Existence of Positive Solutions for Higher Order (p,q)-Laplacian Two-Point Boundary Value Problems

Abstract

We derive sufficient conditions for the existence of positive solutions to higher order $(p,q)$-Laplacian two-point boundary value problem, $(-\mathrm{1}{)}^{m_{\mathrm{1}}+n_{\mathrm{1}}-\mathrm{1}}{[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(t))]}^{(n_{\mathrm{1}})}=f_{\mathrm{1}}(t,u(t),v(t))$, $t\in [\mathrm{0,1}]$, $\mathrm{}(-\mathrm{1}{)}^{m_{\mathrm{2}}+n_{\mathrm{2}}-\mathrm{1}}{[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}^{(\mathrm{2}{n}_{\mathrm{2}})}=f_{\mathrm{2}}(t,u(t),v(t))$, $t\in [\mathrm{0,1}]$, $\mathrm{}\mathrm{}{u}^{(\mathrm{2}i)}(\mathrm{0})=\mathrm{0}=u^{(\mathrm{2}i)}(\mathrm{1})$, $i=\mathrm{0,1},\mathrm{2},\dots ,m_{\mathrm{1}}-\mathrm{1}$, ${[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{0}}^{(j)}=\mathrm{0}$, $j=\mathrm{0,1},\dots ,n_{\mathrm{1}}-\mathrm{2}$; $[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(\mathrm{1}))]=\mathrm{0}$, ${[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{0}}^{(\mathrm{2}i)}=\mathrm{0}={[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{1}}^{(\mathrm{2}i)}$, $i=\mathrm{0,1},\dots ,n_{\mathrm{2}}-\mathrm{1}$, $v^{(j)}(\mathrm{0})=\mathrm{0}$, $j=\mathrm{0,1},\mathrm{2},\dots ,m_{\mathrm{2}}-\mathrm{2}$, and $v(\mathrm{1})=\mathrm{0}$, where $f_{\mathrm{1}},f_{\mathrm{2}}$ are continuous functions from $[\mathrm{0,1}]\times \mathrm{}\mathrm{}\mathrm{}ℝ{\mathrm{}}^{\mathrm{2}}$ to $[\mathrm{0},\mathrm{\infty })$, $m_{\mathrm{1}},n_{\mathrm{1}},m_{\mathrm{2}},n_{\mathrm{2}}\in \mathrm{}\mathrm{}\mathrm{}\mathbb{N}\mathrm{}$ and $\mathrm{1}/p+\mathrm{1}/q=\mathrm{1}$. We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example.