A Note on the Nonlocal Boundary Value Problem for a Third Order Partial Differential Equation

Abstract

\begin{abstract}
The nonlocal boundary-value problem for a third order partial differential
equation
\begin{equation*}
\left\{
\begin{array}{l}
\frac{d^{3}u(t)}{dt^{3}}+A\frac{du(t)}{dt}=f(t),\quad 0<t<1,\vspace{0.2cm}
\\
u(0)=\gamma u\left( \lambda \right) +\varphi ,\quad \vspace{0.1cm}u^{\prime
}(0)=\alpha u^{\prime }\left( \lambda \right) +\psi ,\left\vert \gamma
\right\vert <1,\vspace{0.1cm} \\
u^{\prime \prime }(0)=\beta u^{\prime \prime }\left( \lambda \right) +\xi
,\quad \left\vert 1+\beta \alpha \right\vert >\left\vert \alpha +\beta
\right\vert ,\vspace{0.1cm}0<\lambda \leq 1%
\end{array}%
\right.
\end{equation*}%
in Hilbert space $H$ with a self-adjoint positive definite operator $A$ is
considered. Applying operator approach, the theorem on stability for
solution of this nonlocal boundary value problem is established. In
applications, the stability estimates for the solution of three nonlocal
boundary value problems for third order partial differential equations are
obtained.
\end{abstract}