On a boundary value problem for a Sturm-Liouville type differential inclusion. (English) Zbl 1210.34083

Considered the problem
\[ (p(t)x'(t))'\in F(t,x(t))\quad \text{for a.e. } t\in I=[0,T], \tag{1} \]
\[ \alpha x(0) -\beta\lim_{t\to 0+}p(t)x'(t)=0,\quad \gamma x(T) +\delta\lim_{t\to T-}p(t)x'(t)=0, \tag{2} \]
where \(F:I\times E\to P(E)\) is a set-valued map, \(E\) is a real separable Banach space, \(p:I\to (0,\infty)\) is continuous, \(\alpha,\beta, \gamma,\delta\) are nonnegative reals with
\[ \alpha\delta+\beta\gamma+\gamma\alpha\int_{0}^{T}p^{-1}(t)\,dt\neq 0. \]
Hypothesis. 6mm
\(F\) has nonempty closed values and, for every \(x\in E\), \(F(\cdot,x)\) is measurable.
There exists \(L\in L^{1}(I,E)\) such that, for almost all \(t\in I,F(t,\cdot)\) is \(L(t)\)-Lipschitz in the sense that \(d_{H}(F(t,x),F(t,y))\leq L(t)|x-y|\,\, \forall x,y \in E\) and \(d(0,F(t,0))\leq L(t)\) for a.e. \(t\in I.\)
Theorem. Assume that the Hypothesis is satisfied and \(ML<1.\) Let \(y(\cdot)\in AC_{p}^{1}(I,E)\) be such that there exists \(q(\cdot)\in L^{1}(I,E)\) with \(d((p(t)y(t))',F(t,y(t)))\leq q(t)\) a.e. \((I)\),
\[ \alpha y(0) -\beta\lim_{t\to 0+}p(t)y'(t)=0,\quad \gamma y(T) +\delta\lim_{t\to T-}p(t)y'(t)=0. \]
Then, for every \(\varepsilon >0\), there exists \(x(\cdot)\) a solution of (1), (2) satisfying for all \(t\in I,\)
\[ |x(t) - y(t)|\leq \frac{M}{1-ML}\int_{0}^{T} q(t)\,dt +\varepsilon. \]


34G25 Evolution inclusions
34B15 Nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


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