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Application of properties of the right-hand sides of evolution equations to an investigation of nonlocal evolution problems. (English) Zbl 0933.34064
The author considers nonlocal Cauchy problems in the abstract setting of evolution equations. Two types of equations are dealt with: $$u'(t)+Au(t)=f_1(t,u(t)),\ t_0 < t\le t_0+a, \quad \text{with}\quad u(t_0)+\sum_{k=1}^pc_ku(t_k)=u_0;$$ $$u'(t)+Au(t)=f_2(t,u(t),u(b(t))),\ t_0 < t\le t_0+a, \quad \text{with}\quad u(t_0)+\sum_{k=1}^pc_ku(t_k)=u_0;$$ where $-A$ is the infinitesimal generator of a $C_0$-semigroup, $a>0$, the points $t_k$ lie in the interval $(t_0,t_0+a]$, $c_k\not =0$ and the functions $f_1$, $f_2$ and $b$ satisfy certain conditions. The appropriate concept of a mild solution is introduced and theorems on existence and uniqueness of mild solutions are obtained. Moreover, conditions for the existence of classical solutions are provided. The methods of proof are based upon the general theory of evolution equations and semigroups.

##### MSC:
 34G20 Nonlinear ODE in abstract spaces
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##### References:
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