New Poisson–Sch type inequalities and their applications in quantum calculus

The Poisson type inequalities, which were improved by Shu, Chen, and Vargas-De-Teón (J. Inequal. Appl. 2017:114, 2017), are generalized by using Poisson identities involving modified Poisson kernel functions with respect to a cone. New generalizations of improved Poisson–Sch type inequalities are obtained by using the generalized Montgomery identity associated with the Schrödinger operator. As applications in quantum calculus, we estimate the size of weighted Schrödingerean harmonic Bergman functions in the upper half space.


Introduction
The Poisson-Sch inequality problem has many applications, e.g., second-order irreversible reactions, obstacle problems, the diffusion problem involving Michaelis-Menten, and reservoir simulation, see, for example, [11,[16][17][18] and the references therein for details. In recent years, various extensions and generalizations of the classical variational inequality models and complementarity problems have emerged in mechanics, nonlinear programming, physics, optimization and control, economics, transportation, finance, structural, elasticity, and applied sciences; see [7,12,17,18] and the references therein for more details. And hence there are a number of numerical methods, such as descent and decomposition, neutral differential equations, for the solution of Poisson-Sch inequality models and complementarity problems [16,17].
In general, when this method, or its many Poisson-Sch forms, is used to solve the Poisson-Sch inequality problem, a key element for implementing this is to find the projection operator. And then, based on the assumption of the convex set, the sequence generated by the proposed method converges to the unique solution of the Poisson-Sch inequality problem. However, for some classes of variational inequalities, such as the generalized nonlinear Poisson-Sch inequality systems, there is not a general convergence theorem, owing to the fact that the convex set cannot be built and the projection method is inapplicable [1,5,13,24]. To fix this issue, the auxiliary principle method has been used to the Poisson-Sch inequality problem, the origin of which can be traced back to the reference by Lions and Stampacchia [16]. Moreover, the authors in [12,20,22] used an auxiliary principle method to study the existence of a solution of mixed variational inequalities. In recent years, under the frame of the auxiliary principle, some authors, such as Huang [14], Qiao [20], Shu et al. [21], Wang et al. [22], Zhao and Zhang [23], and so on, introduced some interesting iterative algorithms to solve some classes of Poisson-Sch inequality problems, and built the corresponding convergence theorems.
Due to the rapid advancement of computing resource, there is a growing interest in developing parallel algorithms for the simulation of the Poisson-Schinequality problem. However, most approaches for Poisson-Sch inequality problems are based on the sequential iterative method. Motivated and inspired by the references [21,23], in this paper we introduce and investigate some new Poisson-Sch type inequalities and obtain some applications.
Let R + be a set of all positive real numbers and R n-1 be the n-dimensional Euclidean space, where n ≥ 2. A point z in H is denoted by (z , z n ), where H = R n-1 × R + , z ∈ R n-1 , and z n > 0.
Let ζ > 0 and h be a Schwarz function. Then the positive powers of the Laplace operator can be defined by (see, e.g., [10, p. 102 and It is well known that the definition (1) can be extended to certain negative powers of -, and we can define If we define the inverse Fourier transform of |ξ | -τ by L τ , then it follows that [9, p. 61] where γ τ is a certain constant.
Let 0 < τ < n and g = (-) τ 2 h. Then it is well known that any Schwartz function h can be written as follows: A time scale is defined by T. Then the operators σ : T → T and ρ : T → T are defined as follows: and ρ(t) = sup{s ∈ T : s < t}, respectively, where t ∈ T (see [4,6,24]).
Let a and b be fixed two points in T satisfying a ≤ b. The modified Schrödinger equation is defined by where q : T → C is a continuous function, p : where t ∈ T * (see [8]). Consider the boundary-value problem defined by subject to the boundary conditions where λ is a spectral parameter and υ 1 , υ 2 , υ 1 , υ 2 ∈ R, and υ is defined by Motivated by this Riesz kernel L τ , we shall introduce the modified Riesz kernel function in H. To do this, we first set (see [2]) We define the modified Riesz kernel G τ (z, w) by where z = w, 0 < τ ≤ n and * denotes reflection in the boundary plane ∂H just as w * = (w 1 , w 2 , . . . , w n-1 , -w n ). Let ζ > 0, 0 < p < ∞, ⊂ R n , and 1/p + 1/q = 1. Then the weighted harmonic space ℵ p ζ ( ) can be defined by where u are real-valued harmonic functions on , d℘ ζ (z) = dist(z, ∂ ) ζ dz. Let dist(z, ∂ ) be the distance from z to ∂ and dz denote the Lebesgue measure on R n (see [15,19]

Preliminary results
In this section, we further present some basic definitions, concepts, and some fundamental results that will be used later.
Definition 2.1 A mapping T : H → H goes by the name of (see [3]): where w ∈ H and w = (w , -w n ). We call it the general Poisson kernel. It follows from (6) that for κ = κ 1 + κ 2 + · · · + κ n and ∂H P(z, w) dw = 1 (8) for each z ∈ H and for every w ∈ H, where f is a homogeneous polynomial of degree | κ| + 2 (see [2] for more details).

Main results and their applications
In this section, we present the proposed parallel iterative method with auxiliary principle for the generalized Schrödinger inequality systems. We first prove new Poisson-Sch inequalities associated with the Schrödinger operator in D κ z P(z, w).
Proof It follows that (7), which together with z → (z + w , z n ) gives that So z → w n z.
If we put then it is easy to see that P S i is nonexpansive and monotone. So which yields that It follows that from Ostrowski type inequality (see [20]). So for any t > 0.
Proof To derive local energy estimates, we use ℵ m and its proprieties. It follows from Lemma 3.1 that we have Combining the above identities, we have which together with the facts that ℵ m = 0 for t = 0 yields that In order to estimate the left-hand side of the above equality, we should use the following inequality: which yields that Considering the properties of Q t and taking into account that γ ≤ 0, we have So it can be estimated by Here and in the sequel, we notice that The same inequality, for p = q = 2, yields that Since σ is bounded, Poincaré's inequality yields that where c σ is a positive Poincaré constant. By applying Poincaré's inequality again, we have Considering that ε is small enough, we have t v (t) 2 + ∇v(t) 2 + γ (t) u(t) ρ+2 ℵ 2 m (t) dx This completes the proof.

Conclusions
In this paper, we generalized the Poisson-Sch type inequalities by using new identities involving new Green-Sch's functions. As applications in quantum calculus, we estimated the size of weighted Schrödingerean harmonic Bergman functions and L p -norm size of partial derivatives of extended Poisson-Sch kernel functions associated with the Schrödinger operator in the upper half space.