Without the usual assumption of monotonicity, we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorising interval function for the solution of the hyperbolic initial value problem. Using this function, a variation of parameters formula and interval iterative technique, the existence of solution to the problem is established.

Interval Functions Interval Majorant Interval Extension Interval Operator Nested Sequence
1. Introduction

In this paper, we utilize interval analytic methods in the investigation of the existence of solution of the hyperbolic partial differential equation

with characteristic initial values

prescribed in a two-dimensional rectangle where and , and where means that z is continuous on and possesses continuous partial derivatives on

Without the assumption of monotonicity on the function we establish some results on the theory of hyperbolic differential inequalities which enable us to produce a majorizing interval function for the solution of the equation. With the use of a variation of parameters formula used in  and theorem 5.7 of  on interval iterative technique we generate a nested sequence of interval functions which converges to an interval solution. This interval solution is thus a majorant of the solution of the equation and it coincides with the real valued solution if it is degenerate. Similar interval methods had earlier been used by some authors in  - for solution to differential equation but not for hyperbolic initial value problems. The result in this paper generalizes those of   as the monotonicity condition imposed on the function is not in any way necessary.

The basic results in interval analysis used in this work are found in     - for readers who may not be familiar with them.

2. Differential Inequalities and Majorisation of Solution

Definition 2.1: A function is said to be an upper solution of the hyperbolic initial value problem (1.1) and (1.2) on if   .

Definition 2.2: A function is said to be a lower solution of the hyperbolic initial value problem (1.1) and (1.2) on if the reversed inequalities hold true with in place of in the specified intervals.

Next, we shall consider some results concerning the upper and lower solutions of Equation (1.1) and conditions (1.2).

Theorem 2.1: Suppose that and

Then we have

where the inequality is componentwise.

Proof: We shall establish this theorem by contradiction. From assumption (2.3) we see clearly that the theorem is true for the point (0,0) on

Suppose that inequality (2.4) is not true at a point and assume that

then by assumption (2.3) and cannot both be zero.

Let be such that then and so Thus, we have, for (or), If then (or vice-versa) and for we have If and a similar argument can be advanced to obtain Hence, and this is still a contradiction to our earlier assumption (2.5).

Then otherwise condition (2.3) would immediately give the required contradiction.

Now for let such that, we have so  Similarly, if we assume that, we would also arrive at a contradiction. At and left hand derivatives are used to obtain the result.

Hence, we conclude that, the assertion (2.4) holds true on and this proves the theorem.

Theorem 2.2: Let and be functions defined on which satisfy assumptions (2.1), (2.2) and (2.3) of Theorem 2.1. Suppose in addition that they satisfy the following conditions,

Then the solution of problem (1.1) and (1.2) together with its derivatives satisfy On the rectangle, , where the inclusion is componentwise.

Proof: Notice that the lower endpoints of the intervals in Equation (2.7) satisfy assumption (2.3) of Theorem 2.1 when is replaced by. Therefore and satisfy the hypothesis of Theorem 2.1 and hence

Similarly, replacing by in assumption (2.3) we obtain the upper endpoints of the intervals in conditions (2.7) and so by Theorem 2.1 we also have

Combining inequalities (2.8) and (2.9) we have the desired result.

3. Construction and Existence of Solution

Our purpose in this section is to establish the existence of solution to the problem (1.1) satisfying initial values (1.2) by means of interval analytic method. To this end an integral operator is constructed, the solution of the resulting operator equation is equivalent to the solution of the initial value problem under consideration. An interval extension of this operator is then used to generate a sequence of interval functions which converges to the required solution.

Let be such that on and a function , defined by

where is the function in Equation (1.1) and is a constant suitably chosen such that Clearly it can be seen that is continuous on

With this new function, Equation (1.1) becomes

By using the variation of constant formula of Lemma 4.1 in  , we obtain the solution of Equation (3.2), satisfying initial values (1.2) as: Differentiating with respect to, we obtain and similarly by differentiating with respect to we obtain Eliminating the derivatives and by introducing the function and into the integro-differential equations we obtain the system of integral equations

which is equivalent to the problem (3.2) and initial values (1.2).

Denoting the right hand side of these integral equations by and respectively, we have the following:

With these we prove the following result.

Lemma 3.1: Let and satisfy conditions (2.7) of Theorem 2.2. Suppose that for functions with on, we have

where is the constant appearing in Equation (3.1). Then the following hold true.

for all

Proof: We first consider the lower endpoints of the inclusions and differentiating we have, from Equation (3.3) differentiating again with respect to we obtain This, by Equation (3.1) and assumption (3.7), gives Similarly by differentiating Equation (3.3) with respect to, we obtain By conditions (2.1) and (3.7) we have for and for From these we see that satisfies the assumptions of Lemma 4.2 of  since

Thus      It could similarly be proved that  and Hence the lemma is established.

Theorem 3.1: Let the functions satisfy conditions (2.7). Suppose that the function is such that and

for function satisfying and, constant, suitably chosen in Equation (3.1).

Then there exists a convergent nested sequence of interval functions such that the unique solution of Equations (1.1) and (1.2) satisfies with, degenerate where the initial interval is given by Proof: From the construction earlier considered, we see that any solution of Equation (1.1) which satisfies condition (1.2) solves the integral Equation (3.3). Conversely if solves the integral Equation (3.3) we have that which by Equations (3.1) and (3.6) gives with

.

and these imply that again solves the Equation (1.1) and satisfies condition (1.2). Therefore, we shall seek the solution of the integral equation given by (3.3) which is transformed to the operator equation Let be an interval function defined on such that for and the interval function an interval extension of the function defined in Equation (3.1). Then the interval integral operator defined by is an interval majorant of.

Then the problem reduces to solving the interval operator equation However to determine we need to also determine and which are respectively interval extensions to the function and. This is done by solving the interval operator equations With and defined respectively by and which majorise the real operators and respectively.

Define the sequences by

with with

and

with We have the sequence as required.

We shall show that convergences to a limit. But this can only be so if the sequence and also converge.

By Theorem 5.7 of  , these sequences converge if and Now for  by the first inclusion of Equation (3.8). Hence Similarly we have by the result given in Equation (3.8) of Lemma 3.1 and Since these initial intervals satisfy the hypothesis of Theorem 5.7 of  , the result of the theorem implies that and converge as sequences and are equally nested. Furthermore, the solution of Equation (1.1) satisfying condition (1.2) belongs to the limit function of the sequence that is, and this proves the theorem.

Lemma 3.2: Assume that the functions satisfy conditions (2.7) and in addition they also satisfy conditions (2.1) and (2.2). Suppose further that the function f appearing on the right hand side of Equation (1.1) satisfies:

whenever the functions and are such that for constant suitably chosen. Then we have

for any function satisfying Proof: From inequality (2.1) we have Since From inequality (3.9) we have and so which is the first inequality in (3.7).

Also from inequality (2.2) we have and using inequality (3.9) we have Therefore which also is the second inequality in (3.7). Since all the other conditions of Lemma 3.1 are also satisfied, the proof of this lemma follows as for Lemma 3.1 to obtain the desired result.

Remark 3.1: If in inequality (3.9) then we have for and this implies that is monotone increasing in its domain of definition. Therefore the result of lemma 3.2 also holds for a monotone function

Theorem 3.2: Suppose that the function satisfies conditions (2.1), (2.2) and (2.7). If in addition the function appearing in Equation (1.1) satisfies whenever for some constant, suitably chosen.

Then there exists a nested sequence of interval function with each term majorising the unique solution of Equation (1.1) satisfying condition (1.2) such that the limit of this sequence also contains, that is, Proof: As it has been shown in the proof of Lemma 3.2, the conditions prescribed in this theorem can equally be linked with those of Theorem 3.1. Therefore the proof can be established in a manner similar to that of Theorem 3.1.

ReferencesLakshmikantham, V. and Pandit, S.G. (1985) The Method of Upper and Lower Solution and Hyperbolic Partial Differential Equations. Journal of Mathematical Analysis and Applications, 105, 466-477. http://dx.doi.org/10.1016/0022-247X(85)90062-9Moore, R.E. (1979) Methods and Application of Interval Analysis. SIAM (Studies in Applied and Numerical Mathematics), Philadelphia. http://dx.doi.org/10.1137/1.9781611970906Arawomo P.O. and Akinyele, O. ,et al. (2002)An Interval Analytic Method in Constructive Existence Result for Initial Value Problems Dynamic Systems and Applications 11, 545-556.Chan, C.Y. and Vatsala, A.S. (1990) Method of Upper and Lower Solution and Interval Method for Semilinear Euler-Pision-Darboux Equations. Journal of Mathematical Analysis and Applications, 150, 378-381. http://dx.doi.org/10.1016/0022-247X(90)90110-2Lakshmikantham, V. and Swansundaran, S. (1987) Interval Method for 1st Order Differential Equations. Applied Mathematics and Computation, 23, 1-5. http://dx.doi.org/10.1016/0096-3003(87)90052-XMoore, R.E. (1984) Upper and Lower Bounds on Solutions of Differential, Integral Equations without Using Monotonicity, Convexity or Maximum Principle., In: Vichnevetsky, R. and Stepheman, R., Eds., Advances in Computer Methods for Partial Differential Equations, IMACS, 458-461.Moore, R.E. (1984) A Survey of Interval Methods for Differential Equations. Proceedings of 23rd Conference on Decision and Control, Las Vegas, 12-14 December 1984, 1529-1535.Walter, W. (1970) Differential and Integral Inequalities. Springer-Verlag, Berlin, Heidelberg. http://dx.doi.org/10.1007/978-3-642-86405-6Caprani, O. and Madsen, K. (1975) Contraction Mappings in Interval Analysis. BIT Numerical Mathematics, 15, 362-366. http://dx.doi.org/10.1007/BF01931673Moore R.E. ,et al. (1977)A Test for Existence of Solution to Nonlinear Systems SIAM Journal on Numerical Analysis 14, 611-615.Moore R.E. ,et al. (1978)Bounding Sets in Function Spaces with Application to Nonlinear Operator Equations SIAM Review 20, 492-512.Rall L.B. ,et al. (1983)Mean-Value and Taylor Forms in Interval Analysis SIAM Journal on Mathematical Analysis 14, 223-238.Rall L.B. ,et al. (1982)Integration of Interval Functions II, the Finite Case SIAM Journal on Mathematical Analysis 13, 690-698.