Holder’s Inequality ρ–Mean Continuity for Existence and Uniqueness Solution of Fractional Multi-Integrodifferential Delay System

We herein present the detailed results for the existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition involving several types of semigroups of bounded linear operators, which were established on probability density functions related with the fractional differential equation. Additionally, we present the necessary and sufficient conditions to investigate Schauder’s fixed point theorem with Holder’s inequality ρ–mean continuity and infinite delay parameter to guarantee the uniqueness of a fixed point.


Introduction
e importance of investigating the solution for fractional order derivatives in integrodifferential equations with a nonlocal initial or boundary condition lies in the fact that they include several classes of fractional order integrodifferential equations, as presented in studies on the existence and uniqueness of nonlocal initial fractional order integrodifferential equations in [1][2][3][4][5] and in some other studies with a nonlocal boundary condition [6], as well as fractional order differential equations involving integral conditions as a boundary condition, as found in multiple papers, including those by [7,8].
We consider the impulsive multifractional order integrodifferential equations with nonlocal conditions and finite delay as follows: c D α x(t) � Ax(t) + Bu(t) + f t, x t , I β 1 (x(t)), I β 2 (x(t)) , where c D α and I β 1 and I β 2 are the Caputo fractional derivative and fractional integration, respectively, of order 0 < α, β 1 , β 2 < 1, the state x(·) is defined on the Banach space X with the norm ‖·‖, u(·) is the control function in Banach space L 2 (J, V) of admissible control functions, and V is Banach space, where B: V ⟶ X is a linear bounded operator. T(t) { } t≥0 is a strongly continuous semigroup of operators on X generated by A. PC(J, X) � x: [0, b] { ⟶ X, x(t) is continuous at t ≠ t k and left continuous at t � t k and x(t + k )}, the impulsive functions I i : D ⟶ X, i � 1, 2, . . . , s, 0 < t 1 < t 2 < · · · < t s < t s+1 � b. e functions f: J × D × X × X ⟶ X and g: PC(J, X) ⟶ X are continuous and satisfies some assumptions, where D � ω: is continuous for all t ∈ J except for a finite number of points t i at which ω(t + i ) and ω(t − i ) exist and ω(t i ) � ω(t − i )}, the impulsive functions I i : D ⟶ X, i � 1, 2, . . . , s, { }. e scientific problem of these types of fractional integrodifferential equations that have not easy solvability even some time there are difficult to study also their behaviours for their solutions on the certain space and the description of the equation terms, however it is need more effort and practise.when the solvability of these problems have been investigated will be cover all a particular results from this problem such as [3,10,11].
In this paper, we first present the basic theory of existence and uniqueness of mild solution for multifractional order impulsive integrodifferential control equations with a nonlocal condition and infinite delay parameter (1) by defining several types of semigroups of linear-bounded operators established on probability density functions defined on (0, ∞) and consider the necessary and sufficient estimators conditions, which play an important role in investigating Schauder's fixed point theorem with Holder's inequality ρ-mean continuity to guarantee the existence and uniqueness of a fixed point.

Preliminaries
Definition 1 (see [17]). Let AC[0, ∞) be a space of absolutely continuous function. en, the fractional integral for a function g ∈ AC[0, ∞) of order α is defined as follows: where Γ(·) is the gamma function.
Definition 3 (see [18]). e family of bounded linear operators T(t), 0 ≤ t < ∞ defined on the Banach space X is a semigroup if T(0) � I. Here, I is the identity operator on X and T(t + s) � T(t)T(s), for every t, s ≥ 0.
Definition 4 (see [18]). Let T(t) be a semigroup, then T(t) is called strongly continuous and is denoted by Definition 5 (see [18]). e domain of the linear operator Here, A is the generator of the semigroup T(t).
Remark 1 (see [19]). We define the norm for measurable functions n: J ⟶ R as follows: where μ(J) is the Lebesgue measure on J. e Banach space of all Lebesgue measurable functions is L p (J, R).
Lemma 2 (see [19] Lemma 3 (see [20]). If D is a bounded, closed, and convex subset of a Banach space X and F: D ⟶ D is completely continuous, then F has a fixed point in D.
Lemma 4 (see [21]). Let ω ∈ L p (J, X), 1 ≤ p < +∞, then lim In order to guarantee the existence and uniqueness of mild solution of problem (1), we introduce the following assumptions.

(A 1 ) T(t)
{ } t≥0 is a strongly continuous semigroup generated by A, that is, a compact operator for every t ≥ 0, and ‖T(t)‖ ≤ M 1 .
and there is a positive function μ(·) ∈ L p (J, R + ) for some p, 1 < p < ∞ such that (A 4 ) e function g: PC(J, X) ⟶ X is a continuous compact function, satisfying (A 5 ) e operator I i : D ⟶ X, i � 1, 2, . . . , s, is continuous and there is nondecreasing function where q a positive constant. erefore, D q is a closed, bounded, and convex subset in D.

Definition 6.
e function x(·) ∈ PC(J, X) is the mild . . , s, the restriction of x(·) to the interval J i , i � 1, 2, . . . , s, is continuous and the following equation is satisfied: and Φ α is the probability density function on (0, ∞), with

Theorem 1. Assume that the hypotheses (A 1 )-(A 5 ) with the
, the impulsive multifractional order integrodifferential equations with nonlocal conditions (1) has a mild solution x ∈ D q , where q is a positive constant, for every control u ∈ L 2 (J,V).
Proof. For any positive constant q and x ∈ D q , from (A 1 ) and (A 4 ), we have Now, we define the operators F on D q as follows: For ∅ ∈ D, we define ∅ ∈ PC by Let Clearly, since operator F has one fixed point and thus G too. So, we have where y ∈ PC. Let

Journal of Mathematics
In the following, we will prove that G has a fixed point on D q , and then we get F has a fixed point on D q .

□
Step 1. ‖Gy‖ ≤ q, then ‖Fx‖ ≤ q, where y ∈ PC and x ∈ D q . By using a method similar to the one used in (15) and (16), we have Hence, if ‖Gy‖ ≤ q, then ‖Fx‖ ≤ q, where y ∈ PC and x ∈ D q .
Step 2. G is a completely continuous operator.

Conclusion
e existence and uniqueness of multifractional order impulsive integrodifferential equations with nonlocal conditions and infinite delay using Schauder's fixed point theorem required certain types of semigroups defined on probability density functions and Holder's inequality ρ-mean continuity as well as some necessary and sufficient estimators conditions that play an important role in guaranteeing the solution.
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.