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In this paper, generalized versions of Hadamard and Fejér–Hadamard type fractional integral inequalities are obtained. By using generalized fractional integrals containing Mittag-Leffler functions, some well-known results for convex and harmonically convex functions are generalized. The results of this paper are connected with various published fractional integral inequalities.

In this paper, we introduce a Halpern algorithm and a nonconvex combination algorithm to approximate a solution of the split common fixed problem of quasi- ϕ -nonexpansive mappings in Banach space. In our algorithms, the norm of linear bounded operator does not need to be known in advance. As the application, we solve a split equilibrium problem in Banach space. Finally, some numerical examples are given to illustrate the main results in this paper and compare the computed results with other ones in the literature. Our results extend and improve some recent ones in the literature.

This paper concerns on two types of integral boundary value problems of a nonlinear fractional differential system, i . e ., nonlocal strip integral boundary value problems and coupled integral boundary value problems. With the aid of the monotone iterative method combined with the upper and lower solutions, the existence of extremal system of solutions for the above two types of differential systems is investigated. In addition, a new comparison theorem for fractional differential system is also established, which is crucial for the proof of the main theorem of this paper. At the end, an example explaining how our studies can be used is also given.

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

Recently, a unified integral operator has been introduced by Farid, 2020, which produces several kinds of known fractional and conformable integral operators defined in recent decades (Kwun, 2019, Remarks 6 and 7). The aim of this paper is to establish bounds of this unified integral operator by means of α,m-convex functions. The resulting inequalities provide the bounds of all associated fractional and conformable integral operators in a compact form. Also, the results of this paper hold for different kinds of convex functions connected with α,m-convex functions.

In this paper, the fixed-point theorem for monotone contraction mappings in the setting of a uniformly convex smooth Banach space is studied. This paper provides a version of the Banach fixed-point theorem in a complete metric space.

The main purpose of this paper is to study almost α -cosymplectic pseudo metric manifold satisfying certain η -parallel tensor fields. We first focus on the concept of almost α -cosymplectic pseudo metric manifold and its curvature properties. Then, we obtain some results related to the η -parallelity of h , φ h , and τ . Moreover, the deformation of almost α -Kenmotsu pseudo metric structure is given. We conclude the paper with an illustrative example of almost α -cosymplectic pseudo metric manifold.

Textile dyes are untreated discharge into the environment which results in a significant increase in water pollution levels worldwide. Due to the continuous addition of toxic organic dyes, a necessary strategic model is required for the complete degradation of dyes in textile effluent. This paper considers the possibility of biological synthesis of silver and iron nanoparticles and their use in photocatalytic degradation. The immediate change of silver nitrate solution occurring from colorless to brown is observed after the addition of the aqueous leaf extract, indicating the successive reduction of Ag+ ions to the Ag nanoparticles. These formed Ag nanoparticles were subjected to examine the photocatalytic activity under the solar radiation for the degradation of methyl orange. Green synthesized Ag nanoparticles were found to successfully degrade methyl orange up to 95% between 70 hours than the initial exposure time. The absorbance of methyl orange was measured at 465 nm. The present paper is focused on fractional mathematical modelling of dye degradation in textile effluents using the Caputo–Fabrizio fractional derivative without the singular kernel. The iterative Laplace transform method is employed to obtain an analytic solution for the absorption transport equation. The obtained experimental results showing significant removal of dyes from textile wastewater are compared using modelling results. The innovative approach is in outstanding agreement with the findings of the experiment. The mathematical modelling for the dye removal process helps to design suitable environmental management studies to reduce the adverse effect caused by toxic wastewater. Model validation has been shown by comparing analytical simulated solutions with experimental results for photocatalytic degradation using silver and iron nanoparticles as eco-friendly and low-cost agents.

In this paper, the authors investigated the concept of s , m -exponential-type convex functions and their algebraic properties. New generalizations of Hermite–Hadamard-type inequality for the s , m -exponential-type convex function ψ and for the products of two s , m -exponential-type convex functions ψ and ϕ are proved. Many refinements of the (H–H) inequality via s , m -exponential-type convex are obtained. Finally, several new bounds for special means and new error estimates for the trapezoidal and midpoint formula are provided as well. The ideas and techniques of this paper may stimulate further research in different areas of pure and applied sciences.

This paper introduces the concept of the theta cone metric, studies its various topological properties, and gives some examples of it. Furthermore, it proves some lemmas and then uses them to give further generalizations of some well-known fixed point theorems. Specifically, Theorem 2 of the paper is a generalization of Reich’s fixed point theorem.

The theory of convex functions plays an important role in engineering and applied mathematics. The Caputo–Fabrizio fractional derivatives are one of the important notions of fractional calculus. The aim of this paper is to present some properties of Caputo–Fabrizio fractional integral operator in the setting of h -convex function. We present some new Caputo–Fabrizio fractional estimates from Hermite–Hadamard-type inequalities. The results of this paper can be considered as the generalization and extension of many existing results of inequalities and convex functions. Moreover, we also present some application of our results to special means of real numbers.

The paper is concerned with the dynamics behavior of positive solutions for the following max-type fuzzy difference equation system: xn+1=maxA/xn, A/xn−1, xn−2, n=0,1,2,…, where xn is a sequence of positive fuzzy numbers, and the parameter A and the initial conditions x−2, x−1, x0 are also positive fuzzy numbers. Firstly, the fuzzy set theory is used to transform the fuzzy difference equation into the corresponding ordinary difference equations with parameters. Then, the expression for the periodic solution of the max-type ordinary difference equations is obtained by the iteration, the inequality technique, and the mathematical induction. Moreover, we can obtain the expression for the periodic solution of the max-type fuzzy difference equation. In addition, the boundedness and persistence of solutions for the fuzzy difference equation is proved. Finally, the results of this paper are simulated and verified by using MATLAB 2016 software package.

The purpose of this paper is to introduce and analyze the shrinking projection algorithm with errors for a finite set of costerro bounded linear mappings in the setting of uniformly convex smooth Banach spaces. Here, under finite dimensional or compactness restriction or the error term being zero, the strong limit point of the sequence stated in the iterative scheme for these mappings in uniformly convex smooth Banach spaces was studied. This paper extends Ezearn and Prempeh’s result for nonexpansive mappings in real Hilbert spaces.

The purpose of this paper is to present some existence and uniqueness results for common fixed point theorems for θ-ϕ contraction mappings with two metrics endowed with a directed graph. In addition, by using our main results, we obtain some results about coupled coincidence points endowed with a directed graph. Our results generalize those presented in previous papers.

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p 1 / 2 + ɛ ≤ n < p , there must exist two primitive roots α and β modulo p with 1 < α , β ≤ n − 1 such that the equation n = α + β holds, where 0 < ɛ < 1 / 2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p .

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

In this paper, we establish some new variants of Leray–Schauder-type fixed point theorems for a 2 × 2 block operator matrix defined on nonempty, closed, and convex subsets Ω of Banach spaces. Note here that Ω need not be bounded. These results are formulated in terms of weak sequential continuity and the technique of De Blasi measure of weak noncompactness on countably subsets. We will also prove the existence of solutions for a coupled system of nonlinear equations with an example.

In this paper, using the variational principle, the existence and multiplicity of solutions for p x , q x -Kirchhoff type problem with Navier boundary conditions are proved. At the same time, the sufficient conditions for the multiplicity of solutions are obtained.

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.

In this paper, some variants of strongly normal closure spaces obtained by using binary relation are introduced, and examples in support of existence of the variants are provided by using graphs. The relationships that exist between variants of strongly normal closure spaces and covering axioms in absence/presence of lower separation axioms are investigated. Further, closure subspaces and preservation of the properties studied under mapping are also discussed.

In this paper, new weighted Hermite–Hadamard type inequalities for differentiable h -convex and quasi h -convex functions are proved. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to s ˘ -divergence and to statistics are given.

In this paper, we perform a further investigation for a weighted average of Gauss sums. By making use of some properties of the cotangent function and the Bernoulli polynomials, we explicitly evaluate the weighted average of Gauss sums in terms of the special values of Dirichlet L -functions at positive integers.

In this paper, based on combinatorial methods and the structure of RFMLR-circulant matrices, we study the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions. Firstly, we give some properties of exponential forms and trigonometric functions. Furthermore, we study Frobenius norms, the lower and upper bounds for the spectral norms of RFMLR-circulant matrices involving exponential forms and trigonometric functions by some ingenious algebra methods, and then we obtain new refined results.

The Uzawa method is an iterative approach to find approximated solutions to the Stokes equations. This method solves velocity variables involving augmented Lagrangian operator and then updates pressure variable by Richardson update. In this paper, we construct a new version of the Uzawa method to find optimal numerical solutions of the Stokes equations including corner singularities. The proposed method is based on the dual singular function method which was developed for elliptic boundary value problems. We estimate the solvability of the proposed formulation and special orthogonality form for two singular functions. Numerical convergence tests are presented to verify our assertion.

In this paper, the conformal super-biderivations of two classes of Lie conformal superalgebras are studied. By proving some general results on conformal super-biderivations, we determine the conformal super-biderivations of the loop super-Virasoro Lie conformal superalgebra and Neveu–Schwarz Lie conformal superalgebra. Especially, any conformal super-biderivation of the Neveu–Schwarz Lie conformal superalgebra is inner.

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

This paper presents a new approach of constructing special ruled surfaces and aims to study their developability and minimalist conditions. Our concept opens opportunities for application in engineering, surface modeling, and architectural design. The principle of our study is to introduce the original definitions of Smarandache ruled surfaces according to Darboux frame of a curve lying on an arbitrary regular surface in E 3 . It concerns T g -Smarandache ruled surface, T n -smarandache ruled surface, and g n -Smarandache ruled surface. New theorems giving necessary and sufficient conditions for those surfaces to be developable and minimal are investigated. Finally, an example with illustrations is presented.

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

In this paper, we introduce the concept of a P b r -cone metric space over Banach algebras and prove some fixed point results under various contractive mappings in such a space. Some examples are given to elucidate the results. Our results extend and generalize many existing results in the literature.

In this paper, we propose an iterative scheme for a special split feasibility problem with the maximal monotone operator and fixed-point problem in Banach spaces. The algorithm implements Halpern’s iteration with an inertial technique for the problem. Under some mild assumption of the monotonicity of the related mapping, we establish the strong convergence of the sequence generated by the algorithm which does not require the spectral radius of A T A. Finally, the numerical example is presented to demonstrate the efficiency of the algorithm.

This paper is concerned with the periodicity of entire functions with finite growth order, and some sufficient conditions are given. Let f is a transcendental entire function with finite growth order, zero is a Picard exceptional value of f , and a given differential monomial Q f of f is periodic, then f is also periodic. We are also interested in finding the following: let f is a transcendental entire function with finite growth order, d is a Picard exceptional value of f and f z Δ c n f z is a periodic function, then f is also a periodic function. These results extend Yang’s conjecture.

This paper introduces the concept of q 1 , q 2 -quasi metric-like space X , p and delves into some topological properties of it. A necessary and sufficient condition related to equivalent Cauchy sequences and Cauchy sequences in X , p have been proved. Furthermore, the results of the fixed point are shown in the setting of q 1 , q 2 -quasi metric-like space as applications of conditions of equivalent Cauchy sequences. Besides, some instances are inclined to epitomize the examined consequences.

Graphs are essential tools to illustrate relationships in given datasets visually. Therefore, generating graphs from another concept is very useful to understand it comprehensively. This paper will introduce a new yet simple method to obtain a graph from any finite affine plane. Some combinatorial properties of the graphs obtained from finite affine planes using this graph-generating algorithm will be examined. The relations between these combinatorial properties and the order of the affine plane will be investigated. Wiener and Zagreb indices, spectrums, and energies related to affine graphs are determined, and appropriate theorems will be given. Finally, a characterization theorem will be presented related to the degree sequences for the graphs obtained from affine planes.

In this paper, we discuss one-dimensional differential equation with ω -period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S . The minimum cardinality of a total dominating set of G is the total domination number γ t G of G . The graph G is total domination edge-critical, or γ t EC , if for every edge e in the complement of G , γ t G + e < γ t G . If G is γ t EC and γ t G = k , we say that G is k t EC . In this paper, we show that every 3 t EC graph with δ G ≥ 2 and α G = κ G + 1 has a Hamilton cycle.

In this paper, we firstly introduce a new notion of inverse C k − class functions which extends the notion of inverse C − class functions introduced by Saleem et al., 2018. Secondly, some common fixed point theorems are stated under some compatible conditions such as weak semicompatible of type A , weak semicompatibility, and conditional semicompatibility in metric spaces. Moreover, we introduce a new kind of compatibility called S τ − compatibility which is weaker than E . A . property and also present a common fixed point theorem in metric spaces via inverse C k − class functions. Some examples are provided to support our results.

In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.

The boundedness and compactness of the inclusion mapping from Besov spaces to tent spaces are studied in this paper. Meanwhile, the boundedness, compactness, and essential norm of the Volterra integral operator T g from Besov spaces to a class of general function spaces are also investigated.

Iterative methods for solving variational inclusions and fixed-point problems have been considered and investigated by many scholars. In this paper, we use the Halpern-type method for finding a common solution of variational inclusions and fixed-point problems of pseudocontractive operators. We show that the proposed algorithm has strong convergence under some mild conditions.

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

In this paper, we use the mean value theorem of Dirichlet L -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.

An L h , k -labeling of a graph G = V , E is a function f : V ⟶ 0 , ∞ such that the positive difference between labels of the neighbouring vertices is at least h and the positive difference between the vertices separated by a distance 2 is at least k . The difference between the highest and lowest assigned values is the index of an L h , k -labeling. The minimum number for which the graph admits an L h , k -labeling is called the required possible index of L h , k -labeling of G , and it is denoted by λ k h G . In this paper, we obtain an upper bound for the index of the L h , k -labeling for an inverse graph associated with a finite cyclic group, and we also establish the fact that the upper bound is sharp. Finally, we investigate a relation between L h , k -labeling with radio labeling of an inverse graph associated with a finite cyclic group.

In this paper, we use the analytic methods, the properties of the sixth-order characters, and the classical Gauss sums to study the computational problems of a certain special sixth residues’ modulo p and give two exact calculating formulas for them.

In this paper, some classes of concerned multiplication operators consisting of analytic and hyperbolic functions are defined and considered. Furthermore, some properties such as boundedness and compactness of the new operators are discussed. Finally, a general class of weighted hyperbolic Bloch functions is characterized by metric spaces.

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended b -metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α -fuzzy fixed point theorems in the context of extended b -metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α -fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

In this paper, we introduce fractional Boas transforms and discuss some of their properties. We also introduce the notion of wavelets associated with fractional Boas transforms and give some results related to their vanishing moments. Finally, a comparative study of Hilbert transforms and fractional Boas transforms is done.

In this paper, we discuss about various generalizations of α − admissible mappings. Furthermore, we extend the concept of α − admissible to generalize rational α − Geraghty contraction in G − metric space. With this new contraction mapping, we establish some fixed-point theorems in G − metric space. The obtained result is verified with an example.

In this paper, we propose Fourier–Boas-Like wavelets and obtain sufficient conditions for their higher vanishing moments. A sufficient condition is given to obtain moment formula for such wavelets. Some properties of Fourier–Boas-Like wavelets associated with Riesz projectors are also given. Finally, we formulate a variation diminishing wavelet associated with a Fourier–Boas-Like wavelet.

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for α ∈ ℝ / ℚ , β ∈ ℝ , and 0 < θ < 10 / 1561 , there exist infinitely many primes p , such that α p 2 + β < p − θ and p + 2 = P 4 , which constitutes an improvement upon the previous result.