The existence and uniqueness results to higher-order boundary value problems

The current work aims to demonstrate the existence and uniqueness of the solution to the following problem: $$\begin{aligned} v^{(n)}(s)+g(s, v(s))&=0,\quad s\in [a, d], \end{aligned}$$ fulfilling with the following non-homogeneous type: $$\begin{aligned} v(a)-\curlyvee_1 v(\flat)=0, ~ v^{(i)}(a)=0, ~ \text{for} ~ 1 \le i \le n-2, ~ v^{(n-2)}(d)-\curlyvee_2 v^{(n-2)}(\flat)=\lambda, \end{aligned}$$ where $n\geq3$, $0\leq a< \flat< d$, $g\in C([a,d]\times\R,\R$), $\lambda$ is a parameter, and $\curlyvee_1 $ and $\curlyvee_2$ are real constants. First, the new form of the solution is derived in terms of the kernel, and then finer bounds on the integral kernels are obtained. Using these finer estimates, the uniqueness results for the above problem are proved by applying the Banach and Rus theorems. Finally, we extend our results to the following even-order problem: $$ v^{(2p)}(s)+g(s,v(s))=0, \quad s\in[a,d],$$ $$v(a)-\curlyvee_1 v(\flat)=0, ~ v^{(i)}(a)=0, ~ \text{for} ~ 1 \leq i \leq 2p-2, ~ v^{(2p-2)}(d)-\curlyvee_2 v^{(2p-2)}(\flat)=\lambda.$$