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In this paper we define and study the weak pseudo-BCK algebras as generalizations of weak BCK-algebras, extending some results given by Cı⃖rulis for weak BCK-algebras. We give some characterizations of weak pseudo-BCK algebras and we prove that a weak pseudo-BCK algebra satisfying the right distribution laws is a BCK-algebra. We define the class of commutative weak pseudo-BCK algebras, and we give equivalent definitions and characterization theorems for commutative weak pseudo-BCK algebras. The classes of quasi pseudo-BCK algebras and weak pseudo-BCK(E) algebras are introduced and a characterization theorem for quasi pseudo-BCK algebras is given. We prove that any weak pseudo-BCK(E) algebra is a pseudo-BE algebra and the class of commutative weak pseudo-BCK(E) algebras is equivalent to the class of commutative pseudo-BCK algebras. [ABSTRACT FROM AUTHOR]

In this paper, we improve the famous Reid inequality related to linear operators. Some monotony results for positive operators are also established with a different approach from what is known in the existing literature. Lastly, Reid's (and Halmos-Reid's) inequalities are extended to unbounded operators. [ABSTRACT FROM AUTHOR]

In this paper, several sufficient conditions for oscillation of all solutions of fourth order functional differential equations of neutral type of the form (r (t) (y (t) + p (t) y (t − τ)) ″ ) ″ + q (t) G (y (t − σ)) = 0 $$\begin{array}{} \displaystyle \bigl(r(t)(y(t)+p(t)y(t-\tau))''\bigr)''+q(t)G\bigl(y(t-\sigma)\bigr)=0 \end{array}$$ are studied under the assumption ∫ 0 ∞ t r (t) d t = ∞ $$\begin{array}{} \displaystyle \int\limits^{\infty}_{0}\frac{t}{r(t)}{\rm d} t =\infty \end{array}$$ [ABSTRACT FROM AUTHOR]

In this paper, by using the concept of changing-periodic time scales and composition theorem of time scales introduced in 2015, we establish a local phase space for functional dynamic equations with infinite delay (FDEID) on an arbitrary time scale with a bounded graininess function μ. Through Krasnoseľskiĭ's fixed point theorem, some sufficient conditions for the existence of local-periodic solutions for FDEID are established for the first time. This research indicates that one can extract a local-periodic solution for dynamic equations on an arbitrary time scale with a bounded graininess function μ through some index function. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some Cauchy problems involving a left-sided Hadamard-type fractional derivative. A theorem on the existence of a unique solution to a nonlinear problem is proved. The main result is obtained using a fixed point theorem due to Banach, as well as the Bielecki norm. A Cauchy formula for the solution of the linear problem is derived. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve . Complete asymptotic expansions will be constructed if that order is constant along , and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested. [ABSTRACT FROM AUTHOR]

Recently, the bounded pseudo-hoops with internal states have been defined and studied. Some of results are based on the hypothesis that a bounded pseudo-hoop has the Glivenko and (mN) properties. In this paper we show that this hypothesis is super uous, namely every good pseudo-hoop satisfies the Glivenko and (mN) properties and those results are reformulated without that hypothesis. The pointed pseudo-hoops are introduced and studied and the notion of state operator has been generalized for the case of unbounded pseudo-hoops. We define the Bosbach and Riečan states on pointed pseudo-hoops and we make some considerations regarding the relationship between the state operators and states on these structures. [ABSTRACT FROM AUTHOR]