Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

In this paper, we firstly establish the existence and uniqueness of solutions of the operator equation $A(x,x)+ B(x,x)+C(x)+e = x$ A ( x , x ) + B ( x , x ) + C ( x ) + e = x , where A and B are two mixed monotone operators, C is a decreasing operator, and $e\in P$ e ∈ P with $\theta \leq e \leq h$ θ ≤ e ≤ h . Then, using our abstract theorem, we prove a class of fractional boundary value problems with the derivative term to have a unique solution and construct the corresponding iterative sequences to approximate the unique solution.

Fefferman–Stein Inequalities for the Hardy–Littlewood Maximal Function on the Infinite Rooted k-ary Tree

In this paper, weighted endpoint estimates for the Hardy–Littlewood maximal function on the infinite rooted $k$-ary tree are provided. Motivated by Naor and Tao [ 23], the following Fefferman–Stein estimate $$\begin{align*}& w\left(\left\{ x\in T\,:\,Mf(x)>\lambda\right\} \right)\leq c_{s}\frac{1}{\lambda}\int_{T}|f(x)|M(w^{s})(x)^{\frac{1}{s}}\: \text{d}x\qquad s>1\end{align*}$$is settled, and moreover, it is shown that it is sharp, in the sense that it does not hold in general if $s=1$. Some examples of nontrivial weights such that the weighted weak type $(1,1)$ estimate holds are provided. A strong Fefferman–Stein-type estimate and as a consequence some vector-valued extensions are obtained. In the appendix, a weighted counterpart of the abstract theorem of Soria and Tradacete [ 38] on infinite trees is established.

Note on Limit-Periodic Solutions of the Difference Equation xt1-[h(xt)λ]xt=rt, λ>1++

As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied.

Approximations by differences of lower semicontinuous functions

A classical theorem of W. Sierpiński, S. Mazurkiewicz and S. Kempisty says that the class of all differences of lower semicontinuous functions is uniformly dense in the space of all Baire-one functions. We show a generalization of this result to more general situations and derive an abstract theorem in the case of a binormal topological space.

A Priori Bounds and Existence of Solutions for Slightly Superlinear Elliptic Problems

We consider the semilinear elliptic problemwhere a is a continuous function which may change sign and f is superlinear but does not satisfy the standard Ambrosetti-Rabinowitz condition. We show that if f is regularly varying of index one at infinity then the above problem has a positive solution, provided α satisfies some additional assumptions. Our proof uses an abstract theorem due to L. Jeanjean on critical points of functionals with mountain-pass structure, and it relies on the obtention of a priori bounds for positive solutions..