In this paper, we present results of \(\omega\)-order preserving partial contraction mapping generating a nonlinear Schr\”odinger equation. We used the theory of semigroup to generate a nonlinear Schr\(\ddot{o}\)dinger equation by considering a simple application of Lipschitz perturbation of linear evolution equations. We considered the space \(L^2(\mathbb{R}^2)\) and of linear operator \(A_0$ by $D(A_0)=H^2(\mathbb{R}^2)\) and \(A_0u=-i\Delta u\) for \(u\in D(A_0)\) for the initial value problem, we hereby established that \(A_0\) is the infinitesimal generator of a \(C_0\)-semigroup of unitary operators \(T(t)\), \(-\infty<t<\infty\) on \(L^2(\mathbb{R}^2)\).
Consider the initial value problem for the following nonlinear Schrödinger equation in \(\mathbb{R}^2\)
It follows that the operator \(A_0\) is the infinitesimal generator of a \(C_0\)-semigroup of unitary operators \(T(t)\), \(-\infty< t< \infty\), on \(L^2(\mathbb{R}^2)\). A simple application of the Fourier transform gives the following explicit formula for \(T(t);\)
Suppose \(X\) is a Banach space, \(H\) is Hilbert space, \(X_n\subseteq X\) is a finite set, \(\omega-OCP_n\) the \(\omega\)-order preserving partial contraction mapping, \(M_{m}\) be a matrix, \(L(X)\) be a bounded linear operator on \(X\), \(P_n\) a partial transformation semigroup, \(\rho(A)\) a resolvent set, \(\sigma(A)\) a spectrum of \(A\) and A is a generator of \(C_{0}\)-semigroup. This paper consists of results of \(\omega\)-order preserving partial contraction mapping generating a nonlinear Schrödinger equation.
Akinyele et al., [1], obtained a continuous time Markov semigroup of linear operators and also in [2], Akinyele et al., established results of \(\omega\)-order reversing partial contraction mapping generating a differential operator. Balakrishnan [3], presented an operator calculus for infinitesimal generators of the semigroup. Banach [4], established and introduced the concept of Banach spaces. Brezis and Gallouet [5] generated a nonlinear Schr\(\ddot{o}\)dinger evolution equation. Chill and Tomilov [6], introduced some resolvent approaches to stability operator semigroup. Davies [7] deduced linear operators and their spectra. Engel and Nagel [8] obtained a one-parameter semigroup for linear evolution equations. Omosowon et al., [9], generated some analytic results of the semigroup of the linear operator with dynamic boundary conditions, and also in [10], Omosowon et al., introduced dual properties of \(\omega\)-order reversing partial contraction mapping in semigroup of linear operator. Omosowon et al., [11], established a regular weak*-continuous semigroup of linear operators, and also in [12], Omosowon et al., generated quasilinear equations of evolution on semigroup of a linear operator. Pazy [13] presented the asymptotic behaviour of the solution of an abstract evolution and some applications and also, in [14], obtained a class of semi-linear equations of evolution. Rauf and Akinyele [15] obtained \(\omega\)-order preserving partial contraction mapping and obtained its properties, also in [16], Rauf et al., introduced some results of stability and spectra properties on semigroup of a linear operator. Vrabie [17], proved some results of \(C_{0}\)-semigroup and its applications. Yosida [18] deduced some results on differentiability and representation of one-parameter semigroup of linear operators.
Definition 1.(\(C_0\)-Semigroup) [17] A \(C_0\)-Semigroup is a strongly continuous one parameter semigroup of bounded linear operator on Banach space.
Definition 2. (\(\omega\)-\(OCP_n\)) [15] A transformation \(\alpha\in P_n\) is called \(\omega\)-order preserving partial contraction mapping if \(\forall x,y \in~ \)Dom\(\alpha:x\le y~~\implies~~ \alpha x\le \alpha y\) and at least one of its transformation must satisfy \(\alpha y=y\) such that \(T(t+s)=T(t)T(s)\) whenever \(t,s>0\) and otherwise for \(T(0)=I\).
Definition 3.(Evolution Equation) [13] An evolution equation is an equation that can be interpreted as the differential law of the development (evolution) in time of a system. The class of evolution equations includes, first of all, ordinary differential equations and systems of the form \begin{equation*} u=f(t,u),u=f(t,u,u), \end{equation*} etc., in the case where \(u(t)\) can be regarded naturally as the solution of the Cauchy problem; these equations describe the evolution of systems with finitely many degrees of freedom.
Definition 4. (Mild Solution) [14] A continuous solution \(u\) of the integral equation.
Definition 5. (Schrödinger Equation) [19] The Schr\(\ddot{o}\)dinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject.
Example 1. \(2\times 2\) matrix \([M_m(\mathbb {R}^{n})]\): Suppose \[ A=\begin{pmatrix} 2&0\\ \Delta & 2\\ \end{pmatrix} \] and let \(T(t)=e^{t A}\), then \[ e^{t A}=\begin{pmatrix} e^{2t}& I\\e^{\Delta t} & e^{2t}\\ \end{pmatrix}. \]
Example 2. \(3\times 3\) matrix \([M_m(\mathbb{C})]\): We have for each \(\lambda>0\) such that \(\lambda\in \rho(A)\) where \(\rho(A)\) is a resolvent set on \(X\). Suppose we have \[ A=\begin{pmatrix} 2&2&I\\ 2&2&2\\ \Delta &2&2 \end{pmatrix} \] and let \(T(t)=e^{t A_\lambda}\), then \[ e^{t A_\lambda}=\begin{pmatrix} e^{2t\lambda}&e^{2t\lambda}& I\\ e^{2t\lambda}&e^{2t\lambda}&e^{2t\lambda}\\ e^{\Delta t\lambda}&e^{2t\lambda}&e^{2t\lambda}\end{pmatrix} .\]
Example 3. Let \(X=C_{ub}(\mathbb{N}\cup\{0\})\) be the space of all bounded and uniformly continuous function from \(\mathbb{N}\cup\{0\}\) to \(\mathbb{R}\), endowed with the sup-norm \(\|\cdot\|_\infty\) and let \(\{T(t); t \in \mathbb{R_{+}}\}\subseteq L(X)\) be defined by \[ [T(t)f](s)=f(t+s)\,. \] For each \(f\in X\) and each \(t,s\in \mathbb{R_+}\), one may easily verify that \(\{T(t); t \in \mathbb{R_{+}}\}\) satisfies Examples 1 and 2 above.
Theorem 1. Suppose \(A:D(A)\subseteq L^2(\mathbb{R}^2)\) is the infinitesimal generator of a semigroup \(\{T(t),\ t\geq0\}\) given by (3) where \(A\in\omega-OCP_n\). If \(2\leq p\leq \infty\) and \(\frac{1}{q}+\frac{1}{p}=1\), then \(T(t)\) can be extended in a unique way to an operator from \(L^q(\mathbb{R}^2)\) into \(L^p(\mathbb{R}^2)\) and
Proof. Since \(T(t)\) is a unitary operator on \(L^2(\mathbb{R}^2)\) we have $$ \|T(t)u\|_{0,2}=\|u\|_{0,2}\quad for\ u\in L^2(\mathbb{R}^2). $$ On the other hand it is clear from (3) that \(T(t):L^1(\mathbb{R}^2)\to L^\infty(\mathbb{R}^2)\) and that for \(t>0\), we have $$ \|T(t)u\|_{0,\infty}\leq(4\pi t)^{-1}\|u\|_{0,1}. $$ The Riesz convexity theorem implies in this situation that \(T(t)\) can be extended uniquely to an operator from \(L^q(\mathbb{R}^2)\) into \(L^p(\mathbb{R}^2)\) and that (6) holds. In order to prove the existence of a local solution of the initial value problem (2) for every \(u\in H^2(\mathbb{R}^2)\) and \(A\in\omega-OCP_n\). We note that the graph norm of the operator \(A_0\) in \(L^2(\mathbb{R}^2)\), that is the norm \( \|u\|=\|u\|_{0,2} + \|A_0u\|\), for \(u\in D(A_0)\) and \(A\in\omega-OCP_n\) is equivalent to the norm \(\|\cdot\|_{2,2}\) in \(H^2(\mathbb{R}^2)\). Therefore \(D(A_0)\) equipped with the graph norm is the space \(H^2(\mathbb{R}^2)\). Hence the proof in competed.
Theorem 2. Assume \(A:D(A)\subseteq H^2(\mathbb{R}^2)\to H^2(\mathbb{R}^2)\) is the infinitesimal generator of a \(C_0\)-semigroup \(\{T(t);\ t\geq 0\}\). The nonlinear mapping \(Fu=ik|u|^2u\) maps \(H^2(\mathbb{R}^2)\) into itself and satisfies for \(u,v\in H^2(\mathbb{R}^2)\) and \(A\in\omega-OCP_n\), we have
Proof. From Sobolev’s theorem in \(\mathbb{R}^2\), it follows that \(H^2(\mathbb{R}^2)\subset L^\infty(\mathbb{R}^2)\) and that there is a constant \(C\) such that
Theorem 3. Suppose \(A:D(A)\subseteq H^2(\mathbb{R}^2)\to H^2(\mathbb{R}^2)\) is the infinitesimal generator of a \(C_0\)-semigroup \(\{T(t);\ t\geq 0\}\). Let \(u_0\in H^2\mathbb{R}^2\), \(A\in\omega-OCP_n\) and \(u\) be the solution of initial value problem (2) on \([0,T)\). If \(K\geq 0\), then \(\|u(t)\|_{2,2}\) is bounded on \([0,T)\).
Proof. We will first show that \(\|u(t)\|_{1,2}\) is bounded on \([0,T)\). To this end we multiply the equation