## 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
(i)
$$F$$ has nonempty closed values and, for every $$x\in E$$, $$F(\cdot,x)$$ is measurable.
(ii)
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 $$t\in I,$$
$|x(t) - y(t)|\leq \frac{M}{1-ML}\int_{0}^{T} q(t)\,dt +\varepsilon.$

### MSC:

 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:

### References:

  Y. Liu, J. Wu, and Z. Li, Impulsive boundary value problems for Sturm-Liouville type differential inclusions, Journal of Systems Science & Complexity, 2007, 20: 370–380. · Zbl 1333.34040  A. F. Filippov, Classical solutions of differential equations with multivalued right hand side, SIAM J. Control, 1967, 5: 609–621. · Zbl 0238.34010  H. Covitz and S. B. Nadler Jr., Multivalued contraction mapping in generalized metric spaces, Israel J. Math., 1970, 8: 5–11. · Zbl 0192.59802  Z. Kannai and P. Tallos, Stability of solution sets of differential inclusions, Acta Sci. Math. (Szeged), 1995, 61: 197–207. · Zbl 0851.34015  P. Tallos, A Filippov-Gronwall type inequality in infinite dimensional space, Pure Math. Appl., 1994, 5: 355–362. · Zbl 0827.34008  A. Cernea, Some qualitative properties of the solution set of infinite horizon operational differential inclusions, Revue Roumaine Math. Pures Appl., 1998, 43: 317–328. · Zbl 1005.34052  A. Cernea, Existence for nonconvex integral inclusions via fixed points, Arch. Math (Brno), 2003, 39: 293–298. · Zbl 1113.45014  A. Cernea, An existence theorem for some nonconvex hyperbolic differential inclusions, Mathematica (Cluj), 2003, 45(68): 121–126. · Zbl 1084.34508  A. Cernea, An existence result for nonlinear integrodifferential inclusions, Comm. Applied Nonlin. Anal., 2007, 14: 17–24. · Zbl 1308.45007  A. Cernea, On the existence of solutions for a higher order differential inclusion without convexity, Electron. J. Qual. Theory Differ. Equ., 2007, 8: 1–8. · Zbl 1123.34046  T. C. Lim, On fixed point stability for set-valued contractive mappings with applications to generalized differential equations, J. Math. Anal. Appl., 1985, 110: 436–441. · Zbl 0593.47056  C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, LNM 580, Springer, Berlin, 1977. · Zbl 0346.46038  Y. Sun, B. L. Xu, and L. S. Liu, Positive solutions of singular boundary value problems for Sturm-Liouville equations, Journal of Systems Science & Complexity, 2005, 25: 69–77. · Zbl 1079.34015  Y. K. Chang and W. T. Li, Existence results for second order impulsive functional differential inclusions, J. Math. Anal. Appl., 2005, 301: 477–490. · Zbl 1067.34083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.