Volterra-Hammerstein and the Hammerstein integral equations
The weak semilinear Cauchy problem for ordinary differential equations
Let us consider the Cauchy problem NEWLINE\[NEWLINEx'=f(t,x),\qquad x(0)=x_0\in E, \tag{1}NEWLINE\]NEWLINE where \(t\in I\subset \mathbb{R}\) is a compact interval, \(E\) is a real Banach space, \(B=\{x\in E:\|x-x_0\|\leq b\}\), \(b>0\) and \(f: I\times B\to E\) is a bounded weakly-weakly continuous function, i.e., continuous with respect to the weak topologies of \(E\) and \(B\). The derivative in \((1)\) is considered in the weak sense, i.e., it is defined in the usual way but in the sense of weak topology. NEWLINENEWLINENEWLINEIn 1971, A. Szép proved that, under the above assumptions, problem \((1)\) has a weak solution defined on an interval \(J\subset I\) if \(E\) is reflexive. In 1978, this result was extended by \textit{E. Cramer, V. Lakshmikantham} and \textit{A. R. Mitchell} [Nonlinear Anal., Theory, Methods Appl. 2, 169-177 (1978; Zbl 0379.34041)] to the case when \(E\) is a sequentially complete Banach space; the additional assumption imposed on \(f\) was formulated in terms of the measure of weak noncompactness introduced by \textit{S. F. de Blasi} [Bull. Math. Soc. Sci. Math. Répub. Soc. Roum., Nouv. Sér. 21(69), 259-262 (1977; Zbl 0365.46015)] in 1977. Kneser-type theorems, i.e., theorems which described the topological structure of weak solution sets to \((1)\) were proved by S. Szufla in 1978 for reflexive spaces and next by \textit{I. Kubiaczyk} and \textit{S. Szufla} [Publ. Inst. Math., Nouv. Sér. 32(46), 99-103 (1982; Zbl 0516.34058)] for nonreflexive ones. The existence and the topological structure of weak solutions to Volterra-Hammerstein and the Hammerstein integral equations in weakly sequentially complete Banach spaces have been investigated, e.g., by \textit{D. Bugajewski} [Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 34, 49-58 (1994; Zbl 0824.45014) and Commentat. Math. Univ. Carol. 35, No. 1, 35-41 (1994; Zbl 0816.45012)]. NEWLINENEWLINENEWLINEIn the reviewed paper, the author investigates the initial value problem NEWLINE\[NEWLINEx'(t)=A(t)x(t)+f(t,x(t)),\qquad x(0)=x_0\in E,\tag{2}NEWLINE\]NEWLINE where \(A\) is a continuous mapping from \(I\) into the Banach space of linear bounded operators from \(E\) into \(E\) with the usual norm. The main assumption imposed on \(f\) is a Cellina-type condition formulated in terms of the measure of weak noncompactness. Applying the method of proving due to the author [Analysis 16, No. 4, 347-364 (1996; Zbl 0866.47042)], the author proves that the set of weak solutions to (2) is nonvoid, compact and connected in a suitable function space without assuming the Banach space to be weakly sequentially complete.
scientific article; zbMATH DE number 1647681