The aim of this paper is to study unified integral operators for generalized convex functions namely \((\alpha,h-m)\)-convex functions. We obtained upper as well as lower bounds of these integral operators in diverse forms. The results simultaneously hold for many kinds of well known fractional integral operators and for various kinds of convex functions.

The concept of residuated relational systems ordered under a quasi-order relation was introduced in 2018 by S. Bonzio and I. Chajda. In such algebraic systems, we have introduced and developed the concepts of implicative and comparative filters. In addition, we have shown that every comparative filter is an implicative filter at the same time and that converse it does not have to be. In this article, as a continuation of previous research, we introduce the concept of strong quasi-ordered residuated systems and we show that in such systems implicative and comparative filters coincide. In addition, we show that in such systems the concept of least upper bound for any two pair of elements can be determined.

The paper proves convergence for three uniquely defined recursive sequences, namely, arithmetico-geometric sequence, the Newton-Raphson recursive sequence, and the nested/composite recursive sequence. The three main hurdles for this prove processes are boundedness, monotonicity, and convergence. Oftentimes, these processes lie in the predominant use of prove by mathematical induction and also require some bit of creativity and inspiration drawn from the convergence monotone theorem. However, these techniques are not adopted here, rather, as a novelty, extensive use of basic manipulation of inequalities and useful equations are applied in illustrating convergence for these sequences. Moreover, we established a mathematical expression for the limit of the nested recurrence sequence in terms of its leading term which yields favorable results.

The concept of q-rung orthopair fuzzy sets generalizes the notions of intuitionistic fuzzy sets and Pythagorean fuzzy sets to describe complicated uncertain information more effectively. Their most dominant attribute is that the sum of the \(q^{th}\) power of the truth-membership and the \(q^{th}\) power of the falsity-membership must be equal to or less than one, so they can broaden the space of uncertain data. This set can adjust the range of indication of decision data by changing the parameter \(q, ~q\geq 1\). In this paper, we define the Hamacher operations of q-rung orthopair fuzzy sets and proved some desirable properties of these operations, such as commutativity, idempotency, and monotonicity. Further, we proved De Morgan’s laws for these operations over complement. Furthermore, we defined the Hamacher scalar multiplication \(({n._{h}}A)\) and Hamacher exponentiation \((A^{\wedge_{h}n})\) operations on q-rung orthopair fuzzy sets and investigated their algebraic properties. Finally, we defined the necessity and possibility operators based on q-rung orthopair fuzzy sets and some properties of Hamacher operations that are considered.

In this study a convergence analysis for a fast multi-step Chebyshe-Halley-type method for solving nonlinear equations involving Banach space valued operator is presented. We introduce a more precise convergence region containing the iterates leading to tighter Lipschitz constants and functions. This way advantages are obtained in both the local as well as the semi-local convergence case under the same computational cost such as: extended convergence domain, tighter error bounds on the distances involved and a more precise information on the location of the solution. The new technique can be used to extend the applicability of other iterative methods. The numerical examples further validate the theoretical results.