The main idea in obtaining our bifurcation result is to study the asymptotic behavior of the integral operator J and then to deduce a spectral result for operators that are not necessarily homogeneous.
In this section we deduce a composition result for pseudodifferential operators with symbols in modulation spaces with Lebesgue parameters in ((0,infty ]).
For the first time, a Carleman-type estimate with one observation in parabolic systems was introduced by Ammar Khodja et al.[9], [10] where the authors used this estimate to establish observability inequality and deduce a controllability result by one control force.
Following Corollary 3.8 of Karapınar and Samet [6], we deduce an existing result on generalized metric spaces endowed with a partial order.
Now, we deduce an analogous result to Theorem 3.1 of Alotaibi and Alsulami [34] (Theorem 6) without g-mixed monotone property of F. Corollary 18 Let ( X, ⪯ ) be a partially ordered set and suppose there is a metric d on X such that ( X, d ) is a complete metric space.
In the following theorem, we use the estimate (30) in order to deduce a continuous dependence result upon initial data.
Besides, it is easy to deduce a similar absorption result for eta -1)=(eta -1},eta_{2})= eta_{(-1},u_{t}(-1)+ varepsilon u(-1)bigr)^{T} instead of (psi(-1)).
From McMullen's decomposition of the space V al stated in Corollary 3.1, one immediately deduces a corresponding result for the space BV al : Corollary 6.1 BV al = ⨁ i, j = 0 n BV al i, j.
We deduce a uniqueness and existence result for the solution of the related Dirichlet problem.
As an application of our almost preservation results we deduce a variety of gap and smoothing results of independent interest, including a classification for non-collapsed manifolds with almost non-negative curvature operator and a smoothing result for singular spaces coming from sequences of manifolds with lower curvature bounds.
