Set Integral Equations in Metric Spaces
|
|
- Rosa Wright
- 6 years ago
- Views:
Transcription
1 Mthemtic Morvic Vol , 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 equtions: 1 2 Xt = Xt = Z b Z t Kt, s, Xs d s + X 0 Kt, s, Xs d s + X 0, where K : [, b] [, b] P cp,cvr n P cp,cvr n nd X 0 P cp,cvr n. The purpose of the pper is to study the existence nd dt dependence of the solutions of the set integrl equtions 1 nd 2, by using fixed point pproch. Our results generlize nd extend the results given in [2]. For other similr results see [3] nd [4]. 1. Introduction Let R n be the rel n-dimensionl Euclidin spce nd P cp,cv R n be the fmily of ll nonempty compct, convex subset of R n endowed with the Pompeiu-Husdorff metric H. It is well-known tht P cp,cv R n, H is complete metric spce. We consider the following set integrl equtions: 1 2 Xt = Xt = Kt, s, Xs d s + X 0, Kt, s, Xs d s + X 0, where K : [, b] [, b] P cp,cv R n P cp,cv R n. A solution of 1 or 2 mens continuous function X : [, b] P cp,cv R n which stisfies 1 respectively 2 for ech t [, b]. The purpose of the rticle is to study the existence nd dt dependence of the solutions of the eqution 1 nd 2. The pproch is bsed on the well-known Bnch-Cccioppoli contrction principle. Our results generlize 2000 Mthemtics Subject Clssifiction. Primry: 47H10. Key words nd phrses. Fixed point, set integrl eqution, Pompeiu-Husdorff metric. 95 c 2009 Mthemtic Morvic
2 96 Set Integrl Equtions in Metric Spces nd extend the results given in [2]. For other similr results see [3], [5] nd [4]. The pper is orgnized s follows. Next section, Preliminries, contins some bsic nottions nd notions used through the pper. Third section presents existence nd the dt dependence results of the solution for the equtions 1 nd Preliminries The im of this section is to present some notions nd symbols used in the pper. Let us define the following generlized functionls: D : P R n P R n R +, DA, B = inf{d, b A, b B}. D is clled the gp functionl between A nd B. In prticulr, if x 0 X then Dx 0, B := D{x 0 }, B. ρ : P R n P R n R + {+ }, ρ is clled the generlized excess functionl. ρa, B = sup{d, B A}. H : P R n P R n R + {+ }, HA, B = mx{ρa, B, ρb, A}. H is the generlized Pompeiu-Husdorff functionl. It is known tht P cp,cv R n, H is complete metric spce [1]. Lemm 2.1 [6]. Let X be Bnch spce. Then HA + C, B + D HA, B + HC, D, for A, B, C, D P X. Proof. Let ε > 0. From the definition of H it follows tht there exists + c A + C such tht D + c, B + D HA + C, B + D ε or exists b + d B + D such tht Db + d, A + C HA + C, B + D ε. Let us consider the first cse. For, c we get b B, d D such tht: b HA, B + ε 2, c d HC, D + ε 2. Then HA+C, B +D ε D+c, B +D +c b+d we obtin tht HA + C, B + D ε HA, B + HC, D + ε, proving the desired inequlity. 3. Min results We consider on C[, b], P cp,cv R n the metric: H X, Y := mx HXt, Y t. t [,b] The pir C[, b], P cp,cv R n, H forms complete metric spce. Our first result is n existence nd uniqueness theorem for the solution of the eqution 1.
3 Ion Tişe 97 Theorem 3.1. Let K : [, b] [, b] P cp,cv R n P cp,cv R n be multivlued opertor. Suppose tht: i K is continuous on [, b] [, b] P cp,cv R n nd X 0 P cp,cv R n, ii Kt, s, is Lipschitz, i.e., there exists L K 0 such tht: HKt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd for ll t, s [, b], iii L K b < 1. Then the set integrl eqution 1 Xt = hs unique solution. Kt, s, Xs d s + X 0 Proof. Consider the opertor: Γ : P cp,cv R n P cp,cv R n defined for ech t [, b] by ΓXt = Kt, s, Xs d s + X 0. We need to verify the contrction condition for Γ. H ΓXt, ΓY t = = H Kt, s, Xs d s + X 0, HX 0, X 0 + H Kt, s, Xs d s, H Kt, s, Xs, Kt, s, Y s d s L K H Xs, Y s d s. Tking the mximum for t [, b], then we hve: mx t [,b] H ΓXt, ΓY t L K Kt, s, Y s d s + X 0 Kt, s, Y s d s mx H Xs, Y s d s t [,b] H ΓX, ΓY LK b H X, Y, for ll t [, b], nd X, Y C [, b], P cp,cv R n. Thus, the integrl opertor Γ is Lipschitz with constnt L Γ = L K b < 1. From the contrction principle we get the result. A dt dependence result for the solution of eqution 1 is:
4 98 Set Integrl Equtions in Metric Spces Theorem 3.2. Let K 1, K 2 : [, b] [, b] P cp,cv R n P cp,cv R n, be continuous. Consider the following set equtions: 3 4 Xt = Y t = K 1 t, s, Xs d s + X 0 K 2 t, s, Y s d s + Y 0 Suppose: i there exists L K 0 such tht H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b], with L K b < 1 denote by X the unique solution of the eqution 3; ii there exists η 1, η 2 > 0 such tht: H K 1 t, s, U, K 2 t, s, U η 1, for ll t, s, U [, b] [, b] P cp,cv R n, nd b HX 0, Y 0 η 2 ; iii there exists Y solution of the eqution 4. Then H X, Y η 2 + η 1 b 1 L K b. Proof. We hve: HX t, Y t = = H K 1 t, s, X s d s + X 0, H K 1 t, s, X s d s, H K 1 t, s, X s d s, + H K 1 t, s, Y s d s, + K 2 t, s, Y s d s + Y 0 K 2 t, s, Y s d s + HX 0, Y 0 K 1 t, s, Y s d s + H K 1 t, s, X s, K 1 t, s, Y s d s+ K 2 t, s, Y s d s + η 2 H K 1 t, s, Y s, K 2 t, s, Y s d s + η 2 H K 1 t, s, X s, K 1 t, s, Y s d s + η 1 d s + η 2.
5 Ion Tişe 99 By tking the mximum for t [, b], then we hve: mx H X t, Y t t [,b] LK b H X t, Y t + η 1 b + η 2 mx t [,b] L K b mx t [,b] H X t, Y t + η 1 b + η 2 mx H X t, Y t η 2 + η 1 b t [,b] 1 L K b H X, Y η 2 + η 1 b 1 L K b. We will prove now n existence result nd dt dependence result for the solution of the eqution 2. We consider on C [, b], P cp,cv R n the metric: H B X, Y := mx t [,b] [HXt, Y te τt ], with rbitrry τ > 0. The pir C[, b], P cp,cv R n, H B forms complete metric spce. Theorem 3.3. Let K : [, b] [, b] P cp,cv R n P cp,cv R n be n opertor. Suppose tht: i K is continuous on [, b] [, b] P cp,cv R n nd X 0 P cp,cv R n, ii Kt, s, is Lipschitz, i.e., there exists L K 0 such tht H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b]. Then the set integrl eqution 2 Xt = hs unique solution. Kt, s, Xs d s + X 0 Proof. Consider the opertor Γ : P cp,cv R n P cp,cv R n defined for ech t [, b] by ΓXt = Kt, s, Xs d s + X 0.
6 100 Set Integrl Equtions in Metric Spces We will prove the contrction condition for Γ. H ΓXt, ΓY t = = H Kt, s, Xs d s + X 0, HX 0, X 0 + H Kt, s, Xs d s, Kt, s, Y s d s + X 0 H Kt, s, Xs, Kt, s, Y s d s = L K H Xs, Y s e τs e τs d s L K H B X, Y e τs d s = = L K τ HB X, Y e τt 1 L K τ HB X, Y e τt. Then we hve: Kt, s, Y s d s L K H Xs, Y s d s = H ΓXt, ΓY t e τt L K τ HB X, Y H B L K ΓX, ΓY τ HB X, Y, for ll t [, b], X, Y C [, b], P cp,cv R n, τ > 0. Hence, we cn pply the Bnch contrction principle for Γ, since by choosing τ > L K, we get L Γ := L K τ < 1. By the contrction principle, the proof is complete. Remrk 3.1. Theorem 3.3 in this pper is specil cse of Hmmerstein s equlity. Generl solution of this equlity is given in [7]. A dt dependence result is: Theorem 3.4. Let K 1, K 2 : [, b] [, b] P cp,cv R n P cp,cv R n be continuous. Consider the following set equtions: 5 6 Xt = Y t = K 1 t, s, Xs d s + X 0 K 2 t, s, Y s d s + Y 0 Suppose: i H Kt, s, A, Kt, s, B L K HA, B, for ll A, B P cp,cv R n nd t, s [, b], where L K 0 denote by X the unique solution of the eqution 5; ii there exists η 1, η 2 > 0, such tht H K 1 t, s, U, K 2 t, s, U η 1, for ll t, s, U [, b] [, b] P cp,cv R n nd HX 0, Y 0 η 2 ;
7 Ion Tişe 101 iii There exists Y solution of the eqution 6. Then Proof. We hve: H B X, Y η 2 + η 1 b 1 L K τ e τb where τ > L K. HX t, Y t = = H K 1 t, s, X s d s + X 0, H K 1 t, s, X s d s, H K 1 t, s, X s d s, + H K 1 t, s, Y s d s, + K 2 t, s, Y s d s + Y 0 K 2 t, s, Y s d s + HX 0, Y 0 K 1 t, s, Y s d s + H K 1 t, s, X s, K 1 t, s, Y s d s+ K 2 t, s, Y s d s + η 2 H K 1 t, s, Y s, K 2 t, s, Y s d s + η 2 H K 1 t, s, X s, K 1 t, s, Y s d s + L K H X s, Y s e τs e τs d s + By tking the mximum for t [, b], we hve: mx H X t, Y t e τt e τt t [,b] mx t [,b] L K H X s, Y s e τs e τs ds + H B X, Y e τb = L K H B X, Y L K τ HB X, Y e τt 1 + η 1 b + η 2 L K τ HB X, Y e τb + η 1 b + η 2, H B X, Y η 2 + η 1 b 1 L K τ e τb. η 1 d s + η 2 η 1 d s + η 2. η 1 d s + η 2, e τs d s + η 1 b + η 2 =
8 102 Set Integrl Equtions in Metric Spces For τ > L K, we hve H B X, Y η 2 + η 1 b 1 L K τ e τb. References [1] J.P. Aubin, H. Frnkowsk, Set-Vlued Anlysis, Birkhuser, Bsel, [2] A.J. Brndo, F.S. De Blsi, F. Iervillino, Uniqueness nd Existence Theorems for Differentil Equtions with Compct Convex Vlued Solutions, Boll. Unione Mt. Itl., IV. Ser , [3] F.S. de Blsi, Semifixed sets of mps in hyperspces with ppliction to set differentil equtions, Set-Vlued Anl., , No. 3, [4] G.N. Glnis, T.G. Bhskr, V. Lkshmiknthm nd P.K. Plmides, Set vlued functions in Fréchet spces: continuity, Hukuhr differentibility nd pplictions to set differentil equtions, Nonliner Anl., , No. 4, [5] V. Lkshmiknthm, T. Gnn Bhskr nd J. Vsundhr Devi, Theory of Set Differentil Equtions in Metric Spces, Cmbridge Scientific Publishers, [6] A. Petruşel, G. Petruşel, G. Moț, Topics in Nonliner Anlysis nd Applictions to Mthemticl Economics, House of the Book Science, Cluj-Npoc, [7] M.R. Tskovic, A Chrcteriztion of the Clss of Contrction Type Mppings, Kobe Journl of Mthemtics, , Deprtment of Applied Mthemtics, Bbeş-Bolyi University Kogălnicenu 1, Cluj-Npoc, Romni E-mil ddress: ti_cmeli@yhoo.com
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 informationSome 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 informationON BERNOULLI BOUNDARY VALUE PROBLEM
LE MATEMATICHE Vol. LXII (2007) Fsc. II, pp. 163 173 ON BERNOULLI BOUNDARY VALUE PROBLEM FRANCESCO A. COSTABILE - ANNAROSA SERPE We consider the boundry vlue problem: x (m) (t) = f (t,x(t)), t b, m > 1
More informationOn 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 informationPositive 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 informationASYMPTOTIC 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 informationThe 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 informationSUPERSTABILITY 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 informationResearch 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 informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More informationRegulated 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 informationPOSITIVE 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 informationThe Bochner Integral and the Weak Property (N)
Int. Journl of Mth. Anlysis, Vol. 8, 2014, no. 19, 901-906 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.4367 The Bochner Integrl nd the Wek Property (N) Besnik Bush Memetj University
More informationGENERALIZATIONS 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 informationON A GENERALIZED STURM-LIOUVILLE PROBLEM
Foli Mthemtic Vol. 17, No. 1, pp. 17 22 Act Universittis Lodziensis c 2010 for University of Łódź Press ON A GENERALIZED STURM-LIOUVILLE PROBLEM GRZEGORZ ANDRZEJCZAK AND TADEUSZ POREDA Abstrct. Bsic results
More informationON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More informationON 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 informationA 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 informationAN 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 informationTRAPEZOIDAL 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 informationMultiple 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 informationA HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127-9-56-z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
More informationHouston Journal of Mathematics. c 1999 University of Houston Volume 25, No. 4, 1999
Houston Journl of Mthemtics c 999 University of Houston Volume 5, No. 4, 999 ON THE STRUCTURE OF SOLUTIONS OF CLSS OF BOUNDRY VLUE PROBLEMS XIYU LIU, BOQING YN Communicted by Him Brezis bstrct. Behviour
More informationA 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 informationLYAPUNOV-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 informationInvited Lecture Delivered at Fifth International Conference of Applied Mathematics and Computing (Plovdiv, Bulgaria, August 12 18, 2008)
Interntionl Journl of Pure nd Applied Mthemtics Volume 51 No. 2 2009, 189-194 Invited Lecture Delivered t Fifth Interntionl Conference of Applied Mthemtics nd Computing (Plovdiv, Bulgri, August 12 18,
More informationON 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 informationON 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 informationNEW TYPE OF MULTIVALUED CONTRACTIONS WITH RELATED RESULTS AND APPLICATIONS
U.P.B. Sci. Bull., Series A, Vol. 80, Iss., 018 ISSN 13-707 NEW TYPE OF MULTIVALUED CONTRACTIONS WITH RELATED RESULTS AND APPLICATIONS Hüseyin Işik 1 nd Cristin Ionescu In this pper, we focus on chieving
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationInternational Jour. of Diff. Eq. and Appl., 3, N1, (2001),
Interntionl Jour. of Diff. Eq. nd Appl., 3, N1, (2001), 31-37. 1 New proof of Weyl s theorem A.G. Rmm Mthemtics Deprtment, Knss Stte University, Mnhttn, KS 66506-2602, USA rmm@mth.ksu.edu http://www.mth.ksu.edu/
More informationHERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS
HERMITE-HADAMARD TYPE INEQUALITIES OF CONVEX FUNCTIONS WITH RESPECT TO A PAIR OF QUASI-ARITHMETIC MEANS FLAVIA CORINA MITROI nd CĂTĂLIN IRINEL SPIRIDON In this pper we estblish some integrl inequlities
More informationGeneralized Hermite-Hadamard-Fejer type inequalities for GA-convex functions via Fractional integral
DOI 763/s4956-6-4- Moroccn J Pure nd Appl AnlMJPAA) Volume ), 6, Pges 34 46 ISSN: 35-87 RESEARCH ARTICLE Generlized Hermite-Hdmrd-Fejer type inequlities for GA-conve functions vi Frctionl integrl I mdt
More informationA 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 informationWENJUN LIU AND QUÔ C ANH NGÔ
AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous
More informationAMATH 731: Applied Functional Analysis Fall Additional notes on Fréchet derivatives
AMATH 731: Applied Functionl Anlysis Fll 214 Additionl notes on Fréchet derivtives (To ccompny Section 3.1 of the AMATH 731 Course Notes) Let X,Y be normed liner spces. The Fréchet derivtive of n opertor
More informationWirtinger s Integral Inequality on Time Scale
Theoreticl themtics & pplictions vol.8 no.1 2018 1-8 ISSN: 1792-9687 print 1792-9709 online Scienpress Ltd 2018 Wirtinger s Integrl Inequlity on Time Scle Ttjn irkovic 1 bstrct In this pper we estblish
More informationNEW 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 informationSequential Fractional Differential Equations with Hadamard Derivative
Sequentil Frctionl Differentil Equtions with Hdmrd Derivtive M. Klimek Institute of Mthemtics, Czestochow University of Technoy ul. Dbrowskiego 73, 42-200 Czestochow, Polnd e-mil: klimek@im.pcz.pl). Abstrct:
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationApproximation of functions belonging to the class L p (ω) β by linear operators
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 3, 9, Approximtion of functions belonging to the clss L p ω) β by liner opertors W lodzimierz Lenski nd Bogdn Szl Abstrct. We prove
More informationAn inequality related to η-convex functions (II)
Int. J. Nonliner Anl. Appl. 6 (15) No., 7-33 ISSN: 8-68 (electronic) http://d.doi.org/1.75/ijn.15.51 An inequlity relted to η-conve functions (II) M. Eshghi Gordji, S. S. Drgomir b, M. Rostmin Delvr, Deprtment
More informationThe 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 informationON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES
Volume 8 (2007), Issue 4, Article 93, 13 pp. ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ AMERICAN COLLEGE OF MANAGEMENT AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY
More informationGENERALIZED 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 informationS. 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 informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More informationAn iterative method for solving nonlinear functional equations
J. Mth. Anl. Appl. 316 (26) 753 763 www.elsevier.com/locte/jm An itertive method for solving nonliner functionl equtions Vrsh Dftrdr-Gejji, Hossein Jfri Deprtment of Mthemtics, University of Pune, Gneshkhind,
More informationNew 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 informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationNew Integral Inequalities of the Type of Hermite-Hadamard Through Quasi Convexity
Punjb University Journl of Mthemtics (ISSN 116-56) Vol. 45 (13) pp. 33-38 New Integrl Inequlities of the Type of Hermite-Hdmrd Through Qusi Convexity S. Hussin Deprtment of Mthemtics, College of Science,
More informationJournal 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 informationDYNAMICAL SYSTEMS SUPPLEMENT 2007 pp Natalija Sergejeva. Department of Mathematics and Natural Sciences Parades 1 LV-5400 Daugavpils, Latvia
DISCRETE AND CONTINUOUS Website: www.aimsciences.org DYNAMICAL SYSTEMS SUPPLEMENT 2007 pp. 920 926 ON THE UNUSUAL FUČÍK SPECTRUM Ntlij Sergejev Deprtment of Mthemtics nd Nturl Sciences Prdes 1 LV-5400
More informationLYAPUNOV-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 informationResearch Article Some Extensions of Banach s Contraction Principle in Complete Cone Metric Spaces
Hindwi Publishing Corportion Fixed Point Theory nd Applictions Volume 2008, Article ID 768294, 11 pges doi:10.1155/2008/768294 Reserch Article Some Extensions of Bnch s Contrction Principle in Complete
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationFredholm Integral Equations of the First Kind Solved by Using the Homotopy Perturbation Method
Int. Journl of Mth. Anlysis, Vol. 5, 211, no. 19, 935-94 Fredholm Integrl Equtions of the First Kind Solved by Using the Homotopy Perturbtion Method Seyyed Mhmood Mirzei Deprtment of Mthemtics, Fculty
More informationINEQUALITIES 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 informationThe Delta-nabla Calculus of Variations for Composition Functionals on Time Scales
Interntionl Journl of Difference Equtions ISSN 973-669, Volume 8, Number, pp. 7 47 3) http://cmpus.mst.edu/ijde The Delt-nbl Clculus of Vritions for Composition Functionls on Time Scles Monik Dryl nd Delfim
More informationON CLOSED CONVEX HULLS AND THEIR EXTREME POINTS. S. K. Lee and S. M. Khairnar
Kngweon-Kyungki Mth. Jour. 12 (2004), No. 2, pp. 107 115 ON CLOSED CONVE HULLS AND THEIR ETREME POINTS S. K. Lee nd S. M. Khirnr Abstrct. In this pper, the new subclss denoted by S p (α, β, ξ, γ) of p-vlent
More informationarxiv: 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 informationAN 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 informationMEAN 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 informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 1938-9787 www.commun-mth-nl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKI-GRÜSS INEQUALITY
More informationA Simple Proof of the Jensen-Type Inequality of Fink and Jodeit
Mediterr. J. Mth. 13 (2016, 119 126 DOI 10.1007/s00009-014-0480-4 0378-620X/16/010119-8 published online October 16, 2014 c Springer Bsel 2014 A Simple Proof of the Jensen-Type Inequlity of Fink nd Jodeit
More informationCyclic Generalized ϕ-contractions in b-metric Spaces and an Application to Integral Equations
Filomt 8:10 (014) 047 057 DOI 10.98/FIL1410047N Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt Cyclic Generlized ϕ-contrctions in b-metric
More informationVariational Techniques for Sturm-Liouville Eigenvalue Problems
Vritionl Techniques for Sturm-Liouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationHYERS-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 informationThe inequality (1.2) is called Schlömilch s Inequality in literature as given in [9, p. 26]. k=1
THE TEACHING OF MATHEMATICS 2018, Vol XXI, 1, pp 38 52 HYBRIDIZATION OF CLASSICAL INEQUALITIES WITH EQUIVALENT DYNAMIC INEQUALITIES ON TIME SCALE CALCULUS Muhmmd Jibril Shhb Shir Abstrct The im of this
More informationFRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES
FRACTIONAL DYNAMIC INEQUALITIES HARMONIZED ON TIME SCALES M JIBRIL SHAHAB SAHIR Accepted Mnuscript Version This is the unedited version of the rticle s it ppered upon cceptnce by the journl. A finl edited
More informationAsymptotic 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 informationJournal of Computational and Applied Mathematics. On positive solutions for fourth-order boundary value problem with impulse
Journl of Computtionl nd Applied Mthemtics 225 (2009) 356 36 Contents lists vilble t ScienceDirect Journl of Computtionl nd Applied Mthemtics journl homepge: www.elsevier.com/locte/cm On positive solutions
More informationON 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 informationResearch Article On Hermite-Hadamard Type Inequalities for Functions Whose Second Derivatives Absolute Values Are s-convex
ISRN Applied Mthemtics, Article ID 8958, 4 pges http://dx.doi.org/.55/4/8958 Reserch Article On Hermite-Hdmrd Type Inequlities for Functions Whose Second Derivtives Absolute Vlues Are s-convex Feixing
More informationAn optimal 3-point quadrature formula of closed type and error bounds
Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević,
More informationSome 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 informationThree 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 informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationResearch 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 informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationNew 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 informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using
More informationTHE 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 information2000 Mathematical Subject Classification: 65D32
Generl Mthemtics Vol. 11 No. 4 (200 5 44 On the Tricomi s qudrture formul Dumitru Acu Dedicted to Professor Gheorghe Micul on his 60 th birthdy Abstrct In this pper we obtin new results concerning the
More informationRIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROXIMATION OF CSISZAR S f DIVERGENCE
SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationThe Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f
More informationSpanning tree congestion of some product graphs
Spnning tree congestion of some product grphs Hiu-Fi Lw Mthemticl Institute Oxford University 4-9 St Giles Oxford, OX1 3LB, United Kingdom e-mil: lwh@mths.ox.c.uk nd Mikhil I. Ostrovskii Deprtment of Mthemtics
More informationarxiv: v3 [math.dg] 24 Feb 2018
BOUNDS FOR THE FIRST NON-ZERO STEKLOV EIGENVALUE rxiv:80.03747v3 [mth.dg] 4 Feb 08 SHEELA VERMA Deprtment of Mthemtics nd Sttistics Indin Institute of Technology Knpur Knpur, Indi Abstrct. Let be str-shped
More informationHermite-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 informationResearch Article On The Hadamard s Inequality for Log-Convex Functions on the Coordinates
Hindwi Publishing Corportion Journl of Inequlities nd Applictions Volume 29, Article ID 28347, 3 pges doi:.55/29/28347 Reserch Article On The Hdmrd s Inequlity for Log-Convex Functions on the Coordintes
More informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationMAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL
MAT612-REAL ANALYSIS RIEMANN STIELTJES INTEGRAL DR. RITU AGARWAL MALVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR, INDIA-302017 Tble of Contents Contents Tble of Contents 1 1. Introduction 1 2. Prtition
More informationProblem Set 4: Solutions Math 201A: Fall 2016
Problem Set 4: s Mth 20A: Fll 206 Problem. Let f : X Y be one-to-one, onto mp between metric spces X, Y. () If f is continuous nd X is compct, prove tht f is homeomorphism. Does this result remin true
More informationINNER 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 informationGeometrically Convex Function and Estimation of Remainder Terms in Taylor Series Expansion of some Functions
Geometriclly Convex Function nd Estimtion of Reminder Terms in Tylor Series Expnsion of some Functions Xioming Zhng Ningguo Zheng December 21 25 Abstrct In this pper two integrl inequlities of geometriclly
More information