GLASNIK ",IATE",IATICKI Vol. 34(54)(1999),243- 252 SOLUTION OF THE ULAM STABILITY PROBLEM FOR QUARTIC MAPPINGS

ABSTRACT. In 1940S. M. Ulam proposedat the Universityof Wis-consintheproblem:"Giveconditionsin orderfor a linearmappingnearan approximatelylinearmappingto exist. " In 1968S. Jvl. Ulam proposedthe generalproblem:"When is it truethat by changinga little thehypotheses of a theoremone can still assertthat the thesisof the theoremremains true or approximatelytrue? " In 1978P. 1\1.Gruber proposedthe Ulam typeproblem:"Supposea mathematicalobjectsatisfiesa certainproperty approximately.Is it then possibleto approximatethis objectby objects, satisfyingthe propertyexactly?"Accordingto P. lvl. Gruber this kind of stability problemsis of particularinterestin probabilitytheoryand in the caseof functionalequationsof differenttypes. In 1982-1998wesolvedthe aboveUlam problem,or equivalentlythe Ulam type problemfor linear mappingsand alsoestablishedanalogousstability problemsfor quadratic andcubicmappings.In this paperweintroducethenewquarticmappings F: X---t Y, satisfyingthe newquarticfunctionalequation F(XI +2X2) +F(XI- 2X2) +6F(Xl) "" 4 [F(XI +X2) +F(XI- X2) +6F(X2)] for a1l2-dimensionalvectors(Xl,X2) E X2, with X a linearspace(Y: " " a real completelinearspace),andthensolvethe Ulam stabilityproblemfor the abovemappingsF. 1. QUARTIC FUNCTIONAL EQUATION DEFINITION 1.1. Let X be a linear spaceand let Y be a real complete linear space. Then a mappingF: X--t Y, is calledquartic, if thenew quartic 1991MathematicsSubjectClassification.39B. Key words and phrases. Ulam problem,Ulam type problem, quartic mappings, quartic functionalequation,quartic functionalinequality,approximatelyquartic, stability problem