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Gordjiet al.Journal of Inequalities and Applications2011,2011:81 http://www.journalofinequalitiesandapplications.com/content/2011/1/81
R E S E A R C HOpen Access Stability of the second order partial differential equations 1,2,3 4*5 6,7 M Eshaghi Gordji, YJ Cho, MB Ghaemiand B Alizadeh
* Correspondence: yjcho@gnu.ac.kr 4 Department of Mathematics Education and the Rins, Gyeongsang National University, Chinju 660701, Korea Full list of author information is available at the end of the article
Abstract We say that a functional equation (ξ) is stable if any functiongsatisfying the functional equation (ξ) approximately is near to a true solution of (ξ). In this paper, by using Banachs contraction principle, we prove the stability of nonlinear partial differential equations of the following forms: yx(x,t) =f(x,t,y(x,t)), ayx(x,t) +byt(x,t) =f(x,t,y(x,t)), p(x,t)yxt(x,t) +q(x,t)yt(x,t) +pt(x,t)yx(x,t)px(x,t)yt(x,t) =f(x,t,y(x,t)), p(x,t)yxx(x,t) +q(x,t)yx(x,t) =f(x,t,y(x,t)).
2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99. Keywords:generalized HyersUlam stability, linear differential equation, Banachs contraction principle
1. Introduction LetXbe a normed space over a scalar fieldK, and letIbe an open interval. Assume that, for any functionf:I®X(y=f(x)) satisfying the differential inequality
(n) (n1)||an(t)y(t) +an1(t)y(t) +∙ ∙ ∙+a1(t)y(t) +a0(t)y(t) +h(t)|| ≤ε
for alltÎI, whereε0, there exists a functionf0:I®Xsatisfying y0=f0(x), (n) (n1) an(t)y(t) +an1(t)y(t) +a0(t)y0(t) +h(t) = 0 0 0∙ ∙ ∙+a1(t)y0(t) +
and ||f(t) f0(t)||K(ε) for anytÎI. Then we say that the above differential equation has theHyersUlam stability. If the above statement is also true, then we replaceεandK(ε) by(t) andj(t), where,j:I ®[0,) are functions not depending onfandf0explicitly, then we say that the corre sponding differential equation has theHyersUlamRassias stabilityor thegeneralized HyersUlam stability. In 1998, the HyersUlam stability of differential equationy=ywas first investigated by Alsina and Ger [1]. In 2002, this result has been generalized by Takahasi et al. [2] for the Banach spacevalued differential equationy=ly. In 2005, Jung [3] proved the generalized HyersUlam stability of a linear differential equation of the first order. For
© 2011 Gordji et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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