Existence of Positivity of the Solutions for Higher Order Three-Point Boundary Value Problems involving p-Laplacian

Year 2022, Volume: 6 Issue: 4, 451 - 459, 30.12.2022

Ravi Sankar , Sreedhar Namburi , Kapula Rajendra Prasad

https://doi.org/10.31197/atnaa.845044

Abstract

The present study focusses on the existence of positivity of the solutions to the higher order three-point boundary value problems involving $p$-Laplacian
$$[\phi_{p}(x^{(m)}(t))]^{(n)}=g(t,x(t)),~~t \in [0, 1],$$
$$
\begin{aligned}
x^{(i)}(0)=0, &\text{~for~} 0\leq i\leq m-2,\\
x^{(m-2)}(1)&-\alpha x^{(m-2)}(\xi)=0,\\
[\phi_{p}(x^{(m)}(t))]^{(j)}_{\text {at} ~ t=0}&=0, \text{~for~} 0\leq j\leq n-2,\\
[\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=1}&-\alpha[\phi_{p}(x^{(m)}(t))]^{(n-2)}_{\text {at} ~ t=\xi}=0,
\end{aligned}
$$
where $m,n\geq 3$, $\xi\in(0,1)$, $\alpha\in (0,\frac{1}{\xi})$ is a parameter.
The approach used by the application of Guo--Krasnosel'skii fixed point theorem to determine the existence of positivity of the solutions to the problem.

Keywords

three-point, nonlinear, boundary value problem, p-Laplacian, Green's function, positive solution

References

• [1] H. Afshari, S. Kalantari, E. Karapinar, Solution of fractional differential equations via coupled fixed point, Electron. J. Differ. Equ. 2015 (2015) 1-12.
• [2] H. Afshari, E, Karapinar, A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces, Adv. Differ. Equ. 2020 (2020) 1-11.
• [3] R.P. Agarwal, H. Lü, D. O’Regan, Eigenvalues and the one-dimensional p-Laplacian, J. Math. Anal. Appl. 266 (2002) 383-400.
• [4] R.I. Avery, J. Henderson, Existence of three positive pseudo-symmetric solutions for a one-dimensional p-Laplacian, J. Math. Anal. Appl. 277 (2003) 395-404.
• [5] A.C. Cavalheiroa, Existence results for Navier problems with degenerated (p,q)-Laplacian and (p,q)-Biharmonic operators, Results Nonlinear Anal. 1 (2018) 74-87.
• [6] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985.
• [7] L. Diening, P. Lindqvist, B. Kawohl, Mini-Workshop: The p-Laplacian Operator and Applications, Oberwolfach Reports, Report No. 08/2013, 433-482.
• [8] Y. Ding, J. Xu, X. Zhang, Positive solutions for a 2n th order p-Laplacian boundary value problem involving all derivaties, Electron. J. Differ. Equ. 2013 (2013) 1-14.
• [9] G. Dwivedi, Generalised Picone’s identity and some qualitative properties of p-sub-Laplacian on Heisenberg group, Adv. Theory Nonlinear Anal. Appl. 5 (2) (2021) 232-239.
• [10] H. Feng, H. Pang, W. Ge, Multiplicity of symmetric positive solutions for a multi-point boundary value problem with a one-dimensional p-Laplacian, Nonlinear Anal. 69 (2008) 3050-3059.
• [11] J.R. Graef, B. Yang, Multiple positive solutions to a three-point third order boundary value problem, Discrete Contin. Dyn. Syst. 2005 (2005) (Special) 337-344.
• [12] Y. Guo, W. Ge, Twin positive symmetric solutions for Lidstone boundary value problems, Taiwan. J. Math. 8 (2) (2004) 271-283. [13] Y. Guo, Y. Ji, X. Liu, Multiple positive solutions for some multi-point boundary value problems with p-Laplacian, J. Comput. Appl. Math. 216 (2008) 144-156.
• [14] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Acadamic Press, San Diego, CA, 1988.
• [15] E. Karapinar, H.D. Binh, N.H. Luc, N.H. Can, On continuity of the fractional derivative of the time-fractional semilinear pseudo-parabolic systems, Adv. Differ. Equ. 2021 (2021) 1-24.
• [16] E. Karapinar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large contractions on quasi-metrics spaces with an application to nonlinear fractional differential equations, Mathematics 7 (5) (2019) 1-11.
• [17] M. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
• [18] J.E. Lazreg, S. Abbas, M. Benchohra, E. Karapinar, Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces, Open Mathematics 19 (1) (2021) 363-372.
• [19] C. Li, W. Ge, Existence of positive solutions for p-Laplacian singular boundary value problems, Indian J. Pure Appl. Math. 34 (2003) 187-203.
• [20] J. Li, J. Shen, Existence of three positive solutions for boundary value problems with p-Laplacian, J. Math. Anal. Appl. 311 (2005) 457-465.
• [21] W.C. Lian, F.H. Wong, Existence of positive solutions for higher order generalized p-Laplacian BVPs, Appl. Math. Lett. 13 (2000) 35-43.
• [22] J. Liang, Z.W. Lv, Solutions to a three-point boundary value problem, Adv. Differ. Equ. 2011 (2011) 1-20.
• [23] X. Lin, Z. Fu, Positive solutions for a class of third order three-point boundary value problem, Discrete Dyn. Nat. Soc. 2012 (2012) 1-12.
• [24] X. Lin, Z. Zhao, Iterative technique for a third order differential equation with three-point nonlinear boundary value conditions, Electron. J. Qual. Theory Differ. Equ. 2016 (2016) 1-10.
• [25] Z. Liu, H. Chen, C. Liu, Positive solutions for singular third order non-homogeneous boundary value problems, J. Appl. Math. Comput. 38 (2012) 161-172.
• [26] Y. Liu, W. Ge, Multiple positive solutions to a three-point boundary value problem with p-Laplacian, J. Math. Anal. Appl. 277 (2003) 293-302.
• [27] D. Liu, Z. Ouyang, Solvability of third order three-point boundary value problems, Abstr. Appl. Anal. 2014 (2014) 1-7.
• [28] A. Ourraoui, Existence and uniqueness of solutions for Steklov problem with variable exponent, Adv. Theory Nonlinear Anal. Appl. 5 (1) (2021) 158-166.
• [29] A.P. Palamides, G. Smyrlis, Positive solutions to a singular third order three-point boundary value problem with an indefinitely signed Green’s function, Nonlinear Anal. TMA, 68 (7) (2008) 2104-2118.
• [30] K.R. Prasad, N. Sreedhar, L.T. Wesen, Existence of positive solutions for higher order p-Laplacian boundary value problems, Mediterr. J. Math. 15 (2018) 1-12.
• [31] R.R. Sankar, N. Sreedhar, K.R. Prasad, Existence of positive solutions for 3n th order boundary value problems involving p-Laplacian, Creat. Math. Inform. 31 (1) (2022) 101-108.
• [32] G. Shi, J. Zhang, Positive solutions for higher order singular p-Laplacian boundary value problems, Proc. Indian Acad. Sci. Math. Sci. 118 (2008) 295-305.
• [33] Y. Sun, Positive solutions for third order three-point non-homogeneous boundary value problems, Appl. Math. Lett. 22 (2009) 45-51.
• [34] Y. Sun, Q. Sun, X. Zhang, Existence and non existence of positive solutions for a higher order three-point boundary value problem, Abstr. Appl. Anal. 2014 (2014) 1-7.
• [35] C.X. Wang, H.R. Sun, Positive solutions for a class of singular third order three-point non-homogeneous boundary value problem, Dynam. Syst. Appl. 19 (2010) 225-234.
• [36] Z.L. Wei, Existence of positive solutions for n th order p-Laplacian singular sublinear boundary value problems, Appl. Math. Lett. 36 (2014) 25-30.
• [37] Z.L. Wei, Existence of positive solutions for n th order p-Laplacian singular super-linear boundary value problems, Appl. Math. Lett. 50 (2015) 133-140.
• [38] J. Xu, Z. Wei, Y. Ding, Positive solutions for a 2n th order p-Laplacian boundary value problem involving all even derivatives, Topol. Method Nonlinear Anal. 39 (2012) 23-36.
• [39] L. Zhao, W. Wang, C. Zhai, Existence and uniqueness of monotone positive solutions for a third order three-point boundary value problem, Differ. Equ. Appl. 10 (3) (2018) 251-260.