Arabian Journal of Mathematics

The aim of this paper is to study the existence and uniqueness of periodic solutions for a certain type of nonlinear fractional pantograph differential equation with a $$\psi $$ -Caputo derivative. The proofs are based on the coincidence degree theory of Mawhin. To show the efficiency of the results, some illustrative examples are included.

In this paper, we establish sufficient conditions for the existence of local solutions for a class of Cauchy type problems with arbitrary fractional order. The results are established by the application of the contraction mapping principle and Schaefer’s fixed point theorem. An example is provided to illustrate the applicability of the results.

Let G be a doubly connected domain in the complex plane $$\mathbb {C}$$ , bounded by Ahlfors 1-regular curves. In this study the approximation of the functions by Faber–Laurent rational functions in the $$\omega $$ -weighted generalized grand Smirnov classes $$\mathcal {E}^{p),\theta }(G,\omega )$$ in the term of the rth $$,~r=1,2\ldots ,$$ mean modulus of smoothness are investigated.

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.

In the present paper, we consider a special class of third-order polynomial evolutionary equations. These equations, via Lie theory admit the same one-parameter point transformations which leave the equations invariant. Reductions with these invariant functions lead to highly nonlinear third-order ordinary differential equations. We use a power series to establish interesting solutions to the reduced equations, whereby recurrence relations occur and convergence of the series may be tested. Finally, the conserved vectors of the class are constructed.

In this work, we study cyclic codes that have generators as Fibonacci polynomials over finite fields. We show that these cyclic codes in most cases produce families of maximum distance separable and optimal codes with interesting properties. We explore these relations and present some examples. Also, we present applications of these codes to secret sharing schemes.

In this paper, we study the class of rings in which every flat ideal is finitely projective. We investigate the stability of this property under localizations and homomorphic images, and its transfer to various contexts of constructions such as direct products, amalgamation of rings $${A \bowtie^{f} J}$$ , and trivial ring extensions. Our results generate examples which enrich the current literature with new and original families of non-coherent rings that satisfy this property.

In this paper, we prove the following result: Let $$\mathcal {F}$$ be a family of meromorphic functions on a domain D and let $$S=\left\{ \varphi _i:1\le i \le 5\right\} $$ be a set of five distinct meromorphic functions on D. If for each $$f \in \mathcal {F}$$ and $$z_0 \in D$$ , there is a constant $$M>0$$ such that $$f^{\#}(z_0) \le M$$ whenever $$f(z_0)= \varphi (z_0)$$ for some $$\varphi \in S$$ and if $$f(z_0) \ne \varphi (z_0)$$ for all $$\varphi \in S$$ whenever $$\varphi _i(z_0) = \varphi _j(z_0) $$ for some $$i,j \in \left\{ 1,2,3,4,5\right\} $$ with $$i \ne j$$ , then $$\mathcal {F}$$ is normal on D. Further we extend this result to the case where the set S contains fewer functions. In particular, our result generalizes the most significant theorem of Lappan (i.e. Lappan’s five point theorem).

We study the weak convergence of infinite products of convex combinations of operators in complete CAT(0) spaces. We provide a new approach to this problem by considering a constructive selection of convex combinations in CAT(0) spaces that does not depend on the order of the involved elements and retain continuity properties with respect to them.