Existence solution of a system of differential equations using generalized Darbo's fixed point theorem

In this paper, we proposed a generalized of Darbo's fixed point theorem via the concept of operators $ S(\bullet; .) $ associated with the measure of noncompactness. Using this generalized Darbo fixed point theorem, we have given the existence of solution of a system of differential equations. At the end, we have given an example which supports our findings.


Introduction
The measure of noncompactness (MNC) performs an important character in real world problems. First of all, the fundamental paper of Kuratowski [1] in 1930 open up a new direction of MNC to solve diffent type of Functional equations, which comes from the diffent real life problems. Using the notion of MNC, Darbo [2] in 1955 ensure that the endurance of fixed points, which is obtained by the generalization of Schauder fixed point theorem (SFPT) and banach contraction principle. Many authors using the notion of MNC generalize Darbo fixed point theorem (DFPT) which ensure that the endurance of fixed point to solve various kind of integal or differentail equations. Up to now, many authors have been published several papers using the notion of generalization of DFPT and MNC [3][4][5][6][7][8][9][10][11][12][13][14].
Our purpose of present paper is to extend the DFPT and we aaply our obtained results to find the existence of solutions of the functional differential equations.
At the beginning we provide concepts, notations, definitions and the preliminaries, which will be used all over the present paper.

Definition and preliminaries
The set of real numbers is symbolize by R, R + = [0, ∞) and the set of natural numbers by N. Let (Ξ, . ) be real Banach spaces. If Ω is a nonempty subset of Ξ thenΩ and ConvΩ, symbolize the closure and convex closure of Ω respectively. Also, let M Ξ symbolize the set of all nonempty and bounded subsets of Ξ and N Ξ is the subset of all relatively compact sets.
We are going to define the Concept of operator S (•; .) which was introduced by Altan and Turkoglu [16].
Definition 2.2. Let A(R + ) be the set of fuctions f : R + → R + and let Z be the set of functions S (•; .) : A(R + ) → A(R + ), which fulfills the following constraints: Theorem 2.1. (Schauder) [17] A mapping ∆ : Ω → Ω which is compact and continuous has at least one fixed point for a nonempty, bounded, closed and convex (NBCC) subset Ω of a Banach space Ξ.
DFPT is generalize by resting the compactness of Schauder's mapping and theorem is known as SFPT.
Theorem 2.2. (Darbo) [18] Let ∆ : Ω → Ω be a continuous mapping and χ is an MNC. Then for any nonempty subset ℘ of Ω, there exists a k ∈ [0, 1) having the inequality Then the mapping ∆ have a fixed point in Ω.
Isik et al. [10] introduce a function f to generalize the Banach contraction, we find various type of contraction mapping. Theorem 2.3. Let ∆ : Ω → Ω be a continuous self-mapping, where (Ω, ρ) is a complete metric space. Then for all γ, δ ∈ Ξ there exists a mapping f : Then ∆ contains a unique fixed point.
Parvenah et al. [10] generalized DFPT as follows: Theorem 2.4. Let ∆ : Ω → Ω be a continuous operator defined on a NBCC subet Ω of Ξ having the inequality for all ℘ ⊂ Ξ. Consequently ) for all ℘ ⊂ Ξ. Therefore the Darbo Theorem is a specific case of contraction mapping of Theorem (2.4).

Main results
Let us recall an important theorem in this work which extends DFPT by taking the concept of S (h; .). Theorem 3.1. Let (Ξ, . ) be a Banach space. Suppose ∆ : Ξ → Ξ is a continuous, nondecreasing and bounded mapping fulfills the following inequality Since ℘ n is a closed and bounded subset in Ξ and Taking the limit as n → ∞ on both the sides of this inequality, we have By the virtue of (iii) of Definition S (h; .), we get and therefore lim n→∞ But for any > 0, 0 π(τ)dτ > 0, then χ(℘ n ) → 0 as n → ∞.
Now since ℘ n is nested sequence, by the definition of (MNC) of (M 6 ), we conclude that ℘ ∞ = ∩ ∞ n=1 ℘ n is NBCC of Ξ. Also we aware that ℘ ∞ ∈ kerχ. Therefore ℘ ∞ is compact and invariant under the mapping ∆. Therefore by the SFPT, ∆ has a fixed point in ℘ ∞ .
It is a generalization of the result given by Parvenah et al. [10].
where χ be an MNC in Ξ and Θ j is the natural projections of Z into Ξ j for j = 1, 2, 3.
Corollary 1. Let ∆ : C × C × C → C be a continuous function defined on a NBCC subset C of Ξ in such a way that Then ∆ has a TFP.
Hence, as the number r 0 we can take r 0 = 75. Therefore, all the assumptions of Theorem 4.1 are satisfied. Hence the system of Eq (5.1) have at least one solution which belongs to {C[0, T ]} 3 space.

Conclusions
The present paper concentrated on multiple FPT which is based on the generalization of DFPT via MNC. In this work, by using the concept of operators we extend DFPT by using MNC. We demonstrate the endurance of TFP by our extended DFPT and MNC. At the last we yield an example which fulfills our findings.