On a New Stochastic Space of Solution for the Volterra-Type Summable Equations of Fuzzy Functions

Tis article explains the sufcient requirements for a newly constructed stochastic space by Poisson-like matrices and weighted variable exponent sequence spaces of fuzzy functions, as well as the ideals of their operators, for the Kannan contraction operator to have a unique fxed point. Moreover, we investigate some examples and the numerous applications of solutions to Volterra-type summable equations of fuzzy functions


Introduction
Summable equations come up in many situations in the critical point theory for nonsmooth energy functionals, mathematical physics, control theory, biomathematics, diference variational inequalities, fuzzy set theory [1], probability theory [2], and trafc problems, to mention but a few.In particular, Volterra-type summable equations are known to be of great importance in investigating dynamical systems [3] and stochastic processes [4,5].Some instances are in the felds of granular systems, sweeping processes, oscillation problems, control problems, decision-making problems [6], and so on.Te solution of summable equations is contained in a certain sequence space.So, there is a great interest in mathematics to construct new sequence spaces, see [7].Mursaleen and Noman [8] examined some new sequence spaces of nonabsolute type related to the spaces l p and l ∞ , and Mursaleen and Bas ¸ar [9] constructed and investigated the domain of the C 1 matrix in some spaces of double sequences.Mustafa and Bakery [10] introduced the concept of the private sequence space of fuzzy functions (pssff).Suppose R is the set of real numbers and N 0 is the set of nonnegative integers.We have constructed the space (P qw F ) ς equipped with a defnite function ς by the domain of Poisson-like operator defned in l F ((q b ),(w b )) , where the Poisson-like matrix, P � (c ba ), is defned as follows: where μ ∈ (0, 1] and q a , w a ∈ (0, ∞), for all a ∈ N 0 .For 0 < r < 1, Matloka [11] introduced the r-level set of a fuzzy real number δ as follows: Let us denote R([0, 1]) to the space of every δ r is normal, compact, fuzzy convex, and upper semicontinuous.For δ ∈ R([0, 1]), one has ]), J: N 0 ⟶ R([0, 1]), and r: N 0 ⟶ R([0, 1]), for every J∈ P qw F .Consider the following Volterra-type summable equations of fuzzy functions [12]: and presume L: (P qw F ) ς ⟶ (P qw F ) ς , for certain functional ς, is defned as follows: Mustafa and Bakery [10] investigated the unique solution of (4) of  in the operators' ideal (or in short O.I) formed by a weighted binomial matrix in the variable exponent sequence space of extended s-fuzzy functions.Bakery and Mohammed [13] explained Kannan nonexpansive mappings on the variable exponent Cesàro sequence space of fuzzy functions.We can use the newly constructed stochastic space to explore more spaces of solutions for the fuzzy fractional evolution equations; see the following interesting articles: Abuasbeh et al. [14], Niazi et al. [15], and Iqbal et al.'s [16] studies.In this paper, geometric and topological properties are used to defne the space of fuzzy functions ((P qw F ) ς ) and the ideal space of its operators.Te fxed point for the Kannan contraction operator is demonstrated, and its prequasi operator ideal is proven to be confrmed in this space.In the fnal section of this article, we discuss the various applications of solutions to Volterra-type summable equations of fuzzy functions and demonstrate the practical relevance of our fndings.

Structures of (P qw F ) ς and Its O.I
A few topological and geometric characteristics of (P qw F ) ς and the corresponding O.I have been examined here.
(2) A B B : the space of all bounded linear operators from A into B. (3) B A : the space of all bounded linear operators from A into itself.(4) A F B : the space of fnite rank linear operators from A into B. (5) A R B : the space of approximable operators from A into B. (6) A K B : the space of compact operators from A into B. (7) B: the ideal of bounded operators between any two Banach spaces.(8) F: the ideal of fnite rank operators between any two Banach spaces.(9) R: the ideal of approximable operators between any two Banach spaces.
where w u > 0 and q u , p u ∈ R, for every a ∈ N 0 .
Proof.First is to show that (P qw F ) ς is a p-m-pssff.

Journal of Mathematics
Defnition 20 (see [21]).A function Υ ∈ [0, ∞) V is said to be a p-qN on the ideal V when the next parts are established. ( Theorem 21. (see [21]).Every qN on the ideal V is a p-qN.
We have ofered some properties of the ideal generated by our fuzzy space and extended s− numbers in this part, presuming that the parts of Teorem 15 are satisfed.

Theorem 22. Te conditions of Teorem 15 are sufcient only for
as one has 6

□
Theorem 24.If w (1)  b < w (2)  b and q (2)  b ≤ q (1) b , for each b ∈ N 0 , then Proof.Suppose J∈ A [B s Hence, J ∉ A [B s Clearly, A [B s From Dvoretzky's theorem [23], one has A/W b and Fixing 0 ≤ b ≤ q, one gets So, for some ξ ≥ 1, then As q ⟶ ∞, one has a contradiction.So, A and B both cannot be infnite-dimensional whenever

Journal of Mathematics
Lemma 27 (see [25]).Presume J∈ A B B and J ∉ A R B , then X ∈ B A and Y ∈ B B so that YJXe i � e i , for each i ∈ N 0 .

□
Defnition 31 (see [25]).A Banach space H F is defned as simple if B H F is a unique nontrivial closed ideal.

Kannan's Contraction Fixed Points
Supposing that the parts of Teorem 15 are established, the existence of a fxed point of the Kannan contraction operator acting on this new space and its associated p-q ideal are presented with some numerical examples to show our results.
Te Banach fxed point theorem, as presented in reference [26], provided mathematicians with a means to extend the applicability of contraction operators by generalization, for instances, the Kannan contraction operator [27], Kannan operators in modular vector spaces [28], and Kannan p-qN contraction operator [29].
Defnition 34 (see [13]).A p-qN-pssff ς on H F holds the Fatou property (or in short FPr); presume for every We will use the following notations: for all j∈ P qw F .

□
Defnition 37 (see [29]).A mapping G: Theorem 38.Te space G has a unique fxed point, whenever G: Hence, for all a, b ∈ N 0 so that a > b, then show that G(q) � q.Since ς 1 verifes the FPr, one obtains hence G(q) � q.So, q is a fxed point of G. To show the uniqueness of the fxed point, let one has two diferent fxed points i, q∈ Proof.Given Teorem 38, we obtain a unique fxed point q of G. Hence, □ Defnition 40 (see [13]).Presume Proof.Let k be not a fxed point of G, then Gk ≠ k.From parts (g2) and (g3), we get lim As G is Kannan ς 2 -contraction, then 12

Journal of Mathematics
By q i ⟶ ∞, there is a contradiction.Ten, k is a fxed point of G.For the uniqueness of k, if one gets two diferent fxed points k, r∈ So, A is Kannan ς 1 -contraction, as ς 1 verifes the FPr.By Teorem 38, A has a unique fxed point θ.Presume

c at θ and T q h
has a T q j h   which converges to θ.From Teorem 41, the vector θ is the only fxed point of A.

We will use in this part
Defnition 42 (see [10]).A function Theorem 43.Te function Υ does not hold the FPr.
Terefore, Υ does not hold the FPr.

Theorem 46. If
the unique fxed point of L, if the following parts are confrmed: Proof.Let U be not a fxed point of L, then one gets LU ≠ U. From the parts (h2) and (h3), one gets lim As L is Kannan Υ-contraction operator, then Journal of Mathematics Journal of Mathematics Presume So, L is not Υ-s.c at K (0) .Hence, L is not continuous at K (0) .

Applications
In this section, a solution in (P qw F ) ς 1 to ( 1) is examined such that the parts of Teorem 15 are established.
Theorem 47. System (1) holds one and only solution in Tis completes the proofs.

Journal of Mathematics
From Teorem 47, system (8) has a unique solution in Theorem Tis completes the proofs.

Conclusion
We explained a few topological and geometric properties of (P qw F ) ς of the class B s (P qw F ) ς and of the class (B s (P qw F ) ς ) λ in this article.Te Kannan contraction operator on these spaces is analyzed, and the possibility of a fxed point is considered.We ran many numerical experiments to ensure our theories were correct.Fuzzy functions with nonlinear uncertainty equation implementations are also investigated.Tis novel fuzzy function space is used to investigate the fxed points of all contraction operators, providing a new universal solution space for a wide variety of stochastic nonlinear dynamical systems.

( 19 )
F: the space of fnite sequences of fuzzy numbers.(20) I: the space of all monotonic increasing sequences of positive reals.(21) D: the space of all monotonic decreasing sequences of positive reals.(22) D F : the space of each monotonic decreasing sequence of fuzzy functions.(23) Γ: the Banach space of one dimension.

F
) ς is a p-qB-pssff, if the next parts are established.

□ Theorem 16 .F
Te space B s P qw is an O.I, when the parts of Teorem 15 are established.
Some properties of s− type (P qw F ) ς are explained in the next theorem in view of Teorems 16 and 17.Theorem 18 (a) s− type (P for each b ∈ N 0 .If T b is the quotient operator from A onto A/W b , I b is the identity operator on l b 2 and J b is the natural embedding operator from M b into B. Presume m b is the Bernstein numbers [24].
and for each b ∈ N 0 , then and for each b ∈ N 0 , then 