In Example 1, we describe a subset X of the plane and a function on X which has a ${}^k$-extension to the whole ${ℝ}^2$ for each finite, but has no ${}^{\infty }$-extension to ${ℝ}^2$. In Example 2, we construct a similar example of a subanalytic subset of ${ℝ}^5$; much more sophisticated than the first one. The dimensions given here are smallest possible.

In this paper, we present some results concerning the existence and the attractivity of solutions for some functional integral equations of Riemann-Liouville fractional order, by using an extension of the Burton-Kirk fixed point theorem in the case of a Fréchet space.

The system of nonlinear differential equations $$x^{\text{'}}+p_1\left(t\right)x^{{\alpha }_1}+q_1\left(t\right)y^{{\beta }_1}=0,y^{\text{'}}+p_2\left(t\right)x^{{\alpha }_2}+q_2\left(t\right)y^{{\beta }_2}=0,A$$ is under consideration, where ${\alpha }_i$ and ${\beta }_i$ are positive constants and $p_i\left(t\right)$ and $q_i\left(t\right)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $\left(x\right(t),y(t\left)\right)$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t\to \infty$, which can be...

We study the existence and uniqueness of integrable solutions to fractional Langevin equations involving two fractional orders with initial value problems. Our results are based on Schauder's fixed point theorem and the Banach contraction principle fixed point theorem. Examples are provided to illustrate the main results.

Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary...

This paper is concerned with a class of nonlinear difference inequalities which include many different classes of difference inequalities and equations as special cases. By means of a Riccati type transformation, necessary and sufficient conditions for the existence of eventually positive solutions and positive nonincreasing solutions are obtained. Various type of comparison theorems are also derived as applications, which extends many theorems in the literature.

We consider the existence of at least one positive solution to the dynamic boundary value problem $$\begin{array}{cccc}\hfill -y^{\Delta \Delta }\left(t\right)& =\lambda f(t,y\left(t\right))\text{,}t\in {[0,T]}_{𝕋}y\left(0\right)\hfill & \hfill ={\int }_{{\tau }_1}^{{\tau }_2}{F}_1(s,y\left(s\right))\Delta sy\left({\sigma }^2\left(T\right)\right)& ={\int }_{{\tau }_3}^{{\tau }_4}{F}_2(s,y\left(s\right))\Delta s,\hfill \end{array}$$ where $𝕋$ is an arbitrary time scale with $0<{\tau }_1<{\tau }_2<{\sigma }^2\left(T\right)$ and $0<{\tau }_3<{\tau }_4<{\sigma }^2\left(T\right)$ satisfying ${\tau }_1$, ${\tau }_2$, ${\tau }_3$, ${\tau }_4\in 𝕋$, and where the boundary conditions at $t=0$ and $t={\sigma }^2\left(T\right)$ can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.

In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\Im (u)$, to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\Im (u)$ are reduced to simpler form $H(v)$ and $\mathscr{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u)\le \Im (u)$, $H(v)\le \mathscr{H}(v)$, we obtain the existence of the absolute minimum for the Serrin forms $\Im (u)$ and $\mathscr{H}(v)$, and...

Motivated by Vityuk and Golushkov (2004), using the Schauder Fixed Point Theorem and the Contraction Principle, we consider existence and uniqueness of positive solution of a singular partial fractional differential equation in a Banach space concerning with fractional derivative.