Abstract
We prove that the four-point boundary value problem
\(-[\U{3d5} (u\prime)]\prime=f(t,u,u\prime),u(0)=\U{3b1} u(\U{3be}),u(T)=\U{3b2} u(\U{3b7} ),)\
where \(f:[0,T]\times R^{2}\rightarrow R)\ is continuous, \(\U{3b1} ,\U{3b2} \in\lbrack0,1),0<\U{3be} <\U{3b7} <T)\, and \(\U{3d5} :(-a,a)\rightarrow R)\ \((0<a<\infty))\ is an increasing homeomorphism, which is always solvable. When instead of \(f)\ is some \(g:[0,T]\times\lbrack0,\infty)\rightarrow \lbrack0,\infty))\, we obtain existence, localization, and multiplicity of positive solutions. Our approach relies on Schauder and Krasnoselskii’s fixed point theorems, combined with a Harnack-type inequality.
Authors
Antonia Chinní
Department of Engineering, University of Messina, Messina, Italy
Beatrice Di Bella
Department of Engineering, University of Messina, Messina, Italy
Petru Jebelean
Department of Mathematics, West University of Timişoara, Timisoara, Romania
Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania
Keywords
Singular φ-Laplace equations; four-point boundary value problems; positive solutions; Schauder’s fixed point theorem; Krasnoselskii’s fixed point theorem
Paper coordinates
A. Chinni, B. Di Bella, P. Jebelean, R. Precup, A four-point boundary value problem with singular φ-Laplacian, J. Fixed Point Theory Appl. (2019) 21:66, pp 16, https://doi.org/10.1007/s11784-019-0703-1
About this paper
Journal
Journal of Fixed Point Theory and Applications
Publisher Name
Springer
Print ISSN
1661-7738
Online ISSN
1661-7746
google scholar link
[1] Bai, Z., Du, Z.: Positive solutions for some second-order four-point boundary value problems. J. Math. Anal. Appl. 330, 34–50 (2007), MathSciNet Article Google Scholar
[2] Bai, Z., Li, W., Ge, W.: Existence and multiplicity of solutions for four-point boundary value problems at resonance. Nonlinear Anal. 60, 1151–1162 (2005), MathSciNet Article Google Scholar
[3] Bai, Z., Zhang, Y.: Solvability of fractional three-point boundary value problems with nonlinear growth. Appl. Math. Comput. 218, 1719–1725 (2011) MathSciNet MATH Google Scholar
[4] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for some nonlinear problems involing mean curvature operators in Euclidean and Minkowski spaces. Proc. Am. Math. Soc. 137, 161–169 (2009), Article Google Scholar
[5] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283, 379–391 (2010), MathSciNet Article Google Scholar
[6] Bereanu, C., Jebelean, P., Mawhin, J.: Periodic solutions of pendulum-like perturbations of singular and bounded φ-Laplacians. J. Dyn. Differ. Equ. 22, 463–471 (2010) MathSciNet Article Google Scholar
[7] Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular φ-Laplacian. J. Differ. Equ. 243, 536–557 (2007), MathSciNet Article Google Scholar
[8] Brezis, H., Mawhin, J.: Periodic solutions of the forced relativistic pendulum. Differ. Integr. Equ. 23, 801–810 (2010), MathSciNet MATH Google Scholar
[9] Coelho, I., Corsato, C., Obersnel, F., Omari, P.: Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation. Adv. Nonlinear Stud. 12, 621–638 (2012), MathSciNet Article Google Scholar
[10] Feng, M., Li, P., Sun, S.: Symmetric positive solutions for fourth-order n-dimensional m-Laplace system. Bound. Value Probl. (2018). https://doi.org/10.1186/s13661-018-0981-3, MathSciNet Article Google Scholar
[11] Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988), MATH Google Scholar
[12] Herlea, D.-R.: Positive solutions for second-order boundary-value problems with φ-Laplacian. Electron. J. Differ. Equ. 2016(51), 1–12 (2016), MathSciNet Google Scholar
[13] Herlea, D.-R., Precup, R.: Existence, localization and multiplicity of positive solutions to φ-Laplace equations and systems. Taiwan J. Math. 20, 77–89 (2016), MathSciNet Article Google Scholar
[14] Jebelean, P., Mawhin, J., Şerban, C.: A vector p-Laplacian type approach to multiple periodic solutions for the p-relativistic operator. Commun. Contemp. Math. 19, 1–16 (2017), MathSciNet Article Google Scholar
[15] Jebelean, P., Precup, R.: Symmetric positive solutions to a singular φ-Laplace equation. J. Lond. Math. Soc. 99, 495–515 (2019), MathSciNet Article Google Scholar
[16] Jebelean, P., Şerban, C.: Boundary value problems for discontinuous perturbations of singular φ-Laplacian operator. J. Math. Anal. Appl. 431, 662–681 (2015), MathSciNet Article Google Scholar
[17] Krasnoselskii, M.A.: Positive Solutions of Operator Equations. Noordhoff, Groningen (1964), Google Scholar
[18] Kwong, M.K., Wong, J.S.W.: An optimal existence theorem for positive solutions of a four-point boundary value problem. Electron. J. Differ. Equ. 2009(165), 1–8 (2009), MathSciNet Google Scholar
[19] Ma, R., Lu, Y.: Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator. Adv. Nonlinear Stud. 15, 798–803 (2015), MathSciNet Article Google Scholar
[20] Precup, R., Rodríguez-López, J.: Positive solutions for discontinuous problems with applications to φ-Laplacian equations. J. Fixed Point Theor. Appl. 20, 1–17 (2018). Article 156, MathSciNet Article Google Scholar
[21] Rachunkova, I.: Upper and lower solutions and topological degree. J. Math. Anal. Appl. 234, 311–327 (1999), MathSciNet Article Google Scholar
[22] Schmitt, K.: On the existence of nontrivial solutions of differential equations subject to linear constraints. Rend. Inst. Mat. Univ. Trieste 49, 27–40 (2017), MathSciNet Google Scholar
[23] Sheng, K., Zhang, W., Bai, Z.: Positive solutions to fractional boundary value problems with p-Laplacian on time scales. Bound. Value Probl. 2018(1), 70 (2018), MathSciNet Article Google Scholar
[24] Tian, Y., Wei, Y., Sun, S.: Multiplicity for fractional differential equations with p-Laplacian. Bound. Value Probl. 2018(1), 127 (2018), MathSciNet Article Google Scholar
[25] Zhang, G., Sun, J.: Positive solutions of m-point boundary value problems. J. Math. Anal. Appl. 291, 406–418 (2004), MathSciNet Article Google Scholar