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The problem about periodic solutions for the family of linear differential equation $$ L u\equiv \left(\frac{\partial}{i\partial t} - a\Delta \right)u(x,t)=\nu G(u-f)$$ is considered on the multidimensional sphere $x \in S^n$ under the periodicity condition $u|_{t=0}=u|_{t=b}$ and $\int_{S^n}u(x,t)dx=0.$ Here $a$ is given real, $\nu$ is a fixed complex number, $ G u(x,t) $ is a linear integral operator, and $\Delta$ is the Laplace operator on $S^n.$ It is shown that the set of parameters $(\nu, b)$ for which the above problem admits a unique solution is a measurable set of full measure in ${\Bbb C}\times {\Bbb R}^+.$}. | VNU Journal of Science Mathematics - Physics 25 2009 169-177 On the set of periods for periodic solusions of some linear differential equations on the multidimensional sphere sn Dang Khanh Hoi Hoa Binh University 216 Nguyen Trai Thanh Xuan Hanoi Vietnam Received 3 August 2009 Abstract. The problem about periodic solutions for the family of linear differential equation d . Lu I -Ỳgị aA I u x i vG u f is considered on the multidimensional sphere X G sn under the periodicity condition u 11 0 u t b and fs u x t dx 0. Here a is given real V is a fixed complex number t is a linear integral operator and A is the Laplace operator on sn. It is shown that the set of parameters ỉ b for which the above problem admits a unique solution is a measurable set of full measure in c X R . This work further develops part of the authors result in 1 2 on the problem on the periodic solution to the equation L X u vG u f . Here L is Schrodinger operator on sphere sn and À belongs to the spectrum of L. Particularly the authors consider the case that À is an eigenvalue of L the case which can be always converted to the case À 0 . It is shown that the main results are all right but on the complement of eigenspace of À in the domain of L. 1. We consider the problem on periodic solutions for the nonlocal Schrodinger type equation Q aA u x t J G u 1 with these conditions u t o u x t dx 0. 2 Jsn Here u x t - is a complex function on sn X 0 b sn - is the multidimensional sphere n 2 a ỹỂ 0 V - are given complex numbers f x t - is a given function. The change of variables t br reduces our problem to a problem with a fixed period but with a new equation in which the coefficient of the T derivative is equal to 7 b 1 d 6Ỡ---- T1 vGÍuÍUL br f xi br . 2. Thus problem 1 2 turns into the problem on periodic solution of the equation 1 Ổ . Lu I . 7T- aA I u x i vG u x t f x t 3 E-mail dangkhanhhoi@yahoo.com 169 170 D.K. Hoi VNU Journal of Science Mathematics - Physics 25 2008 169-177 with the following .
