Showing papers in "Open Mathematics in 2020"

Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function

Abstract: Abstract The primary objective of this research is to establish the generalized fractional integral inequalities of Hermite-Hadamard-type for MT-convex functions and to explore some new Hermite-Hadamard-type inequalities in a form of Riemann-Liouville fractional integrals as well as classical integrals. It is worth mentioning that our work generalizes and extends the results appeared in the literature.

On more general forms of proportional fractional operators

Abstract: In this article, we propose new proportional fractional operators generated from local proportional derivatives of a function with respect to another function. We present some properties of these fractional operators which can be also called proportional fractional operators of a function with respect to another function or proportional fractional operators with dependence on a kernel function.

An efficient approach for the numerical solution of fifth-order KdV equations

Abstract: Abstract The main aim of this article is to use a new and simple algorithm namely the modified variational iteration algorithm-II (MVIA-II) to obtain numerical solutions of different types of fifth-order Korteweg-de Vries (KdV) equations. In order to assess the precision, stability and accuracy of the solutions, five test problems are offered for different types of fifth-order KdV equations. Numerical results are compared with the Adomian decomposition method, Laplace decomposition method, modified Adomian decomposition method and the homotopy perturbation transform method, which reveals that the MVIA-II exceptionally productive, computationally attractive and has more accuracy than the others.

Inverse Sturm-Liouville problem with analytical functions in the boundary condition

Abstract: The inverse spectral problem is studied for the Sturm-Liouville operator with a complex-valued potential and arbitrary entire functions in one of the boundary conditions. We obtain necessary and sufficient conditions for uniqueness, and develop a constructive algorithm for the inverse problem solution. The main results are applied to the Hochstadt-Lieberman half-inverse problem. As an auxiliary proposition, we prove local solvability and stability for the inverse Sturm-Liouville problem by the Cauchy data in the non-self-adjoint case.

On multivalued Suzuki-type θ-contractions and related applications

Abstract: Abstract In this study, we develop the concept of multivalued Suzuki-type θ-contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

Abstract: Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

Analysis of F-contractions in function weighted metric spaces with an application

Abstract: Abstract In this work, we show that the existence of fixed points of F-contraction mappings in function weighted metric spaces can be ensured without third condition ( F 3 ) (F3) imposed on Wardowski function F :(0, ∞ ) → ℜ F\\mathrm{:(0,\\hspace{0.33em}}\\infty )\\to \\Re . The present article investigates (common) fixed points of rational type F-contractions for single-valued mappings. The article employs Jleli and Samet’s perspective of a new generalization of a metric space, known as a function weighted metric space. The article imposes the contractive condition locally on the closed ball, as well as, globally on the whole space. The study provides two examples in support of the results. The presented theorems reveal some important corollaries. Moreover, the findings further show the usefulness of fixed point theorems in dynamic programming, which is widely used in optimization and computer programming. Thus, the present study extends and generalizes related previous results in the literature in an empirical perspective.

Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions

Abstract: Abstract: In this paper, we study boundary value problems of fractional integro-differential equations and inclusions involving Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray-Schauder in the single-valued case, while Martelli’s fixed point theorem, nonlinear alternative for multi-valued maps, and Covitz-Nadler fixed point theorem are used in the inclusion case. Examples illustrating the obtained results are also presented.

Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis

Abstract: Abstract We examine the existence and uniqueness of solutions to two-point boundary value problems involving fourth-order, ordinary differential equations. Such problems have interesting applications to modelling the deflections of beams. We sharpen traditional results by showing that a larger class of problems admit a unique solution. We achieve this by drawing on fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem. The idea is to utilize two metrics on a metric space, where one pair is complete. Our theoretical results are applied to the area of elastic beam deflections when the beam is subjected to a loading force and the ends of the beam are either both clamped or one end is clamped and the other end is free. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.

The core inverse and constrained matrix approximation problem

Abstract: In this paper,we study the constrained matrix approximation problem in the Frobenius norm by using the core inverse:\begin{align} onumber \left\|{Mx - b} \right\|_F=\min\ \ {\rm subject\ to} \ \ {x\in\mathcal{R}(M)} ,\end{align} where $M\in\mathbb{C}^{\texttt{CM}}_n$. We get the unique solution to the problem, provide two Cramer's rules for the unique solution, and establish two new expressions for the core inverse.

Hardy’s inequalities and integral operators on Herz-Morrey spaces

Abstract: Abstract We obtain some estimates for the operator norms of the dilation operators on Herz-Morrey spaces. These results give us the Hardy’s inequalities and the mapping properties of the integral operators on Herz-Morrey spaces. As applications of this general result, we have the boundedness of the Hadamard fractional integrals on Herz-Morrey spaces. We also obtain the Hilbert inequality on Herz-Morrey spaces.

On a functional equation that has the quadratic-multiplicative property

Abstract: Abstract In this article, we obtain the general solution and prove the Hyers-Ulam stability of the following quadratic-multiplicative functional equation: ϕ ( s t − u v ) + ϕ ( s v + t u ) = [ ϕ ( s ) + ϕ ( u ) ] [ ϕ ( t ) + ϕ ( v ) ] \\phi (st-uv)+\\phi (sv+tu)={[}\\phi (s)+\\phi (u)]{[}\\phi (t)+\\phi (v)] by using the direct method and the fixed point method.

Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation

Abstract: Abstract In this article, meromorphic exact solutions for the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff (gCBS) equation are obtained by using the complex method. With the applications of our results, traveling wave exact solutions of the breaking soliton equation are achieved. The dynamic behaviors of exact solutions of the (2 + 1)-dimensional gCBS equation are shown by some graphs. In particular, the graphs of elliptic function solutions are comparatively rare in other literature. The idea of this study can be applied to the complex nonlinear systems of some areas of engineering.

Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group

Abstract: Abstract In this article, we study the commutators of Hausdorff operators and establish their boundedness on the weighted Herz spaces in the setting of the Heisenberg group.

Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation

Abstract: Abstract This paper focuses on the existence and the multiplicity of classical radially symmetric solutions of the mean curvature problem: − div ∇ v 1 + | ∇ v | 2 = f ( x , v , ∇ v ) in Ω , a 0 v + a 1 ∂ v ∂ ν = 0 on ∂ Ω , \\left\\{\\begin{array}{ll}-\\text{div}\\left(\\frac{\ abla v}{\\sqrt{1+|\ abla v{|}^{2}}}\\right)=f(x,v,\ abla v)& \\text{in}\\hspace{.5em}\\text{Ω},\\\\ {a}_{0}v+{a}_{1}\\tfrac{\\partial v}{\\partial \ u }=0& \\text{on}\\hspace{.5em}\\partial \\text{Ω},\\end{array}\\right. with Ω \\text{Ω} an open ball in ℝ N {{\\mathbb{R}}}^{N} , in the presence of one or more couples of sub- and super-solutions, satisfying or not satisfying the standard ordering condition. The novel assumptions introduced on the function f allow us to complement or improve several results in the literature.

A standard form in (some) free fields: How to construct minimal linear representations

Abstract: We describe a standard form for the elements in the universal field of fractions of free associative algebras (over a commutative field). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in standard form). This completes "minimal" arithmetics in free fields since "minimal" constructions for the inverse are already known. Although in general it is difficult to transform a minimal linear representation into standard form, using rational operations "carefully" the form can be kept "close" to standard. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians

Abstract: Abstract We consider the classical functional of the Calculus of Variations of the form I(u)=∫ΩF(x,u(x),∇u(x))dx, $$\\begin{array}{} \\displaystyle I(u)=\\int\\limits_{{\\it\\Omega}}F(x, u(x), \ abla u(x))\\,dx, \\end{array}$$ where Ω is a bounded open subset of ℝn and F : Ω × ℝ × ℝn → ℝ is a Carathéodory convex function; the admissible functions u coincide with a prescribed Lipschitz function ϕ on ∂Ω. We formulate some conditions under which a given function in ϕ + W01,p $\\begin{array}{} \\displaystyle W^{1,p}_0 \\end{array}$(Ω) with I(u) < +∞ can be approximated in the W1,p norm and in energy by a sequence of smooth functions that coincide with ϕ on ∂Ω. As a particular case we obtain that the Lavrentiev phenomenon does not occur when F(x, u, ξ) = f(x, u) + h(x, ξ) is convex and x ↦ F(x, 0, 0) is sufficiently smooth.

On the well-posedness of differential quasi-variational-hemivariational inequalities

Abstract: Abstract The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.

Some results in cone metric spaces with applications in homotopy theory

Abstract: Abstract The self-mappings satisfying implicit relations were introduced in a previous study [Popa, Fixed point theorems for implicit contractive mappings, Stud. Cerc. St. Ser. Mat. Univ. Bacău 7 (1997), 129–133]. In this study, we introduce self-operators satisfying an ordered implicit relation and hence obtain their fixed points in the cone metric space under some additional conditions. We obtain a homotopy result as an application.

Detectable sensation of a stochastic smoking model

Abstract: Abstract This paper is related to the stochastic smoking model for the purpose of creating the effects of smoking that are not observed in deterministic form. First, formulation of the stochastic model is presented. Then the sufficient conditions for extinction and persistence are determined. Furthermore, the threshold of the proposed stochastic model is discussed, when noises are small or large. Finally, the numerical simulations are shown graphically with the software MATLAB.

On a predator-prey system interaction under fluctuating water level with nonselective harvesting

Abstract: Abstract A predator-prey model interaction under fluctuating water level with non-selective harvesting is proposed and studied in this paper. Sufficient conditions for the permanence of two populations and the extinction of predator population are provided. The non-negative equilibrium points are given, and their stability is studied by using the Jacobian matrix. By constructing a suitable Lyapunov function, sufficient conditions that ensure the global stability of the positive equilibrium are obtained. The bionomic equilibrium and the optimal harvesting policy are also presented. Numerical simulations are carried out to show the feasibility of the main results.

On the determination of the number of positive and negative polynomial zeros and their isolation

Abstract: A novel method with two variations is proposed with which the number of positive and negative zeros of a polynomial with real coefficients and degree $n$ can be restricted with significantly better determinacy than that provided by the Descartes rule of signs and also isolate quite successfully the zeros of the polynomial. The method relies on solving equations of degree smaller than that of the given polynomial. One can determine analytically the exact number of positive and negative zeros of a polynomial of degree up to and including five and also fully isolate the zeros of the polynomial analytically and with one of the variations of the method, one can analytically approach polynomials of degree up to and including nine by solving equations of degree no more than four. For polynomials of higher degree, either of the two variations of the method should be applied recursively. Full classification of the roots of the cubic equation, together with their isolation intervals, is presented. Numerous examples are given.

(p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group

Abstract: Abstract The paper deals with the existence of solutions for ( p , Q ) (p,Q) coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set in ℍ n {{\\mathbb{H}}}^{n} .

Further new results on strong resolving partitions for graphs

Abstract: Abstract A set W of vertices of a connected graph G strongly resolves two different vertices x, y ∉ W if either d G (x, W) = d G (x, y) + d G (y, W) or d G (y, W) = d G (y, x) + d G (x, W), where d G (x, W) = min{d(x,w): w ∈ W} and d(x,w) represents the length of a shortest x − w path. An ordered vertex partition Π = {U 1, U 2,…,U k } of a graph G is a strong resolving partition for G, if every two different vertices of G belonging to the same set of the partition are strongly resolved by some other set of Π. The minimum cardinality of any strong resolving partition for G is the strong partition dimension of G. In this article, we obtain several bounds and closed formulae for the strong partition dimension of some families of graphs and give some realization results relating the strong partition dimension, the strong metric dimension and the order of graphs.

Explicit determinantal formula for a class of banded matrices

Abstract: Abstract In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

An extension of the method of brackets. Part 2

Abstract: Abstract The method of brackets, developed in the context of evaluation of integrals coming from Feynman diagrams, is a procedure to evaluate definite integrals over the half-line. This method consists of a small number of operational rules devoted to convert the integral into a bracket series. A second small set of rules evaluates this bracket series and produces the result as a regular series. The work presented here combines this method with the classical Mellin transform to extend the class of integrands where the method of brackets can be applied. A selected number of examples are used to illustrate this procedure.

Bounds on F-index of tricyclic graphs with fixed pendant vertices

Abstract: Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in Ωnα $\\begin{array}{} \\displaystyle {\\it\\Omega}^{\\alpha}_n \\end{array}$, where Ωnα $\\begin{array}{} \\displaystyle {\\it\\Omega}^{\\alpha}_n \\end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e 8n+12α+76≤F(G)≤8(n−1)−7α+(α+6)3 for each G∈Ωnα. $$\\begin{array}{} \\displaystyle 8n+12\\alpha +76\\leq F(G)\\leq 8(n-1)-7\\alpha + (\\alpha+6)^3 ~\\mbox{for each}~ G\\in {\\it\\Omega}^{\\alpha}_n. \\end{array}$$

On surrounding quasi-contractions on non-triangular metric spaces

Abstract: Abstract The aim of this paper is to establish some fixed point results for surrounding quasi-contractions in non-triangular metric spaces. Also, we prove the Banach principle of contraction in non-triangular metric spaces. As applications of our theorems, we deduce certain well-known results in b-metric spaces as corollaries.