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A boundary value problem for a functional-differential equation with a linearly transformed argument. (English. Russian original) Zbl 0863.34070

Differ. Equations 31, No. 8, 1294-1299 (1995); translation from Differ. Uravn. 31, No. 8, 1348-1352 (1995).
The paper is concerned with the boundary value problem for the second order functional differential equation \[ (R(\alpha)u)''= f(x),\qquad x\in(0,1), \quad u(0)=u(1)=0, \] where the linear bounded operator \(R(\alpha):L_2(0,1)\to L_2(0,1)\) defined by the formula \[ (R(\alpha)u)(x)= u(x)+ \alpha u(x/2),\quad 0\neq\alpha\in \mathbb{C}, \] and \(f\in L_2(0,1)\). The generalized solution \(u(x)\) of this problem is defined as a function \(u\in\overset\circ W^1(0,1)\) satisfying the condition \(R(\alpha)u\in W^2(0,1)\) and \((R(\alpha)u)''=f\), where \(W^k(0,1)\), \(k=1,2,\) is the space of absolutely continuous functions with absolutely continuous \((k-1)\)st derivative and with the \(k\)th derivative in \(L_2(0,1)\), and \(\overset\circ W^1(0,1)=\{u\in w^1(0,1): u(0)=u(1)=0\}\). The main theorem proved states the existence of infinitely many generalized solutions \(u(x)\) to the above boundary value problem for \(|\alpha|> \sqrt{2}\), and its unique solvability for \(|\alpha|\leq \sqrt{2}\). Moreover, the authors establish that for some values of the parameter \(\alpha\) the kernel of the operator \(R(\alpha)\) is infinite-dimensional and the stated problem has nonsmooth solutions.

MSC:

34K10 Boundary value problems for functional-differential equations
47B38 Linear operators on function spaces (general)
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