Semicontinuity of an integral functional in a Banach space
In the paper under review the author studies lower semicontinuity of an integral functional \(J_f :(x, u) \to \int_T f(x(t),u(t))dt\) that arises in various minimization problems. It is assumed that \(T\) equals \([0,1]\) and \(x\), \(u\) are elements in the space \(L_1(T,X)\) of all Bochner integrable functions taking values in a separable Banach space \(X\). Let a function \(f: X\times Y\to {\mathbb R}^+\cup\{{\infty}\}\) satisfy the following conditions: (a) the mapping \(x\to f^*(x,x^*)\) is upper semicontinuous for all \(x^*\) in the dual space \(X^*\) (\(f^*\) is the Fenchel-Moreau conjugate to f); (b) the mapping \(u\to f(x,u)\) is convex and lower semicontinuous. The author calls a sequence \(\{u_n\}\subset L_1(T,X)\) \(a\)-weakly convergent to \(u\in L_1(T,X)\) if it is bounded in \(L_1(T,X)\) and \(\int_E\langle x^*,u_n(t)\rangle \to \int_E\langle x^*,u(t)\rangle\) for all \(x^*\) in \(X^*\) and for all measurable subsets \(E\) of \([0,1]\). The main theorem is as follows. If \(x_n\) strongly converges to \(x\) and \(u_n\) \(a\)-weakly converges to \(u\), then \(J_f(x,u)\leq \liminf\limits_{n\to\infty} J_f(x_n,u_n)\). In this case, the functional \(J_f\) is called sequentially strong-\(a\)-weak lower semicontinuous. Sequential lower semicontinuity of \(J_f\) in the classical strong-weak definition is implied by the author's theorem (see, for example, \textit{C. Olech} [Bull. Acad. Polon. Sci., Sér. Sci. Math. Astron. Phys. 25, 135-142 (1977; Zbl 0395.46026)]; \textit{G. Bottaro} and \textit{P. Oppezzi} [Boll. Unione Mat. Ital., V. Ser. B 17, 1290-1307 (1980; Zbl 0451.46029)]; \textit{C. Castaing} and \textit{P. Clauzure} [Acta Math. Vietnam. 7, No. 2, 139-170 (1984; Zbl 0557.49005)]).
scientific article; zbMATH DE number 1037969