T : K [right arrow] K is a
completely continuous operator.
Assume that T : X [right arrow] X is a
completely continuous operator and the set V = {u [member of] X | u = [mu]Tu, 0 < [mu] < 1} is bounded.
Assume [[OMEGA].sub.1], [[OMEGA].sub.2] are bounded open subsets of E with 0 [member of] [[OMEGA].sub.1], [[bar.[OMEGA].sub.1]] [subset] [[OMEGA].sub.2], and let S : P [intersection] ([[OMEGA].sub.2]\[[OMEGA].sub.1]) [right arrow] P be a continuous and
completely continuous operator such that, either
Suppose further that T : K [intersection] ([[OMEGA].sub.2]\[[OMEGA].sub.1]) [right arrow] K is
completely continuous operator such that either
Keywords: Time scale, delta and nabla derivatives and integrals, Green's function,
completely continuous operator, eigenfunction expansion.
Then A : K [right arrow] K is a
completely continuous operator.
Suppose that [[OMEGA].sub.1], [[OMEGA].sub.2] are two bounded open sets of E with [mathematical expression not reproducible]2 is a
completely continuous operator such that either
Besides, using the Arzela Ascoli theorem and the standard arguments, one can easily show that T : p [right arrow] p is
completely continuous operator.
We define the
completely continuous operator T by (2.14).
Suppose further that A: K [intersection] ([[bar.[OMEGA]].sub.2]\[[OMEGA].sub.1]) [right arrow] K is
completely continuous operator such that either (i) [parallel]Au[parallel] [less than or equal to] [parallel]u[parallel], u [member of] k [intersection] [partial derivative][[OMEGA].sub.1] and [parallel]Au[parallel] [greater than or equal to] [parallel]u[parallel], u [member of] k [intersection] [partial derivative][[OMEGA].sub.2], or
Let the cone P [subset] E be solid, and A : E [right arrow] E be a
completely continuous operator, and A = BF, where B is a positive completely continuous linear operator satisfying H condition and F is quasi-additive on lattice.
is a
completely continuous operator, and Q([alpha], [beta], r, R) is a bounded set.