Some inequalities of euler-grüss type

Applying the Grüss inequality to the Euler type formulae we prove some Euler-Grüss-type inequalities.     

Improvement and further generalization of inequalities of Ostrowski-Grüss type

We improve some inequalities of Ostrowski-Grüss type and further generalize them. We apply the obtained results to the estimation of error bounds for some numerical quadrature rules. Also, some bounds for the differences of some special means are discussed.     

On pseudo-fractional integral inequalities related to Hermite–Hadamard type

General versions of Hermite–Hadamard type inequality for pseudo-fractional integrals of the order \(\alpha >0\) on a semiring \(\left( \left[ a,b\right] ,\oplus ,\odot \right) \) are studied. These inequalities include both pseudo-integral and fractional integral. The well-known previous results are shown to be special cases of our results. Finally, two open problems for further investigations are given.     

RM approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Syst Appl 36:6833–6842, 2009) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L–R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets Syst 118:375–385, 2001).     

Some relationships between Losonczi’s based OWA generalizations and the Choquet–Stieltjes integral

The number of aggregation operators existing nowadays is rather large. In this paper, we study some of these operators and establish some relationships between them. In particular, we focus on neat operators. We link some of these operators with the Losonczi’s mean. The results permit us to define a Losonczi’s OWA and a Losonczi’s WOWA.     

A new approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. Cheng (Cheng, C. H. (1998). A new approach for ranking fuzzy numbers by distance method. Fuzzy Sets and Systems, 95, 307–317) pointed out that the proof of the statement “Ranking of generalized fuzzy numbers does not depend upon the height of fuzzy numbers” stated by Liou and Wang (Liou, T. S., & Wang, M. J. (1992). Ranking fuzzy numbers with integral value. Fuzzy Sets and Systems, 50, 247–255) is incorrect. In this paper, by giving an alternative proof it is proved that the above statement is correct. Also with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Chen, S. M., & Chen, J. H. (2009). Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads. Expert Systems with Applications, 36, 6833–6842) is incorrect. The main aim of this paper is to modify the Liou and Wang approach for the ranking of L–R type generalized fuzzy numbers. The main advantage of the proposed approach is that the proposed approach provide the correct ordering of generalized and normal fuzzy numbers and also the proposed approach is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfy all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Wang, X., & Kerre, E. E. (2001). Reasonable properties for the ordering of fuzzy quantities (I). Fuzzy Sets and Systems, 118, 375–385).     

Gauss–Turán quadratures of Kronrod type for generalized Chebyshev weight functions

Abstract The existence and uniqueness of a Kronrod-type extension, with possibly complex nodes, to the well-known Gauss–Turán quadrature formula for any positive measure were proved by Li [15]. For generalized Chebyshev weight functions we obtain explicit formulas of the corresponding generalized Stieltjes polynomials. In the case when such quadratures are used for approximating integrals and when f is an analytic function in a region of the complex plane containing the interval [–1,1] in its interior, the remainder term is presented in the form of a contour integral over confocal ellipses. Very precise L ¹-estimates of the remainder term, as well as their upper bounds which are much simpler for evaluation, are obtained. Numerical results and illustrations are included. Mathematics Subject Classification (2000): 65D30, 65D32, 41A55     

The generalized Bohl–Perron principle for the neutral type vector functional differential equations

We consider a vector homogeneous neutral type functional differential vector equation of a certain class. It is proved that, if the corresponding nonhomogeneous equation with the zero initial conditions and an arbitrary free term bounded on the positive half-line, has a bounded solution, then the considered homogeneous equation is exponentially stable.     

A generalized Suzuki–Trotter type method in optimal control of coupled Schrödinger equations

A generalized Suzuki–Trotter (GST) method for the solution of an optimal control problem for quantum molecular systems is presented in this work. The control of such systems gives rise to a minimization problem with constraints given by a system of coupled Schrödinger equations. The computational bottleneck of the corresponding minimization methods is the solution of time-dependent Schrödinger equations. To solve the Schrödinger equations we use the GST framework to obtain an explicit polynomial approximation of the matrix exponential function. The GST method almost exclusively uses the action of the Hamiltonian and is therefore efficient and easy to implement for a variety of quantum systems. Following a first discretize, then optimize approach we derive the correct discrete representation of the gradient and the Hessian. The derivatives can naturally be expressed in the GST framework and can therefore be efficiently computed. By recomputing the solutions of the Schrödinger equations instead of saving the whole time evolution, we are able to significantly reduce the memory requirements of the method at the cost of additional computations. This makes first and second order optimization methods viable for large scale problems. In numerical experiments we compare the performance of different first and second order optimization methods using the GST method. We observe fast local convergence of second order methods.