and Applied Analysis 3 and using 1.9 we have ( tanpt )′ 1 − sinpt ( cospt )′ cospt 1 ∣ ∣tanpt ∣ ∣. 1.10 Like for p 2, tanpt > t for t ∈ 0, πp/2 and tanpt < t for t ∈ −πp/2, 0 , which is equivalent to ∣ ∣sinp ∣ ∣p > tΦ ( sinpt ) cospt 1.11 for t ∈ −πp/2, πp/2 , t / 0. A similar formula to 1.10 for cotp is related to the Riccati equation associated with 1.1 . Namely, if x t / 0 is a solution of 1...

The purpose of this study is to establish Liouville-type results for a three-dimensional incompressible, unsteady flow described by the Eyring–Powell fluid equations. studied in plane ?p while it moves along z-axis. Therefore main functions analyze are given u(x,y,z,t) and v(x,y,z,t), belonging ?p. obtained globally bounded initial data as well their corresponding derivatives, variations veloci...

Iterated Brownian motion Zt serves as a physical model for diffusions in a crack. If τD(Z) is the first exit time of this processes from a domain D ⊂ Rn, started at z ∈ D, then Pz [τD(Z) > t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of Pz[τPα(Z) > t] which gives exponential integrability of τPα(Z) for parabola-shaped domains ...

This report describes the normalized intimal-medial uptakes [uptake {M, mg • cm~)-r-serum concentration (c0, mg • cm" )] of I-albumin, I-low-density lipoprotein (LDL), and in vivo Evans blue dye (EBD)-albumin complex as functions of pressure (P), time (t), molecular species (i), and location (z) along a ventral longitudinal z axis of the normal, intact, aortic endothelial surface in adult normo...

*Correspondence: [email protected] 3Department of Mathematics, Abdulwali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan Full list of author information is available at the end of the article Abstract In this manuscript, we investigate some appropriate conditions which ensure the existence of at least one solution to a class of fractional order differential equations (FDEs) provided ...

Fix an odd prime p, an algebraic closure Qp, and an isomorphism C ∼ → Qp. Fix an integer N ≥ 1 prime to p, and let T be the polynomial algebra over Z generated by the operators T!, ! ! Np, Up and 〈d〉 , d ∈ (Z/NZ). Set W = Spf(Zp[[Zp ]]), and let W = W rig be the rigid analytic space of characters of Zp together with its universal character χW : Z × p → O(W ) ; we embed Z in W (Qp) by mapping k ...

and Applied Analysis 3 here g t ≤ t∗ ≤ t.Hence x l ( g t ) ≥ x n−1 t∗ n − l − 1 ! t − g t n−l−1 ≥ x n−1 t n − l − 1 ! t − g t n−l−1. 1.7 Combining 1.7 and 1.5 , we see that x′ ( g t ) ≥ g l−1 t 2l−1 l − 1 ! n − l − 1 ! n−1 t t − g t n−l−1 1.8 for all large t. Lemma 1.2. Suppose the linear third-order differential equation z′′′ p t z 0, t ∈ I 1.9 has an eventually positive increasing solution on...

Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k, with char k = 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P . Let u = Lie(U) and let z denote the center of Lie(L). Let T be a maximal torus in L with Lie algebra t. Then the root system of (g, t) is a subset of t∗ and by restri...

and Applied Analysis 3 to BVP 1.1 when T ∞, which we call the infinite case. Some explicit examples are also given in the last section to illustrate our main results. 2. Preliminaries For the convenience of the readers, we provide here some definitions and lemmas which are important in the proof of our main results. Ge-Mawhin’s continuation theorem and the modified one are also stated in this s...