Existence and uniqueness of solutions for the Schrödinger integrable boundary value problem

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.


Introduction
The nonlinear Schrödinger differential (NSD) equation is one of the most important inherently discrete models. NSD equations play a crucial role in the modeling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology [1][2][3][4]. For example, they have been successfully applied to the modeling of localized pulse propagation optical fibers and wave guides, to the study of energy relaxation in solids, to the behavior of amorphous material, to the modeling of self-trapping of vibrational energy in proteins or studies related to the denaturation of the NSD double strand [5].
In 1961, Gross considered a NSD equation with Dirac distribution defect (see [6]), iu t + 1 2 u xx + qδ a u + g |u| 2 u = 0 in × R + , where ⊂ R, u = u(x, t) is the unknown solution maps × R + into C, δ a is the Dirac distribution at the point a ∈ , namely, δ a , v = v(a) for v ∈ H 1 ( ), and q ∈ R represents its intensity parameter. Such a distribution is introduced in order to model physically the defect at the point x = a (see [7]). The function g represents a generalization of the classical nonlinear Schrödinger equation (see for example [8]). As for other contributions to the analysis of nonlinear Schrödinger equations, we refer to Refs. [9][10][11][12] and the references therein.
In this paper, we consider the following NSD equation: where 0 ≤ s ≤ S and B is the quadratic variation of the Brownian motion B.
It is worth mentioning that (1) comes from an expansion of the Feynman path integral from Brownian-like to Lévy-like quantum mechanical paths (see [13] for details). When the coefficients b, h and σ are constants in (1), the Lévy dynamics becomes the Brownian dynamics, and (1) reduces to the classical stochastic differential equation under standard Lipschitz conditions on f (s, y, z), g(s, y, z) in y, z and the L p G ( S ) (p > 1) integrability condition on ξ . The solution (Y, Z, K) is universally defined in the space of the Schrödinger framework, in which the processes have a strong regularity property. It should be noted that K is a decreasing Schrödinger martingale.
It is well known that classical stochastic differential equations are encountered when one applies the stochastic maximum principle to optimal stochastic control problems. Such equations are also encountered in the probabilistic interpretation of a general type of systems quasilinear PDEs, as well as in finance (see [13][14][15] for details).
The rest of this paper is organized as follows. In Sect. 2, we introduce some notions and results. In Sect. 3, the main results and their proofs are presented.

Preliminaries
In this section, we introduce some notations and preliminary results in Schrödinger framework which are needed in the following section. More details can be found in [16][17][18][19].
Let S = C 0 ([0, S]; R), the space of real valued continuous functions on [0, S] with w 0 = 0, be endowed with the distance (see [20]) and let B s (w) = w s be the canonical process. Denote by F := {F s } 0≤s≤S the natural filtration generated by B s , let L 0 ( S ) be the space of all F-measurable real functions. Let where C b,L ip (R n ) denotes the set of bounded Lipschitz functions in R n (see [21]).
In the sequel, we will work under the following assumptions.
(H3) For u 1 , u 2 ∈ R 3 , there exists a positive constant C 2 such that is called a sublinear expectation space and E is called a sublinear expectation.
is a viscosity solution to the following PDE: and a ∈ R.
Definition 2.2 (see [23]) We call a sublinear expectationÊ : We can also define the conditional Schrödinger expectationÊ s of ξ ∈ L ip ( S ) knowing L ip ( t) for t ∈ [0, S]. Without loss of generality, we can assume that ξ has the representation with t = s i , for some 1 ≤ i ≤ n, and we put E is a continuous mapping on L ip ( S ) endowed with the norm · 1,G . Therefore, it can be extended continuously to L 1 G ( S ) under the norm X 1,G . Next, we introduce the Itô integral of Schrödinger Brownian motion. Let M 0 G (0, S) be the collection of processes in the following form: for a given partition where ξ k ∈ L ip ( tk ) and k = 0, 1, . . . , N -1 are given. For respectively. It is easy to see that As in [24], for each η ∈ H p G (0, S) with p ≥ 1, we can define Itô integral S 0 η s dB s . Moreover, the following B -D -G inequality holds.
In [26], the authors also got the explicit solution of the following special type of NSD equation. S). Then the NSD equation has an explicit solution, Lemma 2.3 (see [27]) Suppose that a nonnegative real sequence for any i ≥ 1. Then there exists a positive constant c, such that 2 i a i ≤ c for any i ≥ 0.

Main results and their proofs
In this section, we introduce the main results and their proofs. [·, ·] denotes the usual inner product in real number space and | · | denotes the Euclidean norm.
Our first main result can be summarized as follows. So there exists a positive constant C 3 such that |A(s, u (k) )| ≤ C 3 (see [28]), which concludes that It follows from (H1) and (4) that for any |u n | ≤ η, where n ∈ Z and η is a positive real number satisfying η ∈ (0, 1). Then (H2) and (5) immediately give

R E T R A C T E D A R T I C L E
By Lemma 2.3, (6) and (7), we have It is obvious that the nonnegative real sequence {u (k) } k∈N is bounded in E, so there exists a positive constant C 4 such that (see [29]) for any k ∈ N, which gives u (k) u (0) in E as k → ∞. Let ε be a given number. Then there exists a positive number ζ such that g(s, u) ≤ ε|u| (9) for any u ∈ R from (H3), where |u| ≤ ζ . It follows from (H1) that there exists a positive integer C 5 satisfying for any |n| ≥ C 5 . By (8), (9) and (10), we obtain for any |n| ≥ C 5 .
Since u (k) u (0) in E as k → ∞, it is obvious that u (k) n converges to u (0) n pointwise for all n ∈ Z, that is, for any n ∈ Z, which together with (11) gives for any |n| ≥ C 5 .

R E T R A C T E D A R T I C L E
It follows from (12), (13) and the continuity of g(s, u) on u that there exists a positive integer C 6 such that for any k ≥ C 6 . Meanwhile, we have from (H3), (8), (9) and the Hölder inequality.

It follows that
So the proof is complete.
The following lemma provides the main mathematical result in the sequel. for any x ∈ L 0 ( S ), then L E is called the orthogonal projection from L 0 ( S ) onto E. Furthermore, we have the following properties:

R E T R
for any x, y ∈ L 0 ( S ) and z ∈ E.
Consequently, u n is bounded, and so is v n . Let T = 2L S i -I. From Lemma 2.1, one can know that the projection operator L S i is monotone and nonexpansive, and 2L S i -I is nonexpansive. So which yields On the other hand, we have (see [31]) For convenience, let c n = (I -μ n λ n X * X)v n . Using Lemma 2.2, it follows that I -μ n λ n X * X is nonexpansive and averaged. Hence, (1λ n+1 )u n+1 + λ n+1 c n+1 -(1λ n )u n + λ n c n which yields By virtue of lim n→∞ (λ n+1 -Z n ) = 0 (see [28]), it follows that Moreover, {u n } and {v n } are bounded, and so is {c n }. Therefore, (20) reduces to Applying (21) and Lemma 2.3, we get lim n→∞ b nu n = 0.
Applying the G-Itô formula toX sŶs , then we obtain  (24), we know that both M s and N s are Schrödinger martingale. Moreover, we know that (see [32]) from (H3).
Taking the Schrödinger expectation on both sides of (25), together with Lemma 2.2 and the property of the Schrödinger expectation, we know that which implies u = u in the space of M 2 G (0, S). It follows from Lemma 2.2 that the NSD equation has a unique solution, then K = K . Thus (1) has a unique solution.

Conclusions
This paper was 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 were discussed by using new Riesz representations of linear maps and the Schrödinger fixed point theorem.

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