HIGHER ORDER BOUNDARY VALUE PROBLEMS WITH INTEGRAL BOUNDARY CONDITIONS AT RESONANCE ON THE HALF-LINE

S. A. IYASE, O. F. Imaga

Abstract


In this paper, we study the following higher order boundary value problems at resonance on the half-line:
\begin{equation*}
(q(t)u^{(n-1)}(t))^{\prime}=f(t,u(t),u^{\prime}(t), \cdots , u^{(n-1)}(t)), \quad \text{ a.e. } t\in(0,\infty),
\end{equation*}
subject to the boundary conditions
\begin{align*}
&u^{(n-2)}(0) = \sum \limits _{i=1} ^m \alpha _i \int \limits _0 ^{\xi _i} u(t)dt, \ u^{(i)}(0)=0, i=1,2,\cdots,(n-3),\ \\
&\lim \limits _{t\to\infty}q(t)u^{(n-1)}(t)=0.
\end{align*}
By using coincidence degree arguments we establish some existence criteria under the resonant condition \\ $\sum \limits _{i=1}^{m-2}\alpha _i \xi _i^{n-1}=(n-1)!$.

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References


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