Solvability of a Parametric Fractional-Order Integral Equation Using Advance Darbo G-Contraction Theorem

: The existence of a parametric fractional integral equation and its numerical solution is a big challenge in the ﬁeld of applied mathematics. For this purpose, we generalize a special type of ﬁxed-point theorems. The intention of this work is to prove ﬁxed-point theorems for the class of β − G, ψ − G contractible operators of Darbo type and demonstrate the usability of obtaining results for solvability of fractional integral equations satisfying some local conditions in Banach space. In this process, some recent results have been generalized. As an application, we establish a set of conditions for the existence of a class of fractional integrals taking the parametric Riemann–Liouville formula. Moreover, we introduce numerical solutions of the class by using the set of ﬁxed points.


Introduction
Approximately, a measure of noncompactness is a function demarcated on the class of all nonempty and bounded subsets of a definite metric space where it is identical to zero on the entire class of comparatively compact sets [1]. A survey of theory and applications of measures of noncompactness is presented in [2]. The normal measures of noncompactness are deliberated, and their possessions are associated. Some consequences regarding normal measures of noncompactness in altered spaces are offered. Additionally, the authors introduced some applications of the measure of noncompactness notion to functional equations involving nonlinear integral equations of arbitrary orders, implicit arbitrary integral equations and q-integral equations of arbitrary orders. The measure of noncompactness plays very significant role in the theory of fixed points and applications. The term measures of noncompactness were initially formulated in the elementary paper of Kuratowski [3]. Furthermore, G. Darbo [4] defined condensing operator and established a fixed-point theorem that involved the idea of a measure of noncompactness which is abundant of applications in functional analysis, integral equations differential equations approximation theory (see for example [4]. Owing to numerous applications of fixedpoint theory in proving the existence theorems, this theory has been considered to be an evergreen and considered to be indispensable tool in nonlinear analysis. The Darbo fixed-point theorem extends both the Banach and the Schauder fixed-point theorems. In 2012, Wardowski [5] defined F-contraction and generalized Banach contraction principle in various aspects. Furthermore, Jleli et al. [1] define the F-contraction of Darbo type and established a fixed-point theorem.
In our study, we state and prove fixed-point theorems which are generalized Jleli et al. [1] results. Furthermore, as an application, we demonstrate the applicability of our main result in establishing the existence of solutions of an integral equation of fractional order of the form: where u, g, h and θ satisfies certain conditions.

Methods
Let us recall some notations, definitions and theorems which will be used throughout this paper. In what follows E denotes the Banach space with the norm . and throughout this article we use the following notations; We proceed with an axiomatic definition of measure of noncompactness; Definition 1 (Axiomatic Definition of Measure of Noncompactness [6][7][8]). A function σ : M E −→ R + is called M.N.C provided it fulfills the following axioms: (Generalized Cantor's intersection theorem) If W n ∈ M E for n = 1, 2, · · · is a decreasing sequence of closed subsets of E and lim W n is nonempty.
The family defined in axiom (i) is called the kernel of the M.N.C and denoted by kerσ. In fact, by the virtue of axiom (vi) we have σ(Ω ∞ ) ≤ σ(Ω n ) for any n, thus σ(Ω ∞ ) = 0. This yields that Ω ∞ ∈ kerσ.
Theorem 1 (Schauder's fixed-point theorem [9]). Let Ω be the member of the class N.B.C.C of a Banach space E, then every continuous and compact mapping on Ω has at least one fixed point in Ω.
The Darbo s fixed-point theorem with respect to a M.N.C σ can be stated as below.
Theorem 2 (Darbo s fixed-point theorem [4]). Let Ω be the member of the class N.B.C.C of a Banach space E and T be the continuous self-mapping defined on every nonempty subset W of Ω such that σ(T(W)) ≤ λσ(W) for some λ ∈ [0, 1). Then T has at least one fixed point in Ω.
if it satisfies the following conditions; (G 1 ) G is non-decreasing; (G 2 ) For each sequence α n ⊂ (0, η) of positive real numbers lim G(s) = ln(s) for all t ∈ R + .

Definition 3 (S -function).
A function τ : R + → R is said to be a S -function if it satisfies the inequality; lim inf Example 2. The mapping τ : R + → R + described by the rule τ(t) = (2t) −1 ∀ t ∈: R + is an example of S -function.
Definition 4 (Φ-function). A function φ : R + → R + is said to be a Φ-function if it fulfills the following assumptions;

Theorem 3.
Let Ω be the member of the class N.B.C.C and T be the self-mapping defined on Ω.
The mapping T is said to be a φ-condensing if for some φ ∈ Φ and every nonempty subset W of Ω.

Theorem 4 ([10]).
Let Ω be member of the class N.B.C.C of a Banach space E and T is continuous self-operator on Ω. If T is Darbo-type G-contraction for any nonempty subset W ⊂ Ω, then T has a fixed point in the set Ω.
for any τ > 0 is an example of β µ -function.

Results
In this section, we establish new fixed-point theorems for self-mappings in the setting of measure of noncompactness. Therefore, to obtain our first theorem, we use the following class of functions.

Definition 8.
Let Ω be member of the class N.B.C.C and T is continuous self-operator on Ω. The operator T on Ω is called Darbo-type βG-contraction if ∃ G ∈ G and β ∈ β µ and µ = sup o<t<η G(t) > σ(E) such that; for any W ⊂ Ω with σ(W) > 0, σ(TW) > 0, where σ is a measure of noncompactness defined in E.
Next, we establish the existence of at least one fixed point.

Theorem 5.
Let Ω be member of the class N.B.C.C of a Banach space E and T is Darbo-type βG-contraction on Ω, for G ∈ G and β ∈ β µ , then T has at least one fixed point in Ω.
Proof. The proof begins with the construction of the sequence W n of nonempty, closed and convex subset of W such that the following relation holds: TW n ⊂ W n ⊂ W n−1 for all n ∈ N Let W 0 = W, we construct a sequence W n by the rule W n+1 = ConvP(W n ) for n ∈ {0} ∪ N. For n = 0, we can easily check that TW 0 ⊂ TW ⊂ W = W 0 . Now assume that the rule holds for k = 1, 2, 3, · · · n. Then, by the definition of W n we deduce that TW n ⊂ W n implies W n+1 = conv(TA n ) ⊂ A n , Therefore TW n+1 ⊂ TW n ⊂ W n+1 . If there exist a positive integer K ∈ N such that σ(W K ) = 0, then W K is pre-compact set. Since T(W K ) ⊆ conv(TW K ) = W K+1 ⊆ W K , i.e., T is a self-operator on W K . Then Theorem 2.1 concludes that T has a fixed point in W K ⊂ W. On the other hand, we assume that σ(W n ) > 0, ∀n ≥ 1 and prove that σ(W n ) → 0 as n → +∞. Now using assumption v) of Definition 1 we have, G(σ(W n )) for all n ∈ N. i.e., G(σ(W n )) is decreasing sequence of real numbers. Since the sequence G(σ(W n )) decreasing sequence hence it must be bounded above and may or may not be bounded below.
Claim that the sequence G(σ(W n )) is unbounded below. We will prove the claim by assuming the contradiction that the sequence G(σ(W n )) is bounded below. Since the sequence is decreasing and bounded below hence it has a convergent sub-sequence say G σ W n k , and a finite real number r such that G σ W n k → r as k → +∞.
Since β ∈ Γ µ we obtain r = −∞ which is a contradiction. This implies that G(σ(W n )) unbounded below and so lim n→+∞ G(σ(W n )) = −∞. So, from (G 2 ) of Definition 2 we obtain that σ(W n ) → 0 as n → +∞. On the flip side, if G(σ(W n )) is unbounded then obviously σ(W n ) → 0 as n → +∞. Hence from (vi) of the Definition 1 , the countable interaction W ∞ = ∞ n=1 W n is a nonempty set which is convex & closed invariant under T and relatively compact. Hence applying Theorem 1 to the set W ∞ = ∞ n=1 W n we obtain desired result.
then β is member of the family β µ .
Proof. Assume that β(r n ) → 1 then β ∈ Γ µ if r n → −∞. Assume the contradiction that r n is bounded below, hence it has convergent sub-sequence say r n k such that r n k → r 0 as k → +∞, where r 0 is some finite real number. Now, since ψ is upper semi-continuous, we obtain; Hence the sequence r n is unbounded below, it follows that β ∈ β µ .

Definition 9.
Let Ω be the member of the class N.B.C.C and T is continuous self-operator on Ω. The operator T on Ω is called Darbo- Next, we establish the existence of unique fixed point.

Theorem 6.
Let Ω be the member of the class N.B.C.C of a Banach space E and T be continuous self-operator on Ω. If T is Darbo-type ψG-contraction for G ∈ G and ψ ∈ Ψ µ then T has a fixed point in Ω.
Proof. The proof begins with the construction of the sequence W n of nonempty, convex & closed subset of W, such that the sequence W n validate following relation: we construct a sequence W n by the rule W n+1 = ConvT(W n ) for n ∈ {0} ∪ N. For n = 0 we can easily check that TW 0 ⊂ TW ⊂ W = W 0 . Now assume that the rule holds for k = 1, 2, 3, · · · n. Then, by the pattern of W n we deduce that i.e., T is a self-operator on W K . Then Theorem 2.1 concludes that T has a fixed point in On the flip side, if σ(W n ) > 0, ∀n ≥ 1 then by the axiomatic definition of M.N.C we have, Hence, we remain with the inequality, From Equation (3) we assure that G(σ(W n )) is decreasing sequence of real numbers. Since the sequence G(σ(W n )) decreasing sequence hence it must be bounded above and may or may not be bounded below.
Claim that the sequence G(σ(W n )) is unbounded below. We will prove the claim by assuming the contradiction that the sequence G(σ(W n )) is bounded below. Since the sequence is decreasing and bounded below hence it has a convergent sub-sequence say G σ W k , and a finite real number r such that G σ W n k → r as k → +∞. By Equation (3) we have, keep in mind that ψ is lower semi-continuous and apply limit as k → +∞ we obtain, Since ψ(r) < r ∀ r ∈ (−∞, µ), hence by the Definition 5 we obtain r = −∞ which a contradiction. This implies that G(σ(W n )) unbounded below and so lim n→+∞ G(σ(W n )) = −∞. So, from (G 2 ), we obtain that σ(W n ) → 0 as n → +∞. On the flip side, if G(σ(W n )) is unbounded then obviously σ(W n ) → 0 as n → +∞. Hence from (vi) of the Definition 1 , the countable intersection W ∞ = ∞ n=1 W n is a nonempty set which is convex & closed invariant under T and relatively compact. Hence applying Theorem 1 to the set W ∞ = ∞ n=1 W n we obtain the desired result.

Corollary 1 ([1]).
Let Ω be member of class B.N.C.C of the Banach space E. Let a self-operator T on Ω is of Darbo-type G-contraction if there exist G ∈ G and τ ∈ S such that τ(σ(W)) + G(σ(TW)) G(σ(W)), for any W ⊂ Ω with σ(W), σ(TW) > 0, where σ is a M.N.C defined in E. Then T has a fixed point in the set Ω.
Proof. Let us define ψ(t) = t − τ(t) we obtain the required result from Theorem 6

Corollary 2.
Let Ω be member of class B.N.C.C of the Banach space E. Let a self-operator T be a of Darbo-type ψG-contraction, where ψ ∈ Ψ µ and G is continuous and non-decreasing function. Then for φ ∈ Φ such that T is a φ-contraction.
Proof. Since the function G is monotonic and continuous function, hence ∃ G −1 : G(R + ) → R + inverse of G, which is also monotonic. Now using the Definition 9 we can have, For every W ⊂ Ω with σ(W) = 0, G ∈ G and ψ ∈ Ψ µ , if T is Darbo-type ψGcontraction then for φ ∈ Ψ, T is φ contraction.

Discussion
Presently, there are several results available in the literature about study of existence and behavior of solutions of various types of fractional-order integral equations. The fractional-order integral equations have numerous applications in porous media, control theory, rheology, viscoelasticity, elector chemistry, electromagnetism fluid dynamics (see [11][12][13][14]).
In this article, we will use the measures of noncompactness in BC(R + ); the space of all bounded and continuous functions defined on R + [7]. Let z = z(t) be any real valued bounded and continuous function defined on R + , then the norm on BC(R + ) is defined as; ||z|| BC(R + ) = sup{|z(t)| : t ≥ 0}. Furthermore, we define the term ω ∞ (W) in the following fashion: Next, define the quantity B(W) as: Finally, we will define the quantity which satisfies the axioms of M.N.C in the following manner; We have, σ(W) = ω ∞ (W) + B(W) is the M.N.C in the space BC(R + ) and satisfies regularity, monotonically, invariant under closure, invariant under convex hull, generalized Cantor intersection theorem etc. [7].
To do this, fix M > 0 and > 0. Choose the numbers s, t ∈ [0, M] with |t − s| ≤ . For s < t we obtain; Next, from the expression (8) we obtain Since, ℵ ||z|| M, (g) → 0 as → 0, hence we infer that the function (Iz) is continuous on [0, M]. As M is arbitrary, hence we can say that (Iz) is continuous on R + .
Additionally, from the assumption (b) we deduce the following expression; for an arbitrary δ > 0. By the virtue of assumption (c) we have ℵ ||z|| M, (g) → 0 as → 0. Thus, the expression (10) we conclude that (Hz) is continuous on [0, M], and hence continuous on the interval R + . Hence, representation (7) and assumption (a) implies that (Tz) is continuous on the interval R + . Furthermore, let us choose z ∈ BC(R + ) and arbitrarily t, τ ∈ R + , we will derive the following expression; The above estimation shows that the function (Tz) is bounded on the interval R + . Since The operator (Tz) is bounded and continuous on R + therefore we conclude that the operator T transforms the space BC(R + ) into itself. Moreover, form the assumptions we deduce that there exist number r 0 > 0 such that H maps the ball B r 0 into itself.
Next, we will show that the operator T is continuous on the ball B r 0 . To do this, let us fix an arbitrary positive number and choose z, y ∈ B r 0 such that ||z − y|| ≤ . Let us choose an arbitrary t ∈ R + then we can obtain the following expression by the virtue of assumptions (c) & (d); where ω 2 (h, ε) = sup{|h(t, τ, z(τ)) − h(t, τ, y(τ))| : t, τ ∈ R + , z, y ∈ R, ||z − y|| ≤ ε}.
From assumption (e), it follows that ω 2 (h, ε) → 0 as ε → 0. From Equations (7)-(9) we conclude that (Iz) is continuous on the ball B r 0 . The operator Iz is continuous and bounded on the ball B r 0 which implies that the operator I transform the ball B r 0 into itself.
Let us fix an arbitrary nonempty subset W of the ball B r 0 . Choose z ∈ W and fix the positive numbers M and such that t, s ∈ [0, M] and |t − s| ≤ . Then from the above estimated expressions (7)-(10) we obtain Applying the supreme to both sides we obtain Applying ε → 0 we obtain At length as M → ∞ we have with the following expression, In what follows let us take a nonempty set W ⊂ Br 0 . Then, for arbitrary t ∈ R + and z, y ∈ BC(R + ) such that ||z − y|| ≤ using the assumptions (b), (c) & (e) we can derive the following expression, Hence, we can easily deduce the following inequality, Hence, from assumption ( f ) and expression (5) we can deduce the following inequality, From Equations (6), (15) and (16) we obtain By the virtue of assumption ( f ) and properties of functions G & ψ we derive the following expression, G(σ(TW)) ψ(G(σ(W))) where G : (0, η) → R by G(y) = ln(y) and ψ : (−∞, 0) → (−∞, 0) by ψ(w) = w − τ. Linking the expression (18) with Theorem 6 of Section 2 and assuming the properties of G & ψ we obtain the desired result. In the view of the definition of measure of noncompactness we conclude that the solution of an integral Equation (1) has a finite limit at infinity. Now we will discuss an illustrative example for the obtained result.

Numerical Example
Consider the following fractional-order integral equation in the Banach space BC(R + ); Suppose that α = 1/4 then we obtain Notice that the integral Equation (20) is particular case of integral Equation (1). Indeed, if we replace α = 1 4 and u(s) = as In fact, we have functions u(s) = as s 2 +1 and g(s, z(s)) = 1 1+s 6 arctan(|a| + z(s)) satisfies assumption (a) and (b). The function g(s, z(s)) satisfies assumption (b) with ρ(s) = 1 It has been easily seen that all the above-defined functions are bounded on R + among them ψ, ϑ vanishes at infinity i.e., lim To find the set of fixed points of Equation (20), using (c), we have the approximated solution for a = 1 z(s) = 2.93379s 1/4 z(s) arctan(z(s) + 1) A calculation implies that the set of fixed points f ix α (a, s) is f ix 1/4 (1, s) = {s 1 = 0, s 2 = 0.188507, s 3 = 1.35044}.

Conclusions
In our current work, we defined βG-contraction and ψG-contraction of Darbo type and proved corresponding fixed-point theorems using M.N.C. Furthermore, the fixedpoint theorem proved in Section 2 is applied to demonstrate the existence of a solution of fractional-order integral equation. At the end, an example is given to validate the result. We indicate that the values of the fixed-point increase whenever the values of α increase in (0, 1]. Moreover, the set of fixed points imposed the periodicity and stability of the fractional integral Equation (20). All figures are presented with the help of Mathematica 11.2.  The subfamily of M E consisting only relatively compact sets. co(Ω), co(Ω) The convex hull and closed convex hull of Ω respectively.