1 Introduction

Let p(x),q(y) and u(z) be non-decreasing continuous function on [0,∞) such that p(0) = q(0) = u(0) = 0 and p(x), q(y), u(z) → o as x, y, z →∞. For a locally integrable function f (x, y, z) on ℝ³ ₊ = [0,∞) x [0, ∞) x [0, ∞), we denote its triple integral by F (x, y, z) = ∫^x ₀ ∫^y ₀∫^z ₀ (t, s, v) dtdsdv and its weighted mean of type (α, β, γ) determined by the functions p(x),q(y) and u(z) by

where α, β, γ > - 1. An improper triple integral

is said to be integrable to L by weighted mean method of type (α, β, γ) determined by the functions p(x),q(y) and u(z) if

We use the notion of convergence in Pringsheim's sense, that is, x, y and z tend to independently of each other in (3).

If we take p(x) = x, q(y) = y and u(z) = z in (1), we have the definition of (C, α, β, γ) integrability of f (x, y, z) on [0, ∞) x [0, ∞) x [0,∞) given by (3). The (C, 0, 0, 0) integrability of f(x, y, z) is convergence of the improper double integral (2).

It is clear that if F (x, y, z) = L exists and F (x, y, z) is bounded on ℝ³ ₊, then the limit (3) also exists for α, β, γ > - 1. The converse of this implication is not true in general. The converse of this implication may be true only by adding some suitable condition which is called a Tauberian condition. Any theorem which states that convergence of (2) follows from the integrability of f (x, y, z) by the weighted mean method of type (α, β, γ) determined by the function p( x) , q( y) and u( z) and a Tauberian condition is said to be a Tauberian theorem.

In the last few years, there has been an increasing interest on summability methods for functions of one, two and three variables. First, Laforgia [⁶] obtained a sufficient condition under which convergence of the improper integral follows from (C, 1) integrability. Later, Canak and Totur [¹] extended the main results of Laforgia [⁶] to the (C, α) integrability of functions by weighted mean methods where α > -1. Then, taking into account these works, Totur and Canak [¹⁰] obtained some Tauberian theorems in terms of the concept of the general control modulo of non-integer order for functions of one-variable. Recently, Ozsarac and Canak [⁸] obtained Tauberian theorems for the iterations of weighted mean summable integrals. Besides, Totur et al. [⁹] presented some new Tauberian conditions in terms of the weighted general control modulo for the weighted mean method of integrals. For some interesting Tauberian theorems for Cesáro and weighted integrability in quantum calculus, we refer the readers to Canak et al. [²], Fitouhi and Brahim [⁵] and Totur et al. [¹¹] and so on. In [⁷], Moricz obtained one-sided Tauberian conditions which are necessary and suffcient in order that convergence follow from summability (C, 1 , 1 ) of (2). More generally, Canak and Totur [³] obtained a sufficient condition under which convergence of (2) follows from (C, α, β) integrability of (2) where α, β > -1. Later, Findik and Canak [⁴] showed some new Tauberian theorems for weighted mean of double integrals. In the field of study triple sequences spaces, there are some recently studies that can be seen in [¹², ¹³, ¹⁴, ¹⁵, ¹⁶, ¹⁷].

In this paper, we define the notion of weighted mean method of type (α,β, γ) determined by the functions p(x), q(y) and u(z). Besides, we prove that if (3) exists and t _α β y (x, y, z) is bounded on ℝ³ ₊ for some α, β, y > -1, the limit t _α +i,β +jγ+k (x, y, z) = L exists for all i, j, k > 0. As a corollary of this result, we show that if (2) is convergent to L and the function F(x,y,z) is bounded on ℝ³ ₊. Then, lim tm(x, y, z) = L. But, the converse of this implication may true under some conditions imposed on p, q, u and f. Furthermore, we give a Tauberian condition under which convergence of improper triple integrals follows from the existence of t111(x, y, z) = L.

2 Main Results

Theorem 2.1. If(3) exists and t _{α β γ} (x, y, z) is bounded on ℝ³ ₊ for some α, β, γ > -1. Then, t _α +i,β +j,γ+k(x,y,z) = L exists for all i, j, k > 0.

Proof. Consider

where

where B denotes Beta function defined by

We first prove that

Since

by hypothesis, there exist numbers x _ε , y _ε and z _ε for a given ε > 0 such that

It follows from (5) that

To prove (6), we shall show that

provided that x, y and z are large enough. We realize that by hypothesis, the function t _αβγ (x, y, z) is bounded on ℝ³ ₊. Therefore, there exists a constant K such that

Using (5) and (8), we obtain, by (9),

By the substitution we have

which tends to zero when x, y, z → ∞ for any fixed x _ε , γ _ε and z _ε . Therefore, there exist some x _ε ¹ , γ _ε ¹ and z _ε ¹ such that

By the substitution a = we have

which tends to zero when x, y, z →∞ for any fixed x _ε ,γ _ε and z _ε (see that

Similarly, the integral

f ^x f ^y e [ ^z

Similarly, the integral

Hence, we have (10) for x ≥ max{x_ε ,x _ε ¹ , x _ε ² , x _ε ³ , x _ε ⁴}, y ≥ max{ y_ε ,y _ε ¹ , y _ε ² , y _ε ³ , y _ε ⁴ }, z ≥ max{z_ε ,z _ε ¹ , z _ε ² , z _ε ³ , z _ε ⁴} and this proves (6). Therefore, we obtain

where

Now, we write I(a, b, c; x, , z) as

where

and

Substituting p(t) = p(x) - (p(x) - p(a))x in I₁ (a,x), we have

Similarly, we have

and similarly we have

These show that

Corollary 2.2. If F (x, y, z) = L exists and F (x, y, z) is bounded on ℝ³ ₊. Then, t ₁₁₁ (x, y, z) = L exists.

Proof. Take α = β = γ = 0 and i, j, k = 1 in Theorem 2.1.

The proofs of theorems 2.3, 2.4 and 2.5 can be obtained by the similar techniques and steps as in the proof of Theorem 3 in [⁴], and so we omit them.

Theorem 2.3. If (2) is integrable to L by the weighted mean method of type ( 1 , 0, 0) determined by the function p(x) and

holds. Then, (2) converges to L.

Theorem 2.4. If (2) is integrable to L by the weighted mean method oftype ( 0, 1 , 0) determined by the function q( y) and

holds. Then, (2) converges to L.

Theorem 2.5. If (2) is integrable to L by the weighted mean method oftype ( 0, 0, 1 ) determined by the function u( z) and

holds. Then, (2) converges to L.

Theorem 2.6. If (2) is integrable to L by the weighted mean method of type (1,1,1) determined by the function p(x), q(y) and u(z) and

and

Proof. Consider that (2) is integrable to L by the weighted mean of method of type ( 1 , 1 , 1 ) determined by the functions p(x) , q(y) and u(z) , that is

We rewrite G( x, y, z) as

where

It follows from (17),(18) and (19) that G ₁ (x,y,z) is integrable to L by the weighted mean method of type (0,0,1 ) determined by the function u( z) .

Then, we have

where

Now, we have to show that H ₁ (x, y, z) → 0 as x, y, z→∞.

By (18), we find

Besides, we have

Now, we used the substitution u(z) = e ^w and R(x, y,w) = H ₁ (t,s,u ^-1 (e ^w )). So, we need to show that R(x,y,w) = 0.

By the simple calculation, we have

Differentiating the both sides of (24) with respect to z gives

Taking the weighted mean of type (1,0,0) and (0,1,0) of the both sides of (15) (16), we have

which implies that

by (23). Hence, by (23), we attain that R(x,y,z) is bounded.

Since G(x, y, z) is convergent, given any ε > 0 there exists a z _ε such that

when z ₁ , z ₂ > _yε .

Now, suppose ф > lnu(z _ε ) and R(x,y, ф) > 0. Then, R(x,y,z) > 0 for ф - ξ < z < ф and ф < z < ф + ξ where ξ = . If we integrate R(x,y,z) between ф - ξ and ф + ξ, we have

Moreover, by (28) we have

Therefore,

which shows that H ₁ (x y z) → 0 as x y z→∞ . It follows from (17) and (20) that G₁ (x, y, z) → L as x, y, z →∞.

Since G1 (x, y, z) → L as x, y, z → ∞ and the conditions (14) and (15), we conclude F (x, y, z) = L by Theorems 2.3 and 2.4.