‎in this paper‎, ‎we study the boundary-value problem of fractional‎ ‎order dynamic equations on time scales‎, ‎$$‎ ‎^c{delta}^{alpha}u(t)=f(t,u(t)),;;tin‎ ‎[0,1]_{mathbb{t}^{kappa^{2}}}:=j,;;1

this article is devoted to the study of existence and multiplicity of positive solutions to aclass of nonlinear fractional order multi-point boundary value problems of the type−dq0+u(t) = f(t, u(t)), 1 < q ≤ 2, 0 < t < 1,u(0) = 0, u(1) =m−2∑ i=1δiu(ηi),where dq0+ represents standard riemann-liouville fractional derivative, δi, ηi ∈ (0, 1) withm−2∑i=1δiηi q−1 < 1, and f : [0, 1] × [0, ∞) → [0, ∞...

in this article, we verify existence and uniqueness of positive and nondecreasing solution for nonlinear boundary value problem of fractional differential equation in the form d_{0^{+}}^{alpha}x(t)+f(t,x(t))=0, 0x(0)= x&apos;(0)=0, x&apos;(1)=beta x(xi), where $d_{0^{+}}^{alpha}$ denotes the standard riemann-liouville fractional derivative, 0an illustrative example is also presented.

Despite its name, the Fundamental Theorem of Algebra cannot be a result in pure algebra since the real numbers and hence the complex numbers are not algebraically defined. While there are many proofs, most use some basic facts in complex analysis or plane topology. We describe here a proof based on Galois theory as well as some non-trivial finite group theory, namely the Sylow theorems, but whi...

The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem Adams, Bush and Frick [1] that if f : S 1 → R 2 k + $f: S^1\rightarrow {\mathbb {R}}^{2k+1}$ continuous function on the unit circle S1 in C ${\mathbb {C}}$ such ( − z ) = $f(-z)=-f(z)$ for all ∈ $z\in S^1$ , then there finite subset X diameter at most π / $\pi -\pi /(2k+1)$ (in standard metric which has ...