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Weighted space method for the stability of some nonlinear equations. (English) Zbl 1289.39054
Using the weighted space method, introduced by P. Găvruţa and L. Găvruţa [Int. J. Nonlinear Anal. Appl. 1, No. 2, 11–18 (2010; Zbl 1281.39038)], the authors prove the generalized Hyers-Ulam stability of the single variable functional equation \(y(x)=F(x,y(x),y(\eta(x)))\) and apply it to extend some results of L. Cădariu et al. [An. Univ. Vest Timiş., Ser. Mat.-Inform. 47, No. 3, 21–26 (2009; Zbl 1240.39058)] and J. A. Baker [Proc. Am. Math. Soc. 112, No. 3, 729–732 (1991; Zbl 0735.39004)]. As a consequence they prove the stability of the linear equation \(y(x)=g(x)y(\eta(x))+h(x)\). They also discuss the generalized Hyers-Ulam stability for a general class of the nonlinear Volterra integral equations in Banach spaces.

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
45G10 Other nonlinear integral equations
45D05 Volterra integral equations
45J05 Integro-ordinary differential equations
45N05 Abstract integral equations, integral equations in abstract spaces
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