Analysis of the Error in a Numerical Method Used to Solve Nonlinear Mixed Fredholm-Volterra-Hammerstein Integral Equations

Table 1

Properties of the univariate and bivariate Schauder bases.

𝑏1(𝑡)=1	𝐵1(𝑡,𝑠)=1
𝑛≥2⇒𝑏𝑛(𝑡𝑘)=1,if𝑘=𝑛0,if𝑘<𝑛	𝑛≥2⇒𝐵𝑛(𝑡𝑖,𝑡𝑗)=1,if𝜏(𝑛)=(𝑖,𝑗)0,if𝜏^−1(𝑖,𝑗)<𝑛
𝑦∈𝐶([𝛼,𝛼+𝛽])	𝑧∈𝐶([𝛼,𝛼+𝛽]^2)
⇓	⇓
𝑏^∗1(𝑦)=𝑦(𝑡1)	𝐵^∗1(𝑧)=𝑧(𝑡1,𝑡1)
𝑛≥2⇒𝑏^∗𝑛(𝑦)=𝑦(𝑡𝑛)−∑^𝑛−1𝑘=1𝑏^∗𝑘(𝑦)𝑏𝑘(𝑡𝑛)	𝑛≥2𝜏(𝑛)=(𝑖,𝑗)⇒𝐵^∗𝑛(𝑧)=𝑧(𝑡𝑖,𝑡𝑗)−∑^𝑛−1𝑘=1𝐵^∗𝑘(𝑧)𝐵𝑘(𝑡𝑖,𝑡𝑗)
𝑦∈𝐶([𝛼,𝛼+𝛽])	𝑧∈𝐶([𝛼,𝛼+𝛽]^2)
⇓	⇓
𝑘≤𝑛⇒𝑃𝑛(𝑦)(𝑡𝑘)=𝑦(𝑡𝑘)	𝜏^−1(𝑖,𝑗)≤𝑛⇒𝑄𝑛(𝑧)(𝑡𝑖,𝑡𝑗)=𝑧(𝑡𝑖,𝑡𝑗)
{𝑏𝑛}𝑛≥1 is monotone, that is, sup𝑛≥1‖𝑃𝑛‖=1	{𝐵𝑛}𝑛≥1 is monotone, that is, sup𝑛≥1‖𝑄𝑛‖=1
𝑦∈𝐶^1([𝛼,𝛼+𝛽]), 𝑛≥2	𝑧∈𝐶^1([𝛼,𝛼+𝛽]^2), 𝑛≥2
⇓	⇓
‖𝑦−𝑃𝑛(𝑦)‖≤2‖𝑦^′‖Δ𝑇𝑛	‖𝑧−𝑄𝑛2(𝑧)‖≤4max𝜕𝑧,𝜕𝑡𝜕𝑧𝜕𝑠Δ𝑇𝑛