Variational Approach For P-norm Solutions Of The Two-point Boundary Value Problem Of Stochastic Differential Equations

This study focuses on existence and continuous dependence of P-norm solution-s for the two-point boundary value problem of n dimensional stochastic differential equations (SDE in short) by variational approach.Chapter 1 introduces research background, research status, research contents and some preliminaries for the following chapters.Chapter 2 presents the Picard sequence by variational approach. That two-point boundary value problem of stochastic differential equations do have P-norm solution-s is proved and examples are cited to show the form of this solution is not unique. Xia-Lin(2011) proved that two-point boundary value problem of stochastic differen-tial equations have corresponding square interable solutions. But this chapter mainly studies the existence of P-norm solutions for the two-point boundary value problem of stochastic differential equations. So the author applies variational approach for the study of solutions’existence under the influence of them. Proving steps are as follow: Step 1, the Picard iterative sequence{(Xtn,ftn),t∈[0,T]} of the two-point bound-ary value problem of stochastic differential equations was structured and the space of solutions’sequence (Xtn,ftn) ∈Sp×Mp was proved. Step 2, solutions’sequence {(Xtn,ftn), t∈[0, T]} was proved to be Cauchy convergence sequence in Sp and Mp respectively and was converged to the procedure (Xt, ft). Step 3, that (Xt, ft) is P-norm solutions for the two-point boundary value problem of n dimensional stochastic differential equations is proved. Therefore, compared with Xia-Lin(2011) this study not only promotes the space of solutions two-point boundary value problem of stochas-tic differential equations, but also gives another solution to the existence of P-norm solutions for this kind of equation.Chapter 3 makes a further discussion in examples that P-norm solutions is just a form of the stochastic differential equations with good quality on the basis of Chapter 2, but the P-norm solutions are not unique. But for the good form and quality of P-norm solutions in Chapter 2 the author will especially prove the continuous dependence of this kind of P-norm solutions in Chapter 3.Finally, Chapter 4 gives a summary and prospect of this paper.