Generalized C ψβ − rational contraction and fixed point theorem with application to second order differential equation

In this article, generalized C β rational contraction is defined and the existence and uniqueness of fixed points for self map in partially ordered metric spaces are discussed. As an application, we apply our result to find existence and uniqueness of solutions of second order differential equations with boundary conditions.


Introduction
From last 15 years, several authors have studied and derived various fixed point results for many contractions in partially ordered sets.Ran and Reurings [1] derived a fixed point result on partially ordered sets in which contractive condition assumed to be hold on comparable elements.After that, author in [9,10] deduced some results to get fixed point for monotone, non-decreasing operator with partially ordered relation on a set Y without using the continuity of maps.They also discussed few applications of their main findings and gave existence as well as uniqueness theorem ordinary differential equation of first order and first degree with restricted boundary conditions.Number of results after that have been investigated to establish fixed point in partially ordered metric spaces (for more detail see [2,4,7,8,11,12,13,15,18,19,21,22]).
In 1975, Jaggi [23] and Das and Gupta [24] derived some fixed point results for rational type contraction.There exist several results in the literature for self and pair of maps satisfying rational expression in different spaces [20,25].
In 2007, Suzuki [16] introduced the weaker C-contractive condition and proved some fixed point theorems.The existence as well as uniqueness of fixed point of such types of operator have also been extensively studied in [3,17].such that If {u n } is not a Cauchy sequence in Y then there exist an > 0 and sequences of positive integers (m k ) and In this paper, we first define a generalized C ψ β − rational contraction and then prove the existence and uniqueness of fixed points for self monotone map.We also consider a partially ordered set Y with comparable elements, and a complete metric d with set Y to deduce our main result.As application, we give an existence as well as uniqueness theorem for ordinary differential equation of second order and first degree with restricted boundary conditions.

Fixed point result with partial order
We define generalized C ψ β − rational contraction as follows: Definition 2.1.A mapping f on a metric space (Y, d) is said to satisfy generalized where Main finding of this article is the following result.
Theorem 2.1.Let (Y, d, ) be a partially ordered complete metric space and let f : Y → Y be a non-decreasing, monotone map satisfying generalized Also assume that: (4) For every u, v ∈ Y, there exists z ∈ Y, such that u z and v z.
If there exists u 0 ∈ Y such that u 0 f u 0 , then f has a unique fixed point in Y .
Proof.Let u 0 ∈ Y satisfy u 0 f u 0 .We define a sequence {u n } as follows: If u n = u n+1 for some n ∈ N , then, clearly M (u n , u n+1 ) = 0 and so, u n is the fixed point of f .So, assume that u n = u n+1 for all n ∈ N. Let a n = d(u n , u n+1 ).Then, clearly a n > 0. Since u 0 f u 0 = u 1 and f is non-decreasing, then where .
and hence From (7), we have (8) gives a contradiction to condition (3) and hence Since ψ and β are continuous functions, therefore Similarly we get Thus, we get a sequence {d(u n , u n+1 )} of functions, which is non-increasing and r ≥ 0 such that However, by taking lim n→∞ on both side of (8), we get ψ(r) ≤ β(r), which is a contradiction to (2).Thus we have r = 0, and hence Assume on contrary that sequence {u n } is not Cauchy.Then for every > 0, we can find subsequences of positive integers m k and n k , where Also for this > 0, the convergence of sequence {d(u n , u n+1 )} implies, there exists where, On using Lemma 1.2 and letting k → ∞ in ( 12) and ( 13), we obtain ψ( ) ≤ β( ), that's a contradiction to (3) and hence by Lemma 1.1, we get = 0.This contradicts the assumption that > 0. Therefore our assumption is wrong.Hence {u n } is Cauchy.Since Y is complete, so {u n } converges with all its subsequences to some limiting value, say z ∈ Y .Now assume for every n ∈ N and Then we have this is a contradiction.Hence we must have d(u n , z) ≥ 1 2 d(u n , u n+1 ) or d(u n+1 , z) ≥ 1 2 d(u n+1 , u n+2 ), for all n ∈ N .Thus for a sub-sequence {n k } of N , we obatin where Both, on letting k → ∞, and using (15) in ( 14), we get To establish uniqueness, we suppose on contradictory that for all u, v ∈ Y , u = f u and v = f v provided u = v.Now we discuss following two case for both elements.
Case 1.Without loss of generality, suppose that u v are comparable.Then Thus from (2) and Lemma 1.1, we get d(u, v) = 0, i.e, u = v.
Case 2. Assume that u and v are not comparable then from ( 4), there exists some z ∈ Y comparable to u and v such that where Hence, from (17), Consequently, we have ψ(d(u, w) ≤ β(d(u, w)).
On using Lemma 1.1, we have d(u, w) = 0. Similarly, we can obtain d(v, w) = 0.This implies that u = v.This completes the proof of Theorem 2.1.
Theorem 2.2.Let (Y, d, ) be a partially ordered complete metric space and let f : Y → Y be a non-decreasing, monotone map such that for all u, v ∈ Y, and where ψ ∈ Ψ, a i ≥ 0, a i < 1, for all i = 1, 2, 3 and Also assume that, for every u, v ∈ Y, there exists z ∈ Y , such that u z and v z.If there exists u 0 ∈ Y such that u 0 f u 0 , then f has a unique fixed point in Y .
Proof.Given that f : Y → Y be monotone, nondecreasing map such that for all u, v ∈ Y, and Since all a i ≥ 0 and a i < 1, for all i = 1, 2, 3, then Rest of the proof follows directly from main result (Theorem 2.1).
Also assume that for every u, v ∈ Y , there exists z ∈ Y , such that u z and v z.If there exists u 0 ∈ Y such that u 0 f u 0 , then f has a unique fixed point in Y .
Corollary 2.2.Let (Y, d, ) be a partially ordered complete metric space and let f : Y → Y be a non-decreasing map such that for all u, v ∈ Y, Since G(ω, θ) > 0, for ω ∈ L. This proves that H is also weakly increasing mapping.Also, for all u, v ∈ E with u ≥ v implies that and so, in term of metric This implies It is easy to calculate that Also, G(ω, θ)f (θ, 0)dθ ≥ 0.
Thus one by one all assumptions of Theorem 2.1 are satisfied and therefore, the function H has a unique non negative solution.

Conclusion
In this manuscript, we have first defined a generalized C ψ β − rational contraction and then derived our main result Theorem 2.1.Some consequence results (Corollary 2.1, 2.2) and Remarks 2.1, 2.2 flaunted that our result is a proper generalization and extension of some previous existing results.As an application of our main result, we have presented an example to find the existence and uniqueness of solutions of second order boundary value problem.