Some Hardy Type Inequalities with Weighted Functions via Opial Type Inequalities
|
|
- Grace McBride
- 6 years ago
- Views:
Transcription
1 Advnces in Dynmicl Systems nd Alictions ISSN , 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 University Kingsville Dertment of Mthemtics Kingsville, Tes, 78363, USA Donl O Regn Ntionl University of Irelnd School of Mthemtics, Sttistics nd Alied Mthemtics Glwy, Irelnd Smir H. Ser Mnsour University Dertment of Mthemtics, Fculty of Science Mnsour 35516, Egyt shser@mns.edu.eg Abstrct In this er, we will rove severl new inequlities of Hrdy tye with elicit constnts. The min results will be roved using generliztions of Oil s inequlity. AMS Subject Clssifictions: 26A15, 26D10, 26D15, 39A13, 34A40. Keywords: Hrdy s inequlity, Oil s inequlity. 1 Introduction The clssicl Hrdy inequlity (see [10] sttes tht for f 0 integrble over ny finite intervl (0, nd f integrble nd convergent over (0, nd > 1, then ( 1 ( f(tdt d f (d. ( Received Februry 20, 2014; Acceted August 10, 2014 Communicted by Mrtin Bohner 0
2 2 R. Agrwl, D. O Regn nd S. Ser The constnt (/ ( 1 is the best ossible. Some etensions of Hrdy s inequlity were considered in Beesc [5]. Our im in this er is to rove some inequlities with weighted functions of Hrdy tye using Oil tye inequlities. 2 Min Results Throughout the er, ll functions re ssumed to be ositive nd mesurble nd ll the integrls which er in the inequlities re ssumed to eist nd be finite. To obtin inequlities of Hrdy tye we loo t inequlities for R(, b = r(t dt nd F ( = R(, bf (F (d, f(tdt. Ech Oil tye inequlity will give Hrdy tye inequlity. We will use number of Oil tye inequlities to illustrte this oint. Boyd nd Wong [8] roved if > 0 nd if y is n bsolutely continuous function on [, b] with y( = 0 (or y(b = 0, then q(t y(t y (t dt 1 λ 0 ( + 1 w(t y (t +1 dt, (2.1 q nd w re nonnegtive functions in C 1 [, b], nd such tht the boundry vlue roblem hs solution (q(t (u (t = λw (tu (t, with u( = 0 nd q(b [u (b] = λ w(b u (b, for which u > 0 in [, b] (let λ 0 be the smllest eigenvlue of the boundry vlue roblem. Alying the inequlity (2.1 on the term (+1 ( + 1 R(, bf (F (d, we hve R(, bf (F (d 1 s(t(f ( +1 d, (2.2 λ 0 r nd s re nonnegtive functions, r C[, b], s C 1 [, b], nd such tht the boundry vlue roblem hs solution (R(, b = (R(, b (u ( = λs (u (, (2.3 r(tdt nd note R C 1 [, b] since r C[, b] with u( = 0 nd u(b = 0, for which u > 0 in [, b] (let λ 0 be the smllest eigenvlue of the boundry vlue roblem.
3 Hrdy Tye Inequlities 3 Theorem 2.1. Assume tht r, s re nonnegtive functions with r C[, b], s C 1 [, b] nd > 0. Then +1 r( f(tdt d 1 λ 0 s( (f( +1 d, for ll integrble functions f 0 λ 0 is the smllest eigenvlue of the boundry vlue roblem (2.3. Proof. Let F ( = f(tdt. Since f is integrble on [, b] then F is bsolutely continuous on [, b]. Note F ( = 0, F ( = f( > 0 nd Integrtion by rts gives R(, b = +1 r( f(tdt d = r(f +1 (d. +1 r( f(tdt d = R(, bf +1 ( b + ( + 1 R(, bf (F (d r(tdt. Using R(b, b = 0 nd F ( = 0, we hve +1 r( f(tdt d = ( + 1 Now (2.2 estblishes the result. R(, bf (F (d. (2.4 Boyd in [7] etended the results of [8]. In [7, Theorem 2.1] the uthor estblished inequlities (best ossible constnts of the form s(t y(t y (t q dt ( b +q r(t y (t dt, λ 0 ( + q > 0, > 1, 0 q with r, s C 1 (, b nd r > 0, s > 0.e. on (, b; here λ 0 is the smllest eigenvlue of n rorite boundry vlue roblem (ssuming certin conditions re stisfied; see [7]. With these conditions (with q = 1 nd > 1 we obtin using the rocedure before nd in Theorem r( f(tdt d ( +1 r(t (f(t dt, λ 0
4 4 R. Agrwl, D. O Regn nd S. Ser λ 0 is the smllest eigenvlue of n rorite boundry vlue roblem. Insted of this inequlity (nd resenting the conditions to gurntee the eistence of λ 0 we will consider two secil cses of this result, one found in [7] nd the other in [6]. In the following, we ly n inequlity due to Boyd [7] nd the Hölder inequlity. The Boyd inequlity sttes tht: If y is bsolutely continuous on [, b] with y( = 0 (or y(b = 0, then y(t ν y (t η dt N(ν, η, s(b ν ( ν > 0, s > 1, 0 η < s, nd N(ν, η, s := I(ν, η, s := (s η ν ν σ ν+η s (s 1(ν + η (I(ν, η, s ν, σ := 1 0 { 1 + ν+η s y (t s dt, (2.5 { } 1 ν(s 1 + (s η s, (2.6 (s 1(ν + η } (ν+η+sν/sν s(η 1 s η t [1 + (η 1t]t 1/ν 1 dt. Aly the Hölder inequlity nd inequlity (2.5 to obtin R(, bf (F (d ( ( R b (, bd ( N 1 q (q, q, s(b ( (F ( s d +1 s F q ( (F ( q q d R (, bd, (2.7 > 1, 1/ + 1/q = 1, s > 1 nd 1 < q < s; here N(q, q, s is determined from (2.6 by utting ν = q nd η = q. Theorem 2.2. Assume tht r is nonnegtive mesurble function on (, b, > 1, s > 1, 1 < q < s nd 1/ + 1/q = 1. Then +1 ( r( f(tdt +1 d C (f( s s d, for ll integrble functions f 0; here ( C = ( + 1 N 1 q (q, q, s(b R (, bd.
5 Hrdy Tye Inequlities 5 Proof. The result follows from (2.4 nd (2.7. As in the roof of Theorem 2.1, by utting F ( = f(tdt, we hve the following result. Theorem 2.3. Assume tht r is nonnegtive mesurble function on (, b, > 1, 1 < q < s nd 1/ + 1/q = 1. Then ( +1 ( r( f(tdt +1 d C (f( s s d, for ll integrble functions f 0; here ( C = ( + 1 N 1 q (q, q, s(b When η = s eqution (2.5 becomes R (, d y(t ν y (t η dt L(ν, η(b ν ( L(ν, η := ( ν ηνη ν η ν + η ν + η Γ ( Γ ( η η ν η+1 η nd R(, = r(tdt. ν+η η y (t η dt, (2.8 Γ ( 1 ν nd Γ is the Gmm function. Aly inequlity (2.8 to obtin F q ( (F ( q d L(q, q(b q ( Using (2.10, we see tht ( L(q, q = (qq Γ R(, bf (F (d ( ( Γ ( ν, (2.9 q+q (F ( q q d, (2.10 q q q q+1 q Γ ( R b (, bd ( 1 q ( L 1 q (q, q(b R (, bd ( (F ( q d +1 q > 1 nd 1/ + 1/q = 1. This gives us the following results.,. (2.11 F q ( (F ( q q d
6 6 R. Agrwl, D. O Regn nd S. Ser Theorem 2.4. Assume tht r is nonnegtive mesurble function on (, b, > 1, q > 1 nd 1/ + 1/q = 1. Then +1 ( r( f(tdt +1 d C (f( q q d, for ll integrble functions f 0; here nd L(q, q is defined s in (2.11. ( C = ( + 1 L 1 q (q, q(b R (, bd Theorem 2.5. Assume tht r is nonnegtive mesurble function on (, b, > 1, q > 1 nd 1/ + 1/q = 1. Then ( +1 ( r( f(tdt +1 d C (f( q q d, for ll integrble functions f 0; here nd L(q, q is defined s in (2.11. ( C = ( + 1 L 1 q (q, q(b R (, d Finlly we ly n Oil tye inequlity due to Beesc [6] to rove inequlities of Hrdy tye. The inequlity due to Beesc is given in the following theorem. Theorem 2.6. Let r, s be nonnegtive, mesurble functions on (α, τ. Further ssume tht > 1, > 0, 0 < q <, nd let y be bsolutely continuous in [α, τ] such tht y(α = 0. Then τ α [ τ (+q/ r(t y(t y (t q dt K 1 (, q, s(t y (t dt], (2.12 α K 1 (, q, = q ( q q + ( τ (r(y q (s(y α q q ( y s 1 1 (tdt ( 1/( q dy q.
7 Hrdy Tye Inequlities 7 If insted [α, τ] is relced by [τ, β] nd y(α = 0 is relced by y(β = 0, then β τ [ β (+q/ r(t y(t y (t q dt K 2 (, q, s(t y (t dt], (2.13 τ K 2 (, q, = q ( q q + ( β (r(y q (s(y τ q q ( β y s 1 1 (tdt ( 1/( q dy q. Now, we ly inequlity (2.12 nd (2.13. For comleteness we ly (2.12 with > 1 to obtin [ (+1/ R(, b F ( F (d K 1 (, 1, s((f ( d], (2.14 ( 1 K 1 (, 1, = 1 + ( (R(, b 1 (s( s 1 1 (tdt d. (2.15 Theorem 2.7. Let > 0, > 1 nd let r, s be nonnegtive mesurble functions on (, b. Then +1 [ (+1/ r( f(tdt d ( + 1 K 1 (, 1, s((f( d], for ll integrble functions f 0; here K 1 (, 1, is defined s in (2.15. Proof. The result follows from (2.4 nd (2.14. The roof of the following theorem cn be obtined by lying inequlity (2.13 nd hence is omitted. Theorem 2.8. Let > 0, > 1 nd let r, s be nonnegtive mesurble functions on (, b. Then ( +1 [ (+1/ r( f(tdt d ( + 1 K 2 (, 1, s((f( d],
8 8 R. Agrwl, D. O Regn nd S. Ser for ll integrble functionsf 0; here K 2 (, 1, = ( ( (R(, 1 (s( 1 1 ( s 1 1 (tdt d 1. References [1] R. P. Agrwl nd P. Y. H. Png, Oil inequlities with Alictions in Differentil nd Difference Equtions, Kluwer, Dordrechet (1995. [2] R. P. Agrwl, M. Bohner, D. O Regn nd S. H. Ser, Some Wirtinger-tye inequlities on time scles nd their lictions, Pcific J. Mth. 252 (2011, [3] K. F. Andersen nd H. P. Heining, Weighted norm inequlities for certin integrl oertor, Sim. J. Mth. Anl. 14 (1983, [4] J. A. Oguntuse nd C. O. Imoru, New generliztions of Hrdy s integrl inequlties, J. Mth. Anl. Al. 241 (2000, [5] P. R. Bessc, Hrdy s inequlity nd its etensions, Pcific J. Mth. 11 (1961, [6] P. R. Bessc, Elementry roofs of some Oil-tye integrl inequlities, J. d Anlyse Mth. 36 (1979, [7] D. Boyd, Best constnts in clss of integrl inequlities, Pc. J. Mth. 30 (1969, [8] D. Boyd nd J. S. W. Wong, An etension of Oil s inequlity, J. Mth. Anl. Al. 19 (1967, [9] J. Clvert, Some generliztions of Oil s inequlity, Proc. Amer. Mth. So. 18 (1967, [10] G. H. Hrdy, J. E. Littlewood nd G. Poly, Inequlities, 2nd Ed. Cmbridge Univ. Press [11] G. H. Hrdy, Notes on theorem of Hilbert, Mth. Z. 6 (1920, [12] A. Kufner nd Lrs-Eri Persson, Weighted Inequlities of Hrdy Tye, World Scientific Publishing (2003.
9 Hrdy Tye Inequlities 9 [13] A. Kufner, L. Mligrnd nd L. Persson, The Hrdy inequlities: About its History nd Some Relted Results, Pilsen (2007. [14] N. Levinson, On n inequlity of Oil nd Beesc, Proc. Amer. Mth. Soc. 15 (1964, [15] P. Mroni, Sur l ineglité d Oil-Beesc, C. R. Acd. Sci. Pris Ser A-B. 264 (1967, A62 A64. [16] C. Olech, A simle roof of certin result of Z.Oil, Ann. Polon. Mth. 8 (1960, [17] Z. Oil, Sur uné ineglité, Ann. Polon. Mth. 8 (1960, [18] B. Oic nd A. Kufner, Hrdy-tye inequlities, Longmn Scientific& Technicl, Hrlow, ESSe, UK, (1989. [19] G. S. Yng, A note on some integrodifferentil inequlities, Soochow J. Mth. 9 (1983,
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 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 informationOPIAL S INEQUALITY AND OSCILLATION OF 2ND ORDER EQUATIONS. 1. Introduction We consider the second-order linear differential equation.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Numer, Aril 997, Pges 3 9 S 000-993997)03907-5 OPIAL S INEQUALITY AND OSCILLATION OF ND ORDER EQUATIONS R C BROWN AND D B HINTON Communicted y
More informationHERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α, m)-convex
HERMITE-HADAMARD TYPE INEQUALITIES FOR FUNCTIONS WHOSE DERIVATIVES ARE (α -CONVEX İMDAT İŞCAN Dertent of Mthetics Fculty of Science nd Arts Giresun University 8 Giresun Turkey idtiscn@giresunedutr Abstrct:
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 informationHermite-Hadamard and Simpson-like Type Inequalities for Differentiable p-quasi-convex Functions
Filomt 3:9 7 5945 5953 htts://doi.org/.98/fil79945i Pulished y Fculty of Sciences nd Mthemtics University of Niš Seri Aville t: htt://www.mf.ni.c.rs/filomt Hermite-Hdmrd nd Simson-like Tye Ineulities for
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 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 informationOn Hermite-Hadamard type integral inequalities for functions whose second derivative are nonconvex
Mly J Mt 34 93 3 On Hermite-Hdmrd tye integrl ineulities for functions whose second derivtive re nonconvex Mehmet Zeki SARIKAYA, Hkn Bozkurt nd Mehmet Eyü KİRİŞ b Dertment of Mthemtics, Fculty of Science
More informationON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR
Krgujevc ournl of Mthemtics Volume 44(3) (), Pges 369 37. ON THE HERMITE-HADAMARD TYPE INEQUALITIES FOR FRACTIONAL INTEGRAL OPERATOR H. YALDIZ AND M. Z. SARIKAYA Abstrct. In this er, using generl clss
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 informationON 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(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 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 Subclass of Multivalent Functions with Negative Coefficients inanalytic Topology
AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:1991-8178 EISSN: 239-8414 Journl home ge: www.jbsweb.com New Subclss of Multivlent Functions with Negtive Coefficients inanlytic Toology Lieth Mjed
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 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 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 informationLyapunov-type inequalities for Laplacian systems and applications to boundary value problems
Avilble online t www.isr-publictions.co/jns J. Nonliner Sci. Appl. 11 2018 8 16 Reserch Article Journl Hoepge: www.isr-publictions.co/jns Lypunov-type inequlities for Lplcin systes nd pplictions to boundry
More informationOn some inequalities for s-convex functions and applications
Özdemir et l Journl of Ineulities nd Alictions 3, 3:333 htt://wwwjournlofineulitiesndlictionscom/content/3//333 R E S E A R C H Oen Access On some ineulities for s-convex functions nd lictions Muhmet Emin
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WIT ÖLDER CONTINUOUS DERIVATIVES LJ. MARANGUNIĆ AND J. PEČARIĆ Deprtment of Applied Mthemtics Fculty of Electricl
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 informationGENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES ARE GENERALIZED s-convex IN THE SECOND SENSE
Journl of Alied Mthemtics nd Comuttionl Mechnics 6, 5(4), - wwwmcmczl -ISSN 99-9965 DOI: 75/jmcm64 e-issn 353-588 GENERALIZED OSTROWSKI TYPE INEQUALITIES FOR FUNCTIONS WHOSE LOCAL FRACTIONAL DERIVATIVES
More informationSome new integral inequalities for n-times differentiable convex and concave functions
Avilble online t wwwisr-ublictionscom/jns J Nonliner Sci Al, 10 017, 6141 6148 Reserch Article Journl Homege: wwwtjnscom - wwwisr-ublictionscom/jns Some new integrl ineulities for n-times differentible
More informationFamilies of Solutions to Bernoulli ODEs
In the fmily of solutions to the differentil eqution y ry dx + = it is shown tht vrition of the initil condition y( 0 = cuses horizontl shift in the solution curve y = f ( x, rther thn the verticl shift
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 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 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 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 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 informationHadamard-Type Inequalities for s Convex Functions I
Punjb University Journl of Mthemtics ISSN 6-56) Vol. ). 5-6 Hdmrd-Tye Ineulities for s Convex Functions I S. Hussin Dertment of Mthemtics Institute Of Sce Technology, Ner Rwt Toll Plz Islmbd Highwy, Islmbd
More informationSome New Inequalities of Simpson s Type for s-convex Functions via Fractional Integrals
Filomt 3:5 (7), 4989 4997 htts://doi.org/.98/fil75989c Published by Fculty o Sciences nd Mthemtics, University o Niš, Serbi Avilble t: htt://www.m.ni.c.rs/ilomt Some New Ineulities o Simson s Tye or s-convex
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 informationKRASNOSEL 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 informationEigenfunction Expansions for a Sturm Liouville Problem on Time Scales
Interntionl Journl of Difference Equtions (IJDE). ISSN 0973-6069 Volume 2 Number 1 (2007), pp. 93 104 Reserch Indi Publictions http://www.ripubliction.com/ijde.htm Eigenfunction Expnsions for Sturm Liouville
More informationINEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX
INEQUALITIES OF HERMITE-HADAMARD S TYPE FOR FUNCTIONS WHOSE DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI A, M. DARUS A, AND S.S. DRAGOMIR B Astrct. In this er, some ineulities of Hermite-Hdmrd
More informationSOME 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 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 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 informationA New Generalization of Lemma Gronwall-Bellman
Applied Mthemticl Sciences, Vol. 6, 212, no. 13, 621-628 A New Generliztion of Lemm Gronwll-Bellmn Younes Lourtssi LA2I, Deprtment of Electricl Engineering, Mohmmdi School Engineering Agdl, Rbt, Morocco
More informationCalculus 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 informationТеоремы 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 informationS. 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 informationLyapunov-Type Inequalities for some Sequential Fractional Boundary Value Problems
Advnces in Dynmicl Systems nd Applictions ISSN 0973-5321, Volume 11, Number 1, pp. 33 43 (2016) http://cmpus.mst.edu/ds Lypunov-Type Inequlities for some Sequentil Frctionl Boundry Vlue Problems Rui A.
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 informationSome integral inequalities on time scales
Al Mth Mech -Engl Ed 2008 29(1:23 29 DOI 101007/s10483-008-0104- c Editoril Committee of Al Mth Mech nd Sringer-Verlg 2008 Alied Mthemtics nd Mechnics (English Edition Some integrl ineulities on time scles
More informationA 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 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 informationPositive solutions for system of 2n-th order Sturm Liouville boundary value problems on time scales
Proc. Indin Acd. Sci. Mth. Sci. Vol. 12 No. 1 Februry 201 pp. 67 79. c Indin Acdemy of Sciences Positive solutions for system of 2n-th order Sturm Liouville boundry vlue problems on time scles K R PRASAD
More informationarxiv: v1 [math.ca] 2 Jan 2019
TWO-POINT QUADRATURE RULES FOR RIEMANN STIELTJES INTEGRALS WITH L ERROR ESTIMATES M.W. ALOMARI rxiv:1901.01147v1 [mth.ca] Jn 019 Abstrct. In this work, we construct new generl Two-oint qudrtre rules for
More informationFundamental Theorem of Calculus
Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under
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 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 informationTHIELE 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 informationGeneralized Hermite-Hadamard Type Inequalities for p -Quasi- Convex Functions
Ordu Üniv. Bil. Tek. Derg. Cilt:6 Syı: 683-93/Ordu Univ. J. Sci. Tech. Vol:6 No:683-93 -QUASİ-KONVEKS FONKSİYONLAR İÇİN GENELLEŞTİRİLMİŞ HERMİTE-HADAMARD TİPLİ EŞİTSİZLİKLER Özet İm İŞCAN* Giresun Üniversitesi
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 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 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 informationOn 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 informationODE: 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 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 informationCHEBYSHEV 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 informationResearch 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 informationHenstock Kurzweil delta and nabla integrals
Henstock Kurzweil delt nd nbl integrls Alln Peterson nd Bevn Thompson Deprtment of Mthemtics nd Sttistics, University of Nebrsk-Lincoln Lincoln, NE 68588-0323 peterso@mth.unl.edu Mthemtics, SPS, The University
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 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 informationMath 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that
Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under
More informationOn some Hardy-Sobolev s type variable exponent inequality and its application
Trnsctions of NAS of Azerbijn Issue Mthemtics 37 4 2 27. Series of Phsicl-Technicl nd Mthemticl Sciences On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction Frmn I. Mmedov Sli M. Mmmdli
More informationThe logarithmic mean is a mean
Mthemticl Communictions 2(1997), 35-39 35 The logrithmic men is men B. Mond, Chrles E. M. Perce nd J. Pečrić Abstrct. The fct tht the logrithmic men of two positive numbers is men, tht is, tht it lies
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 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 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 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 informationEntrance Exam, Real Analysis September 1, 2009 Solve exactly 6 out of the 8 problems. Compute the following and justify your computation: lim
1. Let n be positive integers. ntrnce xm, Rel Anlysis September 1, 29 Solve exctly 6 out of the 8 problems. Sketch the grph of the function f(x): f(x) = lim e x2n. Compute the following nd justify your
More informationHermite-Hadamard Type Inequalities for the Functions whose Second Derivatives in Absolute Value are Convex and Concave
Applied Mthemticl Sciences Vol. 9 05 no. 5-36 HIKARI Ltd www.m-hikri.com http://d.doi.org/0.988/ms.05.9 Hermite-Hdmrd Type Ineulities for the Functions whose Second Derivtives in Absolute Vlue re Conve
More informationON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS. 1. Introduction. f(a) + f(b) f(x)dx b a. 2 a
Act Mth. Univ. Comenine Vol. LXXIX, (00, pp. 65 7 65 ON SOME NEW INEQUALITIES OF HADAMARD TYPE INVOLVING h-convex FUNCTIONS M. Z. SARIKAYA, E. SET nd M. E. ÖZDEMIR Abstrct. In this pper, we estblish some
More informationLumpability and Absorbing Markov Chains
umbility nd Absorbing rov Chins By Ahmed A.El-Sheih Dertment of Alied Sttistics & Econometrics Institute of Sttisticl Studies & Reserch (I.S.S.R Ciro University Abstrct We consider n bsorbing rov Chin
More informationSEMIBOUNDED PERTURBATION OF GREEN FUNCTION IN AN NTA DOMAIN. Hiroaki Aikawa (Received December 10, 1997)
Mem. Fc. Sci. Eng. Shimne Univ. Series B: Mthemticl Science 31 (1998), pp. 1 7 SEMIBOUNDED PERTURBATION OF GREEN FUNCTION IN AN NTA DOMAIN Hiroki Aikw (Received December 1, 1997) Abstrct. Let L = P i,j
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 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 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 informationDEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b
DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd 5.4. 1. Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx.
More informationBulletin 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 informationMittag-Leffler-Hyers-Ulam stability of Hadamard type fractional integral equations
INTERNATIONA BAKAN JOURNA OF MATHEMATICS IBJM (2018), VO. 1, NO. 1, 34-42 Mittg-effler-Hyers-Ulm stbility of Hdmrd tye frctionl integrl equtions Nsrin Eghbli 1,, Vid Klvndi 2 Dertment of Mthemtics, Fculty
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 informationarxiv: v1 [math.ca] 7 Mar 2012
rxiv:1203.1462v1 [mth.ca] 7 Mr 2012 A simple proof of the Fundmentl Theorem of Clculus for the Lebesgue integrl Mrch, 2012 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics MOMENTS INEQUALITIES OF A RANDOM VARIABLE DEFINED OVER A FINITE INTERVAL PRANESH KUMAR Deprtment of Mthemtics & Computer Science University of Northern
More informationFundamental Theorem of Calculus for Lebesgue Integration
Fundmentl Theorem of Clculus for Lebesgue Integrtion J. J. Kolih The existing proofs of the Fundmentl theorem of clculus for Lebesgue integrtion typiclly rely either on the Vitli Crthéodory theorem on
More informationarxiv: v1 [math.ca] 11 Jul 2011
rxiv:1107.1996v1 [mth.ca] 11 Jul 2011 Existence nd computtion of Riemnn Stieltjes integrls through Riemnn integrls July, 2011 Rodrigo López Pouso Deprtmento de Análise Mtemátic Fcultde de Mtemátics, Universidde
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 informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis - January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More informationOn 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 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 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 informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSome inequalities of Hermite-Hadamard type for n times differentiable (ρ, m) geometrically convex functions
Avilble online t www.tjns.com J. Nonliner Sci. Appl. 8 5, 7 Reserch Article Some ineulities of Hermite-Hdmrd type for n times differentible ρ, m geometriclly convex functions Fiz Zfr,, Humir Klsoom, Nwb
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 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 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 information