Abstract
Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m + n) = f(m, u(m), u(m + 1), …, u(m + n − 1)), m ∈ ℤ.
Highlights
Let Z denote the integers, and given a < b in Z, let [a, oo) {a,a + }, [a,b] {a,a +,b}, [a,b) {a,...,b- 1}, with (a, oo),(a,b), etc., being defined
Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m + n) /(, u(.), u(. + 1) u(, + n- 1)), Z
We will be concerned with solutions of the nth order difference equation, u(m + n) f(m,u(m),...,u(m + n- I)), (1.1)
Summary
Let Z denote the integers, and given a < b in Z, let [a, oo) {a,a + }, [a,b] {a,a +,b}, [a,b) {a,...,b- 1}, with (a, oo),(a,b), etc., being defined. Conditions are given for the continuity and differentiability of solutions of initial value problems and boundary value problems for the nth order finite difference equation, u(m + n) /(-, u(.), u(. We will present results about continuous dependence and differentiation of solutions of (1.1) with respect to initial values and certain boundary values. In the case of initiM value problems for ordinary differential equations, Hzrtm [22] presents & theorem due to Pc&no in which solutions are differentiated with respect to initial conditions.
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