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X∈V Further, according to Corollary 1, h(x) = a(x) + S(x, x) + c x∈V with some c ∈ W , an additive a : V → W , and a biadditive and symmetric S : V 2 → W . Clearly, by Lemma 4, L = S and a(x) = 0 for every x ∈ V . Hence h(x) = L(x, x)+c x ∈ V. , go (x) := g(x) − g( − x) 2 x ∈ V, g(x) + g( − x) x ∈ V. 2 The next theorem is the main result in this chapter. ge (x) := Theorem 6 Let (V , +) be a commutative group, p : V → V be a homomorphism such that p(V ) = p(V ), W be a Banach space, ε ≥ 0 and g : V → W satisfy inequality (25).

The segment is defined by equations: |x| ≤ a,y = 0. The coordinate systems are related by equations x = (x − cx ) cos (β) + (y − cy ) sin (β), y = −(x − cx ) sin (β) + (y − cy ) cos (β). (71) The stresses in the global coordinates have the form σx = σx cos2 (β) − 2τxy sin (β) cos (β) + σy sin2 (β), σy = σx sin2 (β) + 2τxy sin (β) cos (β) + σy cos2 (β), τxy = (σx − σy ) sin (β) cos (β) + τxy cos (2β). (72) Moreover, we have ) 2 (j ) σx(j ) = PX0 (I A(j x (x, y) cos (β) − 2I Axy (x, y) cos (β) sin (β) + ) 2 (j ) 2 I A(j y (x, y) sin (β)) + PY0 (I Bx (x, y) cos (β) − (j ) 2I Bxy (x, y) cos (β) sin (β) + I By(j ) (x, y) sin2 (β)), ) 2 (j ) σy(j ) = PX0 (I A(j x (x, y) sin (β) + 2I Axy (x, y) cos (β) sin (β) + ) 2 (j ) 2 I A(j y (x, y) cos (β)) + PY0 (I Bx (x, y) sin (β) + (j ) 2I Bxy (x, y) cos (β) sin (β) + I By(j ) (x, y) cos2 (β)), (j ) ) (j ) τxy = PX0 ((I A(j x (x, y) − I Ay (x, y)) sin (β) cos (β) + ) 2 2 (j ) I A(j xy (x, y)( cos (β) − sin (β))) + PY0 ((I Bx (x, y) − (j ) I By(j ) (x, y)) sin (β) cos (β) + I Bxy (x, y)( cos2 (β) − sin2 (β))).

Sci. Can. 13, 274–278 (1991) A Note on the Functions that Are Approximately p-Wright Affine 55 15. : On the stability of functional equations. Aequ. Math. 77, 33–88 (2009) 16. : Stability of homomorphisms and generalized derivations on Banach algebras. J. Inequal. Appl. 2009, 1–12 (2009) 17. : On approximately Jensen-convex and Wright-convex functions. C. R. Math. Rep. Acad. Sci. Can. 23, 141–147 (2001) 18. A. ): Nonlinear Analysis and Variational Problems (In Honor of George Isac). Springer Optimization and its Applications, vol.

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