Journal of Inequalities and Applications

Let α , β be positive real numbers and let <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:msub></mml:math> be a Gamma random variable with shape parameter α and scale parameter β . We study infimum values of the function <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi><mml:mo>)</mml:mo><mml:mo>↦</mml:mo><mml:mi>P</mml:mi><mml:mo>{</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>≤</mml:mo><mml:mi>κ</mml:mi><mml:mi>E</mml:mi><mml:mo>[</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mrow><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:mi>β</mml:mi></mml:mrow></mml:msub><mml:mo>]</mml:mo><mml:mo>}</mml:mo></mml:math> for any fixed <mml:math xmlns:mml="https://www...

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q -differential equation <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>D</mml:mi> <mml:mi>q</mml:mi> <mml:mi>σ</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mi>c</mml:mi> </mml:mmultiscripts> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>]</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>w</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>k</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:msup> <mml:mrow/> <mml:mi>c</mml:mi> </mml:msup> <mml:msubsup> <mml:mi>D</mml:mi> <mml:mi>q</mml:mi> <mml:mi>ζ</mml:mi> </mml:msubsup> <mml:mo>[</mml:mo> <mml:mi>k</mml:mi> <mml:mo>]</mml:mo> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:math> with three-point conditions for <mml:math xmlns:mml="https://www...

In this paper we study upper and lower bounds on the Bregman divergence <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Δ</mml:mi> <mml:mi>F</mml:mi> <mml:mi>ξ</mml:mi> </mml:msubsup> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>-</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>-</mml:mo> <mml:mo>〈</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>〉</mml:mo> </mml:math> for some convex functional <mml:math xmlns:mml="https://www...

The degenerate parabolic equations from the reaction-diffusion problems are considered on an unbounded domain <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:math> . It is expected that only a partial boundary should be imposed the homogeneous boundary value, but how to give the analytic expression of this partial boundary seems very difficult...

Employing a generalized Riccati transformation and integral averaging technique, we show that all solutions of the higher order nonlinear delay differential equation <DispFormula> <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>p</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> </DispFormula> will converge to zero or oscillate, under some conditions listed in the theorems of the present paper...

A semismooth Newton method, based on variational inequalities and generalized derivative, is designed and analysed for unilateral contact problem between two membranes. The problem is first formulated as a corresponding regularized problem with a nonlinear function, which can be solved by the semismooth Newton method. We prove the convergence of the method in the function space. To improve the performance of the semismooth Newton method, we use the path-following method to adjust the parameter automatically...

[This corrects the article DOI: 10.1186/s13660-017-1466-0.].

In this study, the family F and F -convex function are given with its properties. In view of this, we establish some new inequalities of Hermite-Hadamard type for differentiable function. Moreover, we establish some trapezoid type inequalities for functions whose second derivatives in absolute values are F -convex. We also show that through the notion of F -convex we can find some new Hermite-Hadamard type and trapezoid type inequalities for the Riemann-Liouville fractional integrals and classical integrals...

In the present paper, we give the global <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>q</mml:mi> </mml:msup> </mml:math> estimates for maximal operators generated by multiparameter oscillatory integral <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Φ</mml:mi> </mml:mrow> </mml:msub> </mml:math> , which is defined by <DispFormula> <mml:math xmlns:mml="https://www...

In this paper, we establish a new iterative algorithm by combining Nadezhkina and Takahashi's modified extragradient method and Xu's algorithm. The mentioned iterative algorithm presents the common solution of the split variational inequality problems and fixed point problems. We show that the sequence produced by our algorithm is weakly convergent. Finally, we give some applications of the main results. This article extends the previous results in this area.

We study the minimum Skorohod distance estimation <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>θ</mml:mi> <mml:mi>ε</mml:mi> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> and minimum <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -norm estimation <mml:math xmlns:mml="https://www...

The authors define a class of functions on Riemannian manifolds, which are called geodesic semilocal E-preinvex functions, as a generalization of geodesic semilocal E-convex and geodesic semi E-preinvex functions, and some of its properties are established. Furthermore, a nonlinear fractional multiobjective programming is considered, where the functions involved are geodesic E- η -semidifferentiability, sufficient optimality conditions are obtained. A dual is formulated and duality results are proved by using concepts of geodesic semilocal E-preinvex functions, geodesic pseudo-semilocal E-preinvex functions, and geodesic quasi-semilocal E-preinvex functions...

An efficient inner approximation algorithm is presented for solving the generalized linear multiplicative programming problem with generalized linear multiplicative constraints. The problem is firstly converted into an equivalent generalized geometric programming problem, then some magnifying-shrinking skills and approximation strategies are used to convert the equivalent generalized geometric programming problem into a series of posynomial geometric programming problems that can be solved globally. Finally, we prove the convergence property and some practical application examples in optimal design domain, and arithmetic examples taken from recent literatures and GLOBALLib are carried out to validate the performance of the proposed algorithm...

There are almost no results in mathematical literature on the exponential stability of third-order delay differential equations. One of the main purposes of the paper is to fill this gap. We propose an approach to the study of stability for third-order delay differential equations. On the basis of these results, new possibilities of stabilization by delay feedback input control are proposed.

This paper puts forward the basic form of stochastic Gronwall's inequality and uses, respectively, the iterative method, the integral method and the martingale representation method to prove it. Then it presents an application to prove a comparison theorem of <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> solutions for one-dimensional backward stochastic differential equations under the stochastic Lipschitz condition...

In this article we give q -analogs of the Opial inequality for q -decreasing functions. Using a closed form of the restricted q -integral (see Gauchman in Comput. Math. Appl. 47:281-300, 2004), we establish a new integral inequality of the q -Opial type.

Let G be a graph with n vertices and m edges. The term energy of a graph G was introduced by I. Gutman in chemistry due to its relevance to the total π -electron energy of a carbon compound. An analogous energy <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>D</mml:mi> </mml:msub> <mml:mo>(</mml:mo> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:math> , called the distance energy, was defined by Indulal et al...

In this paper, we are concerned with the split equality problem (SEP) in Hilbert spaces. By converting it to a coupled fixed-point equation, we propose a new algorithm for solving the SEP. Whenever the convex sets involved are level sets of given convex functionals, we propose two new relaxed alternating algorithms for the SEP. The first relaxed algorithm is shown to be weakly convergent and the second strongly convergent. A new idea is introduced in order to prove strong convergence of the second relaxed algorithm, which gives an affirmative answer to Moudafi's question...

In this paper, a new method is proposed for low-rank matrix completion which is based on the least squares approximating to the known elements in the manifold formed by the singular vectors of the partial singular value decomposition alternatively. The method can achieve a reduction of the rank of the manifold by gradually reducing the number of the singular value of the thresholding and get the optimal low-rank matrix. It is proven that the manifold-alternative approximating method is convergent under some conditions...

In this paper we introduce a new self-adaptive iterative algorithm for solving the variational inequalities in real Hilbert spaces, denoted by <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:mo>VI</mml:mo> <mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> </mml:math> . Here <mml:math xmlns:mml="https://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> <mml:mo>⊆</mml:mo> <mml:mi>H</mml:mi> </mml:math> is a nonempty, closed and convex set and <mml:math xmlns:mml="https://www...