# Holder's Inequalityρ–Mean Continuity for Existence and Uniqueness Solution of Fractional Multi-Integrodifferential Delay System

Authors: Sameer Qasim Hasan
Journal: Journal of Mathematics
 (2020)

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## Abstract

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 inequalityρ–mean continuity and infinite delay parameter to guarantee the uniqueness of a fixed point.


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## Body

## 1. Introduction

The 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 [1–5] 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].Additionally, existence and uniqueness investigated in several studies for impulsive fractional order integrodifferential problems are presented in [3, 9–14]. Integral nonlocal conditions for impulsive fractional order integrodifferential equations are also presented in [15, 16].We consider the impulsive multifractional order integrodifferential equations with nonlocal conditions and finite delay as follows:(1)Dαcxt=Axt+But+ft,xt,Iβ1xt,Iβ2xt,t∈J=0,b,t≠ti,(2)Δxt=ti=Iixti,i=1,2,…,s,(3)xt=∅t+gxt,t∈−r,0,where Dαc and Iβ1 and Iβ2 are the Caputo fractional derivative and fractional integration, respectively, of order 0<α,β1,β2<1, the state x⋅ is defined on the Banach space X with the norm ⋅, u⋅ is the control function in Banach space L2J,V of admissible control functions, and V is Banach space, where B:V⟶X is a linear bounded operator. Ttt≥0 is a strongly continuous semigroup of operators on X generated by A. PCJ,X=x:0,b⟶X, xt is continuous at t≠tk and left continuous at t=tk and xtk+, the impulsive functions Ii:D⟶X, i=1,2,…,s, 0<t1<t2<⋯<ts<ts+1=b. The functions f:J×D×X×X⟶X and g:PCJ,X⟶X are continuous and satisfies some assumptions, where D=ω:−r,0⟶X, ωt is continuous for all t∈J except for a finite number of points ti at which ωti+ and ωti− exist and ωti=ωti−, the impulsive functions Ii:D⟶X, i=1,2,…,s, 0<t1<t2<⋯<ts<ts+1=b, the jump of a function x at ti is denoted by Δxti, which is defined by Δxti=xti+−xti−.Forx∈PCJ,X, we denote xt as the function in D given by xtθ=xt+θ, θ∈−r,0, and ∅ is a given function in D.Furthermore,PC−r,b,X is a Banach space of all piecewise continuous functions with the norm x=supxt:t∈−r,b.The 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.


## 2. Preliminaries

Definition 1 (see [17]).
LetAC0,∞ be a space of absolutely continuous function. Then, the fractional integral for a function g∈AC0,∞ of order α is defined as follows:(4)Iαgt=1Γα∫0tgst−s1−αds,t>0,0<α<1,where Γ⋅ is the gamma function.Definition 2 (see [17]).
The Caputo derivative for a functiong∈AC0,∞ of order α is defined as follows:(5)Dαcgt=1Γ1−α∫0tg′st−sαds,t>0,0<α<1,where Γ⋅ is the gamma function.Definition 3 (see [18]).
The family of bounded linear operatorsTt,0≤t<∞ defined on the Banach space X is a semigroup if T0=I. Here, I is the identity operator on X and Tt+s=TtTs, for every t,s≥0.Definition 4 (see [18]).
LetTt be a semigroup, then Tt is called strongly continuous and is denoted by C0 on a Banach space X if limt↓0Ttx−IxX=0.Definition 5 (see [18]).
The domain of the linear operatorA:DA⊆X⟶X is defined as follows. The domain DA=x∈X:limt↓0Ttx−x/t exists and Ax=dTt/dtt=0=limt↓0Ttx−x/t for x∈DA. Here, A is the generator of the semigroup Tt.Remark 1 (see [19]).
We define the norm for measurable functionsn:J⟶ℝ as follows:(6)nLpJ=∫Jntpdt1/p,1≤p<∞,infμJ¯=0supt∈J−J¯nt,p=∞,where μJ¯ is the Lebesgue measure on J¯. The Banach space of all Lebesgue measurable functions is LpJ,ℝ.Lemma 1 (see [20]).
Iff1∈LpJ,ℝ and f2∈LqJ,ℝ, then f1f2≤f1f2 for p,q≥1 and 1/q+1/p=1, and the inequality is called Holder’s inequality.Lemma 2 (see [19]).
Leth:0,b⟶ℝ be a measurable function; if h is Lebesgue integrable, then h is called the Bochner integrable.Lemma 3 (see [20]).
IfD 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]).
Letω∈LpJ,X, 1≤p<+∞, then limt⟶0∫0bωt+r−ωtpdt=0, and ωs=0 for s∉J.In order to guarantee the existence and uniqueness of mild solution of problem (1), we introduce the following assumptions.Assumptions.
(A1) Ttt≥0 is a strongly continuous semigroup generated by A, that is, a compact operator for every t≥0, and Tt≤M1.(A2) Let M2 and M3 be the positive constants such that B≤M2 and u≤M3.(A3) The function f:J×D×X×X⟶X satisfies that for any t∈J, ft,⋅,⋅,⋅:D×X⟶X is a continuous function and for all ∅,x∈D×X, f⋅,∅,x,y:J⟶X is strongly measurable, where(7)D=τ:−r,∞⟶X:τtis continous every where except for a finite number ofpointstiat whichτti+andτti−exist andτti=τti−,and there is a positive functionμ⋅∈LpJ,ℝ+ for some p, 1<p<∞ such that(8)ft,x,y,z≤μtx+y+z,x,y,z∈PCJ,X,t∈J.(A4) The function g:PCJ,X⟶X is a continuous compact function, satisfying(9)gx≤Lx+L′,x∈PCJ,X,for constants,L,L′>0.(A5) The operator Ii:D⟶X, i=1,2,…,s, is continuous and there is nondecreasing function Ni:ℝ+⟶ℝ+ such that(10)Iix≤NixD,i=1,2,…,s,x∈D.
LetDq=x∈PCJ,X,x≤q, where q a positive constant. Therefore, Dq is a closed, bounded, and convex subset in D.Definition 6.
The functionx⋅∈PCJ,X is the mild solution of problem (1), if xt=∅t+gxt on t∈−r,0 and Δxt=ti=Iixti,i=1,2,…,s, the restriction of x⋅ to the interval Ji,i=1,2,…,s, is continuous and the following equation is satisfied:(11)xt=Sαt∅+gx+∫0tt−sα−1Tαt−sfs,xs,Iβ1xτ,Iβ2xτ+Busds+∑i=1sSαt−tiIixti,where(12)Sαt=∫0∞ΦαθTtαθdθ,(13)Tαt=α∫0∞θΦαθTtαθdθΦαθ=1αθ−1−1/αwαθ−1/αwαθ=1π∑n=1∞−1n−1θ−αn−1Γαn+1n!sinαnπ,θ∈0,∞,and Φα is the probability density function on 0,∞, with properties Φαθ≥0 and(14)∫0∞Φαθdθ=1,∫0∞θΦαθdθ=1Γα+1.


## 3. Main Results

Theorem 1.
Assume that the hypotheses (A1)–(A5) with the condition M1αbα−1/p/Γ1+αp−1/α−1p+p−1p−1/pσLpJ,R++bβ1+p−1/p2/Γβ1+1σLp2J,R++bβ2+p−1/p2/Γβ1+1σLp2J,R++∑i=1sLi<1 holds. Thus, for every nonlocal initial condition xt=∅t+gxt, t∈−r,0, the impulsive multifractional order integrodifferential equations with nonlocal conditions (1) has a mild solution x∈Dq, where q is a positive constant, for every control u∈ L2J,V.Proof.
For any positive constantq and x∈Dq, from (A1) and (A4), we have (15)Sαt∅+gx=∫0∞ΦαθTtαθ∅+gxdθ≤∫0∞ΦαθTtαθ∅+gxdθ≤M1∅+Lq+L¯.
Therefore,∫0∞ΦαθTtαθ∅+gxdθ exists.
From (A1), (12)–(14), and (A3), we obtain(16)∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds≤∫0t∫0∞αθΦαθt−sα−1Tt−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτs+∅^τ+Busds≤αM1Γ1+α∫0tt−sα−1σsq+∅^PC+1Γβ1∫0ss−τβ1−1q+∅^PCdτ+1Γβ2∫0ss−τβ2−1q+∅^PCdτ+M2M3ds≤αM1q+∅^PCΓ1+α∫0tt−sα−1σsds+1Γβ1+1∫0tt−sα−1σssβ1ds+1β2Γβ2+1∫0tt−sα−1σssβ2ds+αM1Γ1+α∫0tt−sα−1M2M3ds≤αM1q+∅^PCΓ1+α∫0tt−sα−1p/p−1dsp−1/p∫0tσspds1/p+1Γβ1+1∫0tt−sα−1p/p−1dsp−1/p∫0tsβ1σspds1/p+1β2Γβ2+1∫0tt−sα−1p/p−1dsp−1/p∫0tsβ2σspds1/p+αM1M2M3Γ1+α∫0tt−sα−1ds≤αM1q+∅^PCΓ1+αp−1α−1p+p−1p−1/ptα−1p+p−1/pσLpJ,R++1Γβ1+1p−1α−1p+p−1p−1/ptα−1p+p−1/p∫0tsβ1p2/p−1dsp−1/p2∫0tσsp2ds1/p2+1Γβ1+1p−1α−1p+p−1p−1/ptα−1p+p−1/p∫0tsβ2p2/p−1dsp−1/p2∫0tσsp2ds1/p2+αM1M2M3Γ1+α∫0tt−sα−1ds≤αM1q+∅^PCΓ1+αp−1α−1p+p−1p−1/ptα−1p+p−1/pσLpJ,R++1Γβ1+111+β1p2/p−1p−1/p2tβ1p2+p−1/p2σLp2J,R++1Γβ1+111+β2p2/p−1p−1/p2tβ2p2+p−1/p2σLp2J,R++M1M2M3bαΓ1+α≤αM1q+∅^PCbα−1/pΓ1+αp−1α−1p+p−1p−1/pσLpJ,R++bβ1+p−1/p2Γβ1+1σLp2J,R++bβ2+p−1/p2Γβ1+1σLp2J,R++M1M2M3bαΓ1+α,for all t∈J. Thus, ∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds is Lebesgue integrable with respect to s∈0,t, for all t∈J.
By Lemma1, we get ∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds is Bochner’s integrable, for s∈0,t, t∈J.
Now, we define the operatorsF on Dq as follows:(17)Fxt=∅t,t∈−r,0,Sαt∅+gx+∫0tt−sα−1Tαt−sfs,xs,Iβ1xτ,Iβ2xτ+Busds+∑i=1sSαt−tiIixti,t∈J.
For∅∈D, we define ∅^∈PC by(18)∅^t=∅t,t∈−r,0,Sαt∅+gx,t∈J.
Letxt=yt+∅^t, the operator G:PC⟶PC defined as(19)Gyt=0,t∈−r,0,∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds+∑i=1sSαt−tiIiyti+∅^ti,t∈J.
Clearly, since operatorF has one fixed point and thus G too. So, we have(20)Gyt=∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds+∑i=1sSαt−tiIiyti+∅^ti,t∈J.where y∈PC. Let(21)q=αM1∅^PCbα−1/p/Γ1+αp−1/α−1p+p−1p−1/pσLpJ,R++bβ1+p−1/p2/Γβ1+1σLp2J,R++bβ2+p−1/p2/Γβ1+1σLp2J,R++M1M2M3bα/Γ1+α+M1∑i=1sLi∅^PC⋅1−M1αbα−1/p/Γ1+αp−1/α−1p+p−1p−1/pσLpJ,R++bβ1+p−1/p2/Γβ1+1σLp2J,R++bβ2+p−1/p2/Γβ1+1σLp2J,R++∑i=1sLi−1.
In the following, we will prove thatG has a fixed point on Dq, and then we get F has a fixed point on Dq.Step 1.
G
y
≤
q, then Fx≤q, where y∈PC and x∈Dq.
By using a method similar to the one used in (15) and (16), we have(22)Gyt=∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds+∑i=1sSαt−tiIiyti+∅^tiGyt=∫0t∫0∞αθΦαθt−sα−1Tt−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds+∑i=1s∫0∞ΦαθTt−tiαθIiyti+∅^ti≤αM1q+∅^PCbα−1/pΓ1+αp−1α−1p+p−1p−1/pσLpJ,R++bβ1+p−1/p2Γβ1+1σLp2J,R++bβ2+p−1/p2Γβ1+1σLp2J,R++M1M2M3bαΓ1+α+M1∑i=1sLiq+∅^PC=q.
Hence, ifGy≤q, then Fx≤q, where y∈PC and x∈Dq.Step 2.
G is a completely continuous operator.
Firstly, we will show thatG is continuous on Dq.
For anyyn,y⊆Dq, n=1,2,…, with limn⟶∞yn⟶y=0, we get limn⟶∞ynt=yt, t∈J.
By (A3), we obtain(23)limn⟶∞ft,ytn+∅^t,Iβ1ynτ+∅^τ,Iβ2ynτ+∅^τ=ft,yt+∅^t,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ,fort∈J.
Then,(24)sups∈0,bfs,ysn+∅^s,Iβ1ynτ+∅^τ,Iβ2ynτ+∅^τ−fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ⟶0,n⟶∞,and from (A5), we obtain(25)limn⟶∞Iiynti+∅^ti=Iiyti+∅^ti,i=1,2,…,s,sups∈0,bIiynsi+∅^si−Iiysi+∅^si⟶0,n⟶∞.
Additionally, from (A2), we obtain(26)limn⟶∞unt=ut,sups∈0,buns−us⟶0,n⟶∞.
Subsequently, we have that(27)Gyn−Gy≤∫0tt−sα−1Tαt−sfs,ysn+∅^s,Iβ1ynτ+∅^τ,Iβ2ynτ+∅^τ−fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τds+∫0tt−sα−1Tαt−sBun−uds+∑i=1sSαt−tiIiynti+∅^ti−Iiyti+∅^ti≤∫0t∫0∞αθΦαθt−sα−1Tt−sαθfs,ysn+∅^s,Iβ1ynτ+∅^τ,Iβ2ynτ+∅^τ−fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τdθds+∫0t∫0∞αθΦαθt−sα−1Tt−sαθBun−udθds+∑i=1s∫0∞ΦαθTt−tiαθIiynti+∅^ti−Iiyti+∅^tidθ≤αM1bαΓ1+αsups∈0,bfs,ysn+∅^s,Iβ1ynτ+∅^τ,Iβ2ynτ+∅^τ−fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+αM1M2bαΓ1+αsups∈0,bun−u+M1∑i=1sIiynti+∅^ti−Iiyti+∅^ti.
Hence, by (16), (22), and (24), we have(28)Gyn−Gy⟶0,n⟶∞.
Thus,G is continuous.
Now, we will prove thatGy,y∈Dq is equicontinuous on Ji, i=1,2,…,s. For any y∈Dq and 0≤t1<t2≤b, we have(29)Gyt2−Gyt1≤∫0t1∫0∞αθΦαθt2−sα−1−t1−sα−1Tt2−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∫0t1∫0∞αθΦαθt1−sα−1Tt2−sαθ−Tt1−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∫t1t2∫0∞αθΦαθt2−sα−1Tt2−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∑i=1s∫0∞ΦαθSt2−tiα−St1−tiαIiyti+∅^tidθ=αI1+I2+I3+∑i=1s∫0∞ΦαθSt2−tiα−St1−tiαIiyti+∅^tidθ,where (30)I1=∫0t1∫0∞θΦαθt2−sα−1−t1−sα−1Tt2−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+BusdθdsI2=∫0t1∫0∞θΦαθt1−sα−1Tt2−sαθ−Tt1−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+BusdθdsI3=∫t1t2∫0∞θΦαθt2−sα−1Tt2−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+BusdθdsI1≤M1Γ1+α∫0t1t2−sα−1−t1−sα−1×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds≤M1q+∅^PCΓ1+α∫0t1t2−sα−1−t1−sα−1pds1/pσLp/p−1J,R++1Γβ1+1∫0t1t2−sα−1−t1−sα−1pds1/p∫0t1sβ1p2/p−12dsp−1/p2∫0t1σsp2/p−1dsp−1/p2+1Γβ1+1∫0t1t2−sα−1−t1−sα−1pds1/p∫0t1sβ2p2/p−12dsp−12/p2∫0t1σsp2/p−1dsp−1/p2+M1M2M3Γ1+α∫0t1t2−sα−1−t1−sα−1ds≤M1q+∅^PCΓ1+ασLp/p−1J,R++bβ1+p−12/p2Γβ1+111+β1p2/p−12p−12/p2σLp2/p−1J,R++bβ2+p−12/p2Γβ2+111+β2p2/p−12p−12/p2σLp2/p−1J,R+×∫0t1t2−sα−1−t1−sα−1pds1/p+M1M2M3Γ1+α∫0t1t2−sα−1−t1−sα−1ds.
By Lagrange’s mean value theorem, we havet2−sα−1−t1−sα−1⟶0 as t2⟶t1 for s∈J. Using Lemma 4, we obtain ∫0t1t2−sα−1−t1−sα−1pds⟶0 as t2⟶t1. Thus, limt2⟶t1Ι1=0.
Now, fort1=0, 0<t2≤b, clearly, I2=0 for t1>0 and ε>0 is sufficiently small. Therefore,(31)I2≤∫0t1−ε∫0∞θΦαθt1−sα−1Tt2−sαθ−Tt1−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∫t1−εt1∫0∞θΦαθt1−sα−1Tt2−sαθ−Tt1−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds≤q+∅^PCΓ1+αsups∈0,t1−εTt2−sαθ−Tt1−sαθ∫0t1−εt1−sα−1σsds+∫0t1−εt1−sα−1σssβ1β1Γβ1ds+∫0t1−εt1−sα−1σssβ2β2Γβ2ds+1Γ1+α∫0t1−εt1−sα−1sups∈0,t1−εTt2−sαθ−Tt1−sαθM2M3ds+2M1q+∅^PCΓ1+α∫t1−εt1t1−sα−1σsds+∫t1−εt1t1−sα−1σssβ1β1Γβ1ds+∫t1−εt1t1−sα−1σssβ2β2Γβ2ds+2M1Γ1+α∫t1−εt1t1−sα−1M2M3ds≤q+∅^PCΓ1+αsups∈0,t1−εTt2−sαθ−Tt1−sαθ∫0t1−εt1−sα−1p/p−1dsp−1/pσLpJ,R++1Γβ1+1∫0t1−εsβ1p2/p−1dsp−1/p2∫0t1−εσsp2ds1/p2+1Γβ2+1∫0t1−εsβ2p2/p−1dsp−1/p2∫0t1−εσsp2ds1/p2+M2M3Γ1+α∫0t1−εt1−sα−1sups∈0,t1−εTt2−sαθ−Tt1−sαθds+2M1q+∅^PCΓ1+α∫t1−εt1t1−sα−1p/p−1dsp−1/pσLpJ,R++1Γβ1+1∫t1−εt1sβ1p2/p−1dsp−1/p2×∫t1−εt1σsp2ds1/p2+1Γβ2+1∫t1−εt1sβ2p2/p−1dsp−1/p2∫t1−ϵt1σsp2ds1/p2+2M1Γ1+α∫t1−εt1t1−sα−1M2M3ds≤q+∅^PCΓ1+αsups∈0,t1−εTt2−sαθ−Tt1−sαθp−1α−1p+p−1p−1/pt1α−1p+p−1/p−1−εα−1p+p−1/p−1p−1/pσLpJ,R++1Γβ1+1t1−εβ1p2+p−1/p211+β1p2/p−1p−1/p2σLp2J,R++1Γβ2+1t1−εβ2p2+p−1/p211+β2p2/p−1p−1/p2σLp2J,R++M2M3αΓ1+αt1α−εαsups∈0,t1−εTt2−sαθ−Tt1−sαθ+2M1q+∅^PCΓ1+αp−1α−1p+p−1p−1/pεα−1p+p−1/pσLpJ,R++εβ1p2+p−1/p2Γβ1+111+β1p2/p−1p−1/p2σLp2J,R++εβ2p2+p−1/p2Γβ2+111+β2p2/p−1p−1/p2σLp2J,R++2M1αΓ1+αεαM2M3,from (A1) implies the continuity of Ttt≥0 in the uniform operator topology, and it is clear that Ι2⟶0 independent of y∈Dq as t2−t1⟶0, ϵ⟶0.
Notably,(32)I3≤M1Γ1+α∫t1t2t2−sα−1fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds≤M1Γ1+α∫t1t2t2−sα−1σsys+∅^s+1Γβ1∫0ss−τβ1−1yτ+∅^τdτ+1Γβ2∫0ss−τβ2−1yτ+∅^τdτ+Busds≤M1q+∅^PCΓ1+α∫t1t2t2−sα−1p/p−1dsp−1/p∫t1t2σspds1/p+1Γβ1+1∫t1t2t2−sα−1p/p−1dsp−1/p∫t1t2sβ1σspds1/p+1Γβ2+1∫t1t2t2−sα−1p/p−1dsp−1/p∫t1t2sβ2σspds1/p+M1M2M3Γ1+α∫t1t2t2−sα−1ds≤M1q+∅^PCΓ1+ασLpJ,R++1Γβ1+111+β1p2/p−1p−1/p2t2β1p2/p−1+1−t1β1p2/p−1+1p−1/p2σLp2J,R++1Γβ2+111+β2p2/p−1p−1/p2t2β2p2/p−1+1−t1β2p2/p−1+1p−1/p2σLp2J,R+×∫t1t2t2−sα−1p/p−1dsp−1/p+M1M2M3Γ1+α∫t1t2t2−sα−1ds≤M1q+∅^PCΓ1+ασLpJ,R++1Γβ1+111+β1p2/p−1p−1/p2t2β1p2/p−1+1−t1β1p2/p−1+1p−1/p2σLp2J,R++1Γβ2+111+β2p2/p−1p−1/p2t2β2p2/p−1+1−t1β2p2/p−1+1p−1/p2σLp2J,R+p−1α−1p+p−1p−1/pt2−t1α−1/p+M1M2M3αΓ1+α,
We have thatlimt2⟶t1I3=0. Thus, Gyt2−Gyt1 tends to zero independent of y∈Dq as t2−t1⟶0, which means that Gy,y∈Dq is equicontinuous.
Now, we need to show thatWt=Gyt,y∈Dq is relatively compact in X and t∈0,b. So, we define an operator Gε,δ on Dq for all ε∈0,t and δ>0 as follows:(33)Gε,δyt=0,t∈−r,0,∫0t−ε∫δ∞αθt−sα−1ΦαθTt−sαθfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∑i=1s∫δ∞ΦαθTt−tiαθIiyti+∅^ti,t∈J.=0,t∈−r,0,Tϵαδ∫0t−ϵ∫δ∞αθt−sα−1ΦαθTt−sαθ−ϵαδfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∑i=1s∫δ∞ΦαθTt−tiαθ−ϵαδIiyti+∅^ti,t∈J.
Subsequently, from the compactness ofTϵαδ,ϵαδ>0, we obtain the set Wϵ,δt=Gϵ,δyt,y∈Dq is relatively compact in X. Obviously, W0 is relatively compact in X, for all ε∈0,t and δ>0. Furthermore, for ∈Dq, we obtain(34)Gyt−Gε,δyt≤α∫0t∫0δθt−sα−1ΦαθTt−sαθ×fs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+α∫0t∫δ∞θt−sα−1ΦαθTt−sαθfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds−∫0t−ϵ∫δ∞θt−sα−1ΦαθTt−sαθfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busdθds+∑i=1s∫0δΦαθTt−tiαθIiyti+∅^tidθ≤αM1q+∅^PC∫0tt−sα−1σs1+sβ1Γβ1+1+sβ2Γβ2+1+M2M3ds∫0δθΦαθdθ+αM1q+∅^PC∫t−εtt−sα−1σs1+sβ1Γβ1+1+sβ2Γβ2+1+M2M3ds∫0∞θΦαθdθ+M1∑i=1sLiq+∅^PC∫0δΦαθdθ≤αM1q+∅^PC∫0tt−sα−1σs1+sβ1Γβ1+1+sβ2Γβ2+1+M2M3ds∫0δθΦαθdθ+αM1q+∅^PC∫t−ϵtt−sα−1σs1+sβ1Γβ1+1+sβ2Γβ2+1+M2M3ds+M1∑i=1sLiq+∅^PC∫0δΦαθdθ≤αM1q+∅^PC∫0tt−sα−1σsds+∫0tt−sα−1σssβ1Γβ1+1ds+∫0tt−sα−1σssβ2Γβ2+1ds+∫0tt−sα−1M2M3ds∫0δθΦαθdθ+αM1q+∅^PC∫t−ϵtt−sα−1σsds+∫t−εtt−sα−1σssβ1Γβ1+1ds+∫t−ϵtt−sα−1σssβ2Γβ2+1ds+∫t−ϵtt−sα−1M2M3ds+M1∑i=1sLiq+∅^PC∫0δΦαθdθ≤αM1q+∅^PCp−1α−1p+p−1p−1/ptα−1p+p−1/pσLpJ,R++1Γβ1+1p−1α−1p+p−1p−1/ptα−1p+p−1/p∫0tsβ1p2/p−1dsp−1/p2∫0tσsp2ds1/p2+1Γβ1+1p−1α−1p+p−1p−1/ptα−1p+p−1/p∫0tsβ1p2/p−1dsp−1/p2∫0tσsp2ds1/p2+M2M3∫0tt−sα−1ds×∫0δθΦαθdθ+αM1q+∅^PCp−1α−1p+p−1p−1/pεα−1p+p−1/pσLpJ,R++1Γβ1+1p−1α−1p+p−1p−1/pεα−1p+p−1/p∫t−εtsβ1p2/p−1dsp−1/p2∫t−εtσsp2ds1/p2+1Γβ1+1p−1α−1p+p−1p−1/pεα−1p+p−1/p∫t−εtsβ1p2/p−1dsp−1/p2∫t−εtσsp2ds1/p2+M2M3∫t−εtt−sα−1ds+M1∑i=1sLiq+∅^PC∫0δΦαθdθ≤M1q+∅^PCp−1α−1p+p−1p−1/ptα−1p+p−1/pσLpJ,R++1Γβ1+1p−1α−1p+p−1p−1/ptα−1p+p−1/pp−1β1p2+p−1tβ1p2/p−1p−1/p2σLp2J,R++1Γβ2+1p−1α−1p+p−1p−1/ptα−1p+p−1/pp−1β2p2+p−1tβ2p2+p−1/p−1p−1/p2σLp2J,R++M2M3tα×∫0δθΦαθdθ+M1q+∅^PCp−1α−1p+p−1p−1/pεα−1p+p−1/pσLpJ,R++1Γβ1+1p−1α−1p+p−1p−1/pεα−1p+p−1/pp−1β1p2+p−1ϵβ1p2+p−1/p−1p−1/p2σLp2J,R++1Γβ2+1p−1α−1p+p−1p−1/pεα−1p+p−1/pp−1β2p2+p−1ϵβ2p2+p−1/p−1p−1/p2σLp2J,R++M2M3tα+M1∑i=1sLiq+∅^PC∫0δΦαθdθ≤M1q+∅^PCp−1α−1p+p−1p−1/ptα−1p+p−1/pσLpJ,R++1Γβ1+1p−1β1p2+p−1tβ1p2+p−1/p−1p−1/p2×σLp2J,R++1Γβ2+1p−1β2p2+p−1tβ2p2+p−1/p−1p−1/p2σLp2J,R++M2M3tα∫0δθΦαθdθ+M1q+∅^PCp−1α−1p+p−1p−1/pεα−1p+p−1/pσLpJ,R++1Γβ1+1p−1β1p2+p−1ϵβ1p2+p−1/p−1p−1/p2×σLp2J,R++1Γβ2+1p−1β2p2+p−1ϵβ2p2+p−1/p−1p−1/p2σLp2J,R++M2M3tα+M1∑i=1sLiq+∅^PC∫0δΦαθdθ.
Thus, the setsWε,δt=Gε,δyt,y∈Dq are relatively compact and arbitrarily close to the set Wt=Gyt,y∈Dq, for t∈0,b.
The setWt=Gyt,y∈Dq is uniformly bounded from (22), and therefore, according to the Ascoli–Arzela theorem, it is relatively compact. Because G is continuous on Dq considered G has been completely continuous. Subsequently, by Schauder’s fixed point theorem, G has a fixed point ∈Dq. Thus, x=y+∅^ is a fixed point of F in Dq, and multifractional nonlocal impulsive control system (1) has a mild solution and completes the proof.
Now, let us obtain a unique result for the impulsive nonlocal control fractional system (1) by using the Banach contraction principle.Theorem 2.
We assume that the hypotheses (A1)–(A5) with the condition M1Lbα/α1+sβ1/Γβ1+1+sβ2/Γβ2+1+M1∑i=1sNi<1 holds. Thus, for every nonlocal initial condition xt=∅t+gxt, t∈−r,0, the impulsive multifractional order integrodifferential equations with nonlocal conditions (1) have a unique mild solution x∈Dq, where q is a positive constant, for every control u∈ L2J,V.Proof.
Consider the operatorF on Dq defined as follows:(35)Fxt=∅t,t∈−r,0,Sαt∅+gx+∫0tt−sα−1Tαt−sfs,xs,Iβ1xτ,Iβ2xτ+Busds+∑i=1sSαt−tiIixti,t∈J.
Letxt=yt+∅^t. We define the operator G:PC⟶PC as follows:(36)Gyt=0,t∈−r,0,∫0tt−sα−1Tαt−sfs,ys+∅^s,Iβ1yτ+∅^τ,Iβ2yτ+∅^τ+Busds+∑i=1sSαt−tiIiyti+∅^ti,t∈J.
Clearly, the operatorF has a unique fixed point and so as G. Thus, it can be seen that, for y1,y2∈PC−r,b,ℝ and for all t∈−r,0, we get Gy1t−Gy2t=0. For t∈tk,tk+1, we have the following:(37)Gy1t−Gy2t=∫0tt−sα−1Tαt−sfs,y1s+∅^s,Iβ1y1τ+∅^τ,Iβ2y1τ+∅^τ+Busds+∑i=1sSαt−tiIiy1ti+∅^ti−∫0tt−sα−1Tαt−sfs,y2s+∅^s,Iβ1y2τ+∅^τ,Iβ2y2τ+∅^τ+Busds−∑i=1sSαt−tiIiy2ti+∅^ti≤M1∫0tt−sα−1Ly1−y2+Iβ1y1τ−y2τ+Iβ2y1τ−y2τds+M1∑i=1sNiy1ti−y2ti≤M1L∫0tt−sα−1y1−y2+sβ1Γβ1+1y1s−y2s+sβ2Γβ2+1y1s−y2sds+M1∑i=1sNiy1ti−y2ti≤M1L∫0tt−sα−11+sβ1Γβ1+1+sβ2Γβ2+1ds+M1∑i=1sNisupt∈−r,by1t−y2t≤M1Lbαα1+sβ1Γβ1+1+sβ2Γβ2+1+M1∑i=1sNiy1t−y2t.
Subsequently,G is a contraction operator on t∈tk,tk+1, k=0,1,2,…,s, and by Banach contraction principle, G has a unique fixed point on Dq, which means that F has a unique fixed point on Dq.


## 4. Illustrative Example

Consider the impulsive multifractional order integrodifferential equations with nonlocal conditions and finite delay of the form:(38)∂α∂tαzt,τ=∂β∂tβzt,τ+n∗τut,τ+ft,zt−t¯,τ,Iβ1k1t,ws,τ−t¯ds,Iβ2k2t,ws,τ−t¯,Forτ∈0,π, t∈0,b, 0<α,β,β1,β2≤1,(39)zt+,τ−zt−,τ=Iizt−,τ,τ∈0,π,i=1,…,k,(40)z0τ=∅τ+∫0bhslog1+zs,τds,t∈−r,0,τ∈0,π,where r>0,Ii>0,i=1,…,k,(41)∅∈D=φ:−r,b⟶R,φis continous every where except for a countablenumber of points at whichφs−,φs+exist withφs−=φs+0=t0<t1<t2<⋯<tk+1=b,zti+=limk,τ⟶0+,τzti+h,τ,zti−=limk,τ⟶0−,τzti+h,τ. Let X = L20,π and A:X⟶X be defined by Aw=wβ with the DA=w∈X:wis absolutely continuouswβ∈X,wτ=w0=0, which satisfies the following:A1 It is well known that A is an infinitesimal generator of a semigroup Tt defined by Ttws=wt+s for each w∈X and t≥0, Tt is a compact strongly continuous semigroup on X.A2 B:X⟶X is a bounded linear control operator which is defined by (Bu) τ=n∗τuτ for τ∈0,π. u∈L2J,X and B≤M2,u≤M3.(42)A3Ft,z,h1,h2τ=ft,zτ,t,h1τ,t,h2τ,th1τ,t=Iβ1k1t,ws,τ−t¯dsh2τ,t=Iβ2k2t,ws,τ−t¯ds.Takeft,zτ,t,h1τ,t,h2τ,t=c0sinzt,c0 is a constant, F is continuous and for all z,h1∈D×X,F.,z,h1,h2:J⟶X is strongly measurable:(43)Ft,z,h1,h2τ=c0sinzt≤c0,whereμtz+h1+h2=c0,z,h1,h2∈PCJ,X.A4g:PCa,b:X⟶X is a continuous function which is defined by(44)gστ=∫0bhslog1+σsτds,σ∈PC0,b:Xσsτ=zs,τ.Then,g is a compact operator and gστ=∫0bhslog1+σsτds≤hslog1+σsτ≤σsτ. Hence, L = 1 and L′ = 0.A5Ii:D⟶X is a continuous for each i = 1, …, k such thatIizτ=∫0,πρiτ,ycos2zydy, z∈X for each i = 1, ..., k. Then,(45)∫0,πρiτ,ycos2zydy≤∫0,πρiτ,ydyzy.LetNi:R+⟶R+ LizyD=∫0,πρiτ,ydy, i = 1, ..., k, nondecreasing function.A6 Choose the Condition(46)M1bαα1+bβ1Γβ1+1+bβ2Γβ2+1+M1∑i=1kNi<1,0<α,β1,β2≤1.Therefore, the nonlocal multifractional partial impulsive differential systems (38)–(40) can be written in the abstract forms (1)–(3), and all conditions of Theorems 1 and 2 are satisfied.


## 5. Conclusion

The 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 inequalityρ-mean continuity as well as some necessary and sufficient estimators conditions that play an important role in guaranteeing the solution.



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