Abstract: Quantiles of random variable are crucial quantities that give more delicate information about distribution than mean and median and so on. We establish Jensen’s inequality for q-quantile ( \(q\geq 0.5\) ) of a random variable, which includes as a special case Merkle (Stat. Probab. Lett. 71(3):277–281, 2005) where Jensen’s inequality about median (i.e. \(q= 0.5\) ) was given. We also refine this inequality in the case where \(q<0.5\) . An application to the confidence interval of parameters in pivotal quantity is also considered by virtue of the rigorous description on the relationship between quantiles and intervals that have required probability. PubDate: 2021-05-01

Abstract: We establish some interesting refinements of the \((p,q)\) -Hölder integral inequality and the \((p,q)\) -power-mean integral inequality. As applications, we show that some existing \((p,q)\) -integral inequalities can be improved by the results obtained in this paper. PubDate: 2021-04-29

Abstract: In this paper, we propose a modified proximal point algorithm based on the Thakur iteration process to approximate the common element of the set of solutions of convex minimization problems and the fixed points of two nearly asymptotically quasi-nonexpansive mappings in the framework of \(\operatorname{CAT}(0)\) spaces. We also prove the Δ-convergence of the proposed algorithm. We also provide an application and numerical result based on our proposed algorithm as well as the computational result by comparing our modified iteration with previously known Sahu’s modified iteration. PubDate: 2021-04-29

Abstract: In the present paper, we extend the study of (Ali et al. in J. Inequal. Appl. 2020:241, 2020) by using differential equations (García-Río et al. in J. Differ. Equ. 194(2):287–299, 2003; Pigola et al. in Math. Z. 268:777–790, 2011; Tanno in J. Math. Soc. Jpn. 30(3):509–531, 1978; Tashiro in Trans. Am. Math. Soc. 117:251–275, 1965), and we find some necessary conditions for the base of warped product submanifolds of cosymplectic space form \(\widetilde{M}^{2m+1}(\epsilon )\) to be isometric to the Euclidean space \(\mathbb{R}^{n}\) or a warped product of complete manifold N and Euclidean space \(\mathbb{R}\) . PubDate: 2021-04-29

Abstract: In this paper, we establish a finiteness theorem for \(L^{p}\) harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on \(L^{2}\) harmonic 1-forms. PubDate: 2021-04-29

Abstract: In this research, we introduce the notions of \((p,q)\) -derivative and integral for interval-valued functions and discuss their fundamental properties. After that, we prove some new inequalities of Hermite–Hadamard type for interval-valued convex functions employing the newly defined integral and derivative. Moreover, we find the estimates for the newly proved inequalities of Hermite–Hadamard type. It is also shown that the results proved in this study are the generalization of some already proved research in the field of Hermite–Hadamard inequalities. PubDate: 2021-04-29

Abstract: In the present work, we study the second-order neutral differential equation and formulate new oscillation criteria for this equation. Our conditions differ from the earlier ones. Also, our results are expansions and generalizations of some previous results. Examples to illustrate the main results are included. PubDate: 2021-04-27

Abstract: In this work, we establish Lyapunov-type inequalities for the fractional boundary value problems with Caputo–Hadamard fractional derivative subject to multipoint and integral boundary conditions. As far as we know, there is no literature that has studied these problems. PubDate: 2021-04-26

Abstract: We investigate the effect of (interface) surface tensor on the linear Rayleigh–Taylor (RT) instability in stratified incompressible viscous fluids. The existence of linear RT instability solutions with largest growth rate Λ is proved under the instability condition (i.e., the surface tension coefficient ϑ is less than a threshold \(\vartheta _{\mathrm{c}}\) ) by the modified variational method of PDEs. Moreover, we find a new upper bound for Λ. In particular, we directly observe from the upper bound that Λ decreasingly converges to zero as ϑ goes from zero to the threshold \(\vartheta _{\mathrm{c}}\) . PubDate: 2021-04-26

Abstract: This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation \({}^{c}\mathcal{D}_{q}^{\sigma }[k](t) = w (t, k(t), {}^{c} \mathcal{D}_{q}^{\zeta }[k](t) )\) with three-point conditions for \(t \in (0,1)\) on a time scale \(\mathbb{T}_{t_{0}}= \{ t : t =t_{0}q^{n}\}\cup \{0\}\) , where \(n\in \mathbb{N}\) , \(t_{0} \in \mathbb{R}\) , and \(0< q<1\) , based on the Leray–Schauder nonlinear alternative and Guo–Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings. PubDate: 2021-04-23

Abstract: The aim of this paper is to prove the superstability of the following functional equations: $$\begin{aligned}& f \bigl(P(x,y) \bigr)= g(x)h(y), \\& f(x+y)=g(x)h(y). \end{aligned}$$ PubDate: 2021-04-23

Abstract: In this paper, we establish some new characterizations of weighted functions of dynamic inequalities containing a Hardy operator on time scales. These inequalities contain the characterization of Ariňo and Muckenhoupt when \(\mathbb{T}=\mathbb{R}\) , whereas they contain the characterizations of Bennett–Erdmann and Gao when \(\mathbb{T}=\mathbb{N}\) . PubDate: 2021-04-23

Abstract: This paper is concerned with the existence of positive radial solutions of the following resonant elliptic system: $$ \textstyle\begin{cases} -\Delta u=uv+f( \vert x \vert ,u), & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ -\Delta v=cg(u)-dv, & 0< R_{1}< \vert x \vert < R_{2}, x\in \mathbb{R}^{N}, \\ \frac{\partial u}{\partial \textbf{n}}=0= \frac{\partial v}{\partial \textbf{n}},& \vert x \vert =R_{1}, \vert x \vert =R_{2}, \end{cases} $$ where \(\mathbb{R}^{N}\) ( \(N\geq 1\) ) is the usual Euclidean space, n indicates the outward unit normal vector, \(f\in C([R_{1},R_{2}]\times [0,\infty ),\mathbb{R})\) , \(g\in C([0,\infty ),[0,\infty ))\) , and c and d are positive constants. By employing the classical fixed point theory we establish several novel existence theorems. Our main findings enrich and complement those available in the literature. PubDate: 2021-04-23

Abstract: In this article we give two possible generalizations of the Hermite–Hadamard integral inequality for the class of twice differentiable functions, where the convexity property of the target function is not assumed in advance. They represent a refinement of this inequality in the case of convex/concave functions with numerous applications. PubDate: 2021-04-22

Abstract: The main goal of this paper is estimating certain new fractional integral inequalities for the extended Chebyshev functional in the sense of synchronous functions by employing proportional fractional integral (PFI) and Hadamard proportional fractional integral. We establish certain inequalities concerning one- and two-parameter proportional and Hadamard proportional fractional integrals. We also discuss certain particular cases. PubDate: 2021-04-13

Abstract: This work extends the theory of Rychkov, who developed the theory of \(A_{p}^{\mathrm{loc}}\) weights. It also extends the work by Cruz-Uribe SFO, Fiorenza, and Neugebauer. The class \(A_{p(\cdot )}^{\mathrm{loc}}\) is defined. The weighted inequality for the local Hardy–Littlewood maximal operator on Lebesgue spaces with variable exponents is proven. Cruz-Uribe SFO, Fiorenza, and Neugebauer considered the Muckenhoupt class for Lebesgue spaces with variable exponents. However, due to the setting of variable exponents, a new method for extending weights is needed. The proposed extension method differs from that by Rychkov. A passage to the vector-valued inequality is realized by means of the extrapolation technique. This technique is an adaptation of the work by Cruz-Uribe and Wang. Additionally, a theory of extrapolation adapted to our class of weights is also obtained. PubDate: 2021-04-09

Abstract: In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided. PubDate: 2021-04-09

Abstract: A prey–predator model with constant-effort harvesting on the prey and predators is investigated in this paper. First, we discuss the number and type of the equilibria by analyzing the equations of equilibria and the distribution of eigenvalues. Second, with the rescaled harvesting efforts as bifurcation parameters, a subcritical Hopf bifurcation is exhibited near the multiple focus and a Bogdanov–Takens bifurcation is also displayed near the BT singularity by analyzing the versal unfolding of the model. With the variation of bifurcation parameters, the system shows multi-stable structure, and the attractive domains for different attractors are constituted by the stable and unstable manifolds of saddles and the limit cycles bifurcated from Hopf and Bogdanov–Takens bifurcations. Finally, a cusp point and two generalized Hopf points are found on the saddle-node bifurcation curve and the Hopf bifurcation curves, respectively. Several phase diagrams for parameters near one of the generalized Hopf points are exhibited through the generalized Hopf bifurcation. PubDate: 2021-04-09

Abstract: This article aims to introduce and analyze the viscosity method for hierarchical variational inequalities involving a ϕ-contraction mapping defined over a common solution set of variational inclusion and fixed points of a nonexpansive mapping on Hadamard manifolds. Several consequences of the composed method and its convergence theorem are presented. The convergence results of this article generalize and extend some existing results from Hilbert/Banach spaces and from Hadamard manifolds. We also present an application to a nonsmooth optimization problem. Finally, we clarify the convergence analysis of the proposed method by some computational numerical experiments in Hadamard manifold. PubDate: 2021-04-07