Some results for a p ( x )- Kirchho ﬀ type variation - inequality problems in non - divergence form

: The author of this article concerns with the existence, uniqueness, and stability of the weak solution to the variation - inequality problem. The Kirchho ﬀ operator is a non - divergence form with space variable parameter. The existence of generalized solution is proved by the Leray - Schauder principle and parabolic regularization. The uniqueness and stability of the solution are also discussed by contradiction.


Introduction
≥ be a smooth bounded domain and T 0 > . Consider the variation-inequality problems sup .
In finance, variational inequality (1) is often used to study the value of convertible bonds (for details, see [1][2][3][4][5]). Because the holders of convertible bonds can converting bonds into shares at any time to obtain benefits, the pricing of convertible bonds is essentially a variational inequality problem. Theoretical research on variational inequalities have received considerable attention in the recent reference. The previous work [6] studies weak solutions of variational inequalities (1) with fourth-order p-Laplacian Kirchhoff operators, in which the existence and uniqueness of solution are studied by the penalty method (for details, see [7]). The author in [8] With the help of monotone iteration technique and the method of regularization to deal with the auxiliary problem, the existence and uniqueness of solution are also considered. Recently, several interesting existence results can be found in [9,10]. Their characteristic is that scholars used the mountain pass theorem to deal with the auxiliary problem.
In this article, we inspired by [1,2] study a kind of variation-inequality initial-boundary value problem with variable parameter Kirchhoff operators not in divergence form such that the parameter is spatial dependent. To overcome the inequality limitation of problem (1), we introduce a penalty function to turn problem (1) into an obstacle problem. Since [11][12][13][14][15][16][17][18][19][20][21][22] study the problem under the parabolic equation model, this article studies the problem of variation-inequality, which is the first innovation. Because of the coupled problem in Kirchhoff operator, it is unrealistic to obtain the norm boundedness of penalty problem. In this article, we construct an operator by Larey-Schauder lemma to prove the existence of solution, which is the second innovation. Further, some estimates obtained from the auxiliary problem constructed by Larey-Schauder lemma also prove the uniqueness and stability of the solution.

Statement of the problem and main results
This article consider a more general case, and we will study the existence, uniqueness, and stability of the solution of model (1). In doing so, we give the maximal monotone operator [1,2,23], which will be used throughout this article: Moreover, the following result will be used repeatedly [7].
where C is a positive constant depending on p + and p − .
Here, we do not expect problem (1) to have a classical solution, so the generalized solution of problem (1) is defined as follows.
We cannot use Minkowski inequality as usual, in proving the existence. In doing so, we will use the Leray-Schauder principle, first define a map is a solution of the equation with an operator So the existence of problem (1) is equivalent to the existence of the fixed point of operator M ,1

Some preliminaries
This section gives some useful preliminaries to prove the main results. From [5], problem (4) admits a unique solution u ε which satisfies In fact, multiplying the first line of (4) by u ε and integrating the value over Ω t , Using Cauchy and Young inequalities, (10) follows immediately. Further, by replacing u ε with u t ε ∂ , (8) can be changed into One, using Cauchy, Holder, and Young inequalities, can prove (7) by amplifying the values of A 1 , A 2 , and A 3 .
Following the similar way of [1,2], we also give the following inequalities: From (6) and (9), one, using Leray-Schauder theorem, can infer that where B R denotes the ball with radius R centred at the origin. And, there exists a small positive constant r satisfying r R < which were used extensively in [5,7]. One can deduce from (11) and aforementioned equation that (6) and (7).
Following a similar way in [6,12], we infer that problem with Dirichlet boundary condition, admits a unique solution in L T W 0, ; Ω . By the properties of the Leray-Schauder degree, together with (11)-(13) and Lemmas 3.1, the following initial boundary problem also has a solution u L which satisfies the following equation: . Furthermore, the solution of (14) satisfies (9) and (10), that is, where weak ⟶ stands for weak convergence.
, the following estimate holds By suing progressive integral, (17) becomes Thus, by adding from both sides of (18), it can be easily verified that Using Cauchy inequality, it follows from (11), (10), and (16) that as ε 0 → , , one, from (6), can obtain By combining (19), (20), and (21), it is easy to see that From Lemma 2.1, we know that for any fixed ε This combining with (22) implies that Lemma 4.1 follows. □ Proof. We first analyze By using Holder inequality, it follows from (6) that as ε 0 → , By using trigonometric inequality, one from Lemma 4.1 obtains By combining (25)-(27), we derive (24) immediately. □ From Lemmas 4.1 and 4.2, we also know that as ε 0 → , . In fact, we can construct trigonometric inequality to complete the proof as in (25)-(27), which is omitted here. Further, following the similar way of Lemma 4.1 in [12], one obtains the following result for all x t , Ω T ( ) ∈ . Moreover, passing the limit ε 0 → in (4) and (9), it is easy to verify that Combining Lemma 4.1 and (29), we infer that u ξ , ( ) satisfies the initial and boundary condition and the integrating expression. Thus u ξ , ( )is a generalized solution of (1), so we summarize the following theorem.
. Then (1) admits a solution u within the class of Definition 2.1.

Proof of the stability and uniqueness
In this section, we study the stability and uniqueness of generalized solution and consider a generalized solution u ξ , Note that λ 0 = in this section and define φ u u 1 2 = − . We, from Definition 2.1, infer that , it is easy to verify that g u g u g u g u g u g u g u g u 1 sgn 0, 0.
Combining the two inequalities above implies that A 0 8 ≥ . Now, we consider and give the following result , then u x t u x , 1 1 , 0 ( ) ( ) > which, combining with (28) and (29), infer that , then (31) follows. On the contrary, if u x t u x t , , still holds. Further, we remove the non-negative term A 8 and non-positive term (31) to arrive at By passing the limitation, one obtains Note that both f and g are increasing functions, and then using g u g u f u f u sgn sgn , This implies that in (32), uniqueness of solution can be found easily. Thus, we summarize the following theorem.
Furthermore, problem (1) admits a unique solution in the sense of Definition 2.1.

Numerical illustration
Here, we discretize problems (20) By neglecting the truncation errors, we obtain the following estimate of u and F can be formulated at point x t , i k ( ) as follows: Now we present some numerical experiments for variation-inequality (1) based on scheme (36). We first consider the stability of variation-inequality (1) at p x 3 ( ) = , u x π x , 0 sin ( ) ( ) = , T 1 = , N 10 = , M 100 = . Figure 1 shows the simulation results at x 0.5 = with initial error 0, 0.1, 0.2, 0.3, and 0.4. It can be seen from Figure 1 that the initial error gradually decreases with time. This shows that variational inequality (1) is stable with respect to initial value, which is consistent with Theorem 5.1. Figure 2 shows the effect of different values of p on the simulation results. It can be seen that with the increase of p, the peak value of simulation results decreases gradually. Some results for a p(x)-Kirchhoff type variation-inequality problems  9 Author contributions: This is a single-author article. The author read and approved the final manuscript.
Conflict of interest: Author states no conflict of interest.
Data availability statement: No applicable.