By Theorem 2.4, there exists a fixed point-free mapping such that it is of type (P).
Since is an infinite sequence of integers, by Dirichlet principle, there exists a fixed and a subsequence of, still denoted by, such that and.
In [34], Arriola and Beyer showed that, if is a continuous mapping for which there exists a fixed such that for all, then there exists a unique additive mapping such that for all.
Thus, all the hypotheses of the Schauder-Tychonoff fixed point theorem are fulfilled and so there exists a fixed point (x t)inmathcal{X} ) of (mathcal{F}), which satisfies integral equation x t)=x_{0}+int_{T_{0}}^{t} varphi^{-1}biggl(p s)^{-1} int_{s}^{infty}q(r psi bigl(x(r bigr),dr biggr),ds,quad tgeq T_{0}.
As shown in the proof of Theorem 2.3, there exists a fixed point (x^) under the hypotheses of Theorem 2.11.
If T is strongly order preserving, then there exists a fixed point in ([!![a,b ]!!]) which is stable relative to ([!![a,b ]!!]).
Thus, (mathcal{A}) satisfies the hypotheses of Lemma 3.2, then there exists a fixed point of (mathcal{A}) which is a solution of (3.1).
Second, and in accordance with the previous assumption, let us consider that for all medical dispositions there exists a fixed and finite number of concerns for health (Fge 2).
Since the index is not 0, the existence property implies that there exists a fixed point (x in K) such that rho < Vert x Vert < r. □.
We can apply the Ky Fan fixed point theorem [33] to Φ, and we obtain that there exists a fixed point λ ∗ ∈ Φ.
This test is accurate when there exists a fixed point with two bumps in which the set of active excitatory cells is the set of excitatory cells that are active in either embedded activity pattern, or (S= overline {S_{1}}cup overline {S_{2}}).
