A Method to Construct Sets of Commuting Matrices

Using the method of upper and lower solutions, we study the existence of solutions of the hyperbolic equation involving the distributional Henstock-Kurzweil integral. Results presented in this paper are extension of the previous results in the literatures.


Introduction
In the present paper, we consider the existence for solutions of the hyperbolic equation where Q = {(x, y) : a < x < b, c < y < d}, a, c ≥ 0, Q is the closure of Q, z ∈ C( Q), C( Q) denotes the space of all continuous functions on Q. ∂ 2 z ∂x∂y denotes the mixed distributional derivative of z and f is distributionally Henstock-Kurzweil integrable on Q.
The hyperbolic equations have been studied by many contributors.In (A.Alexiewicz & W. Orlicz (1956)), the authors use the ordinary derivative to discuss the hyperbolic equation.Moreover, authors use the approximate derivative instead of the usual derivative to discuss a special case of hyperbolic equation in (D. Bugajewska & S. Szufla (1995)) .The classical Carathéodory's existence theorem about the special case is discussed in (F.Deblasi & J. Myjak, (1986)).It is well known that the notion of a distributional derivative is a general concept, including ordinary derivatives and approximate derivatives .In this paper, we use the distributional derivative instead of the approximate derivative and usual derivative to study the hyperbolic equation (1).The distributional Henstock-Kurzweil integral is defined by the distributional derivative.Hence our results get the corresponding results in (A.Alexiewicz & W. Orlicz (1956)) by another way.This paper is organized as follows.In section 2, we define the distributional Henstock-Kurzweil integral in the plane and present its preliminary concepts and properties.In section 3, we apply the method of upper and lower solutions to verify the existence of solutions of hyperbolic equation (1) under weaker hypotheses.In Section 4, two examples are also worked out to illustrate our results.

The Distributional Henstock-Kurzweil Integral
In this section, we present the definition and some basic properties of the distributional Henstock-Kurzweil integral.).Under this generalized derivative, distribution is any differentiable and any derivative of the distribution is distribution.Through this paper, ∂ = ∂ xy = ∂ yx stand for the mixed distributional derivative and " ∫ " denotes the D HK -integral.
Now we are able to introduce the definition of D HK -integral on Ω.
The D HK -integral of f on Ω is given by ∫ Denoted the space of D HK -integrable distributions by In the following, we consider the structure of D HK (Ω).For f ∈ D HK (Ω), the norm is defined as the follows: Now we introduce an ordering in the space D HK (Ω).For f, g ∈ D HK (Ω), we say that f ≽ g if and only if f − g is a positive measure on Ω.We obtain the ordering preserving property of the D HK -integral in (Ang et al., 1997) From this definition, we know that Now we give the Fubini type theorem for the D HK -integral.

Main Results
In this section, we shall prove the existence of solutions of the hyperbolic equation (1).
Assume that f satisfies Let E be an ordered Banach space, K a nonempty subset of E. The mapping A:K → E is increasing if and only if Au ≤ Av, whenever u, v ∈ K and u ≤ v.The symbol "≤ " represents the relationship of size in the space of E.
Theorem 1Under the assumptions (C 1 ) − (C 3 ), there exist monotone sequences where where z * and z * are the minimal and maximal solutions of the hyperbolic equation (1), respectively.
proof After integral operation of the hyperbolic equation (1), we can get that z(x, y) (5) The insertion y = c and x = a in (5) yields z(x, c) + f (x) + f (c) = 0 and z(a, y) + f (a) + f (y) = 0, which together with (1) imply that f Compute the mixed partial derivative of z with respect to x and y in (6).Then we can get the hyperbolic equation (1).Therefore, the hyperbolic equation ( 1) is equivalent to the integral equation (6).
In view of (C 2 ), A is increasing.Now we verify that A satisfies the hypotheses of Lemma 4 in two steps.
Step 1: Since f 1 , f 2 ∈ D HK ( Q), then the primitives F 1 (x, y) and F 2 (x, y) are continuous on Q.
According to (C 3 ) and ( 2), we have It follows from ( 7) and ( 8) that Denote It is clear that u 0 (x, y), v 0 (x, y) are continuous on Q and Step For simplicity, we denote In view of Lemma 2, then By (C 3 )

Examples
In this section, we shall give two examples for the application of Theorem 1.
Example 1 Consider the existence of solutions of the following special hyperbolic equation where denotes the space of all continuous functions on Q. ∂ 2 z ∂x∂y denotes the mixed distributional derivative of z and f is distributionally Henstock-Kurzweil integrable satisfying It is well known that the D HK -integral contains the Lebesgue integral.Obviously , the existence of solutions of equation ( 9) is guaranteed by Theorem 1.
Remark 2 The Riemann function R(x) in ( 10) is continuous everywhere in the interval (0, 1) within the irrational points but continuous nowhere in the interval (0, 1) within the rational points.Then, the distributional derivative r in ( 10) is neither HK nor Lebesgue integrable.Hence, the methods using HK or Lebesgue integral are ineffective for equation (10).
and ϕ has compact support in Ω} where Ω denotes the open rectangle (a, b) × (c, d) in the plane R 2 , a, b, c, d ∈ R, a < b and c < d and the closure of the set on which ϕ does not vanish.The continuous linear functional on D(Ω) are called distribution , the dual space to D(Ω) is denoted by D ′ (Ω) and its elements are called distributions.Define the generalized derivative f Then we have the following convergence theorem.Lemma 3 [D.D. Ang, K. Schmitt & L. K. Vy,(1997),Corollary 5, Dominated convergence theorem of D HK the primitives of them are bounded and uniformly continuous on Q. Hence,for each z ∈ [u 0 , v 0 ], A([u 0 , v 0 ]) is equi-uniformly continuous.In view of Ascoli-Arzelà theorem, A[u 0 , v 0 ] is relatively compact.According to Lemma 4, A has the minimal fixed point z * and the maximal fixed point z * .It follows from (3) and (4) that z * and z * are the minimal and maximal solutions of the hyperbolic equation (1).Remark 1(Ascoli-Arzelà Theorem) If F is bounded and uniformly continuous on [a, b], then exists { f n } ⊂ F is equi-uniformly continuous on [a, b].