Weak solution of non - Newtonian polytropic variational inequality in fresh agricultural product supply chain problem

: In this article, we study a class of variational inequality problems with non - Newtonian polytropic parabolic operators. We introduce a mapping with an adjustable parameter to control the polytropic term, which exactly meets the conditions of Leray - Schauder ﬁ xed point theory. At the same time, we construct a penalty function to transform the variational inequality into a regular parabolic initial boundary value problem. Thus, the existence is treated with a Leray - Schauder ﬁ xed point theory as well as a suitable version of Aubin - Lions lemma. Then, the uniqueness and stability of the solution are analyzed.


Introduction
(2) and the Dirichlet initial-boundary value condition Variational inequality has a good application in the value analysis of financial products with early implementation clauses, for details see [1,2]. Recent years, much attention has been paid to the study of variational inequality with linear, quasi-linear, and degenerate parabolic operator [3][4][5]. In [3], Li and Bi considered two-dimensional variational inequality systems The existence of weak solution is studied by a limit process. In [4], Sun and Wu considered a kind of variational inequality problem with a double degenerate operator Similar methods to those presented in [3] have been used, and the existence and uniqueness of the solutions in the weak sense are proved.
The structure of parabolic initial boundary value problem is simpler than that of variational inequality. When = m 0, initial boundary value problems with parabolic operator Lu have been extensively studied in the last few years, see for details [6][7][8][9][10][11][12][13]. Some articles are focused on the existence of generalized solutions related to this article [6][7][8]. Some results for uniqueness of generalized solutions can be found in [9][10][11].
There are other arguments worth studying, such as stability of boundary output feedback [12,13].
In this article, we extend the corresponding results in [3,6,9] to study a class of variational inequality problems with non-Newtonian polytropic parabolic operators with the Dirichlet initial-boundary value condition. Since m and p are coupled in Lu and Lu is degenerate, we plan to solve this problem with Leray-Schauder fixed point theory by constructing a map. In order to overcome the difficulty of establishing generalized solutions on variational inequalities, we turn the variational inequalities into regular problems through penalty functions. Some estimates of regular problems and continuity, boundedness and compactness of Leray-Schauder map are given by the inequality technique as well as a suitable version of Aubin-Lions lemma. In what follows, we prove the existence, uniqueness, and stability of the solution under the proper setting of the parameters in (1).

Statement of the problem and the main results
Our consideration in this article is motivated by an application model about fresh agricultural product supply chain. Here we consider a fresh agricultural product supply chain formed by a supplier and a retailer in which retailers face uncertain market demand. Assuming that the current time is 0, the time agreed in the contract for retailers to purchase agricultural products is T , and the retail price of agricultural products agreed in the contract meets where μ and σ denote the expected rate and the volatility of return on the retail price of agricultural products, respectively.
t is a Winner process, which drives the random noise of the market. P 0 represents the market price at time 0.
Since fresh agricultural products are easy to deteriorate and decay, retailers will have no residual value for their remaining agricultural products. Therefore, the order of agricultural products must be placed before the sales season T . Retailers can buy a call option contract in which they have the right to purchase a certain amount of fresh agricultural products at the agreed price of K from 0 to T . Of course, retailers need to pay a certain premium of C to obtain such rights. If the retailer finds a more suitable source of goods, it will give up the option contract and lose the option premium C. This means that retailers can decide whether to exercise or hold options based on their own earnings. According to the literature [1][2][3], the value of options meets where B represents the artificially set price ceiling for agricultural products, In addition, if the loss of agricultural products during transportation is considered, then the expression of LC is more complicated that σ in LC is a function of C and ∂ ∂ C P , for details see [14]. More complex case than (5) is considered in this article. In doing so, we give a class of maximal monotone maps defined in [11,13] where M 0 is a positive constant which can be chosen later. Since m and p are coupled in Lu and Lu is degenerate, problem (1) do not have a classical solution. So we consider its generalized case as follows.
Because of coupling in Lu, we cannot prove the existence of solution of problem (1) by using a common limit method. Here, we plan to use the Leray-Schauder fixed point theory as well as a suitable version of Aubin-Lions lemma based on the map such that for every function with an operator Here Then the existence of problem (1) is equivalent to . With a similar method to that in [6,7], we may prove that the regularized problem (7) admits a weak solution as follows: for any ( ) Because of the denseness of ( ) , one can assert that the identities in Definition 2.1 and Definition 2.2 hold for any ( ( ) ) L T W 0, ; Ω p 1, . Furthermore, one can obtain the following inequalities [6,7]: It is clear that the constructed generalized solution in this article is non-negative. Then we give the restriction in (6) that ≥ ω 0.

Some preliminaries
To discuss the existence of weak solution of equation (1), we give certain useful estimates.
Proof. Since (10), it can be easily verified that ∫∫ ∫∫ From the first part of (11), can be estimated as Since ≥ ω 0, ( ) ∈ θ 0, 1 , it is easy from the first part of (12) to see that Using differential transformation method gives (recall that This implies (12) follows.
On the contrary, if we drop non-negative term in (14), Thus (13) follows. If = ε 0, such that estimate (14) is an immediate result of (13). □ Lemma 2. Assume that ≥ γ 0, > m 0, and ≥ p 2. Then there exists a constant C, independent of ε, such that CpT , , Ω . Proof. Since ( ) ∈ θ 0, 1 . It follows from the first part of (11) that Multiplying the first line of (7) by ( ( ) ) ∂ + − θu θ u 1 t ε m ε and integrating both sides of the equality over Ω T , we have where the constant π depends only upon m and | | ∞ u 0 , Using some differential transformation techniques,

It follows by Holder and Cauchy inequalities that
, combining (25), (26), (27), and (28), it is easy to verify that This implies that (21) follows. Using (24) again, , such that u ε k , is the solution of (9) with degenerate parabolic operator Proof. From the first part of (11), it can be seen that is bounded, which together with the uniform estimates in k allow one to extract from the sequence a subsequence (for the sake of simplicity, we assume that it merely coincides with the whole of the sequence) and a function u ε such that So that we remain to prove that . From (12) and (13), one can infer that for any fix ( ) ∈ ε 0, 1 , Hence, it remains to prove that by φ, and integrating over Ω, we have that Since for any ∈ x Ω, T so we drop the non-negative term on the left-hand side and pass the limit → ∞ k to arrive at and ∇φ have the same sign, Subtracting (39) and (40), we infer that Obviously, if we swap u ε k , and u ε , it is easy to obtain another inequality with the same initial boundary condition in (7), admits a unique solution in (