Exponential-cosine operator-valued functional equation in the strong operator topology (Q762749)

Let X be a Banach space and let B(X) denote the family of bounded linear operators on X. Let \(R^+=[0,\infty)\). A one parameter family of operators \(\{\) S(t); \(t\in R^+\}\), \(S: R^+\to B(X),\) is called exponential-cosine operator function if \(S(0)=1\) and \(S(s+t)- 2S(s)S(t)=(S(2s)-2S^ 2(s))S(t-s),\) for all \(s,t\in R^+\), \(s\leq t\). Let \(Af=\lim_{h\to 0}\frac{S(h)f-f}{h},\) \(f\in D(A)\), and \(Bf=\lim_{h\to 0}\frac{S(2h)f-2S(h)f+f}{h^ 2},\) \(f\in D(B)\). It is shown that for a strongly continuous exponential-cosine operator function \(\{\) S(t)\(\}\), \(f\in D(A^ 2)\) implies \(\int^{t}_{0}(t-u)S(u)fdu\in D(B)\) and \[ B\int^{t}_{0}(t-u)S(u)fdu=S(t)f-f+tAf- 2A\int^{t}_{0}S(u)fdu+2A^ 2\int^{t}_{0}(t-u)S(u)fdu. \] D(B) is seen to be dense in \(D(A^ 2)\). Some regularity properties of S(t) have also been obtained.