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A fixed point approach to the stability of functional equations in non-Archimedean metric spaces. (English) Zbl 1237.39022
Let $$X$$ be an arbitrary set, $$Y$$ be a complete ultrametric space, $$f_1,\dots ,f_k:\;X\to X$$, $$\Phi:\;X\times Y^k\to Y$$. The authors find conditions for the solvability of the functional equation $$\Phi (x,\psi (f_1(x)),\dots ,\psi (f_k(x)))=\psi (x)$$, $$x\in X$$ (with respect to $$\psi:\;X\to Y$$) and its generalizations. The result, in the spirit of the Hyers-Ulam stability where a solution is obtained as a limit of approximate solutions, is based on a new fixed point theorem.

##### MSC:
 39B52 Functional equations for functions with more general domains and/or ranges 39B82 Stability, separation, extension, and related topics for functional equations 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
 [1] Khrennikov, A., () [2] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403 [3] Agarwal, R.P.; Xu, B.; Zhang, W., Stability of functional equations in single variable, J. math. anal. appl., 288, 852-869, (2003) · Zbl 1053.39042 [4] Brzdȩk, J.; Sikorska, J., A conditional exponential functional equation and its stability, Nonlinear anal. TMA, 72, 2923-2934, (2010) · Zbl 1187.39032 [5] Czerwik, S., Functional equations and inequalities in several variables, (2002), World Scientific Publishing Co., Inc. River Edge, NJ · Zbl 1011.39019 [6] Forti, G.-L.; Sikorska, J., Variations on the drygas equation and its stability, Nonlinear anal. TMA, 74, 343-350, (2011) · Zbl 1201.39023 [7] Jung, S.-M., Hyers – ulam – rassias stability of functional equations in mathematical analysis, (2001), Hadronic Press, Inc. Palm Harbor, FL · Zbl 0980.39024 [8] Moszner, Z., On the stability of functional equations, Aequationes math., 77, 33-88, (2009) · Zbl 1207.39044 [9] Stević, S., Bounded solutions of a class of difference equations in Banach spaces producing controlled chaos, Chaos solitons fractals, 35, 238-245, (2008) · Zbl 1142.39013 [10] Hayes, W.; Jackson, K.R., A survey of shadowing methods for numerical solutions of ordinary differential equations, Appl. numer. math., 53, 299-321, (2005) · Zbl 1069.65075 [11] Palmer, K., () [12] Pilyugin, S.Yu., () [13] Moslehian, M.S.; Rassias, Th.M., Stability of functional equations in non-Archimedean spaces, Appl. anal. discrete math., 1, 325-334, (2007) · Zbl 1257.39019 [14] Schwaiger, J., Functional equations for homogeneous polynomials arising from multilinear mappings and their stability, Ann. math. sil., 8, 157-171, (1994) · Zbl 0820.39011 [15] Kaiser, Z., On stability of the Cauchy equation in normed spaces over fields with valuation, Publ. math. debrecen, 64, 189-200, (2004) · Zbl 1084.39026 [16] Kaiser, Z., On stability of the monomial functional equation in normed spaces over fields with valuation, J. math. anal. appl., 322, 1188-1198, (2006) · Zbl 1101.39018 [17] Cho, Y.J.; Park, Ch.; Saadati, R., Functional inequalities in non-Archimedean Banach spaces, Appl. math. lett., 23, 1238-1242, (2010) · Zbl 1203.39015 [18] Ciepliński, K., Stability of multi-additive mappings in non-Archimedean normed spaces, J. math. anal. appl., 373, 376-383, (2011) · Zbl 1204.39027 [19] Eshaghi Gordji, M.; Savadkouhi, M.B., Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta appl. math., 110, 1321-1329, (2010) · Zbl 1192.39018 [20] Mirmostafaee, A.K.; Moslehian, M.S., Stability of additive mappings in non-Archimedean fuzzy normed spaces, Fuzzy sets and systems, 160, 1643-1652, (2009) · Zbl 1187.46068 [21] Caˇdariu, L.; Radu, V., Fixed point methods for the generalized stability of functional equations in a single variable, Fixed point theory appl., (2008), Article ID 749392 · Zbl 1146.39040 [22] Jung, S.-M., A fixed point approach to the stability of isometries, J. math. anal. appl., 329, 879-890, (2007) · Zbl 1153.39309 [23] Jung, S.-M.; Kim, T.-S., A fixed point approach to the stability of the cubic functional equation, Bol. soc. mat. mexicana (3), 12, 51-57, (2006) · Zbl 1133.39028 [24] Jung, Y.-S.; Chang, I.-S., The stability of a cubic type functional equation with the fixed point alternative, J. math. anal. appl., 306, 752-760, (2005) · Zbl 1077.39026 [25] Mirzavaziri, M.; Moslehian, M.S., A fixed point approach to stability of a quadratic quation, Bull. braz. math. soc. (NS), 37, 361-376, (2006) · Zbl 1118.39015 [26] Baker, J.A., The stability of certain functional equations, Proc. amer. math. soc., 112, 729-732, (1991) · Zbl 0735.39004 [27] Kuczma, M., () [28] Kuczma, M.; Choczewski, B.; Ger, R., () [29] Brzdȩk, J.; Jung, S.-M., A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences, J. inequal. appl., (2010), Article ID 793947 · Zbl 1198.39036 [30] Brzdȩk, J.; Popa, D.; Xu, B., Hyers – ulam stability for linear equations of higher orders, Acta math. hungar., 120, 1-8, (2008) · Zbl 1174.39012 [31] Brzdȩk, J.; Popa, D.; Xu, B., Remarks on stability of linear recurrence of higher order, Appl. math. lett., 23, 1459-1463, (2010) · Zbl 1206.39019 [32] Brzdȩk, J.; Popa, D.; Xu, B., On nonstability of the linear recurrence of order one, J. math. anal. appl., 367, 146-153, (2010) · Zbl 1193.39006 [33] Forti, G.-L., Comments on the core of the direct method for proving hyers – ulam stability of functional equations, J. math. anal. appl., 295, 127-133, (2004) · Zbl 1052.39031 [34] Popa, D., Hyers – ulam stability of the linear recurrence with constant coefficients, Adv. difference equ., 101-107, (2005) · Zbl 1095.39024 [35] Sikorska, J., On a direct method for proving the hyers – ulam stability of functional equations, J. math. anal. appl., 372, 99-109, (2010) · Zbl 1198.39039 [36] Trif, T., On the stability of a general gamma-type functional equation, Publ. math. debrecen, 60, 47-61, (2002) · Zbl 1004.39023 [37] Trif, T., Hyers – ulam – rassias stability of a linear functional equation with constant coefficients, Nonlinear funct. anal. appl., 11, 881-889, (2006) · Zbl 1115.39028 [38] Hensel, K., Über eine neue begründung der theorie der algebraischen zahlen, Jahresber. Deutsch. math.-verein., 6, 83-88, (1899) · JFM 30.0096.03 [39] Gouvêa, F.Q., () [40] Robert, A.M., ()
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