on B is a mapping from B × B to B and a permutation π on B is a one-one mapping defined on B. In what follows, the following notations are useful: (i) N denotes a natural number ≥ 2. (iii) a binary operation * on I N can be represented by an N × N matrix throughout its ordered image in such a way that the first and second components run over rows and columns respectively, i.e., (iv) a permutation π on I N can be represented by an N-tuple throughout its ordered image, i.e., π = (π(1), π(2), ..., π(N)).
Example 5.1. The following selection of * ∈ B N represent the concept of fixed point of order N given by Berzig and Samet [32]: For more examples, one can see [3].  The following auxiliary results exhibit that multidimensional concepts can be interpreted in terms of F * . Proof. Observe that Proof. Observe that Then: Proof. The proofs of (i), (ii) and (iii) are trivial. To prove (iv) assume that W is S-precomplete. Let is an S-preserving Cauchy sequence in W. By our hypothesis there is On the other hand, assume that (W N , ∆ N ) is S N -precomplete. Let {x (n) } be an S-preserving Cauchy sequence in W. Then {U (n) = (x (n) , ..., x (n) )(N − times)} is an S-preserving Cauchy sequence in W N . Thus our assumption ensures the existence of a point U = (x, x, ..., Hence, W is S-precomplete. The proof of (v) is similar to (iv).

Applications to ordinary differential equations
As applications of our main results, we will examine in this section the existence and of a unique solution for the first-order periodic boundary value problem: where f : I × R → R is a continuous function and T > 0.
In what follows, C(I, R) denotes the space of all real valued continuous functions defined on I. Now, we recall the following definition which will be useful in the sequel: Definition 6.1. (i) A function x ∈ C 1 (I, R) is said to be a solution for (6.1) if it satisfies (6.1).
In the following results Nieto and Rodriguez-Lopez described some suitable conditions to ensure the existence of a unique solution of (6.1).
Theorem 6.1. [24] Consider problem (6.1) such that f is continuous and there exist γ > 0 and δ > 0 with γ < δ such that If (6.1) has a lower (or an upper) solution, then it has a unique solution.
[25] Consider problem (6.1) such that f is continuous and there exist γ > 0 and δ > 0 with γ < δ such that If (6.1) has a lower (or an upper) solution, then it has a unique solution.
Now, under a new condition which unify conditions (6.2) and (6.3), we prove the existence of a unique solution for the first-order periodic problem (6.1) in the presence of a lower solution. Theorem 6.3. Consider problem (6.1) such that f is continuous and non-decreasing in the second variable and there exist γ > 0 and δ > 0 with γ < δ such that , for all x, y ∈ R with x < y. (6.4) If (6.1) has a lower solution, then it has a unique solution.
Proof. Observe that problem (6.1) can be written in the following form: which is equivalent to the following integral equation: where Let us define d on M by: d(x, y) = sup t∈I |x(t) − y(t)|, ∀x, y ∈ M. Then the pair (M, d) forms a metric space which is complete so that every subspace of M is precomplete. Define a binary relation S on M = C(I, R) as follows: xSy ⇔ [x(t) ≤ y(t), for all t ∈ I], for all x, y ∈ M.
Now, define a mapping K : M → M by: Notice that x ∈ M is a fixed point of K iff it is a solution of (6.1).
Since every subspace of M is precomplete and since every precomplete space is S -precomplete, therefore K M is S -precomplete.
Let {x n } ⊆ M be an S -preserving sequence converging to x ∈ M. Then, for each t ∈ I, we have Since {x n (t)} ⊆ R is S -preserving sequence converging to x(t), therefore (6.6) implies that x n (t) < x(t), ∀ t ∈ I, n ∈ N. Observe that x n (t) x(t), for all t ∈ I, n ∈ N. As if x n 0 (t) = x(t), for all t ∈ I and some n 0 ∈ N, then x n = x n+1 , ∀ n ≥ n 0 , a contradiction. Thus, x n S x, ∀n ∈ N. Thus, S is d-self-closed.
Next, we prove that K is S -closed. Let x, y ∈ M be such that xS y. This amounts to saying that x(t) < y(t), for all t ∈ I. As f is nondecreasing in the second variable, we get (for all t ∈ I) f (t, y(t)) + δy(t) > f (t, x(t)) + δx(t). (6.7) As G(t, s) > 0, ∀t, s ∈ I, so (6.7) implies that for all t ∈ I. That is, K xS Ky so that S is K-closed. Now, let x, y ∈ M with xS y. Then x(t) < y(t), for all t ∈ I. Observe that  which shows that K satisfies the corresponding hypothesis (e) in Theorem 4.1 with E(t 1 , t 2 , ..., t 6 ) = t 1 − γ δ t 2 .
Multiplying both sides of this inequality by e δt , we have α(t)e δt ≤ [ f (t, α(t)) + δα(t)]e δt , t ∈ I, or α(t)e δt ≤ α(0) + so that αSK(α). Hence, Theorem 4.2 ensures the existence of a solution of (6.1). Finally, if x, y ∈ Fix(K), then z = max{x, y} ∈ M. As x ≤ z and y ≤ z, we have xSz and ySz so that Theorem 4.3 shows that the fixed point of K is unique. Hence, (6.1) has a unique solution.
Finally, we present an analogous of Theorem 6.3 in the presence of an upper solution.
Proof. Define a binary relation S on M as follows: xSy ⇔ [x(t) ≥ y(t), for all t ∈ I], for all x, y ∈ M.
Using analogous procedure of the proof of Theorem 6.3, one can analogously show that all requirements of Theorem 4.2 are fulfilled. Hence, Theorem 4.2 ensures the existence of a fixed point of K which is unique (due to Theorem 4.3). Thus, (6.1) admits a unique solution.