Existence and uniqueness of solutions for abstract two-point boundary value problems (Q2800647)

The author considers the abstract two-point boundary value problem (BVP) NEWLINE\[NEWLINE\begin{aligned} x'(t)&=A(t)x(t)+F(x(t),N(t)p(t),t),\,x(a)=x_{0}, \\ p'(t)&=-A^{*}(t)p(t)+G(x(t),p(t),t),\,p(b)=\xi(x(b)), \end{aligned}NEWLINE\]NEWLINE where \(x:[a,b] \rightarrow X\), \(p:[a,b] \rightarrow X\) and \(X\) is a Hilbert space; \(F:X\times Y\times [a,b] \rightarrow X\), \(G:X\times X\times [a,b] \rightarrow X\), \(\xi: X\rightarrow X\) are nonlinear operators and \(Y\) is another Hilbert space; \(N(t):X\rightarrow Y\) is a bounded linear operator for each \(t\in[a,b]\); the family \(\{ A(t): a\leq t\leq b \}\) is of linear closed operators.NEWLINENEWLINEBy using homotopy techniques and fixed point theorems, and under monotonic conditions, the author proves an existence and uniqueness result of solutions for the boundary value problems (BVP). As applications, classic quadratic linear control systems and systems of nonlinear differential equations are investigated.