Skip to main content
Log in

On a global implicit function theorem and some applications to integro-differential initial value problems

  • Published:
Acta Mathematica Hungarica Aims and scope Submit manuscript

Abstract

We generalize a recent global implicit function theorem from [8] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to certain auxiliary functional are used in the proof together with the local implicit function theorem and mountain pass geometry. An application to integro-differential systems is given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Chabrowski, Variational Methods for Potential Operator Equations, De Gruyter (Berlin, New York, 1997).

  2. M. Cristea, A note on global implicit function theorem, J. Inequal. Pure and Appl., 8 (2007).

  3. Dinca G., Jebelean P., Mawhin J.: Variational and topological methods for Dirichlet problems with p-Laplacian. Port. Math. (N.S.), 58, 339–378 (2001)

    MathSciNet  MATH  Google Scholar 

  4. Figueredo D.G.: Lectures on the Ekeland Variational Principle with Applications and Detours. Preliminary Lecture Notes, SISSA (1988)

    Google Scholar 

  5. S. Fučik and A. Kufner, Nonlinear Differential Equations, Studies in Applied Mechanics. 2. Elsevier Scientific Publishing Company (Amsterdam, Oxford, New York, 1980), 359 pp.

  6. Galewski M., Koniorczyk M.: On a global diffeomorphism between two Banach spaces and some application. Studia Sci. Math. Hung., 52, 65–86 (2015)

    MathSciNet  MATH  Google Scholar 

  7. Idczak D., Skowron A., Walczak S.: On the diffeomorphisms between Banach and Hilbert spaces. Adv. Nonlinear Stud., 12, 89–100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. D. Idczak, A. Skowron and S. Walczak, Sensitivity of a fractional integrodifferential Cauchy problem of Volterra type, Abstr. Appl. Anal., (2013), Art. ID 129478, 8 pp.

  9. Idczak D.: A global implicit function theorem and its applications to functional equations Contin. Discrete. Dyn. Syst. Ser. B, 19, 2549–2556 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. D. Idczak, On some strengthening of the global implicit function theorem with an application to a Cauchy problem for an integro-differential Volterra system, ArXiV 1401.4049.

  11. Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and Some Application, Encyclopedia of Mathematics and its Applications, 95. Cambridge University Press (Cambridge, 2003).

  12. A. Kristály, V. Rădulescu and Cs. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, No. 136, Cambridge University Press, Cambridge (2010).

  13. V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-differential Equations. Stability and Control: Theory, Methods and Applications, 1, Gordon and Breach Publ. (Philadelphia, PA, 1995), 362 p.

  14. D. Motreanu and V. Rădulescu; Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optimization and its Applications, 67, Kluwer Academic Publishers (Dordrecht, 2003).

  15. R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed. Lecture Notes in Mathematics. 1364, Springer-Verlag (Berlin, 1993).

  16. Rădulescu S., Rădulescu M.: Local inversion theorems without assuming continuous differentiability. J. Math. Anal. Appl., 138, 581–590 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Rheinboldt W.C.: Local mapping relations and global implicit function theorems. Trans. Amer. Math. Soc., 138, 183–198 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang J.R., Wei W.: Nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces. Ann. Polon. Math., 105, 209–223 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang J., Wei W.: An application of measure of noncompactness in the study of integrodifferential evolution equations with nonlocal conditions. Proc. A. Razmadze Math. Inst., 158, 135–148 (2012)

    MathSciNet  MATH  Google Scholar 

  20. M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser (Boston, MA, 1996).

  21. E. Zeidler, Applied functional analysis. Main principles and their applications, Applied Mathematical Sciences. 109, Springer-Verlag (New York, 1995).

  22. Zhang W., Ge S.S.: A global implicit function theorem without initial point and its applications to control of non-affine systems of high dimensions. J. Math. Anal. Appl., 313, 251–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to M. Galewski.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Galewski, M., Koniorczyk, M. On a global implicit function theorem and some applications to integro-differential initial value problems. Acta Math. Hungar. 148, 257–278 (2016). https://doi.org/10.1007/s10474-016-0589-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10474-016-0589-y

Key words and phrases

Mathematics Subject Classification

Navigation