We obtain necessary and sufficient conditions for the existence of solutions of weakly nonlinear boundary-value problems for differential equations in a Banach space. A convergent iterative procedure is proposed for the determination of solutions. We also establish a relationship between necessary and sufficient conditions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yu. M. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).
O. A. Boichuk and E. V. Panasenko, “Boundary-value problems for differential equations in a Banach space,” Nelin. Kolyvannya, 12, No. 1, 16–19 (2009); English translation: Nonlin. Oscillations, 12, No. 1, 15–18 (2009).
A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht (2004).
A. A. Boichuk, V. F. Zhuravlev, and A. M. Samoilenko, Generalized Inverse Operators and Noetherian Boundary-Value Problems [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1995).
A. A. Boichuk, “Construction of solutions of two-point boundary-value problems for weakly perturbed nonlinear systems in critical cases,” Ukr. Mat. Zh., 41, No. 10, 1416–1420 (1989); English translation: Ukr. Math. J., 41, No. 10, 1219–1223 (1989).
A. A. Boichuk, Boundary-Value Problems for Weakly Perturbed Systems in Critical Cases [in Russian], Preprint, Institute of Mathematics, Ukrainian National Academy of Sciences (1988).
E. A. Grebenikov and Yu. A. Ryabov, Constructive Methods for Analysis of Nonlinear Systems [in Russian], Nauka, Moscow (1979).
I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations [in Russian], Gostekhizdat, Moscow (1956).
O. A. Boichuk and O. O. Pokutnyi, “Bounded solutions of weakly nonlinear differential equations in a Banach space,” Nelin. Kolyvannya, 11, No. 2, 151–159 (2008); English translation: Nonlin. Oscillations, 11, No. 2, 158–167 (2008).
S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1971).
V. A. Trenogin, Functional Analysis [in Russian], Nauka, Moscow (1980).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1968).
V. M. Alekseev, V. M. Tikhomirov, and S. V. Fomin, Optimal Control [in Russian], Fizmatlit, Moscow (2005).
A. A. Boichuk, Constructive Methods for Analysis of Boundary-Value Problems [in Russian], Naukova Dumka, Kiev (1990).
A. M. Samoilenko and Yu. V. Teplinskii, Countable Systems of Differential Equations [in Russian], Institute Mathematics, Ukrainian National Academy Sciences, Kiev (1993).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Neliniini Kolyvannya, Vol. 13, No. 4, pp. 483–496, October–December, 2010.
Rights and permissions
About this article
Cite this article
Boichuk, O.A., Panasenko, E. Weakly nonlinear boundary-value problems for differential equations in a Banach space in the critical case. Nonlinear Oscill 13, 515–529 (2011). https://doi.org/10.1007/s11072-011-0127-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11072-011-0127-3