Variational properties of the solutions of singular second-order differential equations and systems

3 years ago

Abstract

This paper is devoted to the existence and variational characterization of the weak solutions of the Dirichlet boundary value problem for singular second-order ordinary differential equations and systems. The solution appears as a minimizer of the energy functional associated with the equation, and in the case of systems, as a Nash-type equilibrium of the set of energy functionals. The results are connected with the recent abstract fixed point theory due to the second author and with its application, given by the first author, to semilinear operator problems of Michlin type.

Authors

Angela Budescu
Department of Mathematics, Babeş-Bolyai University, Cluj, Romania

Radu Precup
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Paper coordinates

A. Budescu, R. Precup, Variational properties of the solutions of singular second-order differential equations and systems, J. Fixed Point Theor. Appl. 18 (2016), 505-518, https://doi.org/10.1007/s11784-016-0284-1

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About this paper

Journal

Journal of Fixed Point Theory and Applications

Publisher Name

Springer

DOI

https://doi.org/10.1007/s11784-016-0284-1

Print ISSN

16617738

Online ISSN

16617746

google scholar link

[1] Agarwal R. P.: Nonlinear superlinear singular and nonsingular second order boundary value problems. J. Differential Equations 143, 60–95 (1998), MathSciNet Article MATH Google Scholar
[2] Baxley J. V.: Some singular nonlinear boundary value problems. SIAM J. Math. Anal. 22, 463–479 (1991) MathSciNet Article MATH Google Scholar
[3] Bolojan O., Precup R.: Implicit first order differential systems with nonlocal conditions. Electron. J. Qual. Theory Differ. Equ. 69, 1–13 (2014) MathSciNet Article MATH Google Scholar
[4] Budescu A.: Semilinear operator equations and systems with potential-type nonlinearities. Stud. Univ. Babeş-Bolyai Math. 59, 199–212 (2014) MathSciNet MATH Google Scholar
[5] S. G. Michlin, Linear Partial Differential Equations. Vysshaya Shkola, Moscow, 1977 (in Russian).
[6] S. G. Michlin, Partielle Differentialgleichungen in der Mathematischen Physik. [Partial Differential Equations of Mathematical Physics], Akademie-Verlag, Berlin, 1978 (in German).
[7] Muzsi D., Precup R.: Non-resonance and existence for systems of non-linear operator equations. Appl. Anal. 87, 1005–1018 (2008) MathSciNet Article MATH Google Scholar
[8] D. O’Regan, Theory of Singular Boundary Value Problems. World Scientific Publishing, Singapore, 1994.
[9] D. O’Regan and R. P. Agarwal, Singular Differential and Integral Equations with Applications. Springer Science and Business Media, The Netherlands, 2003.
[10] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications. Taylor and Francis, London, 2001.
[11] R. Precup, Linear and Semilinear Partial Differential Equations. De Gruyter, Berlin, 2013.
[12] Precup R.: Nash-type equilibria and periodic solutions to nonvariational systems. Adv. Nonlinear Anal. 3, 197–207 (2014) MathSciNet MATH Google Scholar
[13] Precup R.: The role of matrices that are convergent to zero in the study of semilinear operator systems. Math. Comput. Modelling 49, 703–708 (2009) MathSciNet Article MATH Google Scholar
[14] Rachunkova I.: Singular mixed boundary value problem. J. Math. Anal. Appl. 320, 611–618 (2006) MathSciNet Article MATH Google Scholar
[15] I. Rachunková and J. Tomeček, Singular nonlinear problem for ordinary differential equation of the second-order on the half-line. In: Mathematical Models in Engineering, Biology, and Medicine (A. Cabada, E. Liz and J. J. Nieto, eds.), AIP Conf. Proc. 1124, Amer. Inst. Phys., Melville, NY, 2009, 294–303.
[16] Zabrejko P.: Continuity properties of the superposition operator. J. Austral. Math. Soc. 47, 186–210 (1989), MathSciNet Article MATH Google Scholar