×

On a global implicit function theorem and some applications to integro-differential initial value problems. (English) Zbl 1413.47088

Summary: We generalize a recent global implicit function theorem from [D. Idczak et al., Abstr. Appl. Anal. 2013, Article ID 129478, 8 p. (2013; Zbl 1296.45002)] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to a 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.

MSC:

47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
26B10 Implicit function theorems, Jacobians, transformations with several variables
45J05 Integro-ordinary differential equations

Citations:

Zbl 1296.45002
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Chabrowski, Variational Methods for Potential Operator Equations, De Gruyter (Berlin, New York, 1997). · Zbl 1157.35338
[2] M. Cristea, A note on global implicit function theorem, J. Inequal. Pure and Appl., 8 (2007). · Zbl 1134.26003
[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) · Zbl 0991.35023
[4] Figueredo D.G.: Lectures on the Ekeland Variational Principle with Applications and Detours. Preliminary Lecture Notes, SISSA (1988)
[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. · Zbl 0426.35001
[6] Galewski M., Koniorczyk M.: On a global diffeomorphism between two Banach spaces and some application. Studia Sci. Math. Hung., 52, 65-86 (2015) · Zbl 1349.57011
[7] Idczak D., Skowron A., Walczak S.: On the diffeomorphisms between Banach and Hilbert spaces. Adv. Nonlinear Stud., 12, 89-100 (2012) · Zbl 1244.57056 · doi:10.1515/ans-2012-0105
[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. · Zbl 1296.45002
[9] Idczak D.: A global implicit function theorem and its applications to functional equations Contin. Discrete. Dyn. Syst. Ser. B, 19, 2549-2556 (2014) · Zbl 1303.26014 · doi:10.3934/dcdsb.2014.19.2549
[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). · Zbl 1036.49001
[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). · Zbl 1206.49002
[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. · Zbl 0849.45004
[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). · Zbl 1040.49001
[15] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, 2nd ed. Lecture Notes in Mathematics. 1364, Springer-Verlag (Berlin, 1993). · Zbl 0921.46039
[16] Rădulescu S., Rădulescu M.: Local inversion theorems without assuming continuous differentiability. J. Math. Anal. Appl., 138, 581-590 (1989) · Zbl 0745.58008 · doi:10.1016/0022-247X(89)90312-0
[17] Rheinboldt W.C.: Local mapping relations and global implicit function theorems. Trans. Amer. Math. Soc., 138, 183-198 (1969) · Zbl 0175.45201 · doi:10.1090/S0002-9947-1969-0240644-0
[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) · Zbl 1255.26003 · doi:10.4064/ap105-3-1
[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) · Zbl 1302.45018
[20] M. Willem, Minimax Theorems. Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser (Boston, MA, 1996). · Zbl 0856.49001
[21] E. Zeidler, Applied functional analysis. Main principles and their applications, Applied Mathematical Sciences. 109, Springer-Verlag (New York, 1995). · Zbl 0834.46003
[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) · Zbl 1084.26005 · doi:10.1016/j.jmaa.2005.08.072
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.