6f(x + y) - 6f(x - y) + 4f(3y) = 3f(x + 2y) - 3f(x - 2y) + 9f(2y)6f(x + y) - 6f(x - y) + 4f(3y) = 3f(x + 2y) - 3f(x - 2y) + 9f(2y) 相似文献
5.
A comparative study of the functional equations f( x+ y) f( x– y)= f
2( x)– f
2( y), f( y){ f( x+ y)+ f( x– y)}= f( x) f(2 y) and f( x+ y)+ f( x– y)=2 f( x){1–2 f
2( y/2)} which characterise the sine function has been carried out. The zeros of the function f satisfying any one of the above equations play a vital role in the investigations. The relation of the equation f( x+ y)+ f( x– y)=2 f( x){1–2 f
2( y/2)} with D'Alembert's equation, f( x+ y)+ f( x– y)=2 f( x) f( y) and the sine-cosine equation g( x– y)= g( x) g( y) + f( x) f( y) has also been investigated. 相似文献
6.
Summary. Let f : ]0,¥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶ f( x + y) - f( x) - f( y) = f( x-1 + y-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶ f( xy) = f( x) + f( y) f(xy) = f(x) + f(y) ¶are equivalent to each other. 相似文献
7.
The aim of the paper is to deal with the following composite functional inequalities
f(f(x)-f(y)) £ f(x+y) + f(f(x-y)) -f(x) - f(y), f(f(x)-f(y)) £ f(f(x+y)) + f(x-y) -f(x) - f(y), f(f(x)-f(y)) £ f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\begin{gathered}f(f(x)-f(y)) \leq f(x+y) + f(f(x-y)) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(x-y) -f(x) - f(y), \hfill \\ f(f(x)-f(y)) \leq f(f(x+y)) + f(f(x-y)) -f(f(x)) - f(y),\end{gathered} 相似文献
8.
The stability problems of the exponential (functional) equation on a restricted domain will be investigated, and the results will be applied to the study of an asymptotic property of that equation. More precisely, the following asymptotic property is proved: Let X be a real (or complex) normed space. A mapping f : X → C is exponential if and only if f(x + y) - f(x)f(y) → 0 as ||x|| + ||y|| → ∞ under some suitable conditions. 相似文献
9.
In Geoffroy et al, Acceleration of convergence in Dontchev's iterative method for solving variational inclusions Serdica Math. J. 29 (2003), pp. 45–54] we showed the convergence of a cubic method for solving generalized equations of the form 0 ∈ f( x) +- G( x) where f is a function and G stands for a set-valued map. We investigate here the stability of such a method with respect to some perturbations. More precisely, we consider the perturbed equation y ∈ f( x) +- G( x) and we show that the pseudo-Lipschitzness of the map ( f +- G) −1 is closely tied to the uniformity of our method in the sense that the attraction region does not depend on small perturbations of the parameter y. Finally, we provide an enhanced version of the convergence theorem established by Geoffroy, et al. 相似文献
10.
Let X be a real inner product space of dimension greater than 2 and f be a real functional defined on X. Applying some ideas from the recent studies made on the alternative-conditional functional equation
(x, y) = 0 T f(x + y)2 = [f(x) + f(y)]2(x, y) = 0 \Rightarrow f(x + y)^2 = [f(x) + f(y)]^{2} 相似文献
11.
In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed
additive and quadratic functional equation
f(lx + y) + f(lx - y) = f(x + y) + f(x - y) + (l- 1)[(l+2)f(x) + lf(-x)],f(\lambda x + y) + f(\lambda x - y) = f(x + y) + f(x - y) + (\lambda - 1)[(\lambda +2)f(x) + \lambda f(-x)], 相似文献
12.
Summary. Let ( G, +) and ( H, +) be abelian groups such that the equation 2 u = v 2u = v is solvable in both G and H. It is shown that if f1, f2, f3, f4, : G × G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f1( x + t, y + s) + f2( x - t, y - s) = f3( x + s, y - t) + f4( x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f1, f2, f3, and f4 are given by f1 = w + h, f2 = w - h, f3 = w + k, f4 = w - k where w : G × G ? H w : G \times G \longrightarrow H is an arbitrary solution of f ( x + t, y + s) + f ( x - t, y - s) = f ( x + s, y - t) + f ( x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G × G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D y,t3g( x, y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D x,t3g( x, y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G . 相似文献
13.
In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed
additive-cubic functional equation
f(kx + y) + f(kx - y) = kf(x + y) + kf(x - y) + 2f(kx) - 2kf(x)f(kx + y) + f(kx - y) = kf(x + y) + kf(x - y) + 2f(kx) - 2kf(x) 相似文献
14.
The nonlinear hyperbolic equation ∂ 2u( x, y)/∂ x ∂ y + g( x, y) f( u( x, y)) = 0 with u( x, 0) = φ( x) and u(0, y) = Ψ( y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986). 相似文献
15.
We obtain the super stability of Cauchy's gamma-beta functional equation
16.
We solve the equation
f(x+g(y)) - f(y + g(y)) = f(x) - f(y)f(x+g(y)) - f(y + g(y)) = f(x) - f(y) 相似文献
17.
Let k( y) > 0, 𝓁( y) > 0 for y > 0, k(0) = 𝓁(0) = 0 and lim y → 0k( y)/𝓁( y) exists; then the equation L( u) ≔ k( y) uxx – ∂ y(𝓁( y) uy) + a( x, y) ux = f( x, y, u) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f( x, y, u) in G, u| AC = 0, where G is a simply connected domain in ℝ 2 with piecewise smooth boundary ∂ G = AB∪ AC∪ BC; AB = {( x, 0) : 0 ≤ x ≤ 1}, AC : x = F( y) = ∫ y0( k( t)/𝓁( t)) 1/2dt and BC : x = 1 – F( y) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f( x, y, u) satisfies Carathéodory condition and | f( x, y, u)| ≤ Q( x, y) + b| u| with Q ∈ L2( G), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption | f( x, y, u1) – f( x, y, u2| ≤ C| u1 – u2|, where C = const > 0. 相似文献
18.
Solutions are obtained for the boundary value problem, y
(n) + f( x, y) = 0, y
(i)(0) = y(1) = 0, 0 i n – 2, where f( x, y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. 相似文献
19.
Let μ be a probability measure on [− a, a], a > 0, and let x0ε[− a, a], f ε Cn([−2 a, 2 a]), n 0 even. Using moment methods we derive best upper bounds to ¦∫ −aa ([ f( x0 + y) + f( x0 − y)]/2) μ( dy) − f( x0)¦, leading to sharp inequalities that are attainable and involve the second modulus of continuity of f(n) or an upper bound of it. 相似文献
20.
A numerical estimate is obtained for the error associated with the Laplace approximation of the double integral I(λ) = ∝∝ D g( x, y) e−λf(x,y) dx dy, where D is a domain in
, λ is a large positive parameter, f( x, y) and g( x, y) are real-valued and sufficiently smooth, and ∝( x, y) has an absolute minimum in D. The use of the estimate is illustrated by applying it to two realistic examples. The method used here applies also to higher dimensional integrals. 相似文献
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