Opuscula Math. 39, no. 2 (2019), 131-144
https://doi.org/10.7494/OpMath.2019.39.2.131
Opuscula Mathematica
On unique solvability of a Dirichlet problem with nonlinearity depending on the derivative
Michał Bełdziński
Marek Galewski
Abstract. In this work we consider second order Dirichelet boundary value problem with nonlinearity depending on the derivative. Using a global diffeomorphism theorem we propose a new variational approach leading to the existence and uniqueness result for such problems.
Keywords: diffeomorphism, uniqueness, non-potential problems, variational methods, monotone methods, Palais-Smale condition.
Mathematics Subject Classification: 34A12, 47H30, 47J07.
- G.A. Afrouzi, A. Hadjian, V.D. Rădulescu, A variational approach of Sturm-Liouville problems with the nonlinearity depending on the derivative, Bound. Value Probl. 2015 (2015) 81.
- M. Bełdziński, M. Galewski, Global diffeomorphism theorem applied to the solvability of discrete and continuous boundary value problems, J. Diff. Equ. Appl 24 (2018) 2, 277-290.
- D. Bors, A. Skowron, S. Walczak, System described by Volterra type integral operators, Dys. Cont. Dyn System B 19 (2014), 2401-2416.
- H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.
- I. Ekeland, An inverse function theorem in Fréchet spaces, Ann. Inst. Henri Poincaré, Anal. Non Lineaire 28 (2011) 1, 91-105.
- D.G. Figueredo, Lectures on the Ekeland Variational Principle with Applications and Detours, Preliminary Lecture Notes, SISSA, 1988.
- D. Idczak, A global implicit function theorem and its applications to functional equations, Discrete Contin. Dyn. Syst., Ser. B 19 (2014) 8, 2549-2556.
- D. Idczak, On a generalization of a global implicit function theorem, Adv. Nonlinear Stud. 16 (2016) 1, 87-94.
- D. Idczak, A. Skowron, S. Walczak, On the diffeomorphisms between Banach and Hilbert spaces, Adv. Nonlinear Stud. 12 (2012) 1, 89-100.
- G. Katriel, Mountain pass theorems and global homeomorphism theorems, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 11 (1994) 2, 189-209.
- M. Majewski, Control system defined by some integral operator, Opuscula Math. 37 (2017) 2, 313-325.
- J. Mawhin, Problemes de Dirichlet Variationnels Non Linéaires, Séminaire de Mathématiques Supérieures, vol. 104, Montreal, 1987.
- D. Motreanu, M. Tanaka, Multiple existence results of solutions for quasilinear elliptic equations with a nonlinearity depending on a parameter. Ann. Mat. Pura Appl. 193 (2014) 5, 1255-1282.
- M. Rădulescu, S. Rădulescu, Local inversion theorems without assuming continuous differentiability, J. Math. Anal. Appl. 138 (1989) 2, 581-590.
- B. Ricceri, A three critical points theorem revisited, Nonlinear Anal. 70 (2009), 3084-3089.
- C. Torres Ledesma, Existence of solutions for fractional Hamiltonian systems with nonlinear derivative dependence in \(\mathbb{R}\), J. Fract. Calc. Appl. 7 (2016) 2, 74-87.
- E. Zeidler, Applied Functional Analysis. Main Principles and Their Applications, Applied Mathematical Sciences, vol. 109, New York, Springer-Verlag, 1995.
- Michał Bełdziński
https://orcid.org/0000-0002-9653-612X
- Łódź University of Technology, Institute of Mathematics, Wólczańska 215, 90-924 Łódź, Poland
- Marek Galewski
https://orcid.org/0000-0002-3224-2456
- Łódź University of Technology, Institute of Mathematics, Wólczańska 215, 90-924 Łódź, Poland
- Communicated by Vicentiu D. Radulescu.
- Received: 2018-04-27.
- Revised: 2018-11-05.
- Accepted: 2018-11-06.
- Published online: 2018-12-07.