Abstract
This paper is mainly devoted to the study of one kind of nonlinear Schrödinger differential equations. Under the integrable boundary value condition, the existence and uniqueness of the solutions of this equation are discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.
Highlights
The nonlinear Schrödinger differential (NSD) equation is one of the most important inherently discrete models
× R+, Awhere ⊂ R, u = u(x, t) is the unknown solution maps × R+ into C, δa is the Dirac distribution at the point a ∈, namely, δa, v = v(a) for v ∈ H1( ), and q ∈ R represents
Such a distribution is introduced in order to model physically the defect at the point x = a
Summary
The nonlinear Schrödinger differential (NSD) equation is one of the most important inherently discrete models. D Proof Let a nonnegative real sequence {u(k)}k∈N ⊂ F such that {A(s, u(k))}k∈N is bounded E Lipschitz functions in Rn and. R It is obvious that the nonnegative real sequence {u(k)}k∈N is bounded in E, so there exists A a positive constant C4 such that (see [29]). There exists a unique solution (X, Y, Z, K) for the NSD equation (1). Let (u, K) = (X, Y, Z, K) and (u , K ) = (X , Y , Z , K ) be two solutions of the NSD equation (1). R which implies u = u in the space of MG2 (0, S) It follows from Lemma 2.2 that the NSD A equation has a unique solution, K = K.
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