Stability Analysis Of A Class Of Multistep Methods For Nonlinear Functional Differential And Functional Equations In Banach Space

Functional diferential and functional equations（FDFEs） are more complexthan FDEs. They are a class of hybird problem formed by functional diferentialequations and functional equations. However, there exists lots of stif problemsin science and engineering,despite the whole question itself is good state, whenusing the corresponding norm of inner product, it may happen that the one-sidedLipschitz constant is a very large positive number. Therefore, it is urgent andmeaningful to break the restriction of the corresponding norm of inner productand the one-sided Lipschitz constant to study the numerical theory in Banachspace.In view of this, this paper also research directly a class of the initial valueproblems of nonlinear functional diferential and functional equations in Banachspaces. First of all, we introduce test problem classes D_λ （α, β₁, β₂, γ₁, γ₂, γ₃,1,2） and D_λ,δ（α, β₁, β₂, γ₁, γ₂, γ₃,μ₁,μ₂） with respect to the initial value problemsof nonlinear functional diferential and functional equations in Banach spaces.A series of stability results of the analytic solution are obtained and a conditionestimate for the class D₀（α, β₁, β₂, γ₁, γ₂, γ₃,μ₁,μ₂） which based on logarithmicmatrix norm is also obtained. Second, a series of new stability criterias of thevariable coefcient linear multistep methods when applied to the test problemclasses D_λ （α, β₁, β₂, γ₁, γ₂, γ₃,μ₁,μ₂） and D_λ,δ（α, β₁, β₂, γ₁, γ₂, γ₃,μ₁,μ₂） are es-tablished in Banach space. Finally, Numerical tests have given to confirm thetheoretical results.