Comparison Theorems for Completely and Multiply Monotone Functions and Their Applications

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

Download Full-text

CONTROLLED INTERSECTION THEOREMS FOR CONTINUOUS NOWHERE MONOTONE FUNCTIONS

Real Analysis Exchange ◽

10.2307/44153805 ◽

1993 ◽

Vol 19 (1) ◽

pp. 44

Author(s):

Brown ◽

Darji

Keyword(s):

Monotone Functions

Download Full-text

Nowhere Monotone Functions

Real Analysis Exchange ◽

10.2307/44151093 ◽

1976 ◽

Vol 1 (1) ◽

pp. 44

Author(s):

Foran

Keyword(s):

Monotone Functions

Download Full-text

A note on comparison theorems for graphs

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2021.125307 ◽

2021 ◽

pp. 125307

Author(s):

Andrea Adriani

Keyword(s):

Comparison Theorems

Download Full-text

On fixed point theorems of monotone functions in Ordered metric spaces

The Journal of Analysis ◽

10.1007/s41478-021-00308-7 ◽

2021 ◽

Author(s):

K. Kalyani ◽

N. Seshagiri Rao ◽

Belay Mitiku

Keyword(s):

Fixed Point ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Monotone Functions ◽

Ordered Metric Spaces

Download Full-text

Tail Asymptotics for Monotone-Separable Networks

Journal of Applied Probability ◽

10.1017/s002190020011784x ◽

2007 ◽

Vol 44 (02) ◽

pp. 306-320

Author(s):

Marc Lelarge

Keyword(s):

Queueing Network ◽

Network Models ◽

Moment Generating Function ◽

Stochastic Petri Nets ◽

Arrival Process ◽

Polling Systems ◽

Tail Asymptotics ◽

State Variables ◽

Monotone Functions ◽

Jackson Networks

A network belongs to the monotone separable class if its state variables are homogeneous and monotone functions of the epochs of the arrival process. This framework contains several classical queueing network models, including generalized Jackson networks, max-plus networks, polling systems, multiserver queues, and various classes of stochastic Petri nets. We use comparison relationships between networks of this class with independent and identically distributed driving sequences and the GI/GI/1/1 queue to obtain the tail asymptotics of the stationary maximal dater under light-tailed assumptions for service times. The exponential rate of decay is given as a function of a logarithmic moment generating function. We exemplify an explicit computation of this rate for the case of queues in tandem under various stochastic assumptions.

Download Full-text

Comparison theorems for fourth order differential equations

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171286000133 ◽

1986 ◽

Vol 9 (1) ◽

pp. 105-109

Author(s):

Garret J. Etgen ◽

Willie E. Taylor

Keyword(s):

Continuous Function ◽

Differential Equations ◽

Fourth Order ◽

Linear Equations ◽

Oscillation Criterion ◽

Comparison Theorems ◽

Positive Continuous Function ◽

Order Linear ◽

All Solutions

This paper establishes an apparently overlooked relationship between the pair of fourth order linear equationsyiv−p(x)y=0andyiv+p(x)y=0, wherepis a positive, continuous function defined on[0,∞). It is shown that if all solutions of the first equation are nonoscillatory, then all solutions of the second equation must be nonoscillatory as well. An oscillation criterion for these equations is also given.

Download Full-text