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On the solution sets of four-point boundary value problems for nonconvex differential inclusions. (English) Zbl 1220.34021
Consider the problem
\[ \ddot{u}(t) \in F(t,u(t),\dot{u}(t)),\quad u(0) = x_{0},\quad u(\eta)= u(\theta)=u(T),\quad \text{a.e. on }I, \tag{1} \]
where \(I= [0,T]\), \(0 < \eta < \theta < T\), \(F\) is a multifunction from \(I\times \mathbb R^{n}\times \mathbb R^{n}\) to nonempty compact subsets of \(\mathbb R^{n}\).
Theorem. Let \(F\) be a multifunction satisfying the following conditions: 8mm
(i)
for each \((x,y) \in \mathbb R^{n}\times \mathbb R^{n}\), the multifunction \(F(\cdot,x,y)\) is measurable;
(ii)
for each \(t\in I\) a.e., the multifunction \((x,y)\to F(t,x,y)\) is continuous;
(iii)
for each \((t,x,y)\in I\times \mathbb R^{n}\times \mathbb R^{n}\), \[ \|F(t,x,y)\|=\sup\{\|v\|: v\in F(t,x,y)\}\leq a(t) + c_{1}(t)\|x\| +c_{2}(t)\|y\| \]
with \(a,c_{1}, c_{2}\in L^{p}(I,\mathbb R^{+})\), \(2\leq p<\infty;\)
(iv)
the spectral radius \(r(L) < 1,\) where \(L:C(I,\mathbb R^{n})\times C(I,\mathbb R^{n})\to C(I,\mathbb R^{n})\times C(I,\mathbb R^{n})\),
\(L(f,g)=(L_{1}(f,g), L_{2}(f,g))\) with
\[ L_{1}(f,g) =\int_{0}^{T}|G(t,\tau)|(c_{1}(\tau)f(\tau) +c_{2}(\tau)g(\tau))\,d\tau, \]
\[ L_{2}(f,g) =\int_{0}^{T}|\partial{G(t,\tau)}/\partial{\tau}|(c_{1}(\tau)f(\tau) +c_{2}(\tau)g(\tau))\,d\tau, \]
where \(G\) is the Green function.
Then problem (1) admits a solution.

MSC:
34A60 Ordinary differential inclusions
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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