On the existence of solutions of one-dimensional fourth-order equations

  • S. Shokooh Gonbad Kavous Univ., Iran
  • G. A. Afrouzi Univ. Mazandaran, Babolsar, Iran
  • A. Hadjian Univ. Bojnord, Iran
Keywords: Non-trivial solution, Variational methods, Dirichlet problem


UDC 517.9

Using variational methods and critical point theorems, we prove the existence of nontrivial solutions for one-dimensional fourth-order equations. Multiplicity results are also pointed out.


G. A. Afrouzi, S. Heidarkhani, D. O’Regan, Existence of three solutions for a doubly eigenvalue fourth-order Boundary-value problem, Taiwanese J. Math., 15, no. 1, 201 – 210 (2011), https://doi.org/10.11650/twjm/1500406170 DOI: https://doi.org/10.11650/twjm/1500406170

A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14, 349 – 381 (1973), https://doi.org/10.1016/0022-1236(73)90051-7 DOI: https://doi.org/10.1016/0022-1236(73)90051-7

A. Averna, G. Bonanno, Three solutions for a quasilinear two-point boundary-value problem involving the onedimensional p-Laplacian, Proc. Edinburgh Math. Soc., 47, no. 2, 257 – 270 (2004), https://doi.org/10.1017/S0013091502000767 DOI: https://doi.org/10.1017/S0013091502000767

Z. Bai, Positive solutions of some nonlocal fourth-order Boundary-value problem, Appl. Math. and Comput., 215, no. 12, 4191 – 4197 (2010), https://doi.org/10.1016/j.amc.2009.12.040 DOI: https://doi.org/10.1016/j.amc.2009.12.040

G. Bonanno, A critical point theorem via the Ekeland variational principle, Nonlinear Anal., 75, no. 5, 2992 – 3007 (2012), https://doi.org/10.1016/j.na.2011.12.003 DOI: https://doi.org/10.1016/j.na.2011.12.003

G. Bonanno, B. Di Bella, A Boundary-value problem for fourth-order elastic beam equations, J. Math. Anal. and Appl., 343, no. 2, 1166 – 1176 (2008), https://doi.org/10.1016/j.jmaa.2008.01.049 DOI: https://doi.org/10.1016/j.jmaa.2008.01.049

G. Bonanno, B. Di Bella, A fourth-order Boundary-value problem for a Sturm – Liouville type equation, Appl. Math. and Comput., 217, no. 8, 3635 – 3640 (2010), https://doi.org/10.1016/j.amc.2010.10.019 DOI: https://doi.org/10.1016/j.amc.2010.10.019

G. Bonanno, B. Di Bella, D. O’Regan, nontrivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl., 62, no. 4, 1862 – 1869 (2011), https://doi.org/10.1016/j.camwa.2011.06.029 DOI: https://doi.org/10.1016/j.camwa.2011.06.029

G. Chai, Existence of positive solutions for fourth-order Boundary-value problem with variable parameters, Nonlinear Anal., 66, no. 4, 870 – 880 (2007), https://doi.org/10.1016/j.na.2005.12.028 DOI: https://doi.org/10.1016/j.na.2005.12.028

E. Dulacska, ` Soil settlement effects on buildings, Dev. Geotechn. Eng., Vol. 69, Elsevier, Amsterdam, The Netherlands (1992).

R. Livrea, Existence of three solutions for a quasilinear two point Boundary-value problem, Arch. Math., 79, no. 4, 288 – 298 (2002).

P. K. Palamides, Boundary-value problems for shallow elastic membrane caps, IMA J. Appl. Math., 67, no. 3, 281 – 299 (2002), https://doi.org/10.1093/imamat/67.3.281 DOI: https://doi.org/10.1093/imamat/67.3.281

L. A. Peletier, W. C. Troy, R.C.A.M. Van der Vorst, Stationary solutions of a fourth-order nonlinear diffusion equation, Different. Equat., 31, no. 2, 301 – 314 (1995).

P. Pietramala, A note on a beam equation with nonlinear boundary conditions, Boundary-value Problems, 2011, Article ID 376782 (2011), 14 p., https://doi.org/10.1155/2011/376782 DOI: https://doi.org/10.1155/2011/376782

P. Pucci, J. Serrin, The strong maximum principle revisited, J. Different. Equat., 196, 1 – 66 (2004); Erratum, ibid. 207, no. 1, 226 – 227 (2004), https://doi.org/10.1016/j.jde.2004.09.002

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI (1986), https://doi.org/10.1090/cbms/065 DOI: https://doi.org/10.1090/cbms/065

B. Ricceri, A general variational principle and some of its applications, J. Comput. and Appl. Math., 113, no. 1-2, 401 – 410 (2000), https://doi.org/10.1016/S0377-0427(99)00269-1 DOI: https://doi.org/10.1016/S0377-0427(99)00269-1

V. Shanthi, N. Ramanujam, A numerical method for Boundary-value problems for singularly perturbed fourth-order ordinary differential equations, Appl. Math. and Comput., 129, no. 2-3, 269 – 294 (2002), https://doi.org/10.1016/S0096-3003(01)00040-6 DOI: https://doi.org/10.1016/S0096-3003(01)00040-6

W. Soedel, Vibrations of Shells and Plates, Dekker, New York, NY (1993).

S. Tersian, J. Chaparova, Periodic and homoclinic solutions of extended Fisher – Kolmogorov equations, J. Math. Anal. and Appl., 260, no. 2, 490 – 506 (2001), https://doi.org/10.1006/jmaa.2001.7470 DOI: https://doi.org/10.1006/jmaa.2001.7470

S. P. Timoshenko, Theory of elastic stability, McGraw-Hill Book, New York, NY (1961).

Q. L. Yao, Z. B. Bai, Existence of positive solution for the Boundary-value problem $u^{(4)}(t)-lambda h(t)f(u(t))=0$. (Chinese),Chinese Ann. Math., 20, no. 5, 575 – 578 (1999).

E. Zeidler, Nonlinear functional analysis and its applications, Vols. II/B and III, Springer, New York (1990, 1985), https://doi.org/10.1007/978-1-4612-5020-3 DOI: https://doi.org/10.1007/978-1-4612-5020-3

How to Cite
Shokooh, S., G. A. Afrouzi, and A. Hadjian. “On the Existence of Solutions of One-Dimensional Fourth-Order Equations”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 11, Nov. 2020, pp. 1575-88, doi:10.37863/umzh.v72i11.569.
Research articles