A New Generalization of Lemma Gronwall-Bellman

Size: px
Start display at page:

Download "A New Generalization of Lemma Gronwall-Bellman"

Transcription

1 Applied Mthemticl Sciences, Vol. 6, 212, no. 13, A New Generliztion of Lemm Gronwll-Bellmn Younes Lourtssi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco y lourtssi@yhoo.fr El Houssine El Mzoudi LREEER, Deprtment of Economy, Cddi Ayd University Mrrkech, Morocco h mzoudi@yhoo.fr Noureddine Ellmi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco ellmi@emi.c.m Abstrct In this pper, we present new generliztion of the Gronwll- Bellmn lemm constitutes key element in the stbiliztion of nonliner systems considered. This new generliztion cn develop simple commnd to exponentilly stbilize lrge clss of nonliner systems. Mthemtics Subject Clssifiction: 26D15, 26D2, 34A4, 34H15 Keywords: Gronwll-Bellmn ineulity, nonnegtive continuous functions, stbility, output feedbck 1 Introduction Gronwll-Bellmn ineulity, which is usully proved in elementry differentil eutions using continuity rguments see [6], [7], [9], is n importnt tool in the study of boundedness, uniueness nd other spects of ulittive behvior of solutions of differentil nd stbility.

2 622 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi In 1918, T. Gronwll gve the Gronwll-Bellmn ineulity see [5]. After tht, mny uthors gve number of generliztions of this ineulity nd these generliztions hd significnt pplictions in differentil nd integrl eutions. In this pper, we give new generliztion of the Gronwll Bellmn lemm nd n ppliction for type of nonliner dynmic systems. 2 Sttement of results Integrl ineulities ply n importnt role in the study of differentil eutions. In prticulr, there hs been n incresed interest in the following Gronwll-Bellmn ineulity. Lemm 2.1 see [1], [5] Let zt nd ft be nonnegtive continuous functions on t T, for which the ineulity zt c + fszsds, t [,T] 1 holds, where c is constnt. Then zt c exp fsds, t [,T] 2 With different motivtions, mny generliztions nd pplictions of this lemm hs been obtined nd widely used, such s the study of the stbility of solutions of nonliner differentil eutions. The purpose of this work is to estblish generliztions of Lemm Grnwll-Bellmn nd some conseuences of our results re lso given. Our min results re given in the following theorems: Theorem 2.2 see [2], [3], [4] Let zt be positive differentible function stisfying the ineulity: zt c + fszs+gsz n sds, t I =[, b] 3 where c, the functions ft nd gt re continuous in I nd n>1 is constnt. Then: c exp zt 1 n 1c gs exp fsds s 1 n 1 fτdτ ds Under the ssumption, for t, s [, b] s 1 n 1c gs exp n 1 fτdτ ds > 5 4

3 A new generliztion of lemm Gronwll-Bellmn 623 Proof 2.3 Expression 3 cn be written: for ll t [, b] zt c + using 2, we obtin zt c exp the lst ineulity is euivlent to z t c exp n 1 fs+gsz szsds, 6 fs+gsz s ds, 7 fs+gsz s ds, 8 multiply by n 1gt exp n 1 gsz sds, we obtin n 1gtzt exp n 1 gsz sds t n 1c gt exp n 1 fsds, which implies tht d exp n 1 dt gsz sds c gs exp n 1 nd integrting to t, we obtin exp n 1 gsz sds 1 c under the ssumption 5, we get exp n 1 gsz 1 sds 1 n 1c gs exp Hence the ineulity 8 becomes s: c exp n 1 z t 1 n 1c gs exp 9 fsds, 1 s gs exp n 1 fτdτ ds, 11 fsds n 1 s s, n 1 fτdτ ds 12 fτdτ ds Therefore, tking into ccount tht n>1, we will c exp fsds zt s 1 1 n 1c gs exp n 1 fτdτ ds This completes the proof of the theorem

4 624 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi Remrk 2.4 If the ssumption 5 is not verified, there is subdivision t i m 1 i= defined by: t =, t m = b nd for 1 i m 1: i exp n 1 z sgsds c ti+1 s gs exp n 1fτdτ ds = t i 15 then in this cse zt exp fsds i+1 n 1 gs exp n 1 t i s 1 fτdτ ds 16 Remrk 2.5 In cse < n < 1, the hypothesis 5 is lwys stisfied. Which leds to: s 1 zt c exp 1 n 1c gs exp n 1 fτdτ ds 17 Theorem 2.6 Let zt be positive differentible function stisfying the ineulity: zt c + f i sz i sds, t I =[, b] 18 where c is constnt, the functions f i t for i =1,...,n re continuous in I nd n>1 is constnt. Then: zt c exp f 1 sds s 1 n 1 c i 1 f i s exp n 1f 1 σdσ ds Under the ssumption, for t, s [, b] 1 n s c i 1 f i s exp n 1f 1 σdσ ds > 2 Proof 2.7 Ineulity 18 is written s: t zt c + f 1 s+ f i sz i 1 s zsds, 21

5 A new generliztion of lemm Gronwll-Bellmn 625 by pplying the Gronwll-Bellmn Lemm 2.1, we will t zt c exp f 1 s+ f i sz i 1 s ds, 22 then, for ll i =2,...,n t z i 1 t c i 1 exp i 1 f 1 s+ f i sz i 1 s ds, t c i 1 exp n 1 f 1 s+ f i sz i 1 s ds, 23 multiply the lst ineulity by negtive term n 1f i t, then: t n 1f i tz i 1 t exp n 1 f i sz i 1 sds t 24 n 1f i tc i 1 exp n 1 f 1 sds by summing the ineulities i =2to n, we obtin: t n 1 f i tz i 1 t exp n 1 f i sz i 1 sds 25 n 1 f i tc i 1 exp n 1 f 1 sds by integrting to t, we find t exp n 1 f i sz i 1 sds 1 n 1 f i sc i 1 exp n 1 f 1 σdσ ds then t exp f i sz i 1 sds 1 1 n 1 f i sc i 1 exp 1 n 1 f 1 σdσ ds where ineulity 22 becomes c exp f 1 sds zt s 1 n 1 c i 1 f i s exp n 1f 1 σdσ ds

6 626 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi 3 Applictions There re mny pplictions of the ineulities obtined in the previous section. in this section we study the output feedbck stbiliztion of nonliner dynmicl systems tht model mny phenomen from vrious disciplines. Now consider the following nonliner system: m ẋt = Axt+ g i xt u i t+but 29 yt = Cxt x = x where: xt R + n is the vector of stte, x R + n is the initil condition, yt R m is output, ut R p is the vector controls nd A, B, C re constnt mtrices of pproprite size such tht A, B stbilizble nd A, C is detectble. The nonliner function g i xt is mesurble with g i = nd stisfies the following hypothesis: for ll i =1,...,m, there exists n integer >1 s: where α i re positive constnts. g i xt α i xt 3 Remrk 3.1 We ssume tht the gin K exists becuse the fct tht A, B stbilizble nd A, C detectble provide necessry conditions but not sufficient for the existence of the gin K such tht ll eigenvlues of the mtrix A BKC either negtive rel prt. This implies tht there exists M> nd ω< such tht: e A BKCt Me ωt for ll t> 31 In the literture, mny uthors hve proposed sufficient conditions for problem of stbilizing sttic output feedbck for liner systems. Unfortuntely for the nonliner cse the problem remins open. Exponentil stbiliztion by output feedbck sttic system 29 is extension of Theorem 2.2, it is given by the following theorem. 1 m ω Theorem 3.2 Let α = α i, R = αm +1 KC M nd M = 1+ αm +1 KC x 1 ω For x <R, the system 29 controlled by the liner stte output feedbck

7 A new generliztion of lemm Gronwll-Bellmn 627 ut = Kyt stisfies: xt M x e ωt nd ut M KC x e ωt. And then is exponentilly stble. Proof 3.3 If C I n nd p<n, the solution of system 29 is controlled by the stte output feedbck ut = Kyt is given by: xt =e A BKCt x + e A BKCt s m where KCxs i is the i ieme component of vector KCxs. then g i xs KCxs i ds 32 m while tking into ccount 3 nd 31 with α = α i, it will: xt M x e ωt + e ωt αm KC xs +1 ds 33 xte ωt M x + where: c = M x, fs =, gs = αm KC e ωs ppliction of Theorem 2.2 gives: xte ωt αm KC e ωs xse ωs +1 ds 34 M x 1 αm +1 KC x nd n = +1 the 1 e ωs ds M x 1 αm +1 KC x ω e ωt this proves tht: for ll t M x 1+ αm +1 KC x ω 1 xt M x e ωt Under the ssumption of Theorem 2.2: 1 αm +1 KC x e ωs ds > 36

8 628 Y. Lourtssi, El H. El Mzoudi nd N. Ellmi We obtin: then: 1+ αm +1 KC x ω x < This completes the proof of the theorem. ω αm +1 KC > 1 38 References [1] R. Bellmn, Stbility Theory of Differentil Eultions, McGrw-Hill, New York, [2] N. Ellmi, A generlistion of Grnwll s lemm, 2ieme conférence interntionle sur les éutions différentielles, Mrrkech, [3] N. Ellmi, J. Ellmi, Stbilistion robuste pr retour de sorties des systèmes bilineires non homogénes pr l pproche de Gronwll-Bellmn, Congrès ntionl de mthémtiues ppliuées et industrielles, SMAI 21. [4] N. Ellmi, Une nouvelle générlistion du lemme de Gronwll-Bellmn, Congrès ntionl de mthémtiues ppliuées et industrielles, SMAI 21. [5] T. H. Gronwll, Note on the derivtives with respect to prmeter of the solutions of system of differentil eutions, Ann. of Mth., [6] M. W. Hirsch, S. Smlf, Differentil Eutions, Dynmicl Systems, nd Liner Algebr, Acdemic Press, New York/London, [7] B. G. Pchptte, A note on Gronwll-Bellmn ineulity, J. Mth. Anl. Appl., , [8] B. G. Pchptte. On some generliztions of Bellmn s lemm. J. of Mthemticl Anlysis nd Applictions, , [9] J. A. Oguntuse, Remrk on Gronwll type ineulities, An. Stiint. Univ. Al. I. Cuz Isi. Mt. N.S , no. 2, Received: July, 211

(4.1) D r v(t) ω(t, v(t))

(4.1) D r v(t) ω(t, v(t)) 1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

More information

S. S. Dragomir. 2, we have the inequality. b a

S. S. Dragomir. 2, we have the inequality. b a Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS

POSITIVE SOLUTIONS FOR SINGULAR THREE-POINT BOUNDARY-VALUE PROBLEMS Electronic Journl of Differentil Equtions, Vol. 27(27), No. 156, pp. 1 8. ISSN: 172-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp) POSITIVE SOLUTIONS

More information

S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:

S. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality: FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy

More information

The Hadamard s inequality for quasi-convex functions via fractional integrals

The Hadamard s inequality for quasi-convex functions via fractional integrals Annls of the University of Criov, Mthemtics nd Computer Science Series Volume (), 3, Pges 67 73 ISSN: 5-563 The Hdmrd s ineulity for usi-convex functions vi frctionl integrls M E Özdemir nd Çetin Yildiz

More information

SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES

SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES SOME HARDY TYPE INEQUALITIES WITH WEIGHTED FUNCTIONS VIA OPIAL TYPE INEQUALITIES R. P. AGARWAL, D. O REGAN 2 AND S. H. SAKER 3 Abstrct. In this pper, we will prove severl new ineulities of Hrdy type with

More information

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd

More information

ODE: Existence and Uniqueness of a Solution

ODE: Existence and Uniqueness of a Solution Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

More information

PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS

PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS PERFORMANCE ANALYSIS OF HARMONICALLY FORCED NONLINEAR SYSTEMS A.Yu. Pogromsky R.A. vn den Berg J.E. Rood Deprtment of Mechnicl Engineering, Eindhoven University of Technology, P.O. Box 513, 56 MB, Eindhoven,

More information

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir

AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An

More information

LYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN

LYAPUNOV-TYPE INEQUALITIES FOR NONLINEAR SYSTEMS INVOLVING THE (p 1, p 2,..., p n )-LAPLACIAN Electronic Journl of Differentil Equtions, Vol. 203 (203), No. 28, pp. 0. ISSN: 072-669. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu LYAPUNOV-TYPE INEQUALITIES FOR

More information

Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System

Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System Pure nd Applied Mthemtics Journl 017; 6(1): 5-13 http://www.sciencepublishinggroup.com/j/pmj doi: 10.11648/j.pmj.0170601.1 ISSN: 36-9790 (Print); ISSN: 36-981 (Online) Liner nd Non-liner Feedbck Control

More information

Positive Solutions of Operator Equations on Half-Line

Positive Solutions of Operator Equations on Half-Line Int. Journl of Mth. Anlysis, Vol. 3, 29, no. 5, 211-22 Positive Solutions of Opertor Equtions on Hlf-Line Bohe Wng 1 School of Mthemtics Shndong Administrtion Institute Jinn, 2514, P.R. Chin sdusuh@163.com

More information

New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs

New Integral Inequalities for n-time Differentiable Functions with Applications for pdfs Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute

More information

CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS. 1. Introduction

CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS. 1. Introduction Frctionl Differentil Clculus Volume 6, Number 2 (216), 275 28 doi:1.7153/fdc-6-18 CHEBYSHEV TYPE INEQUALITY ON NABLA DISCRETE FRACTIONAL CALCULUS SERKAN ASLIYÜCE AND AYŞE FEZA GÜVENILIR (Communicted by

More information

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.

THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS. THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem

More information

NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a

NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson

More information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation 1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

More information

LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS

LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR DIFFERENTIAL EQUATIONS Electronic Journl of Differentil Equtions, Vol. 2017 (2017), No. 139, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu LYAPUNOV-TYPE INEQUALITIES FOR THIRD-ORDER LINEAR

More information

Multiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales

Multiple Positive Solutions for the System of Higher Order Two-Point Boundary Value Problems on Time Scales Electronic Journl of Qulittive Theory of Differentil Equtions 2009, No. 32, -3; http://www.mth.u-szeged.hu/ejqtde/ Multiple Positive Solutions for the System of Higher Order Two-Point Boundry Vlue Problems

More information

Some estimates on the Hermite-Hadamard inequality through quasi-convex functions

Some estimates on the Hermite-Hadamard inequality through quasi-convex functions Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper

More information

On Stability of Nonlinear Slowly Time-Varying and Switched Systems

On Stability of Nonlinear Slowly Time-Varying and Switched Systems 2018 IEEE Conference on Decision nd Control CDC) Mimi Bech, FL, USA, Dec. 17-19, 2018 On Stbility of Nonliner Slowly Time-Vrying nd Switched Systems Xiobin Go, Dniel Liberzon, nd Tmer Bşr Abstrct In this

More information

Calculus of variations with fractional derivatives and fractional integrals

Calculus of variations with fractional derivatives and fractional integrals Anis do CNMAC v.2 ISSN 1984-820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810-193 Aveiro, Portugl

More information

THIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions

THIELE CENTRE. Linear stochastic differential equations with anticipating initial conditions THIELE CENTRE for pplied mthemtics in nturl science Liner stochstic differentil equtions with nticipting initil conditions Nrjess Khlif, Hui-Hsiung Kuo, Hbib Ouerdine nd Benedykt Szozd Reserch Report No.

More information

On New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals

On New Inequalities of Hermite-Hadamard-Fejer Type for Harmonically Quasi-Convex Functions Via Fractional Integrals X th Interntionl Sttistics Dys Conference ISDC 6), Giresun, Turkey On New Ineulities of Hermite-Hdmrd-Fejer Type for Hrmoniclly Qusi-Convex Functions Vi Frctionl Integrls Mehmet Kunt * nd İmdt İşcn Deprtment

More information

Improvement of Ostrowski Integral Type Inequalities with Application

Improvement of Ostrowski Integral Type Inequalities with Application Filomt 30:6 06), 56 DOI 098/FIL606Q Published by Fculty of Sciences nd Mthemtics, University of Niš, Serbi Avilble t: http://wwwpmfnicrs/filomt Improvement of Ostrowski Integrl Type Ineulities with Appliction

More information

SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL

SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Abstrct Some ineulities for the dispersion of rndom

More information

Green s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall)

Green s function. Green s function. Green s function. Green s function. Green s function. Green s functions. Classical case (recall) Green s functions 3. G(t, τ) nd its derivtives G (k) t (t, τ), (k =,..., n 2) re continuous in the squre t, τ t with respect to both vribles, George Green (4 July 793 3 My 84) In 828 Green privtely published

More information

A unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term

A unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0

More information

ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES

ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES Volume 1 29, Issue 3, Article 86, 5 pp. ON SOME NEW FRACTIONAL INTEGRAL INEQUALITIES SOUMIA BELARBI AND ZOUBIR DAHMANI DEPARTMENT OF MATHEMATICS, UNIVERSITY OF MOSTAGANEM soumi-mth@hotmil.fr zzdhmni@yhoo.fr

More information

ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS

ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 9811 015, 43 49 DOI: 10.98/PIM15019019H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS

More information

ON THE WEIGHTED OSTROWSKI INEQUALITY

ON THE WEIGHTED OSTROWSKI INEQUALITY ON THE WEIGHTED OSTROWSKI INEQUALITY N.S. BARNETT AND S.S. DRAGOMIR School of Computer Science nd Mthemtics Victori University, PO Bo 14428 Melbourne City, VIC 8001, Austrli. EMil: {neil.brnett, sever.drgomir}@vu.edu.u

More information

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei

INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV FUNCTIONAL. Mohammad Masjed-Jamei Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: http://www.pmf.ni.c.rs/filomt Filomt 25:4 20) 53 63 DOI: 0.2298/FIL0453M INEQUALITIES FOR TWO SPECIFIC CLASSES OF FUNCTIONS USING CHEBYSHEV

More information

INTEGRAL INEQUALITIES SIMILAR TO GRONWALL INEQUALITY

INTEGRAL INEQUALITIES SIMILAR TO GRONWALL INEQUALITY Electronic Journl of Differentil Equtions, Vol. 2007(2007, No. 176, pp. 1 14. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu (login: ftp INTEGRAL INEQUALITIES

More information

Improved Results on Stability of Time-delay Systems using Wirtinger-based Inequality

Improved Results on Stability of Time-delay Systems using Wirtinger-based Inequality Preprints of the 9th World Congress he Interntionl Federtion of Automtic Control Improved Results on Stbility of ime-dely Systems using Wirtinger-bsed Inequlity e H. Lee Ju H. Prk H.Y. Jung O.M. Kwon S.M.

More information

INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS

INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS INNER PRODUCT INEQUALITIES FOR TWO EQUIVALENT NORMS AND APPLICATIONS S. S. DRAGOMIR Abstrct. Some inequlities for two inner products h i nd h i which generte the equivlent norms kk nd kk with pplictions

More information

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn

More information

GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)

GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b) GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions

More information

MATH 3795 Lecture 18. Numerical Solution of Ordinary Differential Equations.

MATH 3795 Lecture 18. Numerical Solution of Ordinary Differential Equations. MATH 3795 Lecture 18. Numericl Solution of Ordinry Differentil Equtions. Dmitriy Leykekhmn Fll 2008 Gols Introduce ordinry differentil equtions (ODEs) nd initil vlue problems (IVPs). Exmples of IVPs. Existence

More information

INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION

INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION INEQUALITIES FOR GENERALIZED WEIGHTED MEAN VALUES OF CONVEX FUNCTION BAI-NI GUO AND FENG QI Abstrct. In the rticle, using the Tchebycheff s integrl inequlity, the suitble properties of double integrl nd

More information

Hermite-Hadamard type inequalities for harmonically convex functions

Hermite-Hadamard type inequalities for harmonically convex functions Hcettepe Journl o Mthemtics nd Sttistics Volume 43 6 4 935 94 Hermite-Hdmrd type ineulities or hrmoniclly convex unctions İmdt İşcn Abstrct The uthor introduces the concept o hrmoniclly convex unctions

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.

More information

A basic logarithmic inequality, and the logarithmic mean

A basic logarithmic inequality, and the logarithmic mean Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu

More information

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity

More information

Lecture notes. Fundamental inequalities: techniques and applications

Lecture notes. Fundamental inequalities: techniques and applications Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: m.h.duong@wrwick.c.uk Februry 8, 207 2 Abstrct Inequlities re ubiquitous in

More information

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )

Euler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), ) Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s

More information

ON MIXED NONLINEAR INTEGRAL EQUATIONS OF VOLTERRA-FREDHOLM TYPE WITH MODIFIED ARGUMENT

ON MIXED NONLINEAR INTEGRAL EQUATIONS OF VOLTERRA-FREDHOLM TYPE WITH MODIFIED ARGUMENT STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LIV, Number 1, Mrch 29 ON MIXED NONLINEAR INTEGRAL EQUATIONS OF VOLTERRA-FREDHOLM TYPE WITH MODIFIED ARGUMENT Abstrct. In the present pper we consider the

More information

Set Integral Equations in Metric Spaces

Set Integral Equations in Metric Spaces Mthemtic Morvic Vol. 13-1 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl

More information

New general integral inequalities for quasiconvex functions

New general integral inequalities for quasiconvex functions NTMSCI 6, No 1, 1-7 18 1 New Trends in Mthemticl Sciences http://dxdoiorg/185/ntmsci1739 New generl integrl ineulities for usiconvex functions Cetin Yildiz Atturk University, K K Eduction Fculty, Deprtment

More information

NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX

NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX Journl of Mthemticl Ineulities Volume 1, Number 3 18, 655 664 doi:1.7153/jmi-18-1-5 NEW HERMITE HADAMARD INEQUALITIES VIA FRACTIONAL INTEGRALS, WHOSE ABSOLUTE VALUES OF SECOND DERIVATIVES IS P CONVEX SHAHID

More information

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION

More information

AMATH 731: Applied Functional Analysis Fall Some basics of integral equations

AMATH 731: Applied Functional Analysis Fall Some basics of integral equations AMATH 731: Applied Functionl Anlysis Fll 2009 1 Introduction Some bsics of integrl equtions An integrl eqution is n eqution in which the unknown function u(t) ppers under n integrl sign, e.g., K(t, s)u(s)

More information

SUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS

SUPERSTABILITY OF DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS Electronic Journl of Differentil Equtions, Vol. 01 (01), No. 15, pp. 1. ISSN: 107-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu SUPERSTABILITY OF DIFFERENTIAL

More information

Теоремы типа Бохера для динамических уравнений на временных шкалах

Теоремы типа Бохера для динамических уравнений на временных шкалах Теоремы типа Бохера для динамических уравнений на временных шкалах c Владимир Шепселевич Бурд Ярославский государственный университет им. П.Г. Демидова Аннотация. Найдены условия, при которых все решения

More information

Bulletin of the. Iranian Mathematical Society

Bulletin of the. Iranian Mathematical Society ISSN: 07-060X Print ISSN: 735-855 Online Bulletin of the Irnin Mthemticl Society Vol 3 07, No, pp 09 5 Title: Some extended Simpson-type ineulities nd pplictions Authors: K-C Hsu, S-R Hwng nd K-L Tseng

More information

The presentation of a new type of quantum calculus

The presentation of a new type of quantum calculus DOI.55/tmj-27-22 The presenttion of new type of quntum clculus Abdolli Nemty nd Mehdi Tourni b Deprtment of Mthemtics, University of Mzndrn, Bbolsr, Irn E-mil: nmty@umz.c.ir, mehdi.tourni@gmil.com b Abstrct

More information

Ordinary differential equations

Ordinary differential equations Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

arxiv: v1 [math.ds] 2 Nov 2015

arxiv: v1 [math.ds] 2 Nov 2015 PROPERTIES OF CERTAIN PARTIAL DYNAMIC INTEGRODIFFERENTIAL EQUATIONS rxiv:1511.00389v1 [mth.ds] 2 Nov 2015 DEEPAK B. PACHPATTE Abstrct. The im of the present pper is to study the existence, uniqueness nd

More information

A PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES

A PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. e-mil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL

More information

The Riemann-Lebesgue Lemma

The Riemann-Lebesgue Lemma Physics 215 Winter 218 The Riemnn-Lebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of

More information

ON THE OSCILLATION OF FRACTIONAL DIFFERENTIAL EQUATIONS

ON THE OSCILLATION OF FRACTIONAL DIFFERENTIAL EQUATIONS ON HE OSCILLAION OF FRACIONAL DIFFERENIAL EQUAIONS S.R. Grce 1, R.P. Agrwl 2, P.J.Y. Wong 3, A. Zfer 4 Abstrct In this pper we initite the oscilltion theory for frctionl differentil equtions. Oscilltion

More information

ON THE C-INTEGRAL BENEDETTO BONGIORNO

ON THE C-INTEGRAL BENEDETTO BONGIORNO ON THE C-INTEGRAL BENEDETTO BONGIORNO Let F : [, b] R be differentible function nd let f be its derivtive. The problem of recovering F from f is clled problem of primitives. In 1912, the problem of primitives

More information

CONDITIONS ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS FOR A CERTAIN CLASS OF FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS

CONDITIONS ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS FOR A CERTAIN CLASS OF FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i, Mtemtică,, f.. CONDITIONS ON THE QUALITATIVE BEHAVIOUR OF SOLUTIONS FOR A CERTAIN CLASS OF FIFTH-ORDER NON-LINEAR DIFFERENTIAL EQUATIONS

More information

A General Dynamic Inequality of Opial Type

A General Dynamic Inequality of Opial Type Appl Mth Inf Sci No 3-5 (26) Applied Mthemtics & Informtion Sciences An Interntionl Journl http://dxdoiorg/2785/mis/bos7-mis A Generl Dynmic Inequlity of Opil Type Rvi Agrwl Mrtin Bohner 2 Donl O Regn

More information

GENERALIZED ABSTRACTED MEAN VALUES

GENERALIZED ABSTRACTED MEAN VALUES GENERALIZED ABSTRACTED MEAN VALUES FENG QI Abstrct. In this rticle, the uthor introduces the generlized bstrcted men vlues which etend the concepts of most mens with two vribles, nd reserches their bsic

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics http://jipmvueduu/ Volume, Issue, Article, 00 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT,

More information

arxiv: v1 [math.ca] 28 Jan 2013

arxiv: v1 [math.ca] 28 Jan 2013 ON NEW APPROACH HADAMARD-TYPE INEQUALITIES FOR s-geometrically CONVEX FUNCTIONS rxiv:3.9v [mth.ca 8 Jn 3 MEVLÜT TUNÇ AND İBRAHİM KARABAYIR Astrct. In this pper we chieve some new Hdmrd type ineulities

More information

Review of Calculus, cont d

Review of Calculus, cont d Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

More information

SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE

SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE RGMIA Reserch Report Collection, Vol., No., 998 http://sci.vut.edu.u/ rgmi SOME INTEGRAL INEQUALITIES OF GRÜSS TYPE S.S. DRAGOMIR Astrct. Some clssicl nd new integrl inequlities of Grüss type re presented.

More information

Bounds for the Riemann Stieltjes integral via s-convex integrand or integrator

Bounds for the Riemann Stieltjes integral via s-convex integrand or integrator ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t www.mth.ut.ee/ct/ Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh

More information

Research Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation

Research Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,

More information

Regulated functions and the regulated integral

Regulated functions and the regulated integral Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed

More information

Some Improvements of Hölder s Inequality on Time Scales

Some Improvements of Hölder s Inequality on Time Scales DOI: 0.55/uom-207-0037 An. Şt. Univ. Ovidius Constnţ Vol. 253,207, 83 96 Some Improvements of Hölder s Inequlity on Time Scles Cristin Dinu, Mihi Stncu nd Dniel Dănciulescu Astrct The theory nd pplictions

More information

arxiv: v1 [math.ra] 1 Nov 2014

arxiv: v1 [math.ra] 1 Nov 2014 CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,

More information

HYERS-ULAM STABILITY OF HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS ON TIME SCALES

HYERS-ULAM STABILITY OF HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS ON TIME SCALES Dynmic Systems nd Applictions 23 (2014) 653-664 HYERS-ULAM STABILITY OF HIGHER-ORDER CAUCHY-EULER DYNAMIC EQUATIONS ON TIME SCALES DOUGLAS R. ANDERSON Deprtment of Mthemtics, Concordi College, Moorhed,

More information

Time Optimal Control of the Brockett Integrator

Time Optimal Control of the Brockett Integrator Milno (Itly) August 8 - September, 011 Time Optiml Control of the Brockett Integrtor S. Sinh Deprtment of Mthemtics, IIT Bomby, Mumbi, Indi (emil : sunnysphs4891@gmil.com) Abstrct: The Brockett integrtor

More information

Asymptotic behavior of intermediate points in certain mean value theorems. III

Asymptotic behavior of intermediate points in certain mean value theorems. III Stud. Univ. Bbeş-Bolyi Mth. 59(2014), No. 3, 279 288 Asymptotic behvior of intermedite points in certin men vlue theorems. III Tiberiu Trif Abstrct. The pper is devoted to the study of the symptotic behvior

More information

On the Generalized Weighted Quasi-Arithmetic Integral Mean 1

On the Generalized Weighted Quasi-Arithmetic Integral Mean 1 Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School

More information

Research Article On New Inequalities via Riemann-Liouville Fractional Integration

Research Article On New Inequalities via Riemann-Liouville Fractional Integration Abstrct nd Applied Anlysis Volume 202, Article ID 428983, 0 pges doi:0.55/202/428983 Reserch Article On New Inequlities vi Riemnn-Liouville Frctionl Integrtion Mehmet Zeki Sriky nd Hsn Ogunmez 2 Deprtment

More information

Revista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13

Revista Colombiana de Matemáticas Volumen 41 (2007), páginas 1 13 Revist Colombin de Mtemátics Volumen 4 7, págins 3 Ostrowski, Grüss, Čebyšev type inequlities for functions whose second derivtives belong to Lp,b nd whose modulus of second derivtives re convex Arif Rfiq

More information

Some Hardy Type Inequalities with Weighted Functions via Opial Type Inequalities

Some Hardy Type Inequalities with Weighted Functions via Opial Type Inequalities Advnces in Dynmicl Systems nd Alictions ISSN 0973-5321, Volume 10, Number 1,. 1 9 (2015 htt://cmus.mst.edu/ds Some Hrdy Tye Inequlities with Weighted Functions vi Oil Tye Inequlities Rvi P. Agrwl Tes A&M

More information

Asymptotic Behavior of the Solutions of a Class of Rational Difference Equations

Asymptotic Behavior of the Solutions of a Class of Rational Difference Equations Interntionl Journl of Difference Equtions ISSN 0973-6069, Volume 5, Number 2, pp. 233 249 200) http://cmpus.mst.edu/ijde Asymptotic Behvior of the Solutions of Clss of Rtionl Difference Equtions G. Ppschinopoulos

More information

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a). The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

More information

Three solutions to a p(x)-laplacian problem in weighted-variable-exponent Sobolev space

Three solutions to a p(x)-laplacian problem in weighted-variable-exponent Sobolev space DOI: 0.2478/uom-203-0033 An. Şt. Univ. Ovidius Constnţ Vol. 2(2),203, 95 205 Three solutions to p(x)-lplcin problem in weighted-vrible-exponent Sobolev spce Wen-Wu Pn, Ghsem Alizdeh Afrouzi nd Lin Li Abstrct

More information

MEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR

MEAN VALUE PROBLEMS OF FLETT TYPE FOR A VOLTERRA OPERATOR Electronic Journl of Differentil Equtions, Vol. 213 (213, No. 53, pp. 1 7. ISSN: 172-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu MEAN VALUE PROBLEMS OF FLETT

More information

Improper Integrals, and Differential Equations

Improper Integrals, and Differential Equations Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted

More information

THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithaca, NY

THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithaca, NY THE EXISTENCE OF NEGATIVE MCMENTS OF CONTINUOUS DISTRIBUTIONS WALTER W. PIEGORSCH AND GEORGE CASELLA. Biometrics Unit, Cornell University, Ithc, NY 14853 BU-771-M * December 1982 Abstrct The question of

More information

Some Hermite-Hadamard type inequalities for functions whose exponentials are convex

Some Hermite-Hadamard type inequalities for functions whose exponentials are convex Stud. Univ. Beş-Bolyi Mth. 6005, No. 4, 57 534 Some Hermite-Hdmrd type inequlities for functions whose exponentils re convex Silvestru Sever Drgomir nd In Gomm Astrct. Some inequlities of Hermite-Hdmrd

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARI- ABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR

More information

Research Article Harmonic Deformation of Planar Curves

Research Article Harmonic Deformation of Planar Curves Interntionl Journl of Mthemtics nd Mthemticl Sciences Volume, Article ID 9, pges doi:.55//9 Reserch Article Hrmonic Deformtion of Plnr Curves Eleutherius Symeonidis Mthemtisch-Geogrphische Fkultät, Ktholische

More information

Remark on boundary value problems arising in Ginzburg-Landau theory

Remark on boundary value problems arising in Ginzburg-Landau theory Remrk on boundry vlue problems rising in Ginzburg-Lndu theory ANITA KIRICHUKA Dugvpils University Vienibs Street 13, LV-541 Dugvpils LATVIA nit.kiricuk@du.lv FELIX SADYRBAEV University of Ltvi Institute

More information

FUNCTIONS OF α-slow INCREASE

FUNCTIONS OF α-slow INCREASE Bulletin of Mthemticl Anlysis nd Applictions ISSN: 1821-1291, URL: http://www.bmth.org Volume 4 Issue 1 (2012), Pges 226-230. FUNCTIONS OF α-slow INCREASE (COMMUNICATED BY HÜSEYIN BOR) YILUN SHANG Abstrct.

More information

On the Continuous Dependence of Solutions of Boundary Value Problems for Delay Differential Equations

On the Continuous Dependence of Solutions of Boundary Value Problems for Delay Differential Equations Journl of Computtions & Modelling, vol.3, no.4, 2013, 1-10 ISSN: 1792-7625 (print), 1792-8850 (online) Scienpress Ltd, 2013 On the Continuous Dependence of Solutions of Boundry Vlue Problems for Dely Differentil

More information

A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL. 1. Introduction. f(x)dx a

A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL. 1. Introduction. f(x)dx a Journl of Frctionl Clculus nd Applictions, Vol. 4( Jn. 203, pp. 25-29. ISSN: 2090-5858. http://www.fcj.webs.com/ A NOTE ON SOME FRACTIONAL INTEGRAL INEQUALITIES VIA HADAMARD INTEGRAL VAIJANATH L. CHINCHANE

More information

Frobenius numbers of generalized Fibonacci semigroups

Frobenius numbers of generalized Fibonacci semigroups Frobenius numbers of generlized Fiboncci semigroups Gretchen L. Mtthews 1 Deprtment of Mthemticl Sciences, Clemson University, Clemson, SC 29634-0975, USA gmtthe@clemson.edu Received:, Accepted:, Published:

More information

A Bernstein polynomial approach for solution of nonlinear integral equations

A Bernstein polynomial approach for solution of nonlinear integral equations Avilble online t wwwisr-publictionscom/jns J Nonliner Sci Appl, 10 (2017), 4638 4647 Reserch Article Journl Homepge: wwwtjnscom - wwwisr-publictionscom/jns A Bernstein polynomil pproch for solution of

More information

1 1D heat and wave equations on a finite interval

1 1D heat and wave equations on a finite interval 1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion

More information

COMPARISON PRINCIPLES FOR DIFFERENTIAL EQUATIONS INVOLVING CAPUTO FRACTIONAL DERIVATIVE WITH MITTAG-LEFFLER NON-SINGULAR KERNEL

COMPARISON PRINCIPLES FOR DIFFERENTIAL EQUATIONS INVOLVING CAPUTO FRACTIONAL DERIVATIVE WITH MITTAG-LEFFLER NON-SINGULAR KERNEL Electronic Journl of Differentil Equtions, Vol. 2018 (2018, No. 36, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.mth.txstte.edu or http://ejde.mth.unt.edu COMPARISON PRINCIPLES FOR DIFFERENTIAL EQUATIONS

More information