Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-31T14:16:18.898Z Has data issue: false hasContentIssue false
  • English
  • Français

MEASURES OF NONCOMPACTNESS IN A SOBOLEV SPACE AND INTEGRO-DIFFERENTIAL EQUATIONS

Part of: Nonlinear operators and their properties Miscellaneous special kernels Integro-ordinary differential equations

Published online by Cambridge University Press:  21 July 2016

REZA ALLAHYARI ,
REZA ARAB  and
ALI SHOLE HAGHIGHI
REZA ALLAHYARI
Affiliation:
Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran email rezaallahyari@mshdiau.ac.ir
REZA ARAB*
Affiliation:
Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran email mathreza.arab@iausari.ac.ir
ALI SHOLE HAGHIGHI
Affiliation:
Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran email ali.sholehaghighi@gmail.com
*
✉ mathreza.arab@iausari.ac.ir
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.

The aim of this paper is to introduce a new measure of noncompactness on the Sobolev space $W^{n,p}[0,T]$ . As an application, we investigate the existence of solutions for some classes of functional integro-differential equations in this space using Darbo’s fixed point theorem.

Keywords

measures of noncompactnessDarbo’s fixed point theoremintegro-differential equationsSobolev spaces

MSC classification

Primary: 47H08: Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
Secondary: 45J05: Integro-ordinary differential equations 47H10: Fixed-point theorems

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 3 , December 2016 , pp. 497 - 506
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Aghajani, A., Banaś, J. and Jalilian, Y., ‘Existence of solutions for a class of nonlinear Volterra singular integral equations’, Comput. Math. Appl. 62 (2011), 1215–1227.Google Scholar
Aghajani, A. and Jalilian, Y., ‘Existence and global attractivity of solutions of a nonlinear functional integral equation’, Commun. Nonlinear Sci. Numer. Simul. 15 (2010), 3306–3312.Google Scholar
Akmerov, R. R., Kamenski, M. I., Potapov, A. S., Rodkina, A. E. and Sadovskii, B. N., Measures of Noncompactness and Condensing Operators (Birkhäuser Verlag, Basel, 1992).CrossRefGoogle Scholar
Ayad, A., ‘Spline approximation for first order Fredholm integro-differential equations’, Stud. Univ. Babeş-Bolyai Math. 41(3) (1996), 1–8.Google Scholar
Ayad, A., ‘Spline approximation for first order Fredholm delay integro-differential equations’, Int. J. Comput. Math. 70(3) (1999), 467–476.Google Scholar
Banaś, J., ‘Measures of noncompactness in the study of solutions of nonlinear differential and integral equations’, Cent. Eur. J. Math. 10(6) (2012), 2003–2011.Google Scholar
Banaś, J. and Goebel, K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, 60 (Dekker, New York, 1980).Google Scholar
Banaś, J., O’Regan, D. and Sadarangani, K., ‘On solutions of a quadratic Hammerstein integral equation on an unbounded interval’, Dynam. Systems Appl. 18 (2009), 251–264.Google Scholar
Behiry, S. H. and Hashish, H., ‘Wavelet methods for the numerical solution of Fredholm integro-differential equations’, Int. J. Appl. Math. 11(1) (2002), 27–35.Google Scholar
Bica, A. M., Cǎus, V. A. and Muresan, S., ‘Application of a trapezoid inequality to neutral Fredholm integro-differential equations in Banach spaces’, J. Inequal. Pure Appl. Math. 7 (2006), Art. 173.Google Scholar
Bloom, F., ‘Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory’, J. Math. Anal. Appl. 73(2) (1980), 524–542.Google Scholar
Darwish, M. A., Henderson, J. and O’Regan, D., ‘Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument’, Korean Math. Soc. 48 (2011), 539–553.Google Scholar
Dhage, B. C. and Bellale, S. S., ‘Local asymptotic stability for nonlinear quadratic functional integral equations’, Electron. J. Qual. Theory Differ. Equ. 10 (2008), 1–13.CrossRefGoogle Scholar
Forbes, L. K., Crozier, S. and Doddrell, D. M., ‘Calculating current densities and fields produced by shielded magnetic resonance imaging probes’, SIAM J. Appl. Math. 57(2) (1997), 401–425.Google Scholar
Guo, D., ‘Existence of solutions for nth-order integro-differential equations in Banach spaces’, Comput. Math. Appl. 41(5–6) (2001), 597–606.Google Scholar
Holmaker, K., ‘Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones’, SIAM J. Math. Anal. 24(1) (1993), 116–128.CrossRefGoogle Scholar
Hosseini, S. M. and Shahmorad, S., ‘Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial base’, Appl. Math. Model. 27(2) (2003), 145–154.CrossRefGoogle Scholar
Kuratowski, K., ‘Sur les espaces complets’, Fund. Math. 15 (1930), 301–309.Google Scholar
Mallet-Paret, J. and Nussbam, R. D., ‘Inequivalent measures of noncompactness and the radius of the essential spectrum’, Proc. Amer. Math. Soc. 139(3) (2011), 917–930.Google Scholar
Micula, G. and Fairweather, G., ‘Direct numerical spline methods for first order Fredholm integro-differential equations’, Rev. Anal. Numér. Théor. Approx. 22(1) (1993), 59–66.Google Scholar
Mursaleen, M. and Mohiuddine, S. A., ‘Applications of noncompactness to the infinite system of differential equations in l p spaces’, Nonlinear Anal. Theory Methods Appl. 75(4) (2012), 2111–2115.Google Scholar
Olszowy, L., ‘Solvability of infinite systems of singular integral equations in Fréchet space of coninuous functions’, Comput. Math. Appl. 59 (2010), 2794–2801.Google Scholar
Pour Mahmoud, J., Rahimi-Ardabili, M. Y. and Shahmorad, S., ‘Numerical solution of the system of Fredholm integro-differential equations by the tau method’, Appl. Math. Comput. 168(1) (2005), 465–478.Google Scholar