Numerical Stability Analysis For Nonlinear Volterra Functional Differential Equations In Abstract Spaces

Many real-life phenomena in physics, biology, medicine, economics, control theory and engineering and so on can be modelled as initial value problem in Volterra functional differential equations (VFDEs). It is important for the development of these fields to study theory and application of numerical methods for VFDEs. In the last few decades, the theory of numerical methods for VFDEs, especially for the delay differential equations (DDEs), have been widely discussed by many authors such as Barwell, Bellen, Torelli, Zennaro,Spijker, Watanabe, Roth, in't Hout, Baker, Paul, Koto, Iser-les, Shoufu Li, Jiaoxun Kuang, Mingzhu Liu, Chengming Huang, Chengjian Zhang, Hongjiong Tian, Guangda Hu, Siqing Gan. However, the existing results for functional differential equations are mainly based upon the inner product and the corresponding norm in Euclidean space of finite dimension. But for some stiff problems, it may happen that the one-sided Lipschitz constant is very large. Therefore, it is urgent and meaningful to break through the restriction of the inner product and the corresponding norm.The main results obtained in this paper are as follows:(1) we introduce the test problem classes D_λ* (α, β, μ₁, μ₂) and D_λ*,δ(α,β, μ₁,μ₂) with respect to the initial value problems of nonlinear Volterra functional differential equations in Banach spaces. A series of stability results of the analytic solution are obtained and a condition estimate for the class D₀(α, β, μ₁, μ₂) which based on logarithmic matrix norm is also obtained. The above results extend the existing results for ordinary differential equa-tions(ODEs).(2) A series of numerical stability results of θ—methods when applied to the class D_0,0(α, β, μ₁, μ₂) are obtained, which can be directly applied to the special problem of DDEs, integro-differential equations(IDEs) and VFDEs of other type which appear in practice, and are more general and deeper than the existing results for θ—methods in literature. A series of new asymptotic stability results are also obtained.(3) A series of new stability criteria of the linear multistep methods and...