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Monotone iterative sequences for non-local elliptic problems

Published online by Cambridge University Press:  20 July 2011

MOHAMMED AL-REFAI ,
NIKOS I. KAVALLARIS  and
MOHAMED ALI HAJJI
MOHAMMED AL-REFAI
Affiliation:
Department of Mathematical Science, United Arab Emirates University, P.O. Box 17551, Al Ain, UAE email: m_alrefai@uaeu.ac.ae, mahajji@uaeu.ac.ae
NIKOS I. KAVALLARIS
Affiliation:
Department of Statistics and Actuarial-Financial Mathematics, University of Aegean, TGr-83200 Karlovassi, Samos, Greece email: nkaval@aegean.gr
MOHAMED ALI HAJJI
Affiliation:
Department of Mathematical Science, United Arab Emirates University, P.O. Box 17551, Al Ain, UAE email: m_alrefai@uaeu.ac.ae, mahajji@uaeu.ac.ae
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Abstract

In this paper we establish an existence and uniqueness result for a class of non-local elliptic differential equations with the Dirichlet boundary conditions, which, in general, do not accept a maximum principle. We introduce one monotone sequence of lower–upper pairs of solutions and prove uniform convergence of that sequence to the actual solution of the problem, which is the unique solution for some range of λ (the parameter of the problem). The convergence of the iterative sequence is tested through examples with an order of convergence greater than 1.

Keywords

Non-Local ProblemsMaximum PrincipleElliptic Differential Equations
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 6 , December 2011 , pp. 533 - 552
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Ackleh, A. & Deng, K. (2003) Monotone method for nonlinear, nonlocal hyperbolic problems. Elec. J. Diff. Eqn. 10, 11–22.Google Scholar
[2]Allegretto, W. & Barabanova, A. (1996) Positivity of solutions of elliptic equations with non-local terms. Proc. R. Soc. Edinb. Sec. A 126, 643–663.CrossRefGoogle Scholar
[3]Allegretto, W. & Barabanova, A. (1997) Existence of positive solutions of semi-linear elliptic equations with non-local terms. Funkcialaj Ekvacioj 40, 395–409.Google Scholar
[4]Al-Refai, M. & Kavallaris, N. I.Bounds and critical parameters for a class of non-local problems. Elec. J. Diff. Eqn. 29, 1–16.Google Scholar
[5]Bebernes, J. W. & Talaga, P. (1996) Non-local problems modelling shear banding. Comm. Appl. Nonlinear Anal. 3, 79–103.Google Scholar
[6]Bebernes, J. W. & Lacey, A. A. (1997) Global existence and finite-time blow-up for a class of non-local parabolic problems. Adv. Diff. Eqn. 2, 927–953.Google Scholar
[7]Burns, T. J. (1994) Does a shear band result from a thermal explosion? Mech. Mater. 17, 261–271.CrossRefGoogle Scholar
[8]Carrillo, J. (1998) On non-local elliptic equation with decreasing nonlinearity arising in plasma physics and heat conduction. Nonlinear Anal. TMA. 32 (1), 97–115.CrossRefGoogle Scholar
[9]Correa, F. J. & Filho, D. C. (2005) On a class of non-local elliptic problems via Galerkin method. J. Math. Anal. Appl. 310, 177–187.CrossRefGoogle Scholar
[10]De Coster, C. & Nicaise, S. (2008) Lower and upper solutions for elliptic problems in nonsmooth domains. J. Diff. Eqn. 244, 599–629.CrossRefGoogle Scholar
[11]Dolbeault, J. & Stańczy, R. (2010) Non-existence and uniqueness results for supercritical semilinear elliptic equations. Ann. Henri Poincaré 10, 1311–1313.CrossRefGoogle Scholar
[12]Du, Y. (2002) Bifurcation from infinity in a class of non-local elliptic problems. Diff. Int. Eqn. 15, 587–606.Google Scholar
[13]Evans, L. C. (1998) Partial Differential Equations, AMS, Providence, RI.Google Scholar
[14]Fowler, A. C., Frigaard, I. & Howison, S. D. (1992) Temperature surges in current-limiting circuits devices. SIAM J. Appl. Math. 52, 998–1011.CrossRefGoogle Scholar
[15]Freitas, P. & Sweers, G. (1998) Positivity results for a non-local elliptic equation. Proc. R. Soc. Edinb. Sec. A 128, 697–715.CrossRefGoogle Scholar
[16]Jiaqi, M. & Cheng, O. (2001) A class of nonlocal boundary value problems of nonlinear elliptic systems in unbounded domains. Acta Mathemaica Scientia 21B, 93–97.Google Scholar
[17]Krzywicki, A. & Nadzieja, T. (1991) Some results concerning the Poisson-Bolzmann equation. Zast. Mat. Appl. Math. 21 (2), 265–272.Google Scholar
[18]Lacey, A. A. (1995) Thermal runaway in a non-local problem modeling Ohmic heating. Part I: Model derivation and some special cases. Euro. J. Appl. Math. 6, 127–144.CrossRefGoogle Scholar
[19]Lacey, A. A. (1995) Thermal runaway in a non-local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway. Euro. J. Appl. Math. 6, 201–224.CrossRefGoogle Scholar
[20]López-Gómez, J. (1998) On the structure and stability of the set of solutions of a non-local problem modeling Ohmic heating. J. Dyn. Diff. Eqn. 10, 537–566.CrossRefGoogle Scholar
[21]McGough, J. S. (1994) On solution of supercritical quasi-linear elliptic problems. Diff. Int. Eqn. 7, 1453–1471.Google Scholar
[22]Nikolopoulos, C. V. & Zouraris, G. E. (2007) Numerical solutions of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. In: Proceedings of the 6th AIMS International Conference on Dynamical Systems and Differential Equations, Disc. Cont. Dyn. Systems (supplement), pp. 768–778.Google Scholar
[23]Pao, C. V. (1992) Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York.Google Scholar
[24]Royden, H. L. (1988) Real Analysis, 3rd ed., Macmillan, New York.Google Scholar
[25]Schaaf, R. (2000) Uniqueness for semi-linear elliptic problems supercritical growth and domain geometry. Adv. Diff. Eqn. 5, 1201–1220.Google Scholar
[26]Stańczy, R. (2001) Non-local elliptic equations. Nonlinear Anal. 47, 3579–3584.CrossRefGoogle Scholar
[27]Tzanetis, D. E. (2002) Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. Electron. J. Diff. Eqn. 11, 1–26.Google Scholar
[28]Wolansky, G. (1997) A critical parabolic estimate and application to non-local equations arsing in chemotaxis. Appl. Anal. 66, 291–321.CrossRefGoogle Scholar