Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T17:30:26.567Z Has data issue: false hasContentIssue false
  • English
  • Français

A NEW EXTENSION OF DARBO’S FIXED POINT THEOREM USING RELATIVELY MEIR–KEELER CONDENSING OPERATORS

Part of: Nonlinear operators and their properties General theory in ordinary differential equations

Published online by Cambridge University Press:  19 July 2018

MOOSA GABELEH Open the ORCID record for MOOSA GABELEH [Opens in a new window]  and
CALOGERO VETRO Open the ORCID record for CALOGERO VETRO [Opens in a new window]
MOOSA GABELEH
Affiliation:
Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran email gab.moo@gmail.com, gabeleh@abru.ac.ir
CALOGERO VETRO*
Affiliation:
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy email calogero.vetro@unipa.it
*
✉calogero.vetro@unipa.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.

Keywords

best proximity pointoptimal solutionrelatively Meir–Keeler condensing operatorstrictly convex Banach space

MSC classification

Primary: 47H10: Fixed-point theorems
Secondary: 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 34A12: Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 2 , October 2018 , pp. 286 - 297
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author was partially supported by a grant from the Institute for Research in Fundamental Sciences (IPM) (No. 96470046).

References

Aghajani, A., Mursaleen, M. and Shole Haghighi, A., ‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B 35 (2015), 552–566.Google Scholar
Ambrosetti, A., ‘Un teorema di esistenza per le equazioni differenziali negli spazi di Banach’, Rend. Semin. Mat. Univ. Padova 39 (1967), 349–361.Google Scholar
Ayerbe Toledano, J. M., Dominguez Benavides, T. and López-Acedo, G., Measures of Noncompactness in Metric Fixed Point Theory, Operator Theory: Advances and Applications, 99 (Birkhäuser, Basel, 1997).Google Scholar
Darbo, G., ‘Punti uniti in trasformazioni a codominio non compatto’, Rend. Semin. Mat. Univ. Padova 24 (1955), 84–92.Google Scholar
Dunford, N. and Schwartz, J. T., Linear Operators III (Wiley–Interscience, New York, 1971).Google Scholar
Eldred, A. A., Kirk, W. A. and Veeramani, P., ‘Proximal normal structure and relatively nonexpansive mappings’, Studia Math. 171 (2005), 283–293.Google Scholar
Gabeleh, M., ‘A characterization of proximal normal structure via proximal diametral sequences’, J. Fixed Point Theory Appl. 19 (2017), 2909–2925.Google Scholar
Gabeleh, M. and Markin, J., ‘Optimum solutions for a system of differential equations via measure of noncompactness’, Indag. Math. (N.S.) 29 (2018), 895–906.Google Scholar
Gohberg, I., Goldenštein, L. S. and Markus, A. S., ‘Investigation of some properties of bounded linear operators in connection with their q-norms’, Učen. Zap. Kishinevsk. Un-ta 29 (1957), 29–36.Google Scholar
Kakutani, S., ‘Topological properties of the unit sphere of a Hilbert space’, Proc. Imp. Acad. Tokyo 19 (1943), 269–271.Google Scholar
Schauder, J., ‘Der Fixpunktsatz in Funktionalräumen’, Studia Math. 2 (1930), 171–180.Google Scholar