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Solvability for a class of abstract two-point boundary value problems derived from optimal control. (English) Zbl 1148.34041

Summary: This paper deals with the solvability of the following abstract two-point boundary value problem (BVP):
\[ \begin{aligned} & \dot x(t)=A(t)x(t)+F(x(t),p(t),t),\quad x(a)=x_0,\\ & \dot p(t)=-A^*(t)p(t)+G(x(t),p(t),t),\quad p(b)=\xi(x(b)).\end{aligned} \]
Here, both \(x(t)\) and \(p(t)\) take values in a Hilbert space \(X\) for \(t\in[a,b]\), \(F\), \(G:X\times X\times [a,b]\to X\), and \(\xi:X\to X\) are nonlinear operators. \(\{A(t):a\leq t\leq b\}\) is a family of linear closed operators with adjoint operators \(A^*(t)\) and generates a unique linear evolution system \(\{U(t,s):a\leq s\leq t\leq b\}\).
By homotopy technique existence and uniqueness results are established under some monotonic conditions. Several examples are given to illustrate the application of the obtained results.

MSC:

34G20 Nonlinear differential equations in abstract spaces
34B15 Nonlinear boundary value problems for ordinary differential equations
49K20 Optimality conditions for problems involving partial differential equations
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References:

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