REMARKS ON ULAM STABILITY OF THE OPERATORIAL EQUATIONS

In this paper we present four types of Ulam stability for operatorial equations: Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. The relations of Ulam stability with the c-weakly Picard operators are also studied. Some examples and counterexamples are given.

[1]  I. Rus.  Gronwall Lemma Approach to the Hyers–Ulam–Rassias Stability of an Integral Equation , 2010 .

[2]  A. Petruşel.  MULTIVALUED WEAKLY PICARD OPERATORS AND APPLICATIONS : Dedicated to the memory of Shouro Kasahara , 2004 .

[3]  S. Czerwik,et al.  Functional equations and inequalities in several variables , 2002 .

[4]  T. Rassias.  On the stability of the linear mapping in Banach spaces , 1978 .

[5]  Dorian Popa,et al.  Hyers-Ulam stability of the linear recurrence with constant coefficients , 2005 .

[6]  K. Deimling.  Fixed Point Theory , 2008 .

[7]  T. Miura,et al.  HYERS-ULAM STABILITY OF A CLOSED OPERATOR IN A HILBERT SPACE , 2006 .

[8]  Soon-Mo Jung,et al.  HYERS-ULAM-RASSIAS STABILITY OF THE BANACH SPACE VALUED LINEAR DIFFERENTIAL EQUATIONS y′ = λy , 2004 .

[9]  A STABILITY RESULT IN FIXED POINT THEORY , 2005 .

[10]  KI,et al.  HYERS–ULAM STABILITY OF FIRST ORDER LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS , 2007 .

[11]  George Isac,et al.  Stability of Functional Equations in Several Variables , 1998 .

[12]  J. Brzdȩk,et al.  Hyers-Ulam stability for linear equations of higher orders , 2008 .

[13]  I. Rus.  WEAKLY PICARD OPERATORS AND APPLICATIONS , 2001 .

[14]  Viorel Radu,et al.  THE FIXED POINT ALTERNATIVE AND THE STABILITY OF FUNCTIONAL EQUATIONS , 2003 .

[15]  P. Gruber,et al.  Stability of isometries , 1978 .

[16]  P. Gǎvruţa.  On a Problem of G. Isac and Th. M. Rassias Concerning the Stability of Mappings , 2001 .

[17]  Z. Páles,et al.  Generalized stability of the Cauchy functional equation , 1998 .

[18]  Claudi Alsina,et al.  On Some Inequalities and Stability Results Related to the Exponential Function , 1998 .

[19]  Dorian Popa.  Hyers-Ulam-Rassias stability of a linear recurrence , 2005 .

[20]  Soon-Mo Jung,et al.  Generalized Hyers-Ulam stability of Riccati differential equation , 2008 .

[21]  Mingyong Xu,et al.  HYERS-ULAM-RASSIAS STABILITY OF A SYSTEM OF FIRST ORDER LINEAR RECURRENCES , 2007 .

[22]  Dorian Popa,et al.  Note on nonstability of the linear recurrence , 2006 .

[23]  Ioan A. Rus,et al.  ULAM STABILITY OF ORDINARY DIFFERENTIAL EQUATIONS , 2009 .

[24]  Adrian Petruşel,et al.  Data dependence of the fixed point set of some multivalued weakly Picard operators , 2003 .

[25]  Soon-Mo Jung.  A Fixed Point Approach to the Stability of a Volterra Integral Equation , 2007 .

[26]  Dorian Popa,et al.  The Hyers–Ulam stability of nonlinear recurrences , 2007 .

[27]  Soon-Mo Jung,et al.  Hyers–Ulam stability of a system of first order linear differential equations with constant coefficients☆ , 2006 .