弱拓扑下的非线性随机积分和微分方程组的解*
Solutions for a System of Nonlinear Random Integral and Differential Equations under Weak Topology
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摘要: 在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell-Smith,Cramer-Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.
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关键词:
- 非线性随机Volterra积分 /
- 随机Cauchy问题 /
- 极值随机解 /
- 比较结果 /
- Banach空间中弱拓补
Abstract: In this paper, a Darbao type random fired point theorem for a system of lveak continuous random operators with random domain is first proved Then, by using thetheorem, some existence criteria of random solutions for a systems of nonlinear random Volterra integral equations relative to the weak topology in Banach spaces aregiven. As applications, some existence theorems of weak random sohttions for the random Cauchy problem of a system of nonlinear random differential equations areobtained, as well as the existence of extremal random solutions and random comparison resultsfor these systems of random equations relative to weak topotogy in Banach spaces. The corresponding results of Szep, Mitchell-Smith, Cramer-Lakshmikantham, Lakshmikantham-Leela and Ding are mproved and generalized bythese theorems. -
[1] A. T. Bharucha-Reid, Random Integral Equations, Acad. Press, New York (1972). [2] C. Castaing and. M. Valadier, Convex Analysis and Measurable Multifunctions, Springer-Verlag (1977), 580. [3] E. J. Cramer, V. Lakshmikantham and A. R. Mitchell, On the existence of weaksolutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal., 2(1978), 169~177. [4] F. S. De Blasi and J. Myjak, Random differential equations on closed subsets of Banachspaces, J. Math. Anal. Appl., 90 (1982), 273~285. [5] 丁协平.随机集值映射的不动点定理及其应用,应用数学和力学,5(4) (1984), 561-576, [6] X, P, Ding, Existence, uniqueness and approximation of solutioas for a system of nonlinear ra mdom operator equations,Nonlinear Anal,a(6)(1984),563-576, [7] 丁协平,随机积分和微分方程解的存在性准则,应用数学和力学,g(3) (1985), 265-270, [8] 丁协平.随机积分方程和微分方程解的存在性和比较结果,应用数学和力学,7(7) (1986),597-604. [9] 丁协平,随机积分和微分方程在弱拓扑下解的存在性和比较结果,应用数学和力学, 8(12),(1987),1039-1050, [10] Ding Xieping, Solutions for a system of random Qperator equations and someapplications, Scientia Sinica, 30, 8 (1987), 785~795. [11] Ding Xieping, A general random fixed point theorem of weakly continuous randomoperator with applications, J. Engineering Math., 5, 2 (1988), 1~7. [12] E. Hille and R. S. Phillips, Ftnctional Analvsis and Semi-Groups, Amer. Math. Soc.Providence, RI (1957). [13] V. Lakshmikantham, Existence and comparison results for Volterra integral equations inBanach spaces, Volterra Intngral Equations, Springer-Verlag, 737 (1979), 120~126. [14] V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces,Pergamon Press, New York (1981). [15] A. R. Mitchell and C. Smith, An existence theorem for weak solutions of differentialequations in Banach spaces, Nonlinear Equalions in Abstract Spaces, Acad. Press, NewYork (1978), 387~403. [16] A. Szep, Existence theorem for weak solutions of ordinary differential equations inrenexive Banach spaces, studta sci. MaIh. Himgica, 6 (1971), 197~203. [17] C. J. Tsokos and W. J. Padgett, Random Integral Equations with Applications to LifeScience and Engineering, Acad. Press, New York (1974). [18] R. L. Vaughn, Existence and Comparison results for nonlinear Volterra integralequations in Banach spaces, Appl. Anal., 7 (1978), 337~348. [19] R. L. Vaughn, Criteria for the existence and comparison of solutions to nonlinearVolterra integral equations in Banach spaces, Nonlinear Equations in Abstract Spaces.Acad. Press. New York (1978), 463~468.
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