International Journal of Differential Equations

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 $[\mathrm{0,1}]\times\mathrm{}\mathrm{}\mathrm{}\Bbb R{\mathrm{}}^{\mathrm{2}}$ to $[\mathrm{0},\mathrm{\infty })$, ${m}_{\mathrm{1}},{n}_{\mathrm{1}},{m}_{\mathrm{2}},{n}_{\mathrm{2}}\in \mathrm{}\mathrm{}\mathrm{}\Bbb N\mathrm{}$ 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.

Article information

Source
Int. J. Differ. Equ., Volume 2013 (2013), Article ID 743943, 9 pages.

Dates
Received: 17 April 2013
Revised: 17 July 2013
Accepted: 17 July 2013
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881347

Digital Object Identifier
doi:10.1155/2013/743943

Mathematical Reviews number (MathSciNet)
MR3102796

Zentralblatt MATH identifier
1300.34056

Citation

Kapula, Rajendra Prasad; Murali, Penugurthi; Rajendrakumar, Kona. Existence of Positive Solutions for Higher Order $(p,q)$ -Laplacian Two-Point Boundary Value Problems. Int. J. Differ. Equ. 2013 (2013), Article ID 743943, 9 pages. doi:10.1155/2013/743943. https://projecteuclid.org/euclid.ijde/1484881347

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