### 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}

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