Research article Special Issues

Controllability of fractional stochastic evolution inclusion via Hilfer derivative of fixed point theory

  • In this study, we use the Hilfer derivative to analyze the approximate controllability of fractional stochastic evolution inclusions (FSEIs) with nonlocal conditions. By assuming that the corresponding linear system is approximately controllable, we obtain a novel set of adequate requirements for the approximate controllability of nonlinear FSEIs in meticulous detail. The fixed-point theorem for multi-valued operators and fractional calculus are used to achieve the results. Finally, we use several instances to demonstrate our findings.

    Citation: Abdelkader Moumen, Ammar Alsinai, Ramsha Shafqat, Nafisa A. Albasheir, Mohammed Alhagyan, Ameni Gargouri, Mohammed M. A. Almazah. Controllability of fractional stochastic evolution inclusion via Hilfer derivative of fixed point theory[J]. AIMS Mathematics, 2023, 8(9): 19892-19912. doi: 10.3934/math.20231014

    Related Papers:

    [1] Chenggang Huo, Humera Bashir, Zohaib Zahid, Yu Ming Chu . On the 2-metric resolvability of graphs. AIMS Mathematics, 2020, 5(6): 6609-6619. doi: 10.3934/math.2020425
    [2] Pradeep Singh, Sahil Sharma, Sunny Kumar Sharma, Vijay Kumar Bhat . Metric dimension and edge metric dimension of windmill graphs. AIMS Mathematics, 2021, 6(9): 9138-9153. doi: 10.3934/math.2021531
    [3] Muhammad Ahmad, Muhammad Faheem, Sanaa A. Bajri, Zohaib Zahid, Muhammad Javaid, Hamiden Abd El-Wahed Khalifa . Optimizing SNARK networks via double metric dimension. AIMS Mathematics, 2024, 9(8): 22091-22111. doi: 10.3934/math.20241074
    [4] Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem . The comparative study of resolving parameters for a family of ladder networks. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908
    [5] Meiqin Wei, Jun Yue, Xiaoyu zhu . On the edge metric dimension of graphs. AIMS Mathematics, 2020, 5(5): 4459-4465. doi: 10.3934/math.2020286
    [6] Yuni Listiana, Liliek Susilowati, Slamin Slamin, Fadekemi Janet Osaye . A central local metric dimension on acyclic and grid graph. AIMS Mathematics, 2023, 8(9): 21298-21311. doi: 10.3934/math.20231085
    [7] Juan Alberto Rodríguez-Velázquez . Corona metric spaces: Basic properties, universal lines, and the metric dimension. AIMS Mathematics, 2022, 7(8): 13763-13776. doi: 10.3934/math.2022758
    [8] Syed Ahtsham Ul Haq Bokhary, Zill-e-Shams, Abdul Ghaffar, Kottakkaran Sooppy Nisar . On the metric basis in wheels with consecutive missing spokes. AIMS Mathematics, 2020, 5(6): 6221-6232. doi: 10.3934/math.2020400
    [9] Dalal Awadh Alrowaili, Uzma Ahmad, Saira Hameeed, Muhammad Javaid . Graphs with mixed metric dimension three and related algorithms. AIMS Mathematics, 2023, 8(7): 16708-16723. doi: 10.3934/math.2023854
    [10] Xiaogang Liu, Muhammad Ahsan, Zohaib Zahid, Shuili Ren . Fault-tolerant edge metric dimension of certain families of graphs. AIMS Mathematics, 2021, 6(2): 1140-1152. doi: 10.3934/math.2021069
  • In this study, we use the Hilfer derivative to analyze the approximate controllability of fractional stochastic evolution inclusions (FSEIs) with nonlocal conditions. By assuming that the corresponding linear system is approximately controllable, we obtain a novel set of adequate requirements for the approximate controllability of nonlinear FSEIs in meticulous detail. The fixed-point theorem for multi-valued operators and fractional calculus are used to achieve the results. Finally, we use several instances to demonstrate our findings.



    The notion of resolving sets in general networks is introduced by Slater in 1975 and he called the minimum cardinality of a resolving set location number [1]. In next year Harary and Melter also introduced the same concept with different name and they called it the metric dimension (MD) of the connected networks. They provide a characterization of MD of the trees and they also proved that the MD of wheel W1,z and complete network Kz is 2 and z1 respectively [2]. Later on the results of the MD of W1,z, were improved by S. Khuller et al. and they also characterized the connected networks that those have MD 1 and 2 [3]. Shanmukha et al. improved the results of Harary and Melter and they computed the MD of wheel-related networks [4]. Chartrand et al. established the bounds on MD of connected networks in terms of the order and diameter of a network [5].

    The concept of MD arises in diverse areas including network discovery and verification [6], robot navigation [7], strategies for the Mastermind game [8], combinatorial optimization [9], coin weighting [10], navigation of robots in networks [11] and image processing [12]. There are some new types of MD are discovered in recent times as local MD [13], k- MD [14], edge MD [15], fault tolrent MD [16] and some interesting results of fault-tolerant MD of convex polytope networks have been derived by Raza et al [17].

    The idea of MD to find the solution of specific integer programming (IPP) is introduced by Chartrand et al. [5] and Currie and Ollermann introduced the concept of fractional metric dimension (FMD) to find improved solution of IPP [18]. The concept of FMD in the field of networking theory is formally introduced by Arumugam and Mathew, they developed different combinatorial techniques to find the exact value of FMD of different connected networks. Moreover, they also found the FMD of Petersen, cycle, friendship and cartesian product of different connected networks [19,20]. Feng et al. established a computational technique to find FMD of vertex transitive networks and as an application they computed the FMD of hamming and generalized Johnson networks [21]. Javaid et al. characterize all those connected networks that attain FMD exactly 1 [22,23] and Zafar et al. computed the exact value of FMD of different connected networks [24].

    The notion of latest derived form of FMD known as a local fractional metric dimension (LFMD) is defined by Asiyah et al. and they calculated the exact values of the LFMD of the corona product of connected networks [25]. Javaid et al. purposed a unique methodology to compute the sharp bounds of LFMD for all the connected networks and they also proved that the lower bound of LFMD of non-bipartite networks is greater than 1 [26,27]. Some interesting results of LFMD of different connected networks can be seen in [28,29,30].

    In this paper, the lower and upper bounds of LFMD of generalized modified prism networks have been computed. It is also proved that all the upper bounds of all these networks is less or equal to 2, when the order of these networks approaches to . The rest of the paper is organized as follows: Section 2 deals with preliminaries, Section 3 consists of the main results of LFMD of generalized modified prism network, Section 4 represents the conclusion and comparison among all the main results.

    A network Γ is a pair (V(Γ)×E(Γ)) with V(Γ) is a vertex set and E(Γ)(V(Γ)×V(Γ)) an edge set. A walk is a sequence of edges and vertices of a network. A path is a sequence of vertices with the property that each vertex in the sequence is adjacent to the vertex next to it. For any two vertices x, y of V(Γ) then the distance d(x,y) between them is the number of edges between the shortest path connecting them. A network is called connected if there exist a path between every pair of vertices of Γ. A vertex xV(Γ) resolves a pair (a,b) if d(x,a)d(x,b). Let R={r1,r2,r3,....,rz}V(Γ) be a ordered set is considered as resolving set of Γ if each pair of vertices of Γ is resolved by some vertex in R. A resolving set with minimum cardinality is called the metric dimension of Γ and it is defined as

    dim(Γ)=min{|R|:RisresolvingsetofΓ}.

    For an edge abE(Γ) the local resolving neighbourhood set (RLN) Rx(ab) of ab is defined as Rx(ab)={cV(Γ):d(a,c)d(b,c)}. A local resolving function (LRF) is defined as η:V(Γ)[0,1] such that η(Rx(ab))1 for each Rx(ab) of Γ. A local resolving function η is called minimal if there exists a function μ:V(Γ)[0,1] such that μη and μ(a)η(a) for at least one aΓ(V) that is not a local resolving function of Γ. If |η|=aRx(ab)η(a) then LFMD of Γ is donated by dimLF(Γ) is defined as

    dimLF(Γ)=min{|η|:ηisminimallocalresolvingfunctionofΓ}.

    Throughout the paper, we have used the symbol of local resolving neighbourhood set of an edge abE(Γ) is Rx(ab). For more details about local resolving neighbourhood set and local resolving function, we refer [25].

    Lemma X. [26] Let Γ=(V(Γ)×E(Γ)) be a connected network. If |Rx(e)A|ω, eE(Γ) then

    1dimlf(Γ)|V(Γ)|ω

    where ω=min{|Rx(e)|:eE(Γ)}, where A={Rx(e):|Rx(e)=ω}.

    Lemma Y. [27] Let Γ=(V(Γ)×E(Γ)) be a connected network. Then

    dimlf(Γ)|V(Γ)|σ

    where σ=max{|Rx(e)|:eE(Γ)}.

    For z5 the modified prism network MPz,1,2 with vertex set vertex set V(MPz,1,2)={aj,aj:1jz} and edge set E(MPz,1,2)={ajaj+2:1jz2}{ajaj+1:1jz}{ajaj:1jz}{ajaj+1:1jz}, where |V(MPz,1,2)|=2z and |E(MPz,1,2)|=4z. Fore more details see Figure 1.

    Figure 1.  Modified prism network MP9,1,2.

    For z5 the modified prism network MQz,1,2 with vertex set vertex set V(MPz,1,2)={aj,aj:1jz} and edge set E(MPz,1,2)={ajaj+2:1jz2}{ajaj+1:1jz}{ajaj:1jz}{ajaj+1:1jz}{ajbj:1jz}{ajaj+1:1jz}{bjbj+1:1jz}, where |V(MPz,1,2)|=3z and |E(MPz,1,2)|=6z. Fore more details see Figure 2.

    Figure 2.  Modified prism network MQ9,1,2.

    In this dissertation, our objective is to compute RLN Sets and LFMD of modified prism networks (MPz,1,2,MQz,1,2) in the form of sharp upper and lower bounds.

    In this section, we compute the RLN sets and LFMD of modified prism network (MPz,1,2).

    Lemma 4.1. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    (a) |Rx(ajaj+1)|=z1 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1) =V(MPz,1,2){aj+2,aj+4,aj+6.....,az+i5,az+i3,az+i1}{aj+2,aj+4,aj+6,.....,az+i5,az+i3,az+i1}{az+2i+22}{az+2i+22} and |Rx(ajaj+1)|=z1 and |zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j3,az+j4}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+j3,ai+4,ai+6,ai+8,ai+10,...,az+i5}.

    The cardinalities among all these RLN sets are classified in Table 1.

    Table 1.  Cardinality of each RLN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z1
    Rx(ajaj+2) 2z2>z1
    Rx(ajaj+1) z+3>z1

     | Show Table
    DownLoad: CSV

    It is clear from above Table 1 that cardinality of Rx(ajaj+1) is less then all other RLN sets.

    Theorem 4.2. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    zz1dimLF(MPz,1,2)2zz1.

    Proof. Case 1. For z=5, we have the following RLN sets

    Rx(a1a2)=Rx(a1a2)={a1,a2,a1,a2},

    Rx(a2a3)=Rx(a2a3)={a2,a3,a2,a3},

    Rx(a3a4)=Rx(a3a4)={a3,a4,a3,a4},

    Rx(a4a5)=Rx(a4a5)={a4,a5,a4,a5},

    Rx(a5a1)=Rx(a5a1)={a1,a5,a1,a5},

    Rx(a1a3)={a1,a3,a1,a3},

    Rx(a1a4)={a1,a4,a1,a4},

    Rx(a2a4)={a2,a4,a2,a4},

    Rx(a2a5)={a2,a5,a2,a5},

    Rx(a3a5)={a3,a5,a3,a5},

    Rx(a1a1)=V(MP5,1,2){a3,a4},

    Rx(a2a2)=V(MP5,1,2){a4,a5},

    Rx(a3a3)=V(MP5,1,2){a5,a1},

    Rx(a4a4)=V(MP5,1,2){a1,a2},

    Rx(a5a5)=V(MP5,1,2){a2,a3}.

    For 1j5 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP5,1,2. Therefore, an upper LRF η:V(MP5,1,2)[0,1] is defined as η(y)=14 for each yV(MP5,1,2). In order to show that η is a minimal LRF, we define another LRF η(y):V(MP5,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP5,1,2. Therefore, dimLF(MP5,1,2)10114=52. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj)|=8 and |Rx(ajaj)||Rx(e)|, where Rx(e) are the other RLN sets of MP5,1,2. Therefore, a lower LRF η:V(MP5,1,2)[0,1] is defined as η(y)=121 for all yV(MP5,1,2) hence dimLF(MP5,1,2)10118=54. Consequently,

    54dimLF(MP5,1,2)52.

    Case 2. For 1jz from Lemma 4.1 it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Therefore, an upper LRF η:V(MPz,1,2)[0,1] is defined as η(y)=1z1 for each yV(MPz,1,2). In order to show that η is a minimal RLF, we define another RLF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a RLF of (MPz,1,2). Therefore, by Lemma X dimLF(MPz,1,2)2zj=11z1=2zz1. In the same way, for 1jz it is clear from Lemma 4.1 |Rx(ajaj+1)|=2z2 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other LRN sets of MPz,1,2. Therefore, a lower RLF η:V(MPz,1,2)[0,1] is defined as η(y)=12z4 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z2=zz2. Consequently,

    zz2dimLF(MPz,1,2)2zz1.

    Lemma 4.3. Let MPz,1,2 be a modified prism network, where z3(mod4). Then

    (a) |Rx(ajaj+1)|=z+1 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.

    Proof. Let aj inner, aj be the outer vertices of modified prism network, for 1jz, where z+11(modz), we have following possibilities,

    (a)Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,.....,az+i5,az+i3,az}{aj+2,aj+4,aj+6,.....,az+i5,az+i3,az} and |Rx(ajaj+1)|=z+1 and |zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j3,az+j4}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+j3,ai+4,ai+6,ai+8,ai+10,...,az+i5}.

    The RLN sets are classified in Table 2 and it is clear that |Rx(ajaj+1)| is less then the all other RLN sets of MPz,1,2.

    Table 2.  Cardinality of each RLN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z+1
    Rx(ajaj+2) 2z2>z+1
    Rx(ajaj+1) z+3>z+1

     | Show Table
    DownLoad: CSV

    Theorem 4.4. Let MPz,1,2 be a modified prism network, where z3(mod4). Then

    zz1dimLF(MPz,1,2)2zz+1.

    Proof. Case 1. For z=7, we have the following RLN sets

    Rx(a1a2)=V(MP7,1,2){a3,a5,a7,a3,a5,a7},

    Rx(a2a3)=V(MP7,1,2){a4,a6,a1,a4,a6,a1},

    Rx(a3a4)=V(MP7,1,2){a5,a7,a2,a5,a7,a2},

    Rx(a4a5)=V(MP7,1,2){a6,a1,a3,a6,a1,a3},

    Rx(a5a6)=V(MP7,1,2){a7,a2,a4,a7,a2,a4},

    Rx(a6a7)=V(MP7,1,2){a1,a3,a5,a1,a3,a5},

    Rx(a7a1)=V(MP7,1,2){a2,a4,a6,a2,a4,a6},

    Rx(a1a1)=V(MP7,1,2){a3,a4,a5,a6},

    Rx(a2a2)=V(MP7,1,2){a4,a5,a6,a7},

    Rx(a3a3)=V(MP7,1,2){a5,a6,a7,a1},

    Rx(a4a4)=V(MP7,1,2){a6,a7,a1,a2},

    Rx(a5a5)=V(MP7,1,2){a7,a1,a2,a3},

    Rx(a6a6)=V(MP7,1,2){a1,a2,a3,a4},

    Rx(a7a7)=V(MP7,1,2){a2,a3,a4,a5,},

    Rx(a1a3)=V(MP7,1,2){a2,a2},

    Rx(a2a4)=V(MP7,1,2){a3,a3},

    Rx(a3a5)=V(MP7,1,2){a4,a4},

    Rx(a4a6)=V(MP7,1,2){a5,a5},

    Rx(a5a7)=V(MP7,1,2){a6,a6},

    Rx(a6a1)=V(MP7,1,2){a7,a7},

    Rx(a7a2)=V(MP7,1,2){a1,a1},

    Rx(a1a2)=V(MP7,1,2){a3,a5,a7,a5},

    Rx(a2a3)=V(MP7,1,2){a4,a6,a1,a6},

    Rx(a3a4)=V(MP7,1,2){a5,a7,a2,a7},

    Rx(a4a5)=V(MP7,1,2){a6,a1,a3,a1},

    Rx(a5a6)=V(MP7,1,2){a7,a2,a4,a2},

    Rx(a6a7)=V(MP7,1,2){a1,a3,a5,a3},

    Rx(a7a1)=V(MP7,1,2){a2,a4,a6,a4}.

    For 1j7 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP7,1,2). Therefore, an upper LRF η:V(MP7,1,2)[0,1] is defined as η(y)=18 for each yV(MP7,1,2). In order to show that η(y) is a minimal upper LRF, we define another LRF η(y):V(MP7,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a local resolving function of P7,1,2). Therefore, dimLF(MP7,1,2)14118=74. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MP7,1,2). Therefore, a lower LRF η:V(MP7,1,2)[0,1] is defined as η(y)=121 for each yV(MP7,1,2) hence dimLF(MP7,1,2)141112=76. Since MP7,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    76dimLF(MP7,1,2)74.

    Case 2. For 1jz from Lemma 4.3, it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Therefore, an upper LRF η:V(MPz,1,2)[0,1] is defined as η(y)=23n+6 for each yV(MPz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP7,1,2 hence by Lemma X dimLF2zj=11z+1=2zz+1. In the same way, for 1jz it is clear from Lemma 4.3 |Rx(ajaj+1)|=2z2 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN of MPz,1,2. Therefore, a maximal lower LRF η:V(MPz,1,2)[0,1] is defined as η(y)=12z2 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z2=zz1. Consequently,

    zz1dimLF(MPz,1,2)2zz+1.

    Lemma 4.5. Let MPz,1,2 be a modified generalized prism network, where z0(mod4). Then

    (a) |Rx(ajaj+1)|=z and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2, aj+4,aj+6.....,az+2j2,az+2j+22, az+2j+62, az+2j+102, .....az+i5, az+i3,az+i1}{aj+2, aj+4,aj+6.....,az+2j2, az+2j+22,az+2j+62, az+2j+102, .....az+i5,az+i3, az+i1} and |Rx(ajaj+1)|=z and |zj=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.

    (b)Rx(ajaj)=V(MPz,1,2)-{aj+2,aj+3,az+j2, az+j3}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,aj+1,an+2j+22, an+2j+22}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4, aj+6,....,az+2j2,az+2j+22, az+2j+62,...., az+j6,az+j3,az+j1}.

    The RLN sets are classified in Table 3 and it is clear that cardinality of Rx(ajaj+1) is less then all other RLN sets of MPz,1,2.

    Table 3.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z
    Rx(ajaj+2) 2z2>z
    Rx(ajaj+1) z+3>z

     | Show Table
    DownLoad: CSV

    Theorem 4.6. Let MPz,1,2 be a modified prism network, where z0(mod4). Then

    zz2dimLF(MPz,1,2)2.

    Proof. Case 1. For z=8, we have the following RLN sets;

    Rx(a1a2)=V(MP8,1,2){a3,a5,a6,a8,a3,a5,a6,a8},

    Rx(a2a3)=V(MP8,1,2){a4,a6,a1,a2,a4,a6,a7,a1},

    Rx(a3a4)=V(MP8,1,2){a5,a7,a8,a3,a5,a7,a8,a2},

    Rx(a4a5)=V(MP8,1,2){a6,a8,a1,a4,a6,a8,a1,a3},

    Rx(a5a6)=V(MP8,1,2){a7,a1,a2,a5,a7,a1,a2,a4},

    Rx(a6a7)=V(MP8,1,2){a8,a2,a3,a6,a8,a2,a3,a5},

    Rx(a7a8)=V(MP8,1,2){a1,a3,a4,a7,a1,a3,a4,a6},

    Rx(a1a8)=V(MP8,1,2){a2,a4,a5,a8,a2,a4,a5,a7},

    Rx(a1a1)=V(MP8,1,2){a3,a4,a6,a7},

    Rx(a2a2)=V(MP8,1,2){a4,a5,a7,a8},

    Rx(a3a3)=V(MP8,1,2){a5,a6,a8,a1},

    Rx(a4a4)=V(MP8,1,2){a6,a7,a1,a2},

    Rx(a5a5)=V(MP8,1,2){a7,a8,a2,a3},

    Rx(a6a6)=V(MP8,1,2){a8,a1,a3,a4},

    Rx(a7a7)=V(MP8,1,2){a1,a2,a4,a5},

    Rx(a8a8)=V(MP8,1,2){a2,a3,a5,a6},

    Rx(a1a2)=V(MP8,1,2){a3,a5,a6,a8},

    Rx(a2a3)=V(MP8,1,2){a4,a6,a7,a1},

    Rx(a3a4)=V(MP8,1,2){a5,a7,a8,a2},

    Rx(a4a5)=V(MP8,1,2){a6,a8,a1,a3},

    Rx(a5a6)=V(MP8,1,2){a7,a1,a2,a4},

    Rx(a6a7)=V(MP8,1,2){a8,a2,a3,a5},

    Rx(a7a8)=V(MP8,1,2){a1,a3,a4,a6},

    Rx(a8a1)=V(MP8,1,2){a2,a4,a5,a7},

    Rx(a1a3)=V(MP8,1,2){a2,a6,a2,a6},

    Rx(a2a4)=V(MP8,1,2){a3,a7,a3,a7},

    Rx(a3a5)=V(MP8,1,2){a4,a8,a4,a8},

    Rx(a4a6)=V(MP8,1,2){a5,a1,a5,a1},

    Rx(a5a7)=V(MP8,1,2){a6,a2,a6,a2},

    Rx(a6a8)=V(MP8,1,2){a3,a7,a3,a7},

    Rx(a7a1)=V(MP8,1,2){a8,a1,a8,a1},

    Rx(a8a2)=V(MP8,1,2){a1,a5,a1,a5}.

    For 1j8 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the RLN sets of MP8,1,2. Then there exits an upper LRF η:V(MP8,1,2)[0,1] and it is defined as η(y)=18 for each yV(MP8,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP8,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP8,1,2. Therefore, dimLF(MP8,1,2)16118=2. In the same context, for 1jz it is clear from RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MP8,1,2). Then there exist a lower LRF η:V(MP8,1,2)[0,1] and it is defined η(y)=121 for each yV(MP7,1,2) hence dimLF(MP8,1,2)161112=43. Since MP8,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    43dimLF(MP8,1,2)2.

    Case 2. For 1jz, it is clear from Lemma 4.5 it is that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Then there exits an upper LRF η:V(MPz,1,2)[0,1] an it is defined as η(y)=1z for each yV(MPz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP8,1,2 hence by Lemma X dimLF(MPz,1,2)2zj=11z=2. In the same way, For 1jz it is clear from Lemma 4.5 |Rx(ajaj+1)|=2z4 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2). Then there exits a maximal lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MPz,1,2) hence by Lemma Y dimLF(MPz,1,2)2zj=112z4=zz2. Consequently,

    zz2dimLF(MPz,1,2)2.

    Lemma 4.7. Let MPz,1,2 be a modified prism network, where z2(mod4). Then

    (a) |Rx(ajaj+1)|=z+2 and zj=1Rx(ajaj+1)=V(MPz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let aj inner, aj be the outer vertices of modified generalized Prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6....., az+j12,az+2j+22, az+2j+62,az+2j+102, .....az+i5, az+i3,az+i1}{aj+2,aj+4,aj+6.....,az+j12, az+2j+22,az+2j+62,az+2j+102, .....az+i5,az+i3,az+i1} and |Rx(ajaj+1)|=z and |zj=1Rx(ajaj+1)|=2z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2){aj+2,aj+3,az+j2,az+j3}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1, an+2j+22,an+2j+22}, Rx(ajaj+1)=V(MPz,1,2){aj+2, aj+4,aj+6, ....,az+j12,az+2j+22, az+2j+62,....,az+j6,az+j3,az+j1}.

    The RLN sets are classified in Table 4 and it is clear that |Rx(ajaj+1)| is less then all other RLN sets of MPz,1,2.

    Table 4.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 2z4>z+2
    Rx(ajaj+2) 2z4>z+2
    Rx(ajaj+1) 2z4>z+2

     | Show Table
    DownLoad: CSV

    Theorem 4.8. Let MPz,1,2 be a modified prism network, where z2(mod4). Then

    zz2dimLF(MPz,1,2)2zz+2.

    Proof. Case 1. For z=6, we have the following RLN sets;

    Rx(a1a2)=V(MP6,1,2){a3,a6,a3,a5,a6},

    Rx(a2a3)=V(MP6,1,2){a4,a1,a4,a6,a1},

    Rx(a3a4)=V(MP6,1,2){a5,a2,a5,a1,a2},

    Rx(a4a5)=V(MP6,1,2){a6,a3,a6,a2,a3},

    Rx(a5a6)=V(MP6,1,2){a1,a4,a1,a3,a4},

    Rx(a6a1)=V(MP6,1,2){a2,a5,a2,a4,a5},

    Rx(a1a1)=V(MP6,1,2){a3,a4,a5},

    Rx(a2a2)=V(MP6,1,2){a4,a5,a6},

    Rx(a3a3)=V(MP6,1,2){a5,a6,a1},

    Rx(a4a4)=V(MP6,1,2){a6,a1,a2},

    Rx(a5a5)=V(MP6,1,2){a1,a2,a3},

    Rx(a6a6)=V(MP6,1,2){a2,a3,a4},

    Rx(a1a3)=V(MP6,1,2){a2,a5,a2,a5},

    Rx(a2a4)=V(MP6,1,2){a3,a6,a3,a6},

    Rx(a3a5)=V(MP6,1,2){a4,a1,a4,a1},

    Rx(a4a6)=V(MP6,1,2){a5,a2,a5,a2},

    Rx(a5a1)=V(MP6,1,2){a6,a3,a6,a3},

    Rx(a6a2)=V(MP6,1,2){a1,a4,a1,a4},

    Rx(a1a2)=V(MP6,1,2){a3,a6},

    Rx(a2a3)=V(MP6,1,2){a4,a1},

    Rx(a3a4)=V(MP6,1,2){a5,a2},

    Rx(a4a5)=V(MP6,1,2){a6,a3},

    Rx(a5a6)=V(MP6,1,2){a1,a4},

    Rx(a1a6)=V(MP6,1,2){a2,a5}.

    For 1j6 it is clear that |Rx(ajaj+1)|=7 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MP6,1,2). Then there exits an upper LRF η:V(MP6,1,2)[0,1] is defined as η(y)=17 for each yV(MP6,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP6,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP6,1,2 hence dimLF(MP6,1,2)12118=32. In the same context, for 1jz it is clear that |Rx(ajaj+2)|=12 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other resolving local neighbour sets of MP6,1,2). Then there exits a lower LRF η:V(MP6,1,2)[0,1] and it is defined as η(y)=121 for each yV(MP6,1,2) hence dimLF(MP6,1,2)121110=65. Since MP6,1,2 is a non bipartite network so its lower bound must be greater then 1. Consequently,

    65<dimLF(MP6,1,2)32.

    Case 2. For 1jz from Lemma 4.7 that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2. Then there exits an upper LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z+2 for each yV(MPz,1,2). In order to show that η is a minimal upper LRF, we define another LRF η:V(MPz,1,2)[0,1] as |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP6,1,2 hence by Lemma X dimLF(MPz,1,2)2zj=11z+2=2zz+2. In the same way, for 1jz it is clear from Lemma 4.7 |Rx(ajaj+1)|=2z4 and |Rx(ajaj+2)||Rx(e)|, where Rx(e) are the other RLN sets of MPz,1,2). Then there exits a lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MPz,1,2) hence by Lemma YdimLF(MPz,1,2)2zj=112z4=zz2. Consequently,

    zz2dimLF(MPz,1,2)2zz+2.

    In this section, we compute RLN sets and LFMD of modified prism network MQz,1,2 in the form of bounds.

    Lemma 5.1. Let MQz,1,2 be a modified prism network, where z2(mod4). Then

    (a) |Rx(ajaj+1)|=3z+62 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(e)| are the other possible RLN sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized Prism network, for 1jz, where z+11(modz), we have the following possibilities

    (a) Rx(ajaj+1)=V(MQz,1,2){aj+2,aj+4,aj+6....., az+j12,az+2j+22,az+2j+62,az+2j+102, .....az+i5,az+i3, az+i1}{aj+2,aj+4,aj+6....., az+j12,az+2j+22, az+2j+62, az+2j+102,.....az+i5, az+i3,az+i1}{bj+2,bj+4,bj+6....., bz+j12,bz+2j+22,bz+2j+62, bz+2j+102,.....bz+i5, bz+i3,bz+i1} and |Rx(ajaj+1)|=3z+62 and |3zj=1Rx(ajaj+1)|=3z=|V(MQz,1,2)|.

    (b)Rx(ajaj)=V(MQz,1,2){aj+2,ai+3,az+j3,az+j2, bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MQz,1,2){aj+1,az+2j+22, aj+1,az+2j+22, bj+1,bz+2j+22}, Rx(ajaj+1)=V(MQz,1,2){az+j1}, Rx(bjbj+1)=V(MQz,1,2){aj+2}, Rx(ajbj)=V(MQz,1,2).

    The RLN sets classified in Table 5 and it is clear that |Rx(ajaj+1| is less then all other RLN sets of MQz,1,2.

    Table 5.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z+62
    Rx(ajaj+2) 3z4>3z+62
    Rx(ajbj) 3z>3z+62
    Rx(ajaj+1) 3z1>3z+62
    Rx(bjbj+1) 3z1>3z+62

     | Show Table
    DownLoad: CSV

    Theorem 5.2. Let MQz,1,2 be a modified prism network, where z2(mod4). Then

    1<dimLF(MPz,1,2)2zz+2.

    Proof. Case 1. For z=6, we have the following RLN sets

    Rx(a1a2)=V(MQ6,1,2){a3,a6,a3,a6,b3,b6},

    Rx(a2a3)=V(MQ6,1,2){a4,a1,a4,a1,b4,b1},

    Rx(a3a4)=V(MQ6,1,2){a5,a6,a5,a2,b5,b2},

    Rx(a4a5)=V(MQ6,1,2){a6,a1,a6,a3,b6,b3},

    Rx(a5a6)=V(MQ6,1,2){a1,a2,a1,a2,b1,b4},

    Rx(a6a1)=V(MQ6,1,2){a2,a3,a2,a3,b2,b5},

    Rx(a1a1)=V(MQ6,1,2){a3,a4,a5,b3,b4,b5},

    Rx(a2a2)=V(MQ6,1,2){a4,a5,a6,b4,b5,b6},

    Rx(a3a4)=V(MQ6,1,2){a5,a6,a1,b5,b6,b1},

    Rx(a4a4)=V(MQ6,1,2){a6,a1,a2,b6,b1,b2},

    Rx(a5a5)=V(MQ6,1,2){a1,a2,a3,b1,b2,b3},

    Rx(a6a6)=V(MQ6,1,2){a2,a3,a4,b2,b3,b4},

    Rx(a1a3)=V(MQ6,1,2){a2,a5,a2,a5,b2,b5},

    Rx(a2a4)=V(MQ6,1,2){a3,a6,a3,a6,b3,b6},

    Rx(a3a5)=V(MQ6,1,2){a4,a1,a4,a1,b4,b1},

    Rx(a4a6)=V(MQ6,1,2){a5,a2,a5,a2,b5,b2},

    Rx(a5a1)=V(MQ6,1,2){a6,a3,a6,a3,b6,b3},

    Rx(a6a2)=V(MQ6,1,2){a1,a4,a6,a4,b1,b4},

    Rx(a1a2)=V(MQ6,1,2){a3,a6},

    Rx(a2a3)=V(MQ6,1,2){a4,a5},

    Rx(a3a4)=V(MQ6,1,2){a5,a6},

    Rx(a4a5)=V(MQ6,1,2){a6,a1},

    Rx(a5a6)=V(MQ6,1,2){a1,a2},

    Rx(a6a1)=V(MQ6,1,2){a2,a3},

    Rx(b1b2)=V(MQ6,1,2){a3,a6},

    Rx(b2b3)=V(MQ6,1,2){a4,a1},

    Rx(b3b4)=V(MQ6,1,2){a5,a2},

    Rx(b4b5)=V(MQ6,1,2){a6,a1},

    Rx(b5b6)=V(MQ6,1,2){a1,a2},

    Rx(b1b6)=V(MQ6,1,2){a2,a1},

    Rx(a1b1)=V(MQ6,1,2),

    Rx(a2b2)=V(MQ6,1,2),

    Rx(a3b3)=V(MQ6,1,2),

    Rx(a4b4)=V(MQ6,1,2),

    Rx(a5b5)=V(MQ6,1,2),

    Rx(a6b6)=V(MQ6,1,2),

    Rx(a6b6)=V(MP6,1,2).

    For 1j6 it is clear that |Rx(ajaj+1)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ6,1,2. Then there exits an upper LRF η:V(MP6,1,2)[0,1] and is defined as η(y)=112 for each yV(MQ6,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MP6,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not LRF. Therefore, dimLF(MQ6,1,2)181112=32. For 1j6 it is clear from the above RLN sets that |Rx(bjbj+1)|=18 and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are other RLN sets of MQ6,1,2). Then there exits a lower LRF η:V(MQ6,1,2)[0,1] and it is defined as η(y)=118 for each yV(MQ6,1,2) hence dimLF(MQ6,1,2)181118=1. Since MQ6,1,2 is a non-bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ6,1,2)32.

    Case 2. For 1jz from Lemma 5.1 it is clear from the above resolving local neighbourhood sets that |Rx(ajaj+1)|=23z+6 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exits an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=23n+6 for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ6,1,2. Therefore by Lemma X dimLF(MQz,1,2)3zj=123z+6=2zz+2.

    For 1jz it is clear from Lemma 5.1 |Rx(ajbj)|=3z and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exits a maximal lower LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2). Hence by Lemma Y dimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower of LFMD bound must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz+2.

    Lemma 5.3. Let MQz,1,2 be a modified prism network, where z0(mod4). Then

    (a) |Rx(ajaj+1)|=3z2 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)|, where |Rx(y)| are the other possible RLN sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized Prism network, for 1jz, where z+11(modz), we have following possibilities

    (a) Rx(ajaj+1) = V(MQz,1,2){aj+2,aj+4,aj+6.....,az+j12, az+2j+22, az+2j+62,az+2j+102,.....az+i5, az+i3,az+i1}{aj+2,aj+4,aj+6.....,az+j12, az+2j+22, az+2j+62,az+2j+102, .....az+i5,az+i3,az+i1}{bj+2,bj+4,bj+6....., bz+j12, bz+2j+22, bz+2j+62,bz+2j+102, .....bz+i5,bz+i3, bz+i1} and |Rx(ajaj+1)|=3z2 and |3zj=1Rx(ajaj+1)| = 3z=|V(MQz,1,2)|.

    (b) Rx(ajaj)=V(MQz,1,2){aj+2,ai+3,az+j3,az+j2,bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MQz,1,2)-{aj+1,az+2j+22, aj+1,az+2j+22,bj+1, bz+2j+22}, Rx(ajaj+1)=V(MQz,1,2){az+j1}, Rx(bjbj+1)=V(MQz,1,2){aj+2}, Rx(ajbj)=V(MQz,1,2).

    The RLN sets are classified in Table 6 and it is clear that |Rx(ajaj+1| is less then all other RLN sets of MQz,1,2.

    Table 6.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z2
    Rx(ajaj+2) 3z4>3z2
    Rx(ajbj) 3z>3z2
    Rx(ajaj+1) 3z1>3z2
    Rx(bjbj+1) 3z1>3z2

     | Show Table
    DownLoad: CSV

    Theorem 5.4. Let MQz,1,2 be a modified prism network, where z0(mod4). Then

    1dimLF(MQz,1,2)2.

    Proof. Case 1. For z=8, we have the following RLN sets;

    Rx(a1a2)=V(MQ8,1,2){a3,a5,a6,a8,a3,a5,a6,a8,b3,b5,b7,b8},

    Rx(a2a3)=V(MQ8,1,2){a4,a6,a1,a2,a4,a6,a7,a1,b4,b6,b8,b1},

    Rx(a3a4)=V(MQ8,1,2){a5,a7,a8,a3,a5,a7,a8,a2,b5,b7,b1,b2},

    Rx(a4a5)=V(MQ8,1,2){a6,a8,a1,a4,a6,a8,a1,a3,b6,b8,b2,b3},

    Rx(a5a6)=V(MQ8,1,2){a7,a1,a2,a5,a7,a1,a2,a4,b7,b1,b3,b4},

    Rx(a6a7)=V(MQ8,1,2){a8,a2,a3,a6,a8,a2,a3,a5,b8,b2,b4,b5},

    Rx(a7a8)=V(MQ8,1,2){a1,a3,a4,a7,a1,a3,a4,a6,b1,b3,b5,b6},

    Rx(a1a8)=V(MQ8,1,2){a2,a4,a5,a8,a2,a4,a5,a7,b2,b4,b6,b7},

    Rx(a1a1)=V(MQ8,1,2){a3,a4,a6,a7,b3,b4,b6,b7},

    Rx(a2a2)=V(MQ8,1,2){a4,a5,a7,a8,b4,b5,b7,b8},

    Rx(a3a3)=V(MQ8,1,2){a5,a6,a8,a1,b5,b6,b8,b1},

    Rx(a4a4)=V(MQ8,1,2){a6,a7,a1,a2,b6,b7,b1,b2},

    Rx(a5a5)=V(MQ8,1,2){a7,a8,a2,a3,b7,b8,b2,b3},

    Rx(a6a6)=V(MQ8,1,2){a8,a1,a3,a4,b8,b1,b3,b4},

    Rx(a7a7)=V(MQ8,1,2){a1,a2,a4,a5,b1,b2,b4,b5},

    Rx(a8a8)=V(MQ8,1,2){a2,a3,a5,a6,b2,b3,b5,b6},

    Rx(a1a2)=V(MQ8,1,2){a3,a5,a6,a8,},

    Rx(a2a3)=V(MQ8,1,2){a4,a6,a7,a1},

    Rx(a3a4)=V(MQ8,1,2){a5,a7,a8,a2},

    Rx(a4a5)=V(MQ8,1,2){a6,a8,a1,a3},

    Rx(a5a6)=V(MQ8,1,2){a7,a1,a2,a4},

    Rx(a6a7)=V(MQ8,1,2){a8,a2,a3,a5},

    Rx(a7a8)=V(MQ8,1,2){a1,a3,a4,a6},

    Rx(a8a1)=V(MQ8,1,2){a2,a4,a5,a7},

    Rx(a1a3)=V(MQ8,1,2){a2,a6,a2,a6,b2,b6},

    Rx(a2a4)=V(MQ8,1,2){a3,a7,a3,a7,b3,b7},

    Rx(a3a5)=V(MQ8,1,2){a4,a8,a4,a8,b4,b8},

    Rx(a4a6)=V(MQ8,1,2){a5,a1,a5,a1,b5,b1},

    Rx(a5a7)=V(MQ8,1,2){a6,a2,a6,a2,b6,b2},

    Rx(a6a8)=V(MQ8,1,2){a3,a7,a3,a7,b7,b3},

    Rx(a7a1)=V(MQ8,1,2){a8,a1,a8,a1,b8,b4},

    Rx(a8a2)=V(MQ8,1,2){a1,a5,a1,a5,b1,b5},

    Rx(b1b2)=V(MQ8,1,2){a3,a5,a6,a8},

    Rx(b2b3)=V(MQ8,1,2){a4,a6,a7,a1},

    Rx(b3b4)=V(MQ8,1,2){a5,a7,a8,a2},

    Rx(b4b5)=V(MQ8,1,2){a6,a8,a1,a3},

    Rx(b5b6)=V(MQ8,1,2){a7,a1,a2,a4},

    Rx(b6b7)=V(MQ8,1,2){a8,a2,a3,a5},

    Rx(b7b8)=V(MQ8,1,2){a1,a3,a4,a6},

    Rx(b8b1)=V(MQ8,1,2){a2,a4,a5,a7},

    Rx(a1b1)=V(MQ8,1,2),

    Rx(a2b2)=V(MQ8,1,2),

    Rx(a3b3)=V(MQ8,1,2),

    Rx(a4b4)=V(MQ8,1,2),

    Rx(a5b5)=V(MQ8,1,2),

    Rx(a6b6)=V(MQ8,1,2),

    Rx(a7b7)=V(MQ8,1,2),

    Rx(a8b8)=V(MQ8,1,2).

    For 1j8 it is clear that |Rx(ajaj+1)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ8,1,2). Then there exits an upper LRF η:V(MQ8,1,2)[0,1] and it is defined as η(y)=18 for each yV(MQ8,1,2). In order to show that η(y) is a minimal LRF, we define another resolving function η(y):V(MQ8,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ8,1,2 hence by Lemma XdimLF(MQ8,1,2)241112=2. In the same context, for 1jz it is clear from the above RLN sets that |Rx(ajaj+2)|=12 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ8,1,2. Then there exits a lower LRF η:V(MQ8,1,2)[0,1] such that η(y)=124 for each yV(MQ8,1,2) hence dimLF(MQ8,1,2)241124=1. Since MQ8,1,2 is non bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQ8,1,2)2.

    Case 2. For 1jz it is clear that |Rx(ajaj+1)|=z and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exits an upper LRF η:V(MQz,1,2)[0,1] is defined as η(y)=1z for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ8,1,2 hence by Lemma X dimLF(MPz,1,2)3zj=123z=2. In the same way, for 1jz it is clear from Lemma 5.3 that |Rx(ajbj)|=3z and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other resolving local neighbour sets of MQz,1,2). Then there exits a maximal lower LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2) hence by Lemma YdimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2.

    Lemma 5.5. Let MQz,1,2 be a modified prism network, where z1(mod4). Then

    (a) |Rx(ajaj+1)|=3z32 and zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible RLN sets.

    Proof. Let aj inner, aj middle and bj are be the outer vertices of modified prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1) = V(MPz,1,2){aj+2,aj+4,aj+6.....,az+i5, az+i3,az+i1}{aj+2,aj+4, aj+6,.....,az+i5, az+i3,az+i1}{bj+2,bj+4, bj+6,.....,bz+i5,bz+i3, bz+i1}{az+2i+22}{az+2i+22} and |Rx(ajaj+1)|=z1 and |3zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b) Rx(ajaj)=V(MPz,1,2)-{aj+2,aj+3,az+j3,az+j4,bj+2,bj+3,bz+j3,bz+j4}, Rx(ajaj+2)=V(MPz,1,2)-{aj+1,aj+1,bj+1,az+2j+12,az+2j+12, bz+2j+12}, Rx(bjbj+1)=Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4, aj+6,....,az+j3,az+j1}{az+2j+12}{az+2j+12,bz+2j+12}. Rx(ajbj)=V(MPz,1,2).

    The RLN sets are classified in Table 7 and it is clear |Rx(ajaj+1)| is less then all other RLN sets of MQz,1,2.

    Table 7.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z8>3z32
    Rx(ajaj+2) 3z6>3z32
    Rx(ajaj+1) 5x25>3z32
    Rx(ajbj) 3z>3z32
    Rx(bjaj+1) 5z25>3z32

     | Show Table
    DownLoad: CSV

    Theorem 5.6. Let MPz,1,2 be a modified prism network, where z1(mod4). Then

    1<dimLF(MPz,1,2)2zz1.

    Proof. Case 1. For z=5, we have the following RLN sets

    Rx(a1a2)={a1,a2,a1,a2,b1,b2},

    Rx(a2a3)={a2,a3,a2,a3,b2,b3},

    Rx(a3a4)={a3,a4,a3,a4,b3,b4},

    Rx(a4a5)={a4,a5,a4,a5,b4,b5},

    Rx(a5a1)={a1,a5,a1,a5,b5,b1},

    Rx(a1a3)={a1,a3,a1,a3,b1,b3},

    Rx(a1a4)={a1,a4,a1,a4,b2,b4},

    Rx(a2a4)={a2,a4,a2,a4,b3,b5},

    Rx(a2a5)={a2,a5,a2,a5,b4,b1},

    Rx(a3a5)={a3,a5,a3,a5,b5,b2},

    Rx(a1a1)=V(MP5,1,2){a3,a4,b3,b4},

    Rx(a2a2)=V(MP5,1,2){a4,a5,b4,b5},

    Rx(a3a3)=V(MP5,1,2){a5,a1,b5,b1},

    Rx(a4a4)=V(MP5,1,2){a1,a2,b1,b2},

    Rx(a5a5)=V(MP5,1,2){a2,a3,b2,b3},

    Rx(a1a2)=V(MP5,1,2){a3,a4,a5,a4,b4},

    Rx(a2a3)=V(MP5,1,2){a4,a5,a1,a5,b5},

    Rx(a3a4)=V(MP5,1,2){a5,a1,a2,a1,b1},

    Rx(a4a5)=V(MP5,1,2){a1,a2,a3,a2,b5},

    Rx(a5a1)=V(MP5,1,2){a2,a3,a4,a3,b1},

    Rx(b1b2)=V(MP5,1,2){a3,a4,a5,a4,b4},

    Rx(b2b3)=V(MP5,1,2){a4,a5,a1,a5,b5},

    Rx(b3b4)=V(MP5,1,2){a5,a1,a2,a1,b1},

    Rx(b4b5)=V(MP5,1,2){a1,a2,a3,a2,b2},

    Rx(b5b1)=V(MP5,1,2){a2,a3,a4,a3,b3},

    Rx(a1b1)=V(MP5,1,2),

    Rx(a2b2)=V(MP5,1,2),

    Rx(a3b3)=V(MP5,1,2),

    Rx(a4b4)=V(MP5,1,2),

    Rx(a5b5)=V(MP5,1,2).

    For 1j5 it is clear that |Rx(ajaj+1)|=8 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ5,1,2. Then there exists an upper LRF η:V(MQ5,1,2)[0,1] and it is defined as η(y)=16 for each yV(MQ5,1,2. In order to show that η(y) is a minimal resolving local function, we define another resolving function η(y):V(MQ5,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MP5,1,2 hence dimLF(MQ5,1,2)15116=52. In the same context, for 1jz it is clear that |Rx(ajaj)|=8 and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other RLN sets of MQ5,1,2. Then there exits a maximal lower LRF η:V(MQ5,1,2)[0,1] and it is defined as η(y)=115 for each yV(MQ5,1,2) hence dimLF(MQ5,1,2)151115=1. Since MQ5,1,2) is a non bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ5,1,2)52.

    Case 2. For 1jz from Lemma 5.5 it is clear that |Rx(ajaj+1)|=z+1 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exists an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=1z1 for each yV(MQz,1,2). In order to show that η is a minimal LRF, we define another LRF η:V(MPz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ5,1,2. Therefore, by Lemma X dimLF(MQz,1,2)3zj=123z3=2zz1. In the same context, for 1jz it is clear from Lemma 5.5 that |Rx(ajbj)|=3z and |Rx(ajbj)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exists an upper LRF η:V(MPz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQ) hence by Lemma Y dimLF(MQz,1,2)3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz1.

    Lemma 5.7. Let MQz,1,2 be a modified prism network, where z3(mod4). Then

    (a) |Rx(ajaj+1)|=3z+32 and 3zj=1Rx(ajaj+1)=V(MQz,1,2).

    (b) |Rx(ajaj+1)|<|Rx(y)|, and |3zj=1Rx(ajaj+1)Rx(y)|>|Rx(ajaj+1)| where |Rx(y)| are the other possible resolving local neighbourhood sets.

    Proof. Let ai inner, ai middle and bi be the outer vertices of modified generalized prism network, for 1jz, where z+1(1modz), we have following possibilities

    (a) Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,....,az+i1}{aj+2,aj+4,aj+6,....,az+i1}{bj+2,bj+4,bj+6,....,bz+i1} and |Rx(ajaj+1)|=3z+32 and |3zj=1Rx(ajaj+1)|=3z=|V(MPz,1,2)|.

    (b)Rx(ajaj) =V(MPz,1,2){aj+2,aj+3,az+j3,az+j2,bj+2,bj+3,bz+j3,bz+j2}, Rx(ajaj+2)=V(MPz,1,2){aj+1,aj+1,bj+1}, Rx(ajaj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,...,az+j1,az+2j+12,bz+2j+12}, Rx(bjbj+1)=V(MPz,1,2){aj+2,aj+4,aj+6,...,az+i1,az+2j+12,bz+2j+12}, Rx(ajbj)=V(MPz,1,2).

    The RLN sets are classified in Table 8 and it is clear that |Rx(ajaj+1)| is less then all other RLN sets MQz,1,2.

    Table 8.  Cardinality of each LRN set.
    RLN Set Cardinality
    Rx(ajaj) 3z4>3z+32
    Rx(ajaj+2) 3z4>3z+32
    Rx(ajbj) 3z>3z+32
    Rx(ajaj+1) 3z1>3z+32
    Rx(bjbj+1) 3z1>3z+32

     | Show Table
    DownLoad: CSV

    Theorem 5.8. Let MQz,1,2 be a generalized modified prism network, where z3(mod4). Then

    1<dimLF(MQz,1,2)2zz+2.

    Proof. Case 1. For z=7, we have the following RLN sets

    Rx(a1a2)=V(MQ7,1,2){a3,a5,a7,a3,a5,a7,b3,b5,b7},

    Rx(a2a3)=V(MQ7,1,2){a4,a6,a1,a4,a6,a1,b4,b6,b1},

    Rx(a3a4)=V(MQ7,1,2){a5,a7,a2,a5,a7,a2,b5,b7,b2},

    Rx(a4a5)=V(MQ7,1,2){a6,a1,a3,a6,a1,a3,b6,b1,b3},

    Rx(a5a6)=V(MQ7,1,2){a7,a2,a4,a7,a2,a4,b7,b2,b4},

    Rx(a6a7)=V(MQ7,1,2){a1,a3,a5,a1,a3,a5,b1,b3,b5},

    Rx(a7a1)=V(MQ7,1,2){a2,a4,a6,a2,a4,a6,b2,b4,b6},

    Rx(a1a1)=V(MQ7,1,2){a3,a4,a5,a6,b3,b4,b5,b6},

    Rx(a2a2)=V(MQ7,1,2){a4,a5,a6,a7,b4,b5,b6,b7},

    Rx(a3a3)=V(MQ7,1,2){a5,a6,a7,a1,b5,b6,b7,b1},

    Rx(a4a4)=V(MQ7,1,2){a6,a7,a1,a2,b5,b7,b1,b2},

    Rx(a5a5)=V(MQ7,1,2){a7,a1,a2,a3b6,b1,b2,b3},

    Rx(a6a6)=V(MQ7,1,2){a1,a2,a3,a4,b7,b2,b3,b4},

    Rx(a7a7)=V(MQ7,1,2){a2,a3,a4,a5,b1,b3,b4,b5},

    Rx(a1a3)=V(MQ7,1,2){a2,a2,b2},

    Rx(a2a4)=V(MQ7,1,2){a3,a3,b3},

    Rx(a3a5)=V(MQ7,1,2){a4,a4,b4},

    Rx(a4a6)=V(MQ7,1,2){a5,a5,b5},

    Rx(a5a7)=V(MQ7,1,2){a6,a6,b6},

    Rx(a6a1)=V(MQ7,1,2){a7,a7,b7},

    Rx(a7a2)=V(MQ7,1,2){a1,a1,b1},

    Rx(b1b2)=V(MQ7,1,2){a3,a5,a7,b5},

    Rx(b2b3)=V(MQ7,1,2){a4,a6,a1,b6},

    Rx(b3b4)=V(MQ7,1,2){a5,a7,a2,b7},

    Rx(b4b5)=V(MQ7,1,2){a6,a1,a3,b1},

    Rx(b5b6)=V(MQ7,1,2){a7,a2,a4,b2},

    Rx(b6b7)=V(MQ7,1,2){a1,a3,a5,b3},

    Rx(b7b1)=V(MQ7,1,2){a2,a4,a6,b4},

    Rx(a1a2)=V(MQ7,1,2){a3,a5,a7,a5,b5},

    Rx(a2a3)=V(MQ7,1,2){a4,a6,a1,a6,b6},

    Rx(a3a4)=V(MQ7,1,2){a5,a7,a2,a7,b7},

    Rx(a4a5)=V(MQ7,1,2){a6,a1,a3,a1,b1},

    Rx(a5a6)=V(MQ7,1,2){a7,a2,a4,a2,b2},

    Rx(a6a7)=V(MQ7,1,2){a1,a3,a5,a3,b3},

    Rx(a7a1)=V(MQ7,1,2){a2,a4,a6,a4,b4},

    Rx(a1b1)=V(MQ7,1,2),

    Rx(a2b2)=V(MQ7,1,2),

    Rx(a3b3)=V(MQ7,1,2),

    Rx(a4b4)=V(MQ7,1,2),

    Rx(a5b5)=V(MQ7,1,2),

    Rx(a6b6)=V(MQ7,1,2),

    Rx(a7b7)=V(MQ7,1,2).

    For 1j7 |Rx(ajaj+1)|=13 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ7,1,2). Then there exists an upper LRF η:V(MQ7,1,2)[0,1] and it is defined as η(y)=113 for each yV(MQ7,1,2). In order to show that η(y) is a minimal LRF, we define another LRF η(y):V(MQ7,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQ7,1,2 hence dimLF211112=712. In the same context, for 1j7 it is clear that |Rx(ajaj+1)|=21 and |Rx(ajaj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQ7,1,2). Then there exists a maximal LLRF η:V(MQ7,1,2)[0,1] and it is defined as η(y)=121 for each yV(MQ7,1,2) hence dimLF=211121=1. Since MQ7,1,2) is non-bipartite network so its lower bound must be greater then 1. Consequently,

    1<dimLF(MQ7,1,2)712.

    Case 2. For 1jz from Lemma 5.7 it is clear that |Rx(ajaj)|=23z+6 and |Rx(ajaj)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2). Then there exits an upper LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=23n+6 for each yV(MQz,1,2). In order to show that η is a minimal LRF of MQz,1,2), we define another LRF η:V(MQz,1,2)[0,1] such that |η(y)|<|η(y)| then η(Rx(e))<1 which shows that η is not a LRF of MQz,1,2) hence by Lemma X dimLF3zj=123z+3=2zz+1. In the same context for 1jz it is clear from Lemma 5.7 that |Rx(ajbj)|=3z and |Rx(bjbj+1)||Rx(e)|, where Rx(e) are the other RLN sets of MQz,1,2. Then there exists a maximal lower LRF η:V(MQz,1,2)[0,1] and it is defined as η(y)=13z for each yV(MQz,1,2). Therefore, by Lemma Y dimLF3zj=113z=1. Since MQz,1,2 is a non-bipartite network so its lower bound of LFMD must be greater then 1. Consequently,

    1<dimLF(MQz,1,2)2zz+1.

    In this paper, we have computed the local fractional metric dimension of generalized modified prism networks (MPz,1,2,MQz,1,2) in the form of lower and upper bounds. The lower bounds of all the modified prism networks MQz,1,2 is strictly greater than 1 in all cases. Moreover, all of these modified prism networks remain bounded when z as shown in Table 9.

    Table 9.  Limiting values of LFMDs of modified prism networks.
    z LFMDs Limiting LFMDs as z Comment
    1(mod4) zz1dimLF(MPz,1,2)2zz1 1<dimLF(MPz,1,2)2 Bounded
    3(mod4) zz1dimLF(MPz,1,2)2zz+1 1<dimLF(MPz,1,2)2 Bounded
    0(mod4) zz2dimLF(MPz,1,2)2 1<dimLF(MPz,1,2)2 Bounded
    2(mod4) zz2dimLF(MPz,1,2)2zz+2 1<dimLF(MPz,1,2)2 Bounded
    2(mod4) 1<dimLF(MQz,1,2)2zz+2 1<dimLF(MQz,1,2)2 Bounded
    0(mod4) 1<dimLF(MQz,1,2)2 1<dimLF(MQz,1,2)2 Bounded
    1(mod4) 1<dimLF(MQz,1,2)2zz1 1<dimLF(MQz,1,2)2 Bounded
    3(mod4) 1<dimLF(MQz,1,2)2zz+2 1<dimLF(MQz,1,2)2 Bounded

     | Show Table
    DownLoad: CSV

    The authors appreciate the valuable comments and remarks of anonymous referees which helped to greatly improve the quality of the paper.

    The second author (Hassan Zafar) and the third author (Muhammad Javaid) are supported by the Higher Education Commission of Pakistan through the National Research Program for Universities (NRPU) Grant NO. 20-16188/NRPU/R & D/HEC/2021 2021.

    The authors declare that they have no conflicts of interest.



    [1] Y. K. Chang, J. J. Nieto, W. S. Li, Controllability of semilinear differential systems with nonlocal initial conditions in Banach spaces, J. Optim. Theory Appl., 142 (2009), 267–273. https://doi.org/10.1007/s10957-009-9535-2 doi: 10.1007/s10957-009-9535-2
    [2] L. Górniewicz, S. K. Ntouyas, D. O'Regan, Existence and controllability results for first-and second-order functional semilinear differential inclusions with nonlocal conditions, Numer. Funct. Anal. Optim., 28 (2007), 53–82. https://doi.org/10.1080/01630560600883093 doi: 10.1080/01630560600883093
    [3] L. Górniewicz, S. K. Ntouyas, D. O'Regan, Controllability results for first and second order evolution inclusions with nonlocal conditions, Ann. Pol. Math., 89 (2007), 65–101. https://doi.org/10.4064/ap89-1-5 doi: 10.4064/ap89-1-5
    [4] N. U. Ahmed, Nonlinear stochastic differential inclusions on balance space, Stoch. Anal. Appl., 12 (1994), 1–10. https://doi.org/10.1080/07362999408809334 doi: 10.1080/07362999408809334
    [5] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge: Cambridge university press, 2014.
    [6] X. Fu, Approximate controllability for neutral impulsive differential inclusions with nonlocal conditions, J. Dyn. Control Syst., 17 (2011), 359–386. https://doi.org/10.1007/s10883-011-9126-z doi: 10.1007/s10883-011-9126-z
    [7] N. I. Mahmudov, Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces, SIAM J. Control Optim., 42 (2003), 1604–1622. https://doi.org/10.1137/S0363012901391688 doi: 10.1137/S0363012901391688
    [8] P. Muthukumar, C. Rajivganthi, Approximate controllability of impulsive neutral stochastic functional differential system with state-dependent delay in Hilbert spaces, J. Control Theory Appl., 11 (2013), 351–358. https://doi.org/10.1007/s11768-013-2061-7 doi: 10.1007/s11768-013-2061-7
    [9] R. P. Agarwal, B. de Andrade, G. Siracusa, On fractional integro-differential equations with state-dependent delay, Comput. Math. Appl., 62 (2011), 1143–1149. https://doi.org/10.1016/j.camwa.2011.02.033 doi: 10.1016/j.camwa.2011.02.033
    [10] G. M. Mophou, G. M. N'Guérékata, Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl. Math. Comput., 216 (2010), 61–69. https://doi.org/10.1016/j.amc.2009.12.062 doi: 10.1016/j.amc.2009.12.062
    [11] X. B. Shu, Y. Lai, Y. Chen, The existence of mild solutions for impulsive fractional partial differential equations, Nonlinear Anal. Theory Method. Appl., 74 (2011), 2003–2011. https://doi.org/10.1016/j.na.2010.11.007 doi: 10.1016/j.na.2010.11.007
    [12] J. Cui, L. Yan, Existence result for fractional neutral stochastic integro-differential equations with infinite delay, J. Phys. A Math. Theor., 44 (2011), 335201. https://doi.org/10.1088/1751-8113/44/33/335201 doi: 10.1088/1751-8113/44/33/335201
    [13] J. Wang, Y. Zhou, Existence and controllability results for fractional semilinear differential inclusions, Nonlinear Anal. Real World Appl., 12 (2011), 3642–3653. https://doi.org/10.1016/j.nonrwa.2011.06.021 doi: 10.1016/j.nonrwa.2011.06.021
    [14] Z. Yan, H. Zhang, Existence of solutions to impulsive fractional partial neutral stochastic integro-differential inclusions with state-dependent delay, Electron. J. Differ. Equ., 2013 (2013), 1–21.
    [15] S. Duan, J. Hu, Y. Li, Exact controllability of nonlinear stochastic impulsive evolution differential inclusions with infinite delay in Hilbert spaces, Int. J. Nonlinear Sci. Numer. Simul., 12 (2011), 23–33. https://doi.org/10.1515/ijnsns.2011.023 doi: 10.1515/ijnsns.2011.023
    [16] A. Debbouche, D. Baleanu, Controllability of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems, Comput. Math. Appl., 62 (2011), 1442–1450. https://doi.org/10.1016/j.camwa.2011.03.075 doi: 10.1016/j.camwa.2011.03.075
    [17] A. Debbouche, D. Baleanu, Exact null controllability for fractional nonlocal integrodifferential equations via implicit evolution system, J. Appl. Math., 2012 (2012), 931975. https://doi.org/10.1155/2012/931975 doi: 10.1155/2012/931975
    [18] A. Debbouche, D. F. Torres, Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces, Int. J. Control, 86 (2013), 1577–1585. https://doi.org/10.1080/00207179.2013.791927 doi: 10.1080/00207179.2013.791927
    [19] S. Kumar, N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differ. Equ., 252 (2012), 6163–6174. https://doi.org/10.1016/j.jde.2012.02.014 doi: 10.1016/j.jde.2012.02.014
    [20] N. I. Mahmudov, Approximate controllability of fractional neutral evolution equations in Banach spaces, Abstr. Appl. Anal., 2013 (2013), 531894. https://doi.org/10.1155/2013/531894 doi: 10.1155/2013/531894
    [21] Z. Yan, Approximate controllability of fractional neutral integro-differential inclusions with state-dependent delay in Hilbert spaces, IMA J. Math. Control Informa., 30 (2013), 443–462. https://doi.org/10.1093/imamci/dns033 doi: 10.1093/imamci/dns033
    [22] H. M. Ahmed, M. M. El-Borai, W. El-Sayed, A. Elbadrawi, Null controllability of Hilfer fractional stochastic differential inclusions, Fractal Fract., 6 (2022), 721. https://doi.org/10.3390/fractalfract6120721 doi: 10.3390/fractalfract6120721
    [23] T. Sathiyaraj, J. Wang, P. Balasubramaniam, Controllability and optimal control for a class of time-delayed fractional stochastic integro-differential systems, Appl. Math. Optimi., 84 (2021), 2527–2554. https://doi.org/10.1007/s00245-020-09716-w doi: 10.1007/s00245-020-09716-w
    [24] X. Ma, X. B. Shu, J. Mao, Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay, Stoch. Dynam., 20 (2020), 2050003. https://doi.org/10.1142/S0219493720500033 doi: 10.1142/S0219493720500033
    [25] M. Liu, L. Chen, X. B. Shu, The existence of positive solutions for Φ-Hilfer fractional differential equation with random impulses and boundary value conditions, Wave. Random Complex Media, 2022, 1–19. https://doi.org/10.1080/17455030.2023.2176695 doi: 10.1080/17455030.2023.2176695
    [26] L. Shu, X. B.Shu, J. Mao, Approximate controllability and existence of mild solutions for Riemann-Liouville fractional stochastic evolution equations with nonlocal conditions of order 1<α<2, Fract. Calc. Appl. Anal., 22 (2019), 1086–1112. https://doi.org/10.1515/fca-2019-0057 doi: 10.1515/fca-2019-0057
    [27] Y. Guo, X. B. Shu, F. Xu, C. Yang, HJB equation for optimal control system with random impulses, Optimization, 2022, 1–25. https://doi.org/10.1080/02331934.2022.2154607 doi: 10.1080/02331934.2022.2154607
    [28] L. Byszewski, V. Lakshmikantham, Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space, Appl. Anal., 40 (1991), 11–19. https://doi.org/10.1080/00036819008839989 doi: 10.1080/00036819008839989
    [29] J. Alzabut, A. G. M. Selvam, R. A. El-Nabulsi, V. Dhakshinamoorthy, M. E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry, 13 (2021), 473. https://doi.org/10.3390/sym13030473 doi: 10.3390/sym13030473
    [30] K. Abuasbeh, R. Shafqat, Fractional Brownian motion for a system of fuzzy fractional stochastic differential equation, J. Math., 2022 (2022), 3559035. https://doi.org/10.1155/2022/3559035 doi: 10.1155/2022/3559035
    [31] K. Abuasbeh, R. Shafqat, A. Alsinai, M. Awadalla, Analysis of the mathematical modelling of COVID-19 by using mild solution with delay caputo operator, Symmetry, 15 (2023), 286. https://doi.org/10.3390/sym15020286 doi: 10.3390/sym15020286
    [32] K. Abuasbeh, R. Shafqat, A. Alsinai, M. Awadalla, Analysis of controllability of fractional functional random integroevolution equations with delay, Symmetry, 15 (2023), 290. https://doi.org/10.3390/sym15020290 doi: 10.3390/sym15020290
    [33] K. Abuasbeh, R. Shafqat, A. U. K. Niazi, M. Awadalla, Oscillatory behavior of solution for fractional order fuzzy neutral predator-prey system, AIMS Math., 7 (2022), 20383–20400. https://doi.org/10.3934/math.20221117 doi: 10.3934/math.20221117
    [34] A. Moumen, R. Shafqat, A. Alsinai, H. Boulares, M. Cancan, M. B. Jeelani, Analysis of fractional stochastic evolution equations by using Hilfer derivative of finite approximate controllability, AIMS Math., 7 (2023), 16094–16114. https://doi.org/10.3934/math.2023821 doi: 10.3934/math.2023821
    [35] A. Moumen, R. Shafqat, Z. Hammouch, A. U. K. Niazi, M. B. Jeelani, Stability results for fractional integral pantograph differential equations involving two Caputo operators, AIMS Math., 8 (2023), 6009–6025. https://doi.org/10.3934/math.2023303 doi: 10.3934/math.2023303
    [36] A. A. A. Ghafli, R. Shafqat, A. U. K. Niazi, K. Abuasbeh, M. Awadalla, Topological structure of solution sets of fractional control delay problem, Fractal Fract., 7 (2023), 59. https://doi.org/10.3390/fractalfract7010059 doi: 10.3390/fractalfract7010059
    [37] R. Sakthivel, R. Ganesh, S. M. Anthoni, Approximate controllability of fractional nonlinear differential inclusions, Appl. Math. Comput., 225 (2013), 708–717. https://doi.org/10.1016/j.amc.2013.09.068 doi: 10.1016/j.amc.2013.09.068
    [38] R. Sakthivel, S. Suganya, S. M. Anthoni, Approximate controllability of fractional stochastic evolution equations, Comput. Math. Appl., 63 (2012), 660–668. https://doi.org/10.1016/j.camwa.2011.11.024 doi: 10.1016/j.camwa.2011.11.024
    [39] Y. Ren, L. Hu, R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math., 235 (2011), 2603–2614. https://doi.org/10.1016/j.cam.2010.10.051 doi: 10.1016/j.cam.2010.10.051
    [40] I. Podlubny, Fractional differential equations, Math. Sci. Eng., 1999,340.
    [41] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Cont Dyn. S, 13 (2020), 709–722. https://doi.org/10.3934/dcdss.2020039 doi: 10.3934/dcdss.2020039
    [42] B. C. Dhage, Multi-valued mappings and fixed points Ⅱ, Tamkang J. Math., 37 (2006), 27–46. https://doi.org/10.5556/j.tkjm.37.2006.177 doi: 10.5556/j.tkjm.37.2006.177
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2043) PDF downloads(133) Cited by(2)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog