Research article Special Issues

A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order r(1,2) with impulses

  • In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1<r<2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.

    Citation: Marimuthu Mohan Raja, Velusamy Vijayakumar, Anurag Shukla, Kottakkaran Sooppy Nisar, Wedad Albalawi, Abdel-Haleem Abdel-Aty. A new discussion concerning to exact controllability for fractional mixed Volterra-Fredholm integrodifferential equations of order r(1,2) with impulses[J]. AIMS Mathematics, 2023, 8(5): 10802-10821. doi: 10.3934/math.2023548

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  • In this article, we look into the important requirements for exact controllability of fractional impulsive differential systems of order 1<r<2. Definitions of mild solutions are given for fractional integrodifferential equations with impulses. In addition, applying fixed point methods, fractional derivatives, essential conditions, mixed Volterra-Fredholm integrodifferential type, for exact controllability of the solutions are produced. Lastly, a case study is supplied to show the illustration of the primary theorems.



    Fractional derivatives have lately emerged as a very significant area of research as a result their steadily increasing numerous uses in several sectors of applied science and engineering. For more information, see the books [11,16,31,47,48]. Fractional differential systems produce a better and more accurate realistic scenario for understanding a wide variety of physical phenomena as compared to differential equations represented by regular integer order derivatives. Integer order definitions can be interpolated to non-integer order using a variety of techniques. Riemann-Liouville and Caputo derivatives are two of the most well-known. As a consequence, a large number of scholars have lately played an important role to fields like electromagnetic theory, rheology, image analysis, diffusion, data processing, porous materials, physiological engineering challenges, fluid mechanics, theology, and many others, see [4,9,10,34,35,40,41].

    Recently, [5,39,45] the authors discussed some important methods for long-time anomalous heat flow study to the fractional derivatives, Laplace transformation, singular boundary approach, and dual reciprocity technique. Recently, the study of impulsive functional differential systems has offered a natural framework for the mathematical modelling of a variety of practical situations, particularly in the fields of control, biology, and medicine. The explanation for this applicability is that impulsive differential issues are a suitable model for explaining changes that occur their state rapidly at some points and can't be represented using traditional differential equations. For additional information on this theory as well as its implications, we suggest reading [7,37,38,42,43].

    Mathematical control theory, a subfield of framework mathematics, focuses on the basic ideas that underlie the formulation and evaluation of control systems. While occasionally appearing to move in opposite directions, the two main study fields in control theory have typically been complementary. One of them assumes that a proper model of the object to be managed is offered and that the user wants to improve the behaviour of the object in some way. For instance, to design a spacecraft's trajectory that minimizes the overall trip time or fuel consumption, physical concepts and engineering standards are applied. The methods used here are strongly related to other branches of optimization theory as well as the classical calculus of variations; the result is typically a pre-programmed flight plan. The limitations imposed by uncertainty regarding the model or the environment in which the item operates provide the basis of the other key area of research. The use of feedback to correct for deviations from the anticipated behaviour in this case is the key strategy.

    It is essential to carry out study on the consequences of such controllability of systems utilizing the resources at hand. As controllability is crucial to this theory, it makes sense to seek to generalize finite-dimensional control theory to infinite dimensions. Control system analysis and innovation have proven to benefit from the usage of controllability notation by employing fractional derivatives. It is used in a multitude of industries, as well as biochemistry, physicists, automation, electronics, transport, fields of study include economics, robotics, biology, physics, power systems, chemical outgrowth control, space technology and technology. As indicated by the researchers' papers [7,9,13,29,30,33,49], resolving these challenges has become a important undertaking for young researchers.

    In addition, integrodifferential equations are used in many technological fields, including control theory, biology, medicine, and ecology, where a consequence or delay must be considered. In fact, it is always necessary to characterize a model with hereditary properties using integral-differential systems. Further, many researchers done the fruitful achievements in fractional Volterra-Fredholm integrodifferential systems with or without delay utilizing the mild solutions, semigroup theory, neutral systems, and fixed point theorems in [14,15,17,18,19,21,22,33]. In [1,25,26,27,28,46], the authors discussed the solution of a functional derivatives utilizing to weak and strong convergence, Chebyshev quadrature collocation algorithm, mixed Volterra-Fredholm type, almost contraction mapping, the iterative method, weak w2-stability, and faster iteration method.

    In [35,36], the researchers established the existence and uniqueness for fractional differential equations of α(1,2) by applying the upper and lower solution methods, sectorial operators, and nonlocal conditions. The authors [6,49] established Caputo fractional derivative of 1<α<2 using nonlocal conditions, the Laplace transform, mild solutions, cosine families, measure of noncompactness(MNC), as well as other fixed point techniques. Additionally, using fractional derivatives, cosine functions, and Sobolev type, the authors discussed exact controllability outcomes for fractional differential systems of (1,2) with finite delay in [13].

    In [18], the authors explored the approximate controllability of Caputo fractional differential systems of (1,2) by utilizing the impulsive system, sequence method, and cosine families. Furthermore, [40] proved fractional integrodifferential inclusions of (1,2) using Laplace transforms, Fredholm integro-differential systems, and the fixed point approach. Moreover, in [29,37], the researchers looked at Gronwall's inequality for the semilinear fractional derivatives of (1,2], stochastic systems, asymptotic stability, optimal control concerns, Lipschitz continuity, and impulsive systems. The researchers are currently investigating the optimal controls for fractional derivative of (1,2) with infinite delay in [19,20].

    In [35], the authors looked into the existence and uniqueness outcomes of fractional differential equations of (1,2). In [12,32], the fixed point theorem, Gronwall's inequality, impulsive systems, and sectorial operators are used to analyze optimal control for fractional derivatives of order (1,2). To identify extremal solutions of fractional partial differential equations of order (1,2), the authors of [36] used upper and lower solution approaches, sectorial operators, the Mittag-Leffler function, and mild solutions. The existence of positive mild solutions for Caputo fractional differential systems of order r(1,2) was also addressed by the authors in [34].

    Taking inspiration from the preceding information, let's investigate impulsive fractional integrodifferential systems of mixed type with order r(1,2) with the following form:

    {CDrϱz(ϱ)=Az(ϱ)+f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ))+Bx(ϱ), ϱ in V, ϱϱj,Δz(ϱj)=mj, Δz(ϱj)=˜mj, j=1,2,,n,z(0)=z0, z(0)=z1. (1.1)

    In the above

    (E1z)(ϱ)=ϱ0e1(ϱ,s,z(s))ds, (E2z)(ϱ)=σ0e2(ϱ,s,z(s))ds,

    with CDrϱ represents Caputo fractional derivative of order r in (1,2); A maps from D(A)Q into Q denotes the sectorial operator of type (P,τ,r,ϕ) on the Banach space Q; the function f maps from [0,σ]×Q×Q×Q into Q and e1,e2:S×Q×QQ are appropriate functions, in which S={(ϱ,s):0sϱσ}. The bounded linear operator B:YQ, the control function x in L2(V,Y), in which Y is also a Banach space. The continuous functions mj, ˜mj:QQ and 0=ϱ0<ϱ1<ϱ2<<ϱj<<ϱn=σ; Δz(ϱj)=z(ϱ+j)z(ϱj), where z(ϱ+j)=limϵ+0z(ϱ+ϵ) and z(ϱj)=limϵ0z(ϱ+ϵ) denote the right and lift limits of z(ϱ) at ϱ=ϱj. Δz(ϱj) has also a similar theories.

    The following sections represent the remaining portions of this article: Section 2 starts with a description of some basic concepts and the results of the preparation. In Section 3, we look at the existence of mild solutions for the impulsive fractional Volterra-Fredholm type (1.1). Lastly, an application for establishing the theory of the primary results is shown.

    We will implement some definitions, sectorial operator assumptions, R-L and Caputo fractional derivative definitions, and preliminaries in this section, which will be used throughout the article.

    The Banach space C(V,Q) maps from V into Q is a continuous with zC=supϱVz(ϱ).

    PC(V,Q)={z:VR:zC((ϱj,ϱj+1],R), j=0,n and  z(ϱ+j) and z(ϱj),j=1,,n with z(ϱj)=z(ϱj)},

    with zPC=supϱVz(ϱ). Consider L(Q) represents the Banach space of every linear and bounded operators on Banach space Q.

    Definition 2.1. [31] The integral fractional order β with such a lower limit of zero for f maps from [0,) into R+ is simply referred to as

    Iβf(ϱ)=1Γ(β)ϱ0f(s)(ϱs)1βds,ϱ>0, βR+.

    Definition 2.2. [31] The fractional order β of R-L derivative with the lower limit of zero for f is known as

    LDβf(ϱ)=1Γ(jβ)djdϱjϱ0f(s)(ϱs)jβ1ds,ϱ>0, β(j1,j), βR+.

    Definition 2.3. [31] The fractional derivative of order β in Caputo's approach with the lower limit zero for f is designated just for

    CDβf(ϱ)=LDβ(f(ϱ)j1i=0f(i)(0)i!ϱi),ϱ>0, β(j1,j), βR+.

    Definition 2.4. [35] The closed and linear operator A is called the sectorial operator of type (P,τ,r,ϕ) provided that there exists ϕ in R, τ in (0,π2), and there exists a positive constant P such that the r-resolvent of A exists outside the sector

    ϕ+Sτ={τ+ρr:ρ in C(V,Q), |Arg(ρr)|<τ}, (2.1)

    and

    (ρrIA)1P|ρrϕ|, ρrϕ+Sτ.

    It is also simple to prove that A represents the infinitesimal generator of an r-resolvent family {Gr(ϱ)}ϱ0 in Banach space if one assumes A stands for a sectorial operator of type (P,τ,r,ϕ), where

    Gr(ϱ)=12πiceρrR(ρr,A)dρ.

    Definition 2.5. A function z in PC(V,Q) is called a mild solution of the system (1.1) provdied that

    z(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, 0ϱϱ1,Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, ϱj<ϱϱj+1, (2.2)

    where

    Nr(ϱ)=12πiceρrρr1R(ρr,A)dρ, Mr(ϱ)=12πiceρrρr2R(ρr,A)dρ,Gr(ϱ)=12πiceρrR(ρr,A)dρ,

    with c being a suitable path such that ρrϕ+Sτ for ρ belongs to c.

    We consider now definition of exact controllability.

    Definition 2.6. The system (1.1) is said to be controllable on V iff for all z0,z1,zw in Q, there exists xL2(V,Y) such that a mild solution z of (1.1) fulfills z(σ)=zw.

    Let us recall some notations about the measure of noncompactness (see [2,3]).

    Definition 2.7. The Hausdorff MNC δ discovered on for every bounded subset θ of Q by

    δ(θ)=inf{κ>0:Afinitenumberofballswithradiismallerthan κ cancover θ}.

    Definition 2.8. [7] Suppose that Q+ is the positive cone of an ordered Banach space (Q,). The value of Q+ is called MNC on Q of N defined on for any bounded subsets of the Banach space Q iff N(¯co θ)=N(θ) for every θQ, in which ¯co θ denotes the closed convex hull of θ.

    Definition 2.9. [2,8] For every bounded subsets θ,θ1,θ2 of Q.

    (i) monotone iff for every bounded subsets θ, θ1, θ2 of Q we obtain: (θ1θ2)(N(θ1)N(θ2));

    (ii) non singular iff N({b}θ)=N(θ) for every b in Q, θQ;

    (iii) regular iff N(θ)=0 iff θ in Q, which is relatively compact;

    (iv) δ(θ1+θ2)δ(θ1)+δ(θ2), where θ1+θ2={u+v:u in θ1,v in θ2};

    (v) δ(θ1θ2)max{δ(θ1),δ(θ2)};

    (vi) δ(βθ)|β|δ(θ), for every βR;

    (vii) If the Lipschitz continuous function T maps from G(T)Q into Banach space X along with l>0, then δX(Tθ)lδ(θ), for θG(T).

    Lemma 2.10. [2] If P subset of C([b,σ],Q) is bounded and equicontinuous, in addition, δ(P(ϱ)) is continuous for all bϱσ and

    δ(P)=sup{δ(P(ϱ)),σ[b,σ]},wherebyP(ϱ)={u(ϱ):zP}Q.

    Lemma 2.11. [24] Suppose that the functions {yv}v=1 is a sequence of Bochner integrable from VQ including the evaluation yv(ϱ)δ(ϱ), for every ϱ in V and for all k1, where δL1(V,R), then the function α(ϱ)=δ({yv(ϱ):v1}) in L1(V,R) and fulfills

    δ({ϱ0yv(s)ds:v1})2ϱ0α(s)ds.

    Now, we consider the some conditions of sectorial operator of type (P,τ,r,ϕ).

    Theorem 2.12. [35,36] Assume that A is a sectorial operator of type (P,τ,r,ϕ). In addition, the subsequent on Nr(ϱ) hold:

    (i) For ζ(0,π), and suppose that ϕ0, we get

    Nr(ϱ)M1(τ,ζ)Pe[M1(τ,ζ)(1+ϕϱr)]12[(1+sinζsin(ζτ))1r1]πsin1+1rτ(1+τϱr)+Γ(r)Pπ(1+ϕϱr)|cosπζr|rsinτsinζ,

    for ϱ>0, and M1(τ,ζ)=max{1,sinζsin(ζτ)}.

    (ii) For ζ(0,π), and suppose that ϕ<0, we get

    Nr(ϱ)(eP[(1+sinζ)1r1]π|cosζ|1+1r+Γ(r)Pπ|cosζ||cosπζr|r)11+|ϕ|ϱr, ϱ>0.

    Theorem 2.13. [35,36] Suppose that A is a sectorial operator of type (P,τ,r,ϕ). In addition, the subsequent on Gr(ϱ), and Mr(ϱ) hold:

    (i) For ζ(0,π) and assume that ϕ0. we get

    Gr(ϱ)P[(1+sinζsin(ζτ))1r1]πsinτ(1+τϱr)1rϱr1e[M1(τ,ζ)(1+ϕϱr)]1r+Pϱr1π(1+ϕϱr)|cosπζr|rsinτsinζ,
    Mr(ϱ)P[(1+sinζsin(ζτ))1r1]M1(τ,ζ)πsinτr+2r(1+τϱr)r1rϱr1e[M1(τ,ζ)(1+ϕϱr)]1r+PrΓ(r)π(1+ϕϱr)|cosπζr|rsinτsinζ,

    for ϱ>0, where M1(τ,ζ)=max{1,sinτsin(ζτ)}.

    (ii) For 0<ζ<π, assume that ϕ<0, we get

    Gr(ϱ)(eP[(1+sinζ)1r1]π|cosζ|+Pπ|cosζ||cosπζr|)ϱr11+|ϕ|ϱr,Mr(ϱ)(eP[(1+sinζ)1r1]ϱπ|cosζ|1+2r+rΓ(r)Pπ|cosζ||cosπζr|)11+|ϕ|ϱr,

    for ϱ>0.

    Lemma 2.14. [23] Suppose that T is closed convex subset of Q and zero in T, let the continuous function E maps from T into Q and that fulfills Mönch's condition, which is, (NT is countable,  N¯co({0}E(N))¯N is compact). Hence, E has a fixed point in T.

    The existence of mild solutions for the Eq. (1.1) will be shown in this section. The following assumptions are required: It is straightforward to show that they are bounded because of the estimations on Nr(ϱ), Mr(ϱ) and Gr(ϱ) in Theorems 2.12 and 2.13.

    (H1) The operators Nr(ϱ), Mr(ϱ), and Gr(ϱ). for every ϱV, there exists a ˆP>0 such that

    sup0ϱσNr(ϱ)ˆP, sup0ϱσMr(ϱ)ˆP, sup0ϱσGr(ϱ)ˆP.

    (H2) f:V×Q×Q×QQ fulfills:

    (i) f(,χ,u,z) is measurable for every (χ,u,z) in Q×Q and f(ϱ,,,) is continuous for all ϱV, zQ, f(,χ,u,z) is strongly measurable;

    (ii) there exists p1(0,p) and ς1L1p1([0,σ],R+), and ω:R+R+ is integrable function such that f(ϱ,χ,u,z)ς1(ϱ)ω(χQ+u+z), for all (ϱ,χ,u,z) in S×Q×Q×Q, where ω satisfies limvinfω(v)v=0;

    (iii) there exists 0<p2<p and ς2L1p2(V,R+) such that for every bounded subset S1Y and W1Q,

    δ(f(ϱ,W1,S1,S2))ς2(ϱ)[δ(W1)+δ(S1)+δ(S2)], for a.e. ϱV,

    W1(φ)={e(φ):eW1} and δ is the Hausdorff MNC.

    (H3)e1:S×Q×QQ satisfies:

    (i) e1(,s,z) is measurable for all (s,z)Q×Q, e1(ϱ,,) is continuous for all ϱV;

    (ii) there exists F0>0 such that e1(ϱ,s,z)F0(1+zQ), for every ϱ in V, zQ;

    (iii) there exists p3(0,p) and ς3L1p3(V,R+) such that for every bounded subset S3Q,

    δ(e1(ϱ,s,S3))ς3(ϱ,s)[δ(S3)] for a.e. ϱV,

    with ς3=supsVs0ς3(ϱ,s)ds<.

    (H4)e2:S×Q×QQ satisfies:

    (i) e2(,s,z) is measurable for any (s,z) in Q×Q, e2(ϱ,,) is continuous for all ϱV;

    (ii) there exists F1>0 such that e2(ϱ,s,z)F1(1+zQ), for every ϱV, z in Q;

    (iii) there exists p4(0,p) and ς4L1p4(V,R+) such that for every bounded subset S4S,

    δ(e2(ϱ,s,S4))ς4(ϱ,s)[δ(S4)] for a.e. σV,

    with ς4=supsVs0ς4(ϱ,s)ds<;

    (H5) the operator B maps from L2(V,Y) into L1(V,Q) is bounded and W:L2(V,Y)L1(V,Q) is defined by

    Wx=σ0G(σs)Bx(s)ds,

    fulfills:

    (i) There exist a positive contants Pσ, Px such that BPσ and W1Px when W have an inverse W1 acquires the value belongs to L2(V,Y)/KerW.

    (ii) For p5 in (0,p) and for all bounded subset TQ, there exists ς5L1p5(V,R+) such that δ(W1(T)(ϱ))ς5(ϱ)δ(T).

    Consider the operator Π:PC(V,Q)PC(V,Q) determined by

    (Πz)(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, 0ϱϱ1,Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, ϱj<ϱϱj+1. (3.1)

    Theorem 3.1. If (H1)(H5) are fuflilled. In addition, the system (1.1) is controllable if

    2ˆPς2[1+2ˆPPσς5](1+(ς3+ς4))<1. (3.2)

    Proof. We introduce the control xz() for arbitrary function zPC(V,Q) and using (H5)(i), presented by

    xz(ϱ)=W1{zwNr(ϱ)z0Mr(ϱ)z1ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds, 0ϱϱ1,zwNr(ϱ)z0Mr(ϱ)z1jq=1Nr(ϱϱj)mjjq=1Mr(ϱϱj)˜mjϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds, ϱj<ϱϱj+1. 

    We can see that the operator Π provided in (3.1) has a fixed point by using the control mentioned above. Moreover, if Π allows a fixed point, it is simple to deduce that (Πz)(σ)=zσ, that suggests that xz(ϱ) drives the mild solution of (1.1) from the initial state z0 and z1 to the final state zσ in time σ.

    Step 1: There exists >0 such that Π(G)G, where G={zPC(V,Q): z}. Indeed, if the above assumption is fails, there is a function for every zG and xzL2(V,Y) according to G such that (Πz)(ϱ) not in G, for every ϱ[0,ϱ1], we get

    xz(ϱ)=W1[zwNr(ϱ)z0Mr(ϱ)z1ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds]Px[zw+ˆPz0+ˆPz1+ˆPϱ0f(s,z(s),(E1z)(s),(E2z)(s))ds]Px[zw+ˆPz0+ˆPz1+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))].

    Likewise, for every ϱ(ϱj,ϱj+1], j=1,2,,n, we can have

    xz(ϱ)=W1[zwNr(ϱ)z0Mr(ϱ)z1jq=1Nr(ϱϱj)mjjq=1Mr(ϱϱj)˜mjϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds]Px[zw+ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPϱ0f(s,z(s),(E1z)(s),(E2z)(s))ds]Px[zw+ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))].

    As a result, there exists ȷ1,ȷ2>0 such that

    xz(ϱ)=Px{zw+ˆPz0+ˆPz1+ˆPς1L1p1(V,R+)×ω(+σF0(1+)+σF1(1+))=ȷ1, 0ϱϱ1,zw+ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))=ȷ2, ϱj<ϱϱj+1.

    Using the assumptions (H1)(H5), and for every ϱ[0,ϱ1], we get

    <(Πz)(ϱ)Nr(ϱ)z0+Mr(ϱ)z1+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bxz(s)dsˆPz0+ˆPz1+ˆPϱ0f(s,z(s),(E1z)(s),(E2z)(s))ds+ˆPϱ0Bxz(s)dsˆPz0+ˆPz1+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))+ˆPPσPxσ[zw+ˆPz0+ˆPz1+ˆPς1L1p1(V,R+)×ω(+σF0(1+)+σF1(1+))].

    Similarly, for every ϱ(ϱj,ϱj+1], j=1,2,,n, we can have

    <(Πz)(ϱ)Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bxz(s)dsˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPϱ0f(s,z(s),(E1z)(s),(E2z)(s))ds+ˆPϱ0Bxz(s)dsˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))+ˆPPσPxσ[zw+ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))].

    Therefore

    <{ˆPz0+ˆPz1+ˆPςL1p1(V,R+)ω(+σF0(1+)+σF1(1+))+ˆPPσPxσ[zw+ˆPz0+ˆPz1+ˆPς1L1p1(V,R+)×ω(+σF0(1+)+σF1(1+))], 0ϱϱ1,ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))+ˆPPσPxσ[zw+ˆPz0+ˆPz1+ˆPjq=1mj+ˆPjq=1˜mj+ˆPς1L1p1(V,R+)ω(+σF0(1+)+σF1(1+))], ϱj<ϱϱj+1. (3.3)

    If we divide (3.3) by , and assuming . In addition, by (H2)(ii), one can get 10. This is a contradiction. Therefore, >0, Π(G)G.

    Step 2: We verify that Π is continuous on PC(V,Q). For such a study, let z(v) tends to zPC(V,Q). there exists >0 such that z(v)(ϱ) for any v and ϱV, so z(v)PC(V,Q) and zPC(V,Q). By the hypotheses (H2)(H5), we get

    Fv(s)=f(s,z(s),(Fz(v))(s),(Hz(v))(s)),F(s)=f(s,z(s),(E1z)(s),(E2z)(s)).

    From Lebesgue's dominated convergence theorem, we get

    ϱ0Fv(s)F(s)ds0 as v, ϱj<ϱϱj+1.

    Then,

    (Πz(v))(ϱ)(Πz)(ϱ)=Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)Fv(s)ds+ϱ0Gr(ϱs)Bxz(v)(s)dsNr(ϱ)z0Mr(ϱ)z1jq=1Nr(ϱϱj)mjjq=1Mr(ϱϱj)˜mjϱ0Gr(ϱs)F(s)dsϱ0Gr(ϱs)Bxz(s)dsϱ0Gr(ϱs)[Fv(s)F(s)]ds+ϱ0Gr(ϱs)B[xz(v)(s)xz(s)]dsˆPϱ0Fv(s)F(s)ds+ˆPPσϱ0xz(v)(s)xz(s)ds, (3.4)

    where

    xz(v)(s)xz(s)PxˆP[ϱ0Fv(s)F(s)ds]. (3.5)

    Since the inequality (3.4) and (3.5), we get

    (Πz(v))(ϱ)(Πz)(ϱ)0 as v.

    Then, Π is continuous on PC(V,Q).

    Step 3: Now, we show that {(Πz):zG} is equicontinuous family.

    (Πz)(ϱ)={Nr(ϱ)z0+Mr(ϱ)z1+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, 0ϱϱ1,Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bx(s)ds, ϱj<ϱϱj+1.

    Suppose 01<2ϱ1. In addition, for every ϱ[0,ϱ1], we get

    (Πz)(2)(Πz)(1)=Nr(2)z0+Mr(2)z1+20Gr(2s)f(s,z(s),(E1z)(s),(E2z)(s))ds+20Gr(2s)Bxz(s)dsNr(1)z0Mr(1)z110Gr(1s)f(s,z(s),(E1z)(s),(E2z)(s))ds10Gr(1s)Bxz(s)ds[Nr(2)Nr(1)]z0+[Mr(2)Mr(1)]z1+21Gr(2s)f(s,z(s),(E1z)(s),(E2z)(s))ds+10[Gr(2s)Gr(1s)]f(s,z(s),(E1z)(s),(E2z)(s))ds+21Gr(2s)Bxz(s)ds+10[Gr(2s)Gr(1s)]Bxz(s)dsNr(2)Nr(1)z0+Mr(2)Mr(1)z1+ˆP21ς1(s)ω(+σF0(1+)+σF1(1+))ds+10Gr(2s)Gr(1s)ς1(s)ω(+σF0(1+)+σF1(1+))ds+ˆPPσxzLμ(V,Y)(21)+Pσ10Gr(2s)Gr(1s)xz(s)ds.

    Similarly, for every ϱ(ϱj,ϱj+1], j=1,2,,n, we can have

    (Πz)(2)(Πz)(1)=Nr(2)z0+Mr(2)z1+0<ϱj<2Nr(2ϱj)mj+0<ϱj<2Mr(2ϱj)˜mj+20Gr(2s)f(s,z(s),(E1z)(s),(E2z)(s))ds+20Gr(2s)Bx(s)dsNr(1)z0Mr(1)z10<ϱj<1Nr(1ϱj)mj0<ϱj<1Mr(1ϱj)˜mj10Gr(1s)f(s,z(s),(E1z)(s),(E2z)(s))ds10Gr(1s)Bx(s)ds[Nr(2)Nr(1)]z0+[Mr(2)Mr(1)]z1+1<ϱj<2Nr(2ϱj)mj+0<ϱj<2[Nr(2ϱj)Nr(1ϱj)]mj+1<ϱj<2Mr(2ϱj)˜mj+0<ϱj<2[Mr(2ϱj)Mr(1ϱj)]˜mj+21Gr(2s)f(s,z(s),(E1z)(s),(E2z)(s))ds+10[Gr(2s)Gr(1s)]f(s,z(s),(E1z)(s),(E2z)(s))ds+21Gr(2s)Bx(s)ds+10[Gr(2s)Gr(1s)]Bx(s)dsNr(2)Nr(1)z0+Mr(2)Mr(1)z1+ˆP1<ϱj<2mj+0<ϱj<2Nr(2ϱj)Nr(1ϱj)mj+ˆP1<ϱj<2˜mj+0<ϱj<2Mr(2ϱj)Mr(1ϱj)˜mj+ˆP21ς1(s)ω(+σF0(1+)+σF1(1+))ds+10Gr(2s)Gr(1s)ς1(s)ω(+σF0(1+)+σF1(1+))ds+ˆPPσxLμ(V,Y)(21)+Pσ10Gr(2s)Gr(1s)x(s)ds.

    Thus, we get

    (Πz)(2)(Πz)(1)={Nr(2)Nr(1)z0+Mr(2)Mr(1)z1+ˆP21ς1(s)ω(+σF0(1+)+σF1(1+))ds+10Gr(2s)Gr(1s)ς1(s)×ω(+σF0(1+)+σF1(1+))ds+ˆPPσxzLμ(V,Y)(21)+Pσ10Gr(2s)Gr(1s)xz(s)ds, ϱ[0,ϱ1],Nr(2)Nr(1)z0+Mr(2)Mr(1)z1+ˆP1<ϱj<2mj+0<ϱj<2Nr(2ϱj)Nr(1ϱj)mj+ˆP1<ϱj<2˜mj+0<ϱj<2Mr(2ϱj)Mr(1ϱj)˜mj+ˆP21ς1(s)ω(+σF0(1+)+σF1(1+))ds+10Gr(2s)Gr(1s)ς1(s)×ω(+σF0(1+)+σF1(1+))ds+ˆPPσxLμ(V,Y)(21)+Pσ10Gr(2s)Gr(1s)x(s)ds, ϱ(ϱj,ϱj+1]. (3.6)

    The aforementioned inequality's RHS of the system (3.6) tends to zero independently of zG as 21 by using the continuity of functions ϱNr(ϱ), ϱMr(ϱ), and ϱGr(ϱ). Therefore, Π(G) is equicontinuous.

    Step 4: Next, we prove that Mönch's condition holds.

    Consider UG is countable and Uconv({0}Π(U)), we show that λ(U)=0, where λ is the Hausdorff measure of noncompactness. Let U={zv}v=1. We check that Π(U)(σ) is relatively compact in PC(V,Q), for every ϱ(ϱj,ϱj+1]. From Theorem 2.11, and

    δ({xz(v)(s)}v=1)=δ{W1(zwNr(ϱ)z0Mr(ϱ)z10<ϱj<2Nr(ϱϱj)mj0<ϱj<2Mr(ϱϱj)˜mjϱ0Gr(ϱs)f(s,z(v)(s),(Fz(v))(s),(Hz(v))(s))ds)}v=12ς5(s)ˆP(ϱ0ς2(s)[δ(U(s))+δ({Fz(v)(s)}v=1)+δ({Hz(v)(s)}v=1)]ds)2ς5(s)ˆP(ϱ0ς2(s)δ(U(s))ds+2ϱ0ς2(s)(ς3+ς4)δ(U(s))ds).

    From Lemma 2.11, and assumptions (H1)(H5), we get

    δ({Πz(v)(s)}v=1)=δ({Nr(ϱ)z0+Mr(ϱ)z1+jq=1Nr(ϱϱj)mj+jq=1Mr(ϱϱj)˜mj+ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds+ϱ0Gr(ϱs)Bxz(s)ds}v=1)δ({ϱ0Gr(ϱs)f(s,z(s),(E1z)(s),(E2z)(s))ds}v=1)+δ({ϱ0Gr(ϱs)Bxz(s)ds}v=1)2ˆP(ϱ0ς2(s)[δ(U(s))+δ({Fz(v)(s)}v=1)+δ({Hz(v)(s)}v=1)]ds)+2ˆPPσ(ϱ0δ({xz(v)(s)}v=1)ds)2ˆP(ϱ0ς2(s)δ(U(s))ds+ϱ0ς2(s)(ς3+ς4)δ(U(s))ds)+4ˆP2Pσ×(ϱ0ς5(s)ds)(ϱ0ς2(s)δ(U(s))ds+ϱ0ς2(s)(ς3+ς4)δ(U(s))ds)2ˆPς2[1+2ˆPPσς5](1+(ς3+ς4))δ(U(ϱ)).

    By Lemma 2.10, we get

    δ({Πz(v)(s)}v=1)Mδ(U(ϱ)).

    Therefore by using Mönch's condition, one can obtain

    δ(Π)δ(conv({0}(Π(U)))=δ(Π(U))Mδ(U),

    this implies δ(U)=0. Hence, Π has a fixed point in G. Thus, the fractional integrodifferential equations (1.1) has a fixed point fulfilling z(σ)=zσ. Thus, the fractional integrodifferential equations (1.1) is exact controllable on [0,σ].

    Suppose the impulsive fractional mixed Volterra-Fredholm type integrodifferential systems of the form:

    {rϱrz(ϱ,ω)=2ϱ2z(ϱ,ω)+cos[z(ϱ,ω)+ϱ0(ϱι)2sinz(ι,ω)dι]+ωz(ϱ,ω)+σ0cosz(ϱ,ω)dι+ξα(ϱ,ω),ϱV=[0,1], ω[0,π], ϱϱj, j=1,2,,n,z(ϱ,0)=z(ϱ,1)=0, ϱV,z(ϱ+j,ω)z(ϱj,ω)=mj, z(ϱ+j,ω)z(ϱj,ω)=˜mj, j=1,2,,n,z(0,ω)=z0(ω), z(0,ω)=z1(ω), (4.1)

    where 32ϱ32 denotes fractional partial derivative of r=32. 0=ϱ0<ϱ1<ϱ2<<ϱj<<ϱn=σ; z(ϱ+j)=lim(ϵ+,ω)(0+,ω)z(ϱj+ϵ,ω) and z(ϱj)=lim(ϵ,ω)(0,ω)z(ϱj+ϵ,ω).

    Consider Q=Y=L2([0,π]), and let A maps from D(A)Q into Q be presented as Az=z along with domain D(A), which is

    D(A)={z in Q:z, z are absolutely continuous, z in Q,z(0)=z(π)=0}.

    Further, A stands for infinitesimal generator of an analytic semigroup {G(ϱ), ϱ0} determined by G(ϱ)z(s)=z(ϱ+s), for every z in Q. G(ϱ) is not compact semigroup on Q and δ(G(ϱ)U)δ(U), then δ stands for the Hausdorff MNC.

    In addition, A has discrete spectrum with eigenvalues μ2, μN, and according normalized eigen functions given by yμ(z)=(2/π)sin(μπz). Then, yμ stands for an orthonormal basic of Q. For more details refer to [35].

    G(ϱ)=μ=1eμ2ϱz,yμyμ, zQ,

    G(ϱ) is compact for any ϱ>0 and G(ϱ)eϱ for any ϱ0 [44].

    A=2ϱ2 represents sectorial operator of type (P,τ,r,ϕ) and generates r-resolvent families Nr(ϱ), Mr(ϱ), and Gr(ϱ) for ϱ0. Since A=2ϱ2 is an m-accretive operator on Q with dense domain (H1) fulfilled.

    Az=μ=1μ2z,yμyμ, zD(A).

    Determine

    f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ)(ω))=cos[z(ϱ,ω)+ϱ0(ϱι)2sinz(ι,ω)dι]+ωz(ϱ,ω)+σ0cosz(ϱ,ω)dι,
    (E1z)(ϱ)=ϱ0sinz(ι,ω)dι,(E2z)(ϱ)=10cosz(ϱ,ω)dι.

    Assume that B:QQ is determined by

    (Bx)(ϱ)(ω)=ξα(ϱ,ω), ω[0,π],

    For ω[0,π], the linear operator W specified by

    (Wx)(ω)=10G(1s)ξα(s,ω)ds,

    fulfilling (H2)(H5). Thus, the system (4.1) can be rewritten as

    {CDrϱz(ϱ)=Az(ϱ)+f(ϱ,z(ϱ),(E1z)(ϱ),(E2z)(ϱ))+Bx(ϱ), ϱV,  ϱϱj,Δz(ϱj)=mj, Δz(ϱj)=˜mj, j=1,2,,n,z(0)=z0, z(0)=z1. (4.2)

    As a result, Theorem 3.1's requirements are all fulfilled. The system (4.1) is therefore exact controllable on V according to Theorem 3.1.

    In this study, we mainly concentrated on the exact controllability outcomes for fractional integrodifferential equations of mixed type via sectorial operators of type (P,τ,r,ϕ), employing fractional calculations, impulsive systems, sectorial operators, and fixed point technique. Lastly, an example for clarifying the theory of the important findings is constructed. The effectiveness of such research discoveries can be effectively increased to exact controllability using varied fractional differential structures (Hilfer system, A-B system, stochastic, etc.). Moreover, null controllability outcomes of impulsive fractional stochastic differential systems via sectorial operators will be the subject of future research.

    This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2023/R/1444). Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

    The authors declare no conflict of interest.



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