A FUNCTIONAL EQUATION CHARACTERIZING CUBIC POLYNOMIALS AND ITS STABILITY

We study the generalized Hyers-Ulam stability of the functional equation f[x1,x2,x3]= h(x1+x2+x3). 2000 Mathematics Subject Classification. 39B22, 39B82.


Introduction. Given an operator T and a solution class
{u} with the property that T (u) = 0, when does T (v) ≤ ε for an ε > 0 imply that u − v ≤ δ(ε) for some u and for some δ > 0? This problem is called the stability of the functional transformation. A great deal of work has been done in connection with the ordinary and partial differential equations. If f is a function from a normed vector space into a Banach space, and f (x + y) − f (x)− f (y) ≤ ε, Hyers [3] proved that there exists an additive map A such that f (x)− A(x) ≤ ε. If f (x) is a real continuous function of x over R, and |f (x + y) − f (x) − f (y)| ≤ ε, it was shown by Hyers and Ulam [4] that there exists a constant k such that |f (x) − kx| ≤ 2ε. Taking these results into account, we say that the additive Cauchy equation f (x + y) = f (x) + f (y) is stable in the sense of Hyers and Ulam.
In this paper, we study a generalized Hyers-Ulam stability of a mean value type functional equation.
Let R be the set of real numbers. For distinct points x 1 ,x 2 ,...,x n in R, the divided difference of f : R → R is recursively defined as Then f is a polynomial of degree at most three and h is linear.
where a 0 and a 1 are arbitrary constants. This can be verified by direct substitution into the expansion of the functional equation (1.2), that is, Each term involving an a 0 or an a 1 has an opposite-sign term and therefore cancels by simple algebraic manipulation. Thus we have again the expanded form (2.1) of (1.2).
We are free to pick a 0 = g(0) so that g(x) yields f (0) = 0. In other words, by a suitable choice for a 0 , without loss of generality, we may assume that Now by setting x = α in the definition of g(x) we get Letting a 0 + a 1 α = g(α) we get f (α) = 0 and we may assume, without loss of generality, that f (α) = 0 (2.6) for some α ≠ 0 in R. Note that there are many choices for such an α.
Next, we substitute(x, 0,y) for ( Then (2.8) reduces to Note that (2.10) is valid even for x = y. Now we consider the equation Next, replace y by −y in (2.10) to get Again (2.13) holds if x + y = 0. Thus we conclude that (2.13) holds for x, y ∈ R \{0}. Subtract (2.10) from (2.13) and use (2.12) to get There are plenty of choices for such v. Let Letting (2.16) into (2.14), we get (Here note that v can be zero since x = y is allowed.) Hence for fixed u = u 1 , we get (2.20) Now using (2.20) in (2.7), we have Removing the assumption that f (0) = 0, we get By (2.4), (2.6), and (2.22), we conclude that f is a polynomial of degree at most three for all x ∈ R. This proof is now complete.
For a more general result, the interested reader should refer to Kannappan and Sahoo [5]. (1.2). Let G be an additive subgroup of C and let ϕ : G 3 → [0, ∞) be a control function. In the following theorem, the stability of (1.2) for cubic polynomials will be investigated in a modified form (3.1).

Stability of the functional equation
then there exist constants a, b, c, d such that Moreover, the constants a, b, c, d are explicitly given by (3.4) Proof. If we define a function g : G → C by then g(α) = g(−α) = 0 and g satisfies the inequality If we substitute (x, α, −α) for (x,y,z) in (3.6), then we have Replace z by −y in (3.6) to get for every x, y ∈ G. By making use of (3.7) and (3.8), we obtain or equivalently 2y (3.10) Multiply both sides by (3.12) (We note that the inequality holds true also for x ∈ {−α, α}.) If we replace y in the last inequality by a constant β ∈ G \ {−α, 0,α} and if we consider definition (3.5), then we can easily show the validity of inequality (3.2) by making a tedious calculation. By using (3.2), (3.5), and (3.7), we may obtain  (3.14) If the functions f ,h : G → C satisfy inequality (3.1) for any x, y, z ∈ G, then there exist uniquely determined constants a, b, c, d such that inequalities (3.2) and (3.3) are valid for all x ∈ G and for all x ∈ G \{−α, α}, respectively.

Corollary 3.3. Suppose that the control function
If the functions f ,h : G → C satisfy inequality (3.1) for any x, y, z ∈ G, then there exist constants a, b, c, d such that  ψ(x, y, z) ∀x, y, z ∈ G with x ≠ y, y ≠ z, z ≠ x, (3.17) then there exist constants a, b, c, d such that

19)
for all x ∈ G, where a, b, c, d are explicitly given in Theorem 3.1.
Proof. If we multiply both sides of (3.17) by |x −y||y −z||z −x|, then f satisfies inequality (3.1) with ϕ(x, y, z) = |x − y||y − z||z − x|ψ(x, y, z) ∀x, y, z ∈ G. (3.20) (We note that (3.1) is also true for x, y, z ∈ G with x = y, y = z, or z = x for our case with (3.20).) According to Theorem 3.1, there exist constants a, b, c, d such that inequalities (3.18) and (3.19) are valid for all x ∈ G and for all x ∈ G \ {−α, α}, respectively. The only reason for excepting −α and α from the domain of validity of inequality (3.3) is that the denominator of the first term on the right-hand side contains a factor |x 2 − α 2 |. However, inequality (3.19) contains no denominator which vanishes at x = α or x = −α. Therefore, we can include −α and α in the domain of validity of inequality (3.19), which completes the proof.

Call for Papers
Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both considerations have been followed by an increase in attention toward intermodal freight transportation research.
Various intermodal freight transport decision problems are in demand of mathematical models of supporting them. As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challenging opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport.
The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support decisions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators.
Topics of relevance to this type of decision-making both in time horizon as in terms of operators are: • Intermodal terminal design • Infrastructure network configuration • Location of terminals • Cooperation between drayage companies • Allocation of shippers/receivers to a terminal • Pricing strategies • Capacity levels of equipment and labour • Operational routines and lay-out structure • Redistribution of load units, railcars, barges, and so forth • Scheduling of trips or jobs • Allocation of capacity to jobs • Loading orders • Selection of routing and service Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/jamds/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/, according to the following timetable:

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