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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.

34G20Nonlinear ODE in abstract spaces
Full Text: DOI
[1] Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. math. Anal. appl. 162, No. 2, 494-505 (1991) · Zbl 0748.34040
[2] Byszewski, L.: Existence and uniqueness of mild and classical solutions of semilinear functional-differential evolution nonlocal Cauchy problem, selected problems of mathematics, Cracow university of technology. Anniversary issue 6, 25-33 (1995)
[3] L. Byszewski, Differential and Functional-Differential Problems with Nonlocal Conditions, Cracow University of Technology, Monograph 184, Cracow 1995. · Zbl 0890.35163
[4] Byszewski, L.; Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Applicable analysis 40, 11-19 (1990) · Zbl 0694.34001
[5] Byszewski, L.: Existence, uniqueness and asymptotic stability of solutions of abstract nonlocal Cauchy problems. Dynamic systems and applications 5, 595-606 (1996) · Zbl 0869.47034
[6] Byszewski, L.: On weak solutions of functional - differential abstract nonlocal Cauchy problem. Annales polonici mathematici 65, 163-170 (1997) · Zbl 0874.47026
[7] Balachandran, K.; Ilamaran, S.: Existence and uniqueness of mild and strong solutions of a semilinear evolution equation with nonlocal conditions, indian. J. pure appl. Math. 25, No. 4, 411-418 (1994) · Zbl 0808.47047
[8] Lin, Y.; Liu, J. H.: Semilinear integrodifferential equations with nonlocal Cauchy problem. Nonlinear analysis 26, No. 5, 1023-1033 (1996) · Zbl 0916.45014
[9] T. Winiarska, Differential Equations with Parameter, Cracow University of Technology, Monograph 68, Cracow, 1988. · Zbl 0669.35009
[10] Pazy, A.: Semigroups of linear operators and applications to partial differential equations. (1983) · Zbl 0516.47023