Abstract
AbstractIn the present paper, the Schrödinger type involutory differential equation is considered which is stated as $$ i\frac{dv(t)}{dt}+Av(t)+bAv(-t)=f(t),t\in I=(-\infty ,\infty ),v\left( 0\right) =\varphi $$in a Hilbert space H with a self-adjoint positive definite operator A. Here, operator approach enables us to apply the results on abstract problem on multi-dimensional or nonlocal problems which deserve a studious treatment. Throughout the paper, the main theorem on stability estimates for the solution of the abstract problem under the condition \(\left| b\right| <1\) is established. Furthermore, the main theorem is applied to a one-dimensional problem with nonlocal condition and involution and a multi-dimensional problem with Dirichlet and Neumann conditions on the boundary.KeywordsInvolutory differential equationStabilityHilbert spacePositive operator
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