Existence of local fractional integral equation via a measure of non-compactness with monotone property on Banach spaces

In this paper, we discuss fixed point theorems for a new χ -set contraction condition in partially ordered Banach spaces, whose positive coneK is normal, and then proceed to prove some coupled fixed point theorems in partially ordered Banach spaces. We relax the conditions of a proper domain of an underlying operator for partially ordered Banach spaces. Furthermore, we discuss an application to the existence of a local fractional integral equation.


Introduction and preliminaries
A measure of non-compactness (MNC) for the first time was given by Kuratowski [1]. It is combined with the algebraically and analytically studies for establishing the existence of nonlinear problems [2]. The fractional calculus is a subject of a long history and has gained great interest in different fields of applied science, and many authors considered this topic [3][4][5][6][7].
Let (X, · ) be an infinite dimensional Banach space and θ be its zero element. B(ϑ, ζ ) will denote the closed ball with center ϑ are radius ζ and B ζ will stand for B(θ, ζ ). Moreover, M X will denote the family of nonempty bounded subsets of X and N X is its subfamily consisting of all relatively compact sets. Definition 1.1 ([8]) A mapping μ : M X → R + is said to be a measure of non-compactness (MNC, for short) in X if it satisfies the following conditions (Y, Y 1 , Y 2 ∈ M X ): (1 • ) ker μ := {Y ∈ M X : μ(Y) = 0} = ∅ and ker μ ⊂ N X , © The Author(s) 2020. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
is a decreasing sequence of nonempty closed sets in M X and if lim n→∞ μ(Y n ) = 0, then the set Y ∞ = ∞ n=1 Y n is nonempty and compact.
A map α : M X → R + is said to be a Kuratowski MNC [1] if We denote by (X) a nonempty, bounded, closed and convex set on Banach space X.
The following extensions of the topological Schauder fixed point theorem and classical Banach fixed point theorem were proved by Darbo (DFPT, in short) in 1955.
Then we can conclude that F has a fixed point.
With the above discussion in mind, an attempt has been made to give a monotone version of Lemma 1.4 with the relaxed conditions of domain of an underlying operator into partially ordered Banach spaces. To achieve the proposed results in partially ordered Banach spaces, we define a notion of MNC. Then we use this notion to prove some FPTs for χ -set contraction condition in partially ordered Banach spaces whose positive cone K is norm. We will relax the conditions of bounds, closed and convexity of the domain of operator at the expense of the operator being monotone and bounded. Next, we use the obtained FPTs to establish the existence of the solution of local fractional integral equation.

FPTs
Let X be a Banach space with the norm · whose positive cone is defined by K = {x ∈ X : x ≥ 0}. (X, · ) is a partially ordered Banach space with the order relation induced by cone K.
Denote by a collection of continuous and strictly increasing function ω : R + → R + . We now discuss our results in partially ordered Banach spaces.

Theorem 2.1
Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: for all bounded subset in X, where χ denotes the arbitrary MNC, ∈ H(R + ), (•; ·) ∈ , ψ ∈ , ω ∈ . If ∃ an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a fixed point * and the sequence {F n ς 0 } of successive iterations converges monotonically to * .
Proof Assume ς 0 ∈ X and define a sequence {ς n } ⊂ X by Since F is non-decreasing and ς 0 Fς 0 , we have Denote B n = conv{ς n , ς n+1 , . . .} for n ∈ N * . By (2.2) and (2.3), each B n is a bounded and closed subset in X and Taking the limit n → ∞ in (2.5), we have by the virtue of ψ ∈ By the virtue of (iii) of Definition 1.1, we get and therefore Hence, for every > 0 there exists an n 0 ∈ N such that β(B n ) < , ∀n ≥ n 0 .
From this we conclude that B n 0 and consequently B 0 is a compact chain in X. Hence, {ς n } has a convergent subsequence. Applying the monotone property of F and the normality of cone K , the whole sequence {ς n } = {F n ς 0 } converges monotonically to a point, say * ∈ B 0 . Finally, from the continuity of F, we get On different setting of functions ∈ H(R + ), (•; ·) ∈ , ω : R + → R + satisfying the condition (2.1) in Theorems 2.1, we can get some new DFPTs. For example, if we set first ω(t) = 0 and secondly ψ(ζ ) = λζ (λ ∈ (0, 1)) and finally = identity map with ( ; ζ ) = ζ , then we have following DFPTs, respectively. Theorem 2.2 Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: for all bounded subset B in X, where χ denotes the arbitrary MNC, ∈ H(R + ). (•; ·) ∈ , ψ ∈ . If ∃ an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a fixed point * and the sequence {F n ς 0 } converges monotonically to * .

Theorem 2.3
Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: If ∃ an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a fixed point * and the sequence {F n ς 0 } of successive iterations converges monotonically to * . Theorem 2.4 Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: If ∃ an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a fixed point * and the sequence {F n ς 0 } of successive iterations converges monotonically to * .
If we take diam(B) = diameter of B, then we have the following.

Proposition 2.5
Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: If there exists an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a fixed point * and the sequence {F n ς 0 } of successive iterations converges monotonically to * .
Proof Theorem 2.1 and Proposition 3.2 [12] claim the existence of a F-invariant nonempty closed convex subset B with diam(B ∞ ) = 0, that is, B ∞ has a singleton element, hence we have a fixed point of F = ∅.
The following is the generalized classical fixed point result derived from Proposition 2.3.

Theorem 2.6
Let (X, · , ) be a partially ordered Banach space, whose positive cone K is normal. Suppose that F : X → X is a continuous, non-decreasing and bounded mapping satisfying the following contraction: for all ζ , ξ ∈ X, where ψ ∈ , ω ∈ . If there exists an element ς 0 ∈ X such that ς 0 Fς 0 , then F has a unique fixed point * and the sequence {F n ς 0 } of successive iterations converges monotonically to * .
Proof Let χ : M X → R + be a set quantity defined by the formula χ(X) = diam X, where diam X = sup{ ζξ : ζ , ξ ∈ X} stands for the diameter of X . It is easily seen that χ is a MNC in a space X in the sense of Definition 1.1. Therefore from (2.11) we have Thus following Proposition 2.3, F has an unique fixed point.

Coupled FPTs
In this section, we prove some coupled fixed point theorems. We begin our discussion by recalling some definitions and notions.
Proof We consider the following map G : X 2 → X 2 : Due to the assumption, G is also a continuous and bounded mapping, having the monotone property. Following Lemma 3.3, for B = B 1 × B 2 , we define a new MNC as where B i , i = 1, 2, denote the natural projections of B. Now let B = B 1 × B 2 ⊂ X 2 be a nonempty bounded subset. Due to (3.1) we conclude that that is, Next, we show that there is a 0 ∈ B such that 0 G( 0 ). Indeed, there exist two elements 0 , σ 0 ∈ X such that 0 G( 0 , σ ) for any σ ∈ X and σ 0 G(σ 0 , ) for any ∈ X, set 0 = ( 0 , σ 0 ). Then, by the definition of G, we have Theorem 2.1 implies that G has a fixed point, and hence G has a coupled fixed point.
Proof We consider the map G : X 2 → X 2 defined by Then G is a continuous and bounded mapping, having the monotone property.
For any B = B 1 × B 2 , we define a new MNC in the space X 2 as where B i , i = 1, 2, denote the natural projections of B. Now let B ⊂ X 2 with B = B 1 × B 2 be a nonempty bounded subset. We can conclude That is, Next, we show that there is a 0 ∈ B such that 0 G( 0 ). There exist elements 0 , σ 0 ∈ X such that 0 G( 0 , σ ) for any σ ∈ X and σ 0 G(σ 0 , ) for any ∈ X, set 0 = ( 0 , σ 0 ). Then, by the definition of G, we have Theorem 2.1 implies that G has a fixed point, and hence G has a coupled fixed point.

Fractals
Recently, a fractional derivative without singular kernel with its details was given in [15,16]. The local fractional derivative of K( ) of order 0 < γ ≤ 1 is inserted by and we have the integral operator as follows: The operator in (4.1) is well defined and it is represented to the classical fractional calculus. The function K is called local fractional continuous at 0 if for all ε > 0 there is a κ that satisfies We denote the space of all local fractional continuous functions by C γ . For K ∈ C γ , the local fractional integral is defined by where (see [17]) The goal of this part is to study the existence and uniqueness of the generalized fractional integral equation For this investigation, we shall apply Theorem 2.6. For our setting, we make the following assumptions: 1] satisfying that there occurs a positive constant > 0 such that (A2) There is a positive constant L satisfying (4.4) Set K(ς) = K(ς, 0) and the ball B r = { ∈ C γ [0, 1] : ≤ r}. Now we subdivide the operator into two operator 1 and 2 on B r as follows: where λ is a positive constant. Since K is a non-decreasing and continuous function, this leads to being also a non-decreasing and continuous mapping.
The proof is as follows.
Step 1. (Boundedness) := 1 + 2 ∈ B r for every ∈ B r . In view of [A1], we have This implies that Hence, is bounded, continuous and non-decreasing in B r .
Step 2. is ψ-contraction mapping (condition (2.11)). For any , η ∈ B r , we obtain This gives Define two continuous functions ϕ and ψ as follows: From the last inequality, we obtain In view of [A2], the operator is a ψ-contraction mapping. Taking the sup. over B r , we have Thus, obeys all the conditions of Theorem 2.6. That is, has a unique fixed point in B r .  where = ( 1 , 2 ) and 1 (0) = 2 (0) = 0 , has at least one fixed point. • All the above fixed point theorems are applicable for both convex and non-convex domains.