On Generalization of Petryshyn’s Fixed Point Theorem and its application to the product of n -nonlinear integral equations

: Regarding the Hausdor ff measure of noncompactness, we provide and demonstrate a generalization of Petryshyn’s fixed point theorem in Banach algebras. Comparing this theorem to Schauder and Darbo’s fixed point theorems, that it enables us to skip demonstrating closed, convex, and compactness properties of the investigated operators. We employ our fixed point theorem to provide the existence findings for the product of n -nonlinear integral equations in the Banach algebra of continuous functions C ( I a ), which is a generalization of various types of integral equations in the literature. Lastly, a few specific instances and informative examples are provided. Our findings can successfully be extended to several Banach algebras, including AC , C 1 , or BV -spaces.


Introduction
Different types of integral equations are crucial to the study of economics, biology, mechanics, mathematical physics, control theory, vehicular traffic, population dynamics, and other fields (cf.[1,2]).
Based on this methodology, we first offer and demonstrate a generalization of Petryshyn's F.P.T. connected with the Hausdorff M.N.C., which is a generalization of numerous F.P.T. types, including Darbo's, Schauder's, and traditional Petryshyn's F.P.T.s [11].The benefit of the proposed F.P.T. is that it enables us to skip demonstrating closed, convex, and compactness properties of the investigated operators.These enable us to investigate various varieties of differential and integral equations under a weaker and more general set of presumptions.
Second, we employ the presented F.P.T. to solve the product of n-nonlinear Volterra integral equations, which are a generalization of the classical and quadratic integral equations of the form for n ≥ 2, in the Banach algebra C(I a ).
In [13] the authors utilized the F.P.T. approach to establish the existence of C[a, b]-solutions of the equation The authors in [14] presented an extension of Darbo F.P.T. in Banach algebra to solve the q-integral equation A generalization of Darbo F.P.T. was used to investigate the existence results for the equation in ideal spaces (not be Banach algebras) in [15] see also [16][17][18].This study focuses on applying a generalization of Petryshyn's F.P.T. to solve a general form of product-type integral problems in the Banach algebra C(I a ).

Preliminaries
We employ the following symbols in the sequel: We recall some theorems & definitions that are required for the sequel.For more information about the properties of the M.N.C. see [11,20].The space C[0, a] yields to a Banach space under the norm ∥z∥ = sup{|z(v)| : v ∈ I a } and we shall write the modulus of continuity of a function z ∈ C(I a ) as (2.1) Definition 2.5.[21] Let P : E → E be a continuous map.P is said to be a contraction map if for all Z ⊂ C(I a ) be bounded, P(Z) be bounded and Moreover, P is said to be condensing (densifying) map if Note that a contraction map yields condensing (densifying) but not vice versa.

Main Results
In order to solve equation (1.1), we first give a fixed point z ∈ Br of the problem where Moreover, P is said to be condensing (densifying) map if Proof.Since Z and P(Z) are bounded sets in C(I a ) and by using Theorem 2.3, we have The above inequality with 0 < k < 1 2 finishes the proof.□ Note that a contraction map related to the M.N.C. µ yields condensing (densifying) with 0 < k < 1 2 but not vice versa.
The following Proposition can be presented and proven by us.
Proposition 3.2.Suppose that the operators P i : Br → E, i = 1, • • • n and that: (B2) There exist k i > 0 such that P i fulfill: then the set Fix(P) of fixed points of P in Br is nonempty.
2 is a general form of the F.P.T. presented in [14,15].Now, we will apply Proposition 3.2 to check the solvability of Eq. (1.1) under the assumptions: (A1) Assume that α i , β i , γ i : I a → I a , and φ i : , where there exist constants k i > 0, s.t.
(A3) There exists M i ≥ 0 and r 0 ≥ 0 such that Proof.First, let us define the operators P i : B r 0 → C(I a ), as follows Next, we will divide the proof into some steps according to Proposition 3.2.
Step 1.The operator P is well defined on C(I a ).Obviously from assumptions (A1) and (A2), we have P : C(I a ) → C(I a ).
Step 2. We will demonstrate that the operators P, P i , i = 1, • • • , n are continuous on the ball B r 0 .Take arbitrary z, y ∈ B r 0 and ε > 0 s.t.∥z − y∥ ≤ ε, then for v ∈ I a , we obtain where From assumption (A2), the functions h i = h i (v, s, z) are uniformly continuous on [0, a] × [0, B] × R, we indicate that ω(h i , ε) → 0 as ε → 0. Thus, the operators P i , i = 1, • • • , n are continuous on B r 0 and consequently, the operator P = n i=1 P i is continuous on B r 0 .
Step 3. We will demonstrate that the operator P fulfills the densifying condition in view of µ.Take arbitrary ρ > 0 and z ∈ M ⊂ C(I a ) is bounded set and for v 1 , v 2 ∈ I a s.t.v 1 ≤ v 2 with v 2 −v 1 ≤ ρ, we obtain where From the above relations we get Let ρ → 0, we get ω(P i z, ρ) ≤ 2k i ω(z, ρ).
This yields the following estimation: Therefore, From assumption (A4), we get P is a condensing map with K < 1 2 .
Step 5.The proof is completed when Proposition 3.2 is applied.□

Applications and examples
To demonstrate the value of our results, we provide a few examples and instances of integral equations.

Conclusion and Perspective
In this article, a generalization of Petryshyn F.P.T. and the MNC idea were used to analyze the solutions for products of n-nonlinear integral equations in the Banach algebra C(I a ).The presented F.P.T. is a generalization of Darbo, Schauder, and the classical Petryshyn F.P.T. Examples are provided to demonstrate the usefulness of our findings.The upcoming work in this field shall consider different Banach algebras, including AC, C 1 , or BV spaces.

Definition 3 . 1 .
are known operators.Definition 2.5 should be rewritten in view of the M.N.C. µ in C(I a ).The operator P : C(I a ) → C(I a ) is said to be a contraction map if for all Z ⊂ C(I a ) be bounded set, P(Z) be bounded set and