Applied Mathematics and Computation 213 (2009) 322–328 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc Some mean value theorems for integrals on time scales ^c-Anh Ngô * Quo Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam Department of Mathematics, National Univesity of Singapore, Science Drive 2, Singapore 117543, Singapore a r t i c l e i n f o Keywords: Inequality Time scales Integral Mean value theorem a b s t r a c t In this short paper, we present time scales version of mean value theorems for integrals in the single variable case Ó 2009 Elsevier Inc All rights reserved Introduction and preliminaries The following two mean value theorems for time scales are due to M Bohner and G Guseinov Theorem A (See [1], Theorem 4.1) Suppose that f is continuous on ½a; b and has a delta derivative at each point of ẵa; bị If f aị ẳ f bị, then there exist points n; g ẵa; bị such that f D nị 5 f D ðgÞ: Theorem B (See [1], Theorem 4.2) Suppose that f is continuous on ½a; b and has a delta derivative at each point of ½a; bị If f aị ẳ f bị, then there exist points n; g ẵa; bị such that f D ðnÞðb À aÞ f ðbÞ À f ðaÞ f D ðgÞðb À aÞ: Motivated by Theorem A, the main aim of this paper is to present time scale version of mean value results for integrals in the single variable case We first introduce some preliminaries on time scales (see [2,3,5] for details) Definition A time scale T is an arbitrary nonempty closed subset of real numbers The calculus of time scales was initiated by Stefan Hilger in his PhD thesis [4] in order to create a theory that can unify discrete and continuous analysis Let T be a time scale T has the topology that it inherits from the real numbers with the standard topology Definition Let rðtÞ and qðtÞ be the forward and backward jump operators in T, respectively For t T, we define the forward jump operator r : T ! T by rtị ẳ inf fs T : s > tg; while the backward jump operator q : T ! T is dened by qtị ẳ sup fs T : s < tg: If rðtÞ > t, then we say that t is right-scattered, while if qðtÞ < t then we say that t is left-scattered * Address: Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam E-mail address: bookworm_vn@yahoo.com 0096-3003/$ - see front matter Ó 2009 Elsevier Inc All rights reserved doi:10.1016/j.amc.2009.03.025 ^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328 Quo 323 In this definition we put inf ; ¼ sup T (i.e., rtị ẳ t if T has a maximum t) and sup ; ẳ inf T (i.e., qtị ẳ t if T has a minimum t), where ; denotes the empty set Points that are right-scattered and left-scattered at the same time are called isolated If rtị ẳ t and t– sup T, then t is called right-dense, and if qtị ẳ t and t inf T, then t is called left-dense Points that are right-dense and left-dense at the same time are called dense Definition Let t T, then two mappings l; m : T ! ẵ0; ỵ1ị satisfying ltị :ẳ rtị t; mtị :ẳ t À qðtÞ are called the graininess functions We now introduce the set Tj which is derived from the time scales T as follows If T has a left-scattered maximum t, then T :¼ T À ftg, otherwise Tj :¼ T j Definition Let f : T ! R be a function on time scales Then for t Tj , we define f D ðtÞ to be the number, if one exists (finite), such that for all e > there is a neighborhood U of t such that for all s U f ðrðtÞÞ À f ðsÞ À f D ðtÞðrðtÞ À sÞ ejrðtÞ À sj: We say that f is D-differentiable on Tj provided f D ðtÞ exists for all t Tj Assume that f : T ! R is a function and let t Tj (t– T) Then we have the following (i) If f is D-differentiable at t, then f is continuous at t (ii) If f is left continuous at t and t is right-scattered, then f is D-differentiable at t with f D tị ẳ f ðrðtÞÞ À f ðtÞ : lðtÞ (iii) If t is right-dense, then f is D-differentiable at t if and only if lims!t f ð t Þ À f ðsÞ ; tÀs exists a finite number In this case f D tị ẳ lims!t f tị f sị : tÀs (iv) If f is D-differentiable at t, then f rtịị ẳ f tị ỵ ltịf D tị: Proposition (See [2], Theorem 1.20) Let f ; g : T ! R be differentiable at t Tj Then D fg ị tị ẳ f D tịg tị ỵ f rt ịịg D t ị ẳ f t ịg D t ị ỵ f D t ịg rt ịị: Definition A mapping f : T ! R is called rd-continuous provided if it satisfies (1) f is continuous at each right-dense point (2) The left-sided limit lims!tÀ f sị ẳ f tị exists at each left-dense point t of T Remark It follows from Theorem 1.74 of Bohner and Peterson [2] that every rd-continuous function has an antiderivative Definition A function F : T ! R is called a D-antiderivative of f : T ! R provided f D tị ẳ f tị holds for all t Tj Then the D-integral of f is dened by Z b f tịDt ẳ F bị F ðaÞ: a Proposition (See [2], Theorem 1.77) Let f ; g be rd-continuous, a; b; c T and a; b R Then Rb Rb ðaf ðtÞ ỵ bgtịịDt ẳ a a f tịDt ỵ b a gtịDt; Ra (2) a f tịDt ẳ b f tịDt; Rc Rb Rb (3) a f tịDt ẳ a f tịDt ỵ c f tịDt; Ra (4) a f tịDt ẳ 0: (1) Rb a Rb ^c-Anh Ngụ / Applied Mathematics and Computation 213 (2009) 322–328 Quo 324 Definition We say that a function p : T ! R is regressive provided 8t T j ỵ lðt Þpðt Þ – 0; holds Definition If a function p is regressive, then we define the exponential function by ep t; sị ẳ exp Z nlsị psịịDs ; t 8s; t T s where nh ðzÞ is the cylinder transformation which is defined by ( n h sị ẳ Log1 h s; ỵ shị; if h > 0; if h ¼ 0; where Log is the principal logarithm function Remark It is obviously to see that e1 ðt; sÞ is well-defined and e1 ðt; sÞ > for all t; s T We now list here two properties of ep ðt; sÞ which we will use in the rest of this paper Theorem C (See [2], Theorem 2.33) If p is regressive, then for each t T fixed, ep ðt; sÞ is a solution of the initial value problem yD ¼ pðt Þy; yðt0 Þ ¼ on T Theorem D (See [2], Theorem 2.36) If p is regressive, then (1) ep t; sị ẳ ep s;tị Dt (2) ep t;sị ị ẳ ep rptị;sị Throughout this paper, we suppose that T is a time scale, a; b T with a < b and an interval means the intersection of real interval with the given time scale Main results Theorem Let f be a continuous function on ½a; b such that Z b f xịDx ẳ 0: a Then there exist n; g ½a; bÞ so that f ðn Þ Z Z n f ðxÞDx; g f ðxÞDx f ðgÞ: a a Proof of Theorem Let hxị ẳ e1 a; xị Z x f tịDt; x ẵa; bị: a Then Z D h xị ẳ e1 a; xị f t ịDt D ẳ e1 a; xịịD a ¼ x e1 ðx; aÞ x Z f ðt ịDt ỵ e1 a; rxịị a D Z ẳ e1 a; rxịị Z x f tịDt ỵ e1 a; rxịị a Z Z f ðt ÞDt a x f ðt ịDt ỵ e1 a; rxịịf xị: a Since haị ẳ hbị then there exists n; g ẵa; bị such that D D h ðnÞ 5 h ðgÞ: a D x x ẳ e1 rxị; xị f ð t Þ Dt Z a D x f ðtÞDt þ e1 ða; rðxÞÞf ðxÞ ^c-Anh Ngơ / Applied Mathematics and Computation 213 (2009) 322–328 Quo 325 Hence Àe1 ða; rnịị Z n f tịDt ỵ e1 a; rnịịf nị 5 À e1 ða; rðgÞÞ a Z g f t ịDt ỵ e1 a; rgịịf gị; a which implies that f ðnÞ Z Z n f ðxÞDx; g f ðxÞDx f ðgÞ: a a The proof is complete h Theorem Let f be a continuous function on ẵa; b such that Z b f xịDx ¼ 0: a Then there exist n; g ½a; bÞ so that e1 ða; nÞ f ðnÞ e1 ða; rðnÞÞ Z rðnÞ f ðtÞDt; a and Z rðgÞ e1 ða; gÞ f ðgÞ: e1 ða; rðgÞÞ f ðt ÞDt a Proof of Theorem Let hðxÞ ¼ e1 ða; xÞ Z x f ðtÞDt: a Then D h xị ẳ e1 a; xịf xị e1 a; rxịị Z rxị f t ịDt: a Since haị ẳ hbị then there exists n; g ẵa; bị such that g D ðnÞ 5 g D ðgÞ: Hence e1 ða; nÞf ðnÞ À e1 ða; rðnÞÞ Z rðnÞ f ðtÞDt 5 e1 ða; gÞf ðgÞ À e1 ða; rðgÞÞ a Z rðgÞ f ðt ÞDt; a which implies that Z rðnÞ e1 ða; nÞ f ðt ÞDt; f ðnÞ e1 ða; rðnÞÞ a Z rðgÞ e1 ða; gÞ f ðt ÞDt f ðgÞ: e1 ða; rðgÞÞ a The proof is complete h Corollary Let T ¼ R, from Theorems and together with the continuity of f we deduce that the existence of c ẵa; b such that f cị ¼ Z c f ðxÞdx a provided Z b f xịdx ẳ 0: a Theorem Let f be a continuous function on ½a; b such that Z b f xịDx ẳ 0: a Then for each T c < a, there exist n; g ẵa; bị so that ^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328 Quo 326 f ðnÞðn À cÞ Z Z n f ðt ÞDt; a g f ðt ÞDt f ðgÞðg À cÞ: a Proof of Theorem Let xc hxị ẳ Z x f t ịDt; x ẵa; bị; T c < a: a Therefore rxị cịx cị D h xị ẳ Z x f t ịDt ỵ a rxị c f xị: Since haị ẳ hbị then there exists n; g ẵa; bị such that D D h ðnÞ 5 h ðgÞ: Hence Z À1 ðrðnÞ cịn cị n f t ịDt ỵ a À1 f ðnÞ 5 rðnÞ À c ðrðgÞ À cÞðg À cÞ Z a g f ð t ị Dt ỵ rgị c f gị; which implies Rn f ð t Þ Dt f ðnÞ a ; rðnÞ À c ðrðnÞ À cÞðn À cÞ Rg f ðt ÞDt f ðgÞ a : ðrðgÞ À cÞðg À cÞ rðgÞ À c Thus f ðnÞðn À cÞ Z Z n f ðt ÞDt; g f ðt ÞDt f ðgÞðg À cÞ: a a The proof is complete h Corollary Let T ¼ R, from Theorem together with the continuity of f we deduce the existence of c ½a; b such that f cịn cị ẳ Z c f ðxÞdx; a for each c < a provided Z b f xịDx ẳ 0: a Theorem Let f ; g be a continuous function on ½a; b Then there exist n; g ẵa; bị so that f n Þ Z ! b g ðt ÞDt Z rðnÞ f ðt ÞDt g ðnÞ; a n and f ðgÞ Z ! b g ðt ÞDt Z rðgÞ = g f ðtÞDt g ðgÞ: a Proof of Theorem Z hxị ẳ x f t ịDt a Z x g t ịDt ; x ẵa; bị: b Then Z D h xị ẳ f xị x b Z rðxÞ g ð t Þ Dt f tịDt gxị: a Since haị ẳ hbị then there exist n; g ẵa; bị such that D D h ðnÞ 5 h ðgÞ: ^c-Anh Ngô / Applied Mathematics and Computation 213 (2009) 322–328 Quo 327 Hence Àf ðnÞ Z n Z rðnÞ Z g ðt ÞDt À f ðtÞDt g ðnÞ 5 À f ðgÞ b a g Z rðgÞ g ð t Þ Dt À f ðt ÞDt g ðgÞ; a b which implies Z f nị Z rnị g t ịDt ỵ f ðtÞDt g ðnÞ; n b Z = f gị a g Z rgị g t ịDt ỵ f ðtÞDt g ðgÞ; a b or equivalently f ðnÞ Z ! b g ðt ÞDt n f ðgÞ Z Z rðnÞ f ðt ÞDt g ðnÞ; a ! b Z rðgÞ g ð t Þ Dt = g f ðt ÞDt g ðgÞ: a The proof is complete h Corollary Let T ¼ R, from Theorem together with the continuity of f and g we deduce the existence of c ½a; b such that Z f cị ! b gxịdx ẳ Z c c f ðxÞdx g ðcÞ: a Theorem Let f ; g be continuous functions on ½a; b Then there exist n; g ẵa; bị so that Z n f ðt ÞDt Z a n g ðt ÞDt Z f ðn Þ b n Z rnị g t ị Dt ỵ f ðtÞDt g ðnÞ b a and Z g f ðt ÞDt Z a g Z g ðt ÞDt = f ðgÞ b g Z rðgÞ g ðt ịDt ỵ f t ịDt g gị: b a Proof of Theorem Let hxị ẳ e1 a; xị Z x f ð t Þ Dt Z a x g t ịDt : b Then D h xị ẳ e1 ða; rðxÞÞ Z x f ð t Þ Dt a ẳ e1 a; rxịị Z Z x b x f ð t Þ Dt a Z x Z g ðt ÞDt À e1 ða; rðxÞÞ x f ð t Þ Dt Z a g ðt ÞDt b Z g ðt ÞDt À e1 ða; rðxÞÞ f xị b x x g t ịDt ỵ b D Z rðxÞ f ðtÞDt gðxÞ : a Since haị ẳ hbị then there exists n; g ½a; bÞ such that D D h ðnÞ 5 h ðgÞ: Hence e1 ða; rðnÞÞ Z n f ðt ÞDt Z a n Z g ðt ÞDt À e1 ða; rðnÞÞ f ðnÞ b n Z rnị g t ịDt ỵ f t ÞDt g ðnÞ 0; b a and Z e1 ða; rðgÞÞ g f ðt ÞDt Z a g Z g ðt ÞDt À e1 ða; rðgÞÞ f ðgÞ b g Z rðgÞ g ðt ịDt ỵ f t ịDt g gị = 0: b Thus Z a n f ðt ÞDt Z b n g ðt ÞDt Z f ðn Þ n b Z rðnÞ g ð t Þ Dt þ f ðtÞDt g ðnÞ a a ^c-Anh Ngơ / Applied Mathematics and Computation 213 (2009) 322–328 Quo 328 and Z g f ðt ÞDt Z a g g ðt ÞDt b = f ðgÞ Z g a b The proof is complete Z rðgÞ g ð t ị Dt ỵ f t ịDt g gị: h Corollary Let T ¼ R, from Theorem together with the continuity of f and g we deduce the existence of c ½a; b such that Z c f xịdx Z a b c Z gxịdx ẳ f cị c b Z gxịdx ỵ c f ðxÞdx g ðcÞ: a Acknowledgements The author wishes to express gratitude to the anonymous referee(s) for a number of valuable comments and suggestions which helped to improve the presentation of the present paper from line to line References [1] [2] [3] [4] [5] M Bohner, G Guseinov, Partial differential equation on time scales, Dynamic Systems and Applications 13 (2004) 351–379 M Bohner, A Peterson, Dynamic Equations on Time Series, Birkhäuser, Boston, 2001 M Bohner, A Peterson, Advances in Dynamic Equations on Time Series, Birkhäuser, Boston, 2003 S Hilger, Ein Mabkettenkalkül mit Anwendung auf Zentrmsmannigfaltingkeiten, Ph.D Thesis, Univarsi, Würzburg, 1988 V Lakshmikantham, S Sivasundaram, B Kaymakcalan, Dynamic Systems on Measure Chains, Kluwer Academic Publishers, 1996 ... differential equation on time scales, Dynamic Systems and Applications 13 (2004) 351–379 M Bohner, A Peterson, Dynamic Equations on Time Series, Birkhäuser, Boston, 2001 M Bohner, A Peterson, Advances... : T ! R be a function on time scales Then for t Tj , we define f D ðtÞ to be the number, if one exists (finite), such that for all e > there is a neighborhood U of t such that for all s U f ðrðtÞÞ... / Applied Mathematics and Computation 213 (2009) 322–328 Quo 324 Definition We say that a function p : T ! R is regressive provided 8t T j ỵ lt ịpt ị 0; holds Denition If a function p is regressive,