A fixed point theorem of Leggett-Williams type with applications to single- and multivalued equations (Q2726694)

In [\textit{M. Meehan} and \textit{D. O'Regan}, Appl. Anal. 74, No. 3-4, 413-427 (2000; Zbl 1021.45007)] the authors studied the existence of nonnegative solutions to the integral equation NEWLINE\[NEWLINEy(t)= h(t)+ \int^1_0 k(t,s) f(s,y(s)) ds\quad\text{for }t\in [0,1].\tag{1}NEWLINE\]NEWLINE In this paper the authors strengthen the conditions of the nonlinearity \(f\) and weaken the conditions on the kernel \(k\), and again establish the existence of a nonnegative solution to (1). Also, the authors establish the existence to general discrete and multivalued equations. To establish the existence the authors use the fixed point theorem which was established by \textit{R. W. Leggett} and \textit{L. R. Williams} [J. Math. Anal. Appl. 76, 91-97 (1980; Zbl 0448.47044)]. The authors establish a general fixed point theorem for multivalued maps defined on cones in Banach spaces which is a multivalued version of the Leggett-Williams theorem.