A note on the topological structure of solution sets of Sturm-Liouville problems in Banach spaces

The authors prove that the set of solutions of the multivalued Sturm-Liouville problem $x''(t)\in F(t,x(t), x'(t))$ a.e. on $J=[a,b]$, $x\in X$; $c_1x(a)-d_1x'(a)=x_a$, $c_2x(b)-d_2x'(b)=x_b$, where $X$ is a Banach space, $F\colon J\times X^2\to 2^X\sbs\emptyset$, $d_i\in{0,1}$, $i=1,2$, $c_i=1$ if $d_i=0$, $i=1,2$, is a retract of $W^{2,1}(J,X)$ or $C^1(J,X)$ under some assumptions, where $W^{2,1}(J,X)$ is the Banach space of the functions from $J$ to $X$ with absolutely continuous first derivative and $C^1(J,X)$ is the Banach space of the continuously differentiable functions from $J$ to $X$.