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A necessary and sufficient condition for to existence of solutions the differential inclusions. (English) Zbl 0910.34029
The author gives necessary and sufficient conditions for the existence of a solution to scalar differential inclusions $x'\in F(x)$, $x\in\bbfR$. The proof is based on a selection of a single-valued function $f(x)\in F(x)$ which satisfies a necessary and sufficient condition for the existence of a solution to the differential equation $x'=f(x)$ given by {\it P. Binding} [J. Differ. Equations, 31, 183-199 (1979; Zbl 0363.34001)].

##### MSC:
 34A60 Differential inclusions
Full Text:
##### References:
 [1] Bressan, A.: Directionally continuous selections and differential inclusions. Funkcialaj ekvacioj 31, 459-470 (1988) · Zbl 0676.34014 [2] Bressan, A.: On the qualitative theory of lower semicontinuous differential inclusions. Journal of differential equations 77, 379-391 (1989) · Zbl 0675.34011 [3] Bressan, A.: Upper and lower semicontinuous differential inclusions. A unified approach. Controllability and optimal control (1990) · Zbl 0704.49011 [4] Bressan, A.: Selections of Lipschitz multifunctions generating a continuous flow. Differential and integral equations 4, No. 3, 483-490 (1991) · Zbl 0722.34009 [5] Bressan, A.; Cortesi, A.: Directionally continuous selections in Banach spaces. Nonlinear analysis, theory, methods and applications 13, No. 8, 987-992 (1989) · Zbl 0687.34013 [6] Cambini, A.; Querci, S.: Equazioni differenziali del primo ordine con secondo membro discontinuo rispetto all’incognita. Rend. ist. Mat. univ. Trieste 1, 89-97 (1969) · Zbl 0193.04203 [7] Colombo, G.: Weak flow-invariance for nonconvex differential inclusions. Differential and integral equations 5, No. 1, 173-180 (1992) · Zbl 0757.34017 [8] Pucci, A.: Sistemi di equazioni differenziali con secondo membro discontinuo rispetto all’incognita. Rend. ist. Mat. univ. Trieste 3, 75-80 (1971) · Zbl 0238.34008 [9] Binding, P.: The differential equation x \dot{} = f $\circ$x. Journal of differential equations 31, 183-199 (1979) · Zbl 0363.34001 [10] Coddington, E. A.; Levinson, N.: Theory of ordinary differential equations. (1955) · Zbl 0064.33002 [11] Aubin, J. P.; Cellina, A.: Differential inclusions. (1984) · Zbl 0538.34007 [12] Jarnik, J.: Multivalued mappings and Filippov’s operation. Czechoslovak math. J. 31, 275-288 (1981) · Zbl 0473.34018