## 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
Revised: 17 July 2013
Accepted: 17 July 2013
First available in Project Euclid: 20 January 2017

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

#### References

• R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999.MR1680024
• P. V. S. Anand, P. Murali, and K. R. Prasad, “Multiple symmetric positive solutions for systems of higher order boundary-value problems on time scales,” Electronic Journal of Differential Equations, vol. 2011, no. 102, pp. 1–12, 2011.MR2832278
• D. R. Anderson, “Eigenvalue intervals for even-order Sturm-Liouville dynamic equations,” Communications on Applied Nonlinear Analysis, vol. 12, no. 4, pp. 1–13, 2005.MR2163174
• R. I. Avery, “A generalization of the Leggett-Williams fixed point theorem,” Mathematical Sciences Research Hot-Line, vol. 3, no. 7, pp. 9–14, 1999.MR1702612
• R. I. Avery, J. M. Davis, and J. Henderson, “Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem,” Electronic Journal of Differential Equations, vol. 2000, no. 40, pp. 1–15, 2000.MR1764708
• J. M. Davis, J. Henderson, and P. J. Y. Wong, “General Lidstone problems: multiplicity and symmetry of solutions,” Journal of Mathematical Analysis and Applications, vol. 251, no. 2, pp. 527–548, 2000.MR1794756
• L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 120, no. 3, pp. 743–748, 1994.MR1204373
• D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, vol. 5 of Notes and Reports in Mathematics in Science and Engineering, Academic Press, Boston, Mass, USA, 1988.MR959889
• J. Henderson, P. Murali, and K. R. Prasad, “Multiple symmetric positive solutions for two-point even order boundary value problems on time scales,” Mathematics in Engineering, Science and Aerospace, vol. 1, no. 1, pp. 105–117, 2010.
• J. Henderson and H. B. Thompson, “Multiple symmetric positive solutions for a second order boundary value problem,” Proceedings of the American Mathematical Society, vol. 128, no. 8, pp. 2373–2379, 2000.MR1709753
• K. R. Prasad, P. Murali, and N. V. V. S. Suryanarayana, “Multiple positive solutions for the system of higher order two-point boundary value problems on time scales,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 2009, no. 32, pp. 1–13, 2009.MR2506153
• K. R. Prasad and P. Murali, “Solvability of p-Laplacian singular boundary value problems on time scales,” Advances in Pure and Applied Mathematics, vol. 3, no. 4, pp. 377–391, 2012.MR3024011