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On Fractional Ultra-Hyperbolic Kernel Related to the Spectrum |
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PP: 25-29 |
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Author(s) |
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A. S. Abdel-Rady,
S. Z. Rida,
H. M. Abo El-Majd,
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Abstract |
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| In this paper, we study the equation (I −)
α2
u(x) = f (x);xεRn;0 < α < n: The operator is named ultra-hyperbolic
operator defined by
= (
∂ 2
∂ x2
1
+
∂ 2
∂ x2
2
+· · ·+
∂ 2
∂ x2
p
− ∂ 2
∂ x2
p+1
−· · ·− ∂ 2
∂ x2
p+q
);
p+q = n is the dimension of Euclidean spaceRn; f (x)is given generalized function.We define the fractional ultra-hyperbolic kernel
Eα and obtain the solution of such equation which is related to the spectrum of Eα .Moreover,such Eα and u(x) are estimated,and then
we show that they are bounded.Then we study the non linear equation
(I−)
α2
u(x) = f (x;u(x)):
And on suitable conditions for f ;u and for the spectrum of the kernel Eα we can obtain a unique bounded solution for the nonlinear
equation in a compact subset of Rn. |
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