Positive solutions for 2nth order p-Laplacian problem with Sturm–Liouville type boundary conditions

This paper focusses on establishing the existence of positive solutions to 2nth order p-Laplacian problem \(\begin{aligned}{} & {} (-1)^n[\phi _{p}(\vartheta ^{(2n-2)}(t))]''=f(t,\vartheta (t)), ~~t \in [c, d],\\{} & {} \alpha _{i+1}\vartheta ^{(2i)}(c)-\beta _{i+1}\vartheta ^{(2i+1)}(c)=0,~ \gamma _{i+1}\vartheta ^{(2i)}(d)+\delta _{i+1}\vartheta ^{(2i+1)}(d)=0,\\{} & {} \alpha _{n}[\phi _{p}(\vartheta ^{(2n-2)}(c))]-\beta _{n}[\phi _{p}(\vartheta ^{(2n-2)}(t))]'_{\text {at}~t=c}=0,~\\{} & {} \gamma _{n}[\phi _{p}(\vartheta ^{(2n-2)}(d))]+\delta _{n}[\phi _{p}(\vartheta ^{(2n-2)}(t))]'_{\text {at}~t=d}=0, \end{aligned}\) $$\begin{array}{cc}\hfill & {(-1)}^n{[{\varphi }_p({\vartheta }^{(2n-2)}(t))]}^{\prime \prime }=f(t,\vartheta (t)),t\in [c,d],\hfill \\ \hfill & {\alpha }_{i+1}{\vartheta }^{(2i)}(c)-{\beta }_{i+1}{\vartheta }^{(2i+1)}(c)=0,{\gamma }_{i+1}{\vartheta }^{(2i)}(d)+{\delta }_{i+1}{\vartheta }^{(2i+1)}(d)=0,\hfill \\ \hfill & {\alpha }_n[{\varphi }_p({\vartheta }^{(2n-2)}(c))]-{\beta }_n{[{\varphi }_p({\vartheta }^{(2n-2)}(t))]}_{\text{at}t=c}^{\prime }=0,\hfill \\ \hfill & {\gamma }_n[{\varphi }_p({\vartheta }^{(2n-2)}(d))]+{\delta }_n{[{\varphi }_p({\vartheta }^{(2n-2)}(t))]}_{\text{at}t=d}^{\prime }=0,\hfill \end{array}$$ where \(0\le i \le n-2\) $0\le i\le n-2$ with \(n\ge 2\) $n\ge 2$ , \( 0\le c<d \) $0\le c<d$ and \(\alpha _i\) ${\alpha }_i$ , \(\beta _i\) ${\beta }_i$ , \(\gamma _i\) ${\gamma }_i$ , \(\delta _i\) ${\delta }_i$ are positive real numbers such that \(\gamma _i\beta _i+\alpha _i\delta _i+\alpha _i\gamma _i(d-c)>0\) ${\gamma }_i{\beta }_i+{\alpha }_i{\delta }_i+{\alpha }_i{\gamma }_i(d-c)>0$ , for \(1\le i\le n\) $1\le i\le n$ . The main findings are established using the Guo–Krasnosel’skii fixed point theorem.