On the Stability of Schrödinger Type Involutory Differential Equations

In the present paper, the Schrödinger type involutory differential equation is considered which is stated as $$ i\frac{dv(t)}{dt}+Av(t)+bAv(-t)=f(t),t\in I=(-\infty,\infty ),v\left(0\right) =\varphi $$idv(t)dt+Av(t)+bAv(-t)=f(t),t∈I=(-∞,∞),v(0)=φ 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 | b| < 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. © 2021, Springer Nature Switzerland AG.