\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\\n"}]',published:!0,mainMedia:{caption:"Highly Cited",originalUrl:"/media/original/117"}},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\n'}],latestNews:[{slug:"webinar-introduction-to-open-science-wednesday-18-may-1-pm-cest-20220518",title:"Webinar: Introduction to Open Science | Wednesday 18 May, 1 PM CEST"},{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"}]},book:{item:{type:"book",id:"5878",leadTitle:null,fullTitle:"Learning Disabilities - An International Perspective",title:"Learning Disabilities",subtitle:"An International Perspective",reviewType:"peer-reviewed",abstract:"Learning disabilities are conditions that are associated with difficulties in knowledge and skill acquisition to the level expected of same-age peers. The current book is an international examination of assessment methods, preventative measures, intervention, and research with those individuals with learning disabilities obtained from authors in the United States of America, Europe, Asia, and Africa.",isbn:"978-953-51-3652-1",printIsbn:"978-953-51-3651-4",pdfIsbn:"978-953-51-4600-1",doi:"10.5772/65822",price:119,priceEur:129,priceUsd:155,slug:"learning-disabilities-an-international-perspective",numberOfPages:228,isOpenForSubmission:!1,isInWos:1,isInBkci:!1,hash:"233671027a40a86828b81f5f6094c4df",bookSignature:"Carolyn S. Ryan",publishedDate:"November 29th 2017",coverURL:"https://cdn.intechopen.com/books/images_new/5878.jpg",numberOfDownloads:15153,numberOfWosCitations:4,numberOfCrossrefCitations:13,numberOfCrossrefCitationsByBook:0,numberOfDimensionsCitations:18,numberOfDimensionsCitationsByBook:0,hasAltmetrics:1,numberOfTotalCitations:35,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 17th 2016",dateEndSecondStepPublish:"November 7th 2016",dateEndThirdStepPublish:"March 6th 2017",dateEndFourthStepPublish:"June 12th 2017",dateEndFifthStepPublish:"September 20th 2017",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6",editedByType:"Edited by",kuFlag:!1,featuredMarkup:null,editors:[{id:"28738",title:"Dr.",name:"Carolyn S",middleName:null,surname:"Ryan",slug:"carolyn-s-ryan",fullName:"Carolyn S Ryan",profilePictureURL:"https://mts.intechopen.com/storage/users/28738/images/6072_n.png",biography:"Dr. Carolyn S. Ryan is a NYS Licensed Psychologist, NYS Licensed Behavior Analyst, and Board Certified Behavior Analyst-Doctorate. She is the President of Ryan Psychological, P.C., and serves the field of applied behavior analysis and autism as well as developmental and learning disabilities and other related disorders for over 25 years. She is the author and co-author of several research articles, book chapters, and books. Dr. Ryan made numerous presentations at local, invited addresses, state, national, and international conferences. Dr. Ryan serves as an editor for journals and book publishing companies on topics of expertise. She serves others by providing trainings, designing curriculum for students, supporting staff and students in educational and home settings, teaching graduate-level coursework, supervising those pursuing the BCBA, assisting on boards, and providing behavior analytic services.",institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:null}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"1239",title:"Learning Disability",slug:"learning-disability"}],chapters:[{id:"55805",title:"Classification and Detection of Specific Language Impairments in Children Based on their Speech Skills",doi:"10.5772/intechopen.69464",slug:"classification-and-detection-of-specific-language-impairments-in-children-based-on-their-speech-skil",totalDownloads:1679,totalCrossrefCites:2,totalDimensionsCites:5,hasAltmetrics:0,abstract:"The ability to use the spoken language is one of the most important characteristics in child development. Speech is difficult to replace in real life, although there are several other options for communication. Inabilities to communicate with speech skills can isolate children from society, especially children with specific language impairments. This research study focused on a specific disorder, known as specific language impairment (SLI); in the Czech language, it is specifically known as developmental dysphasia (DD). One major problem is that this disorder is detected at a relatively late age. Early diagnosis is critical for successful speech therapy in children. The current chapter presents several different approaches to solve this issue, including a simple test for detecting this disorder. One approach involves the use of an original iPad application for detecting SLI based on the number of pronunciation errors in utterances. One advantage of this method is its simplicity; anyone can use it, including parents.",signatures:"Pavel Grill and Jana Tučková",downloadPdfUrl:"/chapter/pdf-download/55805",previewPdfUrl:"/chapter/pdf-preview/55805",authors:[{id:"199652",title:"Ph.D.",name:"Pavel",surname:"Grill",slug:"pavel-grill",fullName:"Pavel Grill"}],corrections:null},{id:"55945",title:"Learning Disability in RASopathies",doi:"10.5772/intechopen.69571",slug:"learning-disability-in-rasopathies",totalDownloads:3190,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"Learning disabilities are relatively common conditions in pediatric population. The incidence of learning disability ranges from 1% to 17%, reflecting that learning disability may be not a single clinical entity but a wide distribution of cognitive traits in the population. As reported by the American Association on Intellectual and Developmental Disabilities (AAIDD), among the prenatal learning disability causes, chromosomal disorders, genetic syndromes, and inborn errors of metabolism must be taken into account. In this chapter, we will focus the attention on RASopathies, genetic disorders characterized by germline mutations in the RAS-MAPK pathway whose role is crucial in the regulation of the cell cycle, differentiation, growth, and cell senescence. This group of disorders includes Noonan syndrome, neurofibromatosis type 1, Costello syndrome (CS), Legius syndrome, Noonan syndrome with multiple lentigines, and cardiofaciocutaneous syndrome. Mutations in RAS-MAPK pathways lead to impairments in synaptic plasticity, necessary for normal brain function, especially for learning and memory. Variation across the RAS/MAPK pathway syndromes suggests that different gene mutations affecting this pathway can have markedly different developmental effects.",signatures:"Ilaria Maccora, Matteo Della Monica, Giovanna Traficante,\nGianpaolo De Filippo and Stefano Stagi",downloadPdfUrl:"/chapter/pdf-download/55945",previewPdfUrl:"/chapter/pdf-preview/55945",authors:[{id:"28293",title:"Dr.",name:"Stefano",surname:"Stagi",slug:"stefano-stagi",fullName:"Stefano Stagi"},{id:"183639",title:"Dr.",name:"Matteo",surname:"Della Monica",slug:"matteo-della-monica",fullName:"Matteo Della Monica"},{id:"200590",title:"Dr.",name:"Ilaria",surname:"Maccora",slug:"ilaria-maccora",fullName:"Ilaria Maccora"},{id:"204488",title:"Dr.",name:"Giovanna",surname:"Traficante",slug:"giovanna-traficante",fullName:"Giovanna Traficante"},{id:"204489",title:"Dr.",name:"Gianpaolo",surname:"De Filippo",slug:"gianpaolo-de-filippo",fullName:"Gianpaolo De Filippo"}],corrections:null},{id:"55379",title:"Developmental Dyslexia in Spain",doi:"10.5772/intechopen.69009",slug:"developmental-dyslexia-in-spain",totalDownloads:1142,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:1,abstract:"Spanish‐speaking children learn to read words written in a relatively transparent orthography. Variations in orthographic transparency may shape the manifestation of developmental dyslexia. In Spanish, as in other transparent orthographies, reading speed/fluency seems to be more evident and relevant than accuracy problems. In addition, the prevalence of dyslexia is much lower in Spanish than in less consistent or less transparent orthographies. Spanish students with developmental dyslexia have numerous lags in several cognitive (e.g., phonological awareness, rapid naming, verbal and visual‐spatial working memory, and executive functioning), academic (e.g., pseudoword reading, spelling, and vocabulary), and emotional (e.g., reading self‐concept, engagement, and reading motivation) areas. Intervention programs developed with Spanish children with dyslexia have ranged from phonological‐based programs to fluency‐based programs, with and without a computer.",signatures:"Manuel Soriano‐Ferrer and Manuel R. Morte‐Soriano",downloadPdfUrl:"/chapter/pdf-download/55379",previewPdfUrl:"/chapter/pdf-preview/55379",authors:[{id:"199542",title:"Dr.",name:"Manuel",surname:"Soriano-Ferrer",slug:"manuel-soriano-ferrer",fullName:"Manuel Soriano-Ferrer"},{id:"200649",title:"Mr.",name:"Manuel",surname:"Morte-Soriano",slug:"manuel-morte-soriano",fullName:"Manuel Morte-Soriano"}],corrections:null},{id:"55102",title:"Current Perspectives on Prevention of Reading and Writing Learning Disabilities",doi:"10.5772/intechopen.68627",slug:"current-perspectives-on-prevention-of-reading-and-writing-learning-disabilities",totalDownloads:1144,totalCrossrefCites:2,totalDimensionsCites:2,hasAltmetrics:0,abstract:"This chapter intends firstly to analyze the problem of identifying learning disabilities, from the standpoint of competing diagnostic models. The controversy between different models for identifying learning disabilities was presented, contrasting the characteristics of diagnostic models and models based on response to intervention. Second, an analysis of the main predictive factors of reading and writing was offered, using recent results from research carried out in different languages. The most often studied predictors—phonological awareness, speech perception, the alphabetic principle, rapid automatic naming, and vocabulary—were analyzed for their relationship to reading and writing. Finally, a discussion follows on the effects of certain programs that have been developed in different countries to prevent reading and writing learning disabilities. Most of these programs have been developed in the United States or Spain; they have also been implemented in other countries such as Canada, Australia, Mexico, Chile, and Israel.",signatures:"Maria-José González-Valenzuela",downloadPdfUrl:"/chapter/pdf-download/55102",previewPdfUrl:"/chapter/pdf-preview/55102",authors:[{id:"201295",title:"Dr.",name:"Maria-Jose",surname:"Gonzalez-Valenzuela",slug:"maria-jose-gonzalez-valenzuela",fullName:"Maria-Jose Gonzalez-Valenzuela"}],corrections:null},{id:"55182",title:"Multiple Intelligences and Videogames: Intervention Proposal for Learning Disabilities",doi:"10.5772/intechopen.68837",slug:"multiple-intelligences-and-videogames-intervention-proposal-for-learning-disabilities",totalDownloads:1446,totalCrossrefCites:1,totalDimensionsCites:2,hasAltmetrics:0,abstract:"In recent years, there has been much research into the possibilities offered by digital tools for intervention in learning disabilities. The most recent studies have found that these tools can have positive effects on diverse aspects of learning, such as the acquisition of reading, writing, vocabulary and mathematics, as well as improvement of executive functioning and behavioural control skills. Despite the results showing the positive effects of using digital tools for students with learning disabilities, it remains necessary to widen their use in areas such as identification, assessment and intervention as early as possible. Within the current chapter, the application of the conceptual framework of multiple intelligences to the design of educational video games is proposed to facilitate diagnosis and improve intervention success in cases of learning disability. In this regard, a proposed novel tool is presented that may be used for the evaluation and intervention for students with learning disabilities.",signatures:"Patricia García‐Redondo, Trinidad García, Debora Areces, Pablo\nGarmen and Celestino Rodríguez",downloadPdfUrl:"/chapter/pdf-download/55182",previewPdfUrl:"/chapter/pdf-preview/55182",authors:[{id:"85114",title:"Dr.",name:"Celestino",surname:"Rodríguez Pérez",slug:"celestino-rodriguez-perez",fullName:"Celestino Rodríguez Pérez"},{id:"206197",title:"Mrs.",name:"Patricia",surname:"García-Redondo",slug:"patricia-garcia-redondo",fullName:"Patricia García-Redondo"},{id:"206198",title:"Dr.",name:"Trinidad",surname:"García",slug:"trinidad-garcia",fullName:"Trinidad García"},{id:"206199",title:"Ph.D. Student",name:"Pablo",surname:"Garmen",slug:"pablo-garmen",fullName:"Pablo Garmen"},{id:"206200",title:"Mrs.",name:"Debora",surname:"Areces",slug:"debora-areces",fullName:"Debora Areces"}],corrections:null},{id:"57054",title:"Specific Learning Disabilities: Response to Intervention",doi:"10.5772/intechopen.70862",slug:"specific-learning-disabilities-response-to-intervention",totalDownloads:2119,totalCrossrefCites:1,totalDimensionsCites:2,hasAltmetrics:1,abstract:"The content included in the current chapter centers around the screening and identification of students who experience learning challenges in an educational setting in the United States of America. The specific learning challenges discussed will focus on students who may have a specific learning disability (SLD). Legislation that brought about concepts such as response to intervention (RTI) is discussed in detail. The various levels of intensity of interventions, or tiers, provided to students are explained by more than one discipline. The new regulations guiding access to special education services are based on the identification, intervention, and close monitoring of student progress. The overarching goal of RTI is to provide support to students who may be experiencing difficulty, before they experience failure by falling too far behind their peers.",signatures:"Kimberly A. Heinemann, Heather Bolanos and Jennifer S. Griffin",downloadPdfUrl:"/chapter/pdf-download/57054",previewPdfUrl:"/chapter/pdf-preview/57054",authors:[{id:"205622",title:"Mrs.",name:"Kimberly",surname:"Heinemann",slug:"kimberly-heinemann",fullName:"Kimberly Heinemann"},{id:"208681",title:"Mrs.",name:"Heather",surname:"Bolanos",slug:"heather-bolanos",fullName:"Heather Bolanos"},{id:"208691",title:"Mrs.",name:"Jennifer S.",surname:"Griffin",slug:"jennifer-s.-griffin",fullName:"Jennifer S. Griffin"}],corrections:null},{id:"57592",title:"Developing Automaticity in Children with Learning Disabilities: A Functional Perspective Part One: Theory and Assessment",doi:"10.5772/intechopen.70807",slug:"developing-automaticity-in-children-with-learning-disabilities-a-functional-perspective-part-one-the",totalDownloads:1487,totalCrossrefCites:3,totalDimensionsCites:3,hasAltmetrics:0,abstract:"The current chapter is the first of two chapters in this book to describe an instructional programme based on Luria’s theories, which can be used to develop basic skills and automaticity in reading, as well as basic skills and automaticity in writing and spelling. This chapter focuses on the theory behind the programme and then describes how assessment is used to develop an individual programme relating to both basic skill and fluency needs in reading, writing and spelling. The process is illustrated with one detailed case study, in which instructional needs identified in the assessment process are linked to particular areas of the programme. The results of this child and other case studies are then presented in the second chapter, Developing Automaticity in Children with Learning Disabilities: A Functional Perspective. Part Two: Programme Methods and Materials, in which the results of children exposed to the programme are analysed to identify key implementation variables affecting the development of reading, writing and spelling fluency.",signatures:"Charles Potter",downloadPdfUrl:"/chapter/pdf-download/57592",previewPdfUrl:"/chapter/pdf-preview/57592",authors:[{id:"93190",title:"Dr.",name:"Charles",surname:"Potter",slug:"charles-potter",fullName:"Charles Potter"}],corrections:null},{id:"57766",title:"Developing Automaticity in Children with Learning Disabilities: A Functional Perspective Part Two: Programme Methods and Materials",doi:"10.5772/intechopen.72052",slug:"developing-automaticity-in-children-with-learning-disabilities-a-functional-perspective-part-two-pro",totalDownloads:1392,totalCrossrefCites:4,totalDimensionsCites:4,hasAltmetrics:0,abstract:"This chapter is the second of two chapters in this book to describe an instructional programme based on Luria's theories, which can be used to develop basic skills and automaticity in reading, writing, and spelling. The first chapter focused on the programme's theory, and then described how assessment is used to develop an individual programme relating to both basic skill and fluency needs in reading, writing, and spelling. The process was illustrated with one detailed case study. The results of this case study (Child 1) are presented in the current chapter, together with the results of 13 children exposed to similar fluency-based interventions. The results of six children exposed to one or more systematic variations in programme implementation are then discussed. Case contrast analysis is used to highlight three variables affecting successful programme implementation, namely: consistent and regular exposure to phonological and phonic instruction to provide a foundation of basic skills on which the fluency interventions in the programme can be built; consistent implementation of methods designed to improve both reading fluency, and writing and spelling fluency, to produce the greatest likelihood of positive effects; and consistent support from parents in programme implementation to produce the greatest likelihood of positive effects.",signatures:"Charles Potter",downloadPdfUrl:"/chapter/pdf-download/57766",previewPdfUrl:"/chapter/pdf-preview/57766",authors:[{id:"93190",title:"Dr.",name:"Charles",surname:"Potter",slug:"charles-potter",fullName:"Charles Potter"}],corrections:null},{id:"56046",title:"Exploring Community Attitudes to People with Learning Disabilities: Using a Micro-Neighbourhood Design",doi:"10.5772/intechopen.69388",slug:"exploring-community-attitudes-to-people-with-learning-disabilities-using-a-micro-neighbourhood-desig",totalDownloads:1557,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"People with learning disabilities living in the community strive for meaningful social inclusion and integration. The attitudes of society to such individuals living in communities continues to be the catalyst that will enable them to achieve genuine social inclusion and integration. Identified barriers to social inclusion are a lack of change in societal attitudes towards people with learning disabilities. People with learning disabilities should live in a socially, not just functionally, integrated manner in community settings. Hence, attitudes of the community are extremely influential in achieving social integration and inclusion. It is important therefore to plan how best to capture attitudes in fact. An exemplar of a completed comparative study in Ireland, which used a micro-neighbourhood design combined with a random survey, illustrates how attitudes can be researched effectively. Study findings show that while attitudes were generally positive towards people with learning disabilities, there was no evidence of social integration, only functional integration. 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The variety of papers dedicated to FD is multiplied swiftly in the mid-twentieth century and later decades. One of the motives for the full-size interest within the discipline of FD is that it’s far feasible to describe the variety of physical [1], synthetic [2], and organic [3] occurrence with fractional differential equations (FDEs). As a new branch of applied mathematics, the field of FD can be seen in many applications. Nevertheless, more and more compelling implementations have been found in various engineering and science fields over the past few decades (see [4]). It is noted that the existing theory of FDEs is committed to a larger part of the research works (see [5, 6, 7, 8]). While modeling functional structures, ambient noise and time delays need to be taken into account, which might be very beneficial in building extra sensible fashions of sciences, and so on [9]. It is referred to as the pattern direction houses of the stochastic fractional partial differential gadget powered by way of area time white noise [10].
The problems in a stochastic environment replicate the modeling of single-sever
However, only few introductions and discussions exist on the definition of finite-time stability in stochastic finite space using fixed point theorem approach. Burton [19] started to analyze the stability characters of dynamical systems broadly using fixed point theorems. Subsequently, few authors applied fixed point approach to establish sufficient conditions for stability of the differential systems (see [20, 21, 22, 23, 24]). Based on the above discussions, this chapter provides finite-time stability of the Caputo sense FDEs via fixed point theorems.
The primary contribution of this chapter is defined as follows:
A fractional higher-order stochastic delay system (FSDS) is considered in finite-dimensional stochastic settings.
Weaker hypothesis on nonlinear terms and appropriate fixed-point analysis are utilized to obtain the existence and uniqueness of solution.
A new set of generalized sufficient conditions for finite-time stability of a certain FSDS is established by using Generalized Gronwall-Bellman inequality.
Novelties and challenges of this chapter are described through the subsequent statements:
Finite-time stability analysis for FSDS is new in literature of finite-dimensional fractional stochastic settings.
It is a challenge to tackle the proposed system with a norm estimation on nonlinear stochastic terms as described in this chapter.
It is more complex to verify the weaker assumptions of the system and the derived result is new, has not been analyzed with the existing literature.
Obtained result is proved in stochastic nature with square norm settings.
Organization of this chapter is as follows: system description and preliminaries are provided in Section 2. Existence and uniqueness of solution are provided in Section 3. Finite-time stability result is proved in Section 4 and Section 5 consists of a numerical example.
define a column-wise matrix sum
Further, let as define the matrix norm
Consider the following system:
where
where
We have the following square norm estimations:
and then
where
for zero initial value has the below form:
where
Substitute the above expression into (2), one can get
since
Using [25] and Lemma 2.2, the solution of (1) is
Define the nonlinear operator
Multiply by
First, we estimate (i):
By Burkholder-Davis-Gundy inequality and Assumption 1, the estimate for (ii) is given by
From the above two estimates of (i) and (ii), Eq. (3) becomes
This implies that
Hence, from statement of the Theorem 3.1, the nonlinear operator (
According to Lemma 2.1, let as take
and
Moreover,
Then from the statement of Theorem 4.1, we get
Hence the system (1) is finite-time stable in mean square. Hence the proof. □
Consider Eq. (1) in the below matrix form:
where
Further, we have the following fractional delayed
From Eq.(5) and using basic calculation, one can get
Hence we verified Theorem 3.1. Further, it is easy to verify that for any
Hence,
The system
The system
In this chapter, we have derived some meaningful and general results for finite-time stability of nonlinear fractional stochastic delay systems. Existence, uniqueness of solution and stability analysis of FSDS have been proved in finite-dimensional stochastic fractional higher-order differential system. Finally, a numerical simulation test is carried out to validate the obtained theoretical results. Derived result generalizes many existing results with integer and fractional-order systems.
34A08; 43A15; 37C25; 37A50
Fractional derivatives (FD) initiative concept is quite old and its history spans three centuries. The variety of papers dedicated to FD is multiplied swiftly in the mid-twentieth century and later decades. One of the motives for the full-size interest within the discipline of FD is that it’s far feasible to describe the variety of physical [1], synthetic [2], and organic [3] occurrence with fractional differential equations (FDEs). As a new branch of applied mathematics, the field of FD can be seen in many applications. Nevertheless, more and more compelling implementations have been found in various engineering and science fields over the past few decades (see [4]). It is noted that the existing theory of FDEs is committed to a larger part of the research works (see [5, 6, 7, 8]). While modeling functional structures, ambient noise and time delays need to be taken into account, which might be very beneficial in building extra sensible fashions of sciences, and so on [9]. It is referred to as the pattern direction houses of the stochastic fractional partial differential gadget powered by way of area time white noise [10].
The problems in a stochastic environment replicate the modeling of single-sever
However, only few introductions and discussions exist on the definition of finite-time stability in stochastic finite space using fixed point theorem approach. Burton [19] started to analyze the stability characters of dynamical systems broadly using fixed point theorems. Subsequently, few authors applied fixed point approach to establish sufficient conditions for stability of the differential systems (see [20, 21, 22, 23, 24]). Based on the above discussions, this chapter provides finite-time stability of the Caputo sense FDEs via fixed point theorems.
The primary contribution of this chapter is defined as follows:
A fractional higher-order stochastic delay system (FSDS) is considered in finite-dimensional stochastic settings.
Weaker hypothesis on nonlinear terms and appropriate fixed-point analysis are utilized to obtain the existence and uniqueness of solution.
A new set of generalized sufficient conditions for finite-time stability of a certain FSDS is established by using Generalized Gronwall-Bellman inequality.
Novelties and challenges of this chapter are described through the subsequent statements:
Finite-time stability analysis for FSDS is new in literature of finite-dimensional fractional stochastic settings.
It is a challenge to tackle the proposed system with a norm estimation on nonlinear stochastic terms as described in this chapter.
It is more complex to verify the weaker assumptions of the system and the derived result is new, has not been analyzed with the existing literature.
Obtained result is proved in stochastic nature with square norm settings.
Organization of this chapter is as follows: system description and preliminaries are provided in Section 2. Existence and uniqueness of solution are provided in Section 3. Finite-time stability result is proved in Section 4 and Section 5 consists of a numerical example.
define a column-wise matrix sum
Further, let as define the matrix norm
Consider the following system:
where
where
We have the following square norm estimations:
and then
where
for zero initial value has the below form:
where
Substitute the above expression into (2), one can get
since
Using [25] and Lemma 2.2, the solution of (1) is
Define the nonlinear operator
Multiply by
First, we estimate (i):
By Burkholder-Davis-Gundy inequality and Assumption 1, the estimate for (ii) is given by
From the above two estimates of (i) and (ii), Eq. (3) becomes
This implies that
Hence, from statement of the Theorem 3.1, the nonlinear operator (