Difference and Difference Quotient. Part II

Difference and Difference Quotient. Part II

The articles [8], [1], [4], [2], [3], [5], [7], [12], [13], [6], [9], and [10] provide the notation and terminology for this paper. We follow the rules: h, r, r1, r2, x0, x1, x2, x3, x4, x5, x, a, b, c, k denote real numbers and f , f1, f2 denote functions from R into R. Next we state a number of propositions: (1)1 ∆[f ](x, x+ h) = ( ~ ∆h[f ])(1)(x) h . (2) If h 6= 0, then ∆[f ](x, x+ h, x+ 2 ·...

Difference and Difference Quotient. Part IV

The papers [2], [7], [13], [3], [1], [6], [9], [4], [14], [8], [5], [15], [11], [12], and [10] provide the notation and terminology for this paper. We adopt the following rules: n denotes an element of N, h, k, x, x0, x1, x2, x3 denote real numbers, and f , g denote functions from R into R. Next we state a number of propositions: (1) If x0 > 0 and x1 > 0, then loge x0 − loge x1 = loge( x1 ). (2...

Difference and Difference Quotient. Part III

(3) (δh[f ])(x) = (∇h 2 [f ])(x)− (∇−h2 [f ])(x). (4) (~ ∆h[r f1 + f2])(n+ 1)(x) = r · (~ ∆h[f1])(n+ 1)(x) + (~ ∆h[f2])(n+ 1)(x). (5) (~ ∆h[f1 + r f2])(n+ 1)(x) = (~ ∆h[f1])(n+ 1)(x) + r · (~ ∆h[f2])(n+ 1)(x). (6) (~ ∆h[r1 f1 − r2 f2])(n+ 1)(x) = r1 · (~ ∆h[f1])(n+ 1)(x)− r2 · (~ ∆h[f2])(n+ 1)(x). (7) (~ ∆h[f ])(1) = ∆h[f ]. (8) (~ ∇h[r f1 + f2])(n+ 1)(x) = r · (~ ∇h[f1])(n+ 1)(x) + (~ ∇h[f2])(...

Making A Difference: Part One

Generalized Cauchy Difference Equations. Ii

The main result is an improvement of previous results on the equation f(x) + f(y)− f(x+ y) = g[φ(x) + φ(y)− φ(x+ y)] for a given function φ. We find its general solution assuming only continuous differentiability and local nonlinearity of φ. We also provide new results about the more general equation f(x) + f(y)− f(x+ y) = g(H(x, y)) for a given function H. Previous uniqueness results required ...