Solutions of self-differential functional equations.
Y Dimitrov, GA Edgar - 2006 - projecteuclid.org
2006•projecteuclid.org
The system of functional differential equations (1) has a continuously differentiable solution
for every value of the parameter a. The boundary values and a are related with d(2-
a)=c(2+a). When a∈S where S=\left {2^ 2n+ 1: n= 1, 2, 3, ...\right\}, the system (1) has
infinitely many solutions with boundary values c=0 and d=0. For all other values of a, the
system equation1_1 has a unique solution. 1 equation1_1\left {F^ ′ (x)= a F (2x) & if\: 0 ≦ x
≦ 1 2\F^ ′ (x)= a F (2-2x) & if\: 1 2 ≦ x ≦ 1\F (0)= c, F (1)= d. &.
for every value of the parameter a. The boundary values and a are related with d(2-
a)=c(2+a). When a∈S where S=\left {2^ 2n+ 1: n= 1, 2, 3, ...\right\}, the system (1) has
infinitely many solutions with boundary values c=0 and d=0. For all other values of a, the
system equation1_1 has a unique solution. 1 equation1_1\left {F^ ′ (x)= a F (2x) & if\: 0 ≦ x
≦ 1 2\F^ ′ (x)= a F (2-2x) & if\: 1 2 ≦ x ≦ 1\F (0)= c, F (1)= d. &.
Abstract
The system of functional differential equations (1) has a continuously differentiable solution for every value of the parameter . The boundary values and are related with . When where the system (1) has infinitely many solutions with boundary values and . For all other values of , the system \eqref{equation1_1} has a unique solution. \begin{equation} \tag{}\label{equation1_1} \left \{ \begin{array} {l l } F^{\prime}(x)=a F(2x) & \mathrm{ if} \: 0\leq x\leq \dfrac{1}{2} \\ F^{\prime}(x)=a F(2-2x) & \mathrm{ if} \: \dfrac{1}{2} \leq x\leq 1 \\ F(0)=c, F(1)=d. & \end{array} \right. \end{equation}
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