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Fractional Differential Equations

In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

On the Stability of Functional Equations and a Problem of Ulam

On the Stability of Functional Equations in Banach Spaces

The paper is a survey about Hyers—Ulam stability of functional equations and systems in several variables.

Hyers—Ulam stability of functional equations in several variables

On approximation of approximately linear mappings by linear mappings

Solution of a problem of Ulam

In 1940 S. M. Ulam proposed at the University of Wisconsin the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist." In 1968 S. Jvl. Ulam proposed the general problem: "When is it true that by changing a little the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true?" In 1978 P. 1\1. Gruber proposed the Ulam type problem: "Suppose a mathematical object satisfies a certain property approximately. Is it then possible to approximate this object by objects, satisfying the property exactly?" According to P. lvl. Gruber this kind of stability problems is of particular interest in probability theory and in the case of functional equations of different types. In 1982-1998 we solved the above Ulam problem, or equivalently the Ulam type problem for linear mappings and also established analogous stability problems for quadratic and cubic mappings. In this paper we introduce the new quartic mappings F : X ---t Y , satisfying the new quartic functional equation F(XI + 2X2) + F(XI 2X2) + 6F(Xl) "" 4 [F(XI + X2) + F(XI X2) + 6F(X2)] for a1l2-dimensional vectors (Xl,X2) E X2, with X a linear space (Y: "" a real complete linear space), and then solve the Ulam stability problem for the above mappings F. 1. QUARTIC FUNCTIONAL EQUATION DEFINITION 1.1. Let X be a linear space and let Y be a real complete linear space. Then a mapping F : X --t Y, is called quartic, if the new quartic 1991 Mathematics Subject Classification. 39B.

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Solution of the Ulam stability problem for quartic mappings

In this paper, J. M. Rassias introduces the general Euler Lagrange type functional equation of the form.

/pdf/ulam-stability-for-the-orthogonally-general-euler-lagrange-2zy974frbc.pdf

John Michael Rassias

Papers

On approximation of approximately linear mappings by linear mappings

Solution of a problem of Ulam

Solution of the Ulam stability problem for quartic mappings

Ulam Stability for the Orthogonally General Euler - Lagrange Type Functional Equation

On the stability of the euler-lagrange functional equation