亚洲免费av电影一区二区三区,日韩爱爱视频,51精品视频一区二区三区,91视频爱爱,日韩欧美在线播放视频,中文字幕少妇AV,亚洲电影中文字幕,久久久久亚洲av成人网址,久久综合视频网站,国产在线不卡免费播放

        ?

        STABILITY ANALYSIS OF CAUSAL INTEGRAL EVOLUTION IMPULSIVE SYSTEMS ON TIME SCALES?

        2021-06-17 13:59:16徐家發(fā)
        關(guān)鍵詞:徐家

        (徐家發(fā))

        School of Mathematical Sciences,Chongqing Normal University,Chongqing 401331,China

        E-mail:xujiafa292@sina.com

        Bakhtawar PERVAIZ Akbar ZADA

        Department of Mathematics,University of Peshawar,Peshawar 25000,Pakistan

        E-mail:Bakhtawar@uop.edu.pk;zadababo@yahoo.com

        Syed Omar SHAH

        Department of Physical and Numerical Sciences,Qurtuba University of Science and Information Technology Peshawar,Dera Ismail Khan,Pakistan

        E-mail:omarshah89@yahoo.com

        Abstract In this article,we present the existence,uniqueness,Ulam-Hyers stability and Ulam-Hyers-Rassias stability of semilinear nonautonomous integral causal evolution impulsive integro-delay dynamic systems on time scales,with the help of a fixed point approach.We use Gr?nwall’s inequality on time scales,an abstract Gr?wall’s lemma and a Picard operator as basic tools to develop our main results.To overcome some difficulties,we make a variety of assumptions.At the end an example is given to demonstrate the validity of our main theoretical results.

        Key words Time scale;Ulam-Hyers stability;impulses;semilinear nonautonomous system;Gr?nwall’s inequality;dynamic system

        1 Introduction

        The study of functional equations with causal operators has seen rapid development in the last several years;some recent results are assembled in the monograph[8].The term of causal operator is adopted from engineering literature and the theory of these operatorsunifies ordinary differential equations,integro-differential equations with finite or infinite delay-Volterra integral equations,neutral functional equations etc.Many articles have addressed various aspects of the notion of causal operators.In[17],a general and new definition of Volterra operators in the sense of Toneli was given.Control problems having casual operators were studied in[2,9,16,38].Some properties of solutions of the differential equations with causal operators can be seen in[13–15,17,18].

        The famous stability of functional equations was introduced by Ulam in 1940 at Wisconsin University[47,48].The problem involved discussing the relationship between an approximate solution of a homomorphism from a group G1to a metric group G2.In the following year,Hyers partially gave the solution to Ulam’s problem by taking G1and G2as a Banach spaces(BS’s)[25].Since then,this notion has been called Ulam-Hyers(UH)stability.The Hyers result was then generalized by Rassias[35],which gave rise to the notion of Ulam-Hyers-Rassias(UHR)stability.Ulam’s stability analysis plays an important role in various fields including numerical analysis,control theory,etc.[4–6,26–29,32,33,36,37,39–46,49,50,53–56,58–65,67–69].

        Impulsive differential equations turned out to be the best tools for modeling physical problems which are subject to abrupt changes of state at a certain instant.These equations are suitable for describing biotechnology processes,biological systems,mathematical economy,population dynamics,medicine,pharmacokinetics,chemical energy,etc.[11,12,31,50].Due to these fruitful applications in different fields,impulsive differential equations have received considerable attention from researchers.

        Ulam-type stability of impulsive differential equations was proved by Wang et al.[51]in 2012.They followed their own work and studied the UHR stability and generalized UHR stability for impulsive evolution equations on a compact interval[52].Recently,Agarwal et al.[1]studied the existence of causal functional evolution equations with respect to the Hausdorffmeasure of noncompactness.

        The notion of time scales analysis has recently been attracting a lot of interest.This theory was introduced by Hilger[24]at the end of twentieth century as a means of unifying the difference and the differential calculus,and is now a well-established subject.For details on time scales,see[3,7,10,20–23,30,34,42–44,57,66].In 2010,Lupulescu and Zada[30]proved the fundamental concepts of linear impulsive systems on time scales.

        Motivated by the work done in[1],in this paper,we present the existence,uniqueness(EU)and the stability of the solution of the following nonlinear causal evolution integro-delay dynamic system:

        and of a nonlinear integro-delay dynamic system with integral impulses of fractional order of the form:

        2 Preliminaries

        3 Existence,Uniqueness and Ulam-Hyers Stability of the Solution of Equation(1.1)

        In order to obtain EU and stability results for(1.1),we need to introduce the following assumptions:

        4 Existence,Uniqueness and Ulam-Hyers Stability of a Solution of Equation(1.2)

        In order to establish the EU and stability results of Equation(1.2),we introduce the following assumptions:

        Utilizing(A5),the above,operator Λ is clearly strictly contractive on PC1(D,Rn),and hence it is a PO on PC1(D,Rn),so it has only one point,which is clearly the only one solution of(1.2)in PC1(D,Rn). □

        Theorem 4.2If Assumptions(A1)–(A5)hold,then Equation(1.2)has UH stability on D.

        ProofIf Y∈PC1(D,Rn)satisfies(2.3),then the unique solution of system

        is

        Conclusion

        In this article,an attempt has been made to establish EU and stability results by using the FP method for(1.1)and(1.2).We established our results by using Lemma 2.7 and an abstract Gr?nwall’s lemma.When it is difficult to find the exact solution,then the notion of UH stability is fruitful;that is why,in approximation theory,our results are important.

        猜你喜歡
        徐家
        Enhancing terahertz photonic spin Hall effect via optical Tamm state and the sensing application
        The existence and blow-up of the radial solutions of a (k1, k2)-Hessian system involving a nonlinear operator and gradient
        Broadband low-frequency acoustic absorber based on metaporous composite
        徐家玨作品
        美術(shù)界(2022年4期)2022-04-26 11:07:00
        Synthesis and thermoelectric properties of Bi-doped SnSe thin films?
        南京市棲霞區(qū)徐家村M4 出土器物
        南京市棲霞區(qū)徐家村M1 出土器物
        徐家柱 用愛(ài)喚醒沉睡12年的妻子
        “多多益善”的政協(xié)主席
        徐家河尾礦庫(kù)潰壩分析
        日韩精品中文一区二区三区在线 | 人妻仑乱a级毛片免费看| 牛鞭伸入女人下身的真视频| 久久亚洲日本免费高清一区| 精品国产3p一区二区三区| 国产伦码精品一区二区| japanese无码中文字幕| 国产成人av综合亚洲色欲| 日本高清二区视频久二区| 亚洲女同高清精品一区二区99 | 人人妻人人澡人人爽久久av| 老色鬼永久精品网站| 成人av在线免费播放| 丰满人妻一区二区三区视频| 97在线观看| √新版天堂资源在线资源| 亚洲人成无码网www| 亚洲中文字幕久久精品蜜桃| 国产一级av理论手机在线| 精品一区二区av天堂色偷偷| 伊人久久久精品区aaa片| 欧美成人www免费全部网站| 国产一区二区三区在线观看蜜桃 | 男女搞事在线观看视频| 人妻色综合网站| 欧美日韩亚洲成人| 日本久久精品国产精品| 亚洲高清在线天堂精品| 精品深夜av无码一区二区老年| 色伊人国产高清在线| 国产亚洲精品视频网站| 国产精品天干天干综合网 | 亚洲毛片网| 亚洲国产一区中文字幕| 亚洲熟女精品中文字幕| 亚洲av一宅男色影视| 国产男女乱婬真视频免费| 男女av免费视频网站| 国产成人精品一区二区不卡| 巨臀中文字幕一区二区| 日本高清一区二区在线观看|