The Existence And Calculation Of A Class Of Solutions With Stieltjes Integral Equations

Since the integral equation has a wide range of application background, there are many scholars and engineers paying more attention to this field. During the past 20 years, with the fractional integral equations and fractional differential integral equations becoming popular, Volterra-Stieltjes integral equations become a hot issue. Through summarizing recent works about solutions of Volterra-Stieltjes integral equations, we discuss the qualitative theory of a class of Stieltjes quadratic functional integral equations.we prove the existence of solution via the Darbo's fixed point theorem and Hausdorff measure of noncompactness. After introducing the definition of generalized Hyers-UlamRassias stability, the stability of the solution has been proved. Then, we give three numerical algorithms to calculate the Stieltjes integral equation. In addition, we analyze the convergence and stability of the three algorithms, and give some examples to illustrate the effectiveness of algorithms.The main innovative points of the paper as follows:1, We use the idea that the concepts of Ulam in the algebra can be applied to the discussion about the Ulam-Hyers and Hyers–Ulam–Rassias Stability of fractional differential integral equations, to prove Ulam-Hyers stability of the Stieltjes integral equation.2, Because of the non-local property of the Stieltjes integral equation, calculated quantity is too large. we give some algorithms based on the piecewise linear interpolation and the difference method. Then, establishing improved algorithm, we obtain a simple algorithm.3, The difficult part of fractional differential equations and integral equations in numerical computation is the research of stability. In the most of literatures, the stability of the numerical scheme is too abstract and hard to obtain, even some schemes are unstable. In Chapter 6, some easy detected conditions which can guarantee the stability of equations will be given, because of the better stability of improved algorithm. The research about the algorithms of nonlinear equations, in this paper, can be used for fractional differential integral equations.