An application of Schauder’s fixed point theorem with respect to higher order BVPs

Source abstract

We shall provide conditions on the function f ( t , u 1 , ⋯ , u n − 1 ) f(t,u_{1},\cdots , u_{n-1}) . The higher order boundary value problem ({BVP}) { ( E ) u ( n ) ( t ) + f ( t , u ( t ) , u ( 1 ) ( t ) , ⋯ , u ( n − 2 ) ( t ) ) = 0 f o r t ∈ ( 0 , 1 ) a n d n ≥ 2 , ( B C ) { u ( i ) ( 0 ) = 0 , 0 ≤ i ≤ n − 3 , α u ( n − 2 ) ( 0 ) − β u ( n − 1 ) ( 0 ) = 0 , γ u ( n − 2 ) ( 1 ) + δ u ( n − 1 ) ( 1 ) = 0 $$\begin{equation*}\begin {cases}(E)~~ u^{(n)}(t)+ f(t, u(t),u^{(1)}(t),\cdots ,u^{(n-2)}(t))=0~~~~~\mathrm {~for~}~~~~~t\in (0,1)~~~~\mathrm {and}~~~~~~n\ge 2,\ (BC)~~ \begin {cases}u^{(i)}(0)=0,~~~~~0\le i \le n-3,\ \alpha u^{(n-2)}(0)-\beta u^{(n-1)}(0)=0,\ \gamma u^{(n-2)}(1)+\delta u^{(n-1)}(1)=0\end{cases} \end{cases} \tag {{BVP}}\end{equation*}$$ has at least one solution.