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### A note on existence of viable solutions for a third order differential inclusion

#### Abstract

We prove the existence of viable solutions for the problem

$x'''\in F(t,x,x',x'')+G(x,x',x'')$, $x(0)=x_0$, $x'(0)=x^1_0$,

$x''(0)=x^2_0$, $x''(t)\in K$, where $K\subset {\mathbf{R}}^n$ is

a locally compact set, $F$ is a continuous set-valued map and $G$

is an upper semicontinuous set-valued map contained in the

Fr\'{e}chet subdifferential of a $\phi $- convex function of order two.

$x'''\in F(t,x,x',x'')+G(x,x',x'')$, $x(0)=x_0$, $x'(0)=x^1_0$,

$x''(0)=x^2_0$, $x''(t)\in K$, where $K\subset {\mathbf{R}}^n$ is

a locally compact set, $F$ is a continuous set-valued map and $G$

is an upper semicontinuous set-valued map contained in the

Fr\'{e}chet subdifferential of a $\phi $- convex function of order two.

#### Keywords

Viable solutions; $\phi $-convex functions of order two; Nonconvex differential inclusions

#### Full Text:

PDFDOI: http://dx.doi.org/10.14510%2Flm-ns.v32i2.18