The diamond-alpha Riemann integral and mean value theorems on time scales
aa r X i v : . [ m a t h . C A ] A p r The diamond-alpha Riemann integral and meanvalue theorems on time scales
Agnieszka B. Malinowska † [email protected] Delfim F. M. Torres ‡ [email protected] † Faculty of Computer ScienceBia lystok Technical University15-351 Bia lystok, Poland ‡ Department of MathematicsUniversity of Aveiro3810-193 Aveiro, Portugal
Abstract
We study diamond-alpha integrals on time scales. A diamond-alphaversion of Fermat’s theorem for stationary points is also proved, as wellas Rolle’s, Lagrange’s, and Cauchy’s mean value theorems on time scales.
Mathematics Subject Classification 2000:
Keywords: time scales, diamond-alpha integral, Fermat’s theorem for station-ary points, mean value theorems.
The calculus on time scales has been initiated by Aulbach and Hilger in order tocreate a theory that can unify and extend discrete and continuous analysis [1].Two versions of the calculus on time scales, the delta and nabla calculus, are nowstandard in the theory of time scales [4, 5]. In 2006, a combined diamond-alphadynamic derivative was introduced by Sheng, Fadag, Henderson, and Davis [9],as a linear combination of the delta and nabla dynamic derivatives on timescales. The diamond-alpha derivative reduces to the standard delta-derivativefor α = 1 and to the standard nabla derivative for α = 0. On the other hand,it represents a weighted dynamic derivative formula on any uniformly discretetime scale when α = . We refer the reader to [6, 7, 8, 9, 10] for a completeaccount of the recent diamond-alpha calculus on time scales. In Section 2 webriefly review the necessary definitions and calculus on time scales; our resultsare given in Section 3. 1he diamond-alpha integral on time scales is defined in [7, 8, 9, 10] by meansof a linear combination of the delta and nabla integrals. In the present paper weuse a Darboux approach to define the Riemann diamond-alpha integral on timescales and to prove the corresponding main theorems of the diamond-alpha in-tegral calculus (Section 3.1). In addition, we briefly investigate diamond-alphaimproper integrals (Section 3.2), and prove some new versions of mean valuetheorems on time scales via diamond- α derivatives and integrals (Section 3.3).A new notion of local extremum on time scales is also proposed, which leads toa diamond-alpha Fermat’s theorem for stationary points (Theorem 3.19) moresimilar in aspect to the classical condition than those of delta or nabla deriva-tives. In this section we introduce basic definitions and results from the theory ofdelta, nabla, and diamond-alpha time scales [4, 7, 9].A nonempty closed subset of R is called a time scale and is denoted by T .The forward jump operator σ : T → T is defined by σ ( t ) = inf { s ∈ T : s > t } , for all t ∈ T , while the backward jump operator ρ : T → T is defined by ρ ( t ) = sup { s ∈ T : s < t } , for all t ∈ T , with inf ∅ = sup T (i.e., σ ( M ) = M if T has a maximum M ), and sup ∅ = inf T (i.e., ρ ( m ) = m if T has a minimum m ).A point t ∈ T is called right-dense , right-scattered , left-dense and left-scattered if σ ( t ) = t , σ ( t ) > t , ρ ( t ) = t and ρ ( t ) < t , respectively.Throughout the paper we let T = [ a, b ] ∩ T with a < b and T a time scalecontaining a and b .The delta graininess function µ : T → [0 , ∞ ) is defined by µ ( t ) = σ ( t ) − t, for all t ∈ T ;the nabla graininess function is defined by η ( t ) := t − ρ ( t ).We introduce the sets T k , T k , and T kk , which are derived from the timescale T , as follows. If T has a left-scattered maximum t , then T k = T − { t } ,otherwise T k = T . If T has a right-scattered minimum t , then T k = T − { t } ,otherwise T k = T . Finally, we define T kk = T k ∩ T k .We say that a function f : T → R is delta differentiable at t ∈ T k if thereexists a number f ∆ ( t ) such that for all ε > U of t (i.e., U = ( t − δ, t + δ ) ∩ T for some δ >
0) such that | f ( σ ( t )) − f ( s ) − f ∆ ( t )( σ ( t ) − s ) | ≤ ε | σ ( t ) − s | , for all s ∈ U .
We call f ∆ ( t ) the delta derivative of f at t and say that f is delta differentiable on T k provided f ∆ ( t ) exists for all t ∈ T k .2e define f ∇ ( t ) to be the number value, if one exists, such that for all ǫ > V of t such that for all s ∈ V , (cid:12)(cid:12) f ( ρ ( t )) − f ( s ) − f ∇ ( t ) ( ρ ( t ) − s ) (cid:12)(cid:12) ≤ ǫ | ρ ( t ) − s | . We say that f is nabla differentiable on T k , provided that f ∇ ( t ) exists for all t ∈ T k .For delta differentiable functions f and g , the next formula holds:( f g ) ∆ ( t ) = f ∆ ( t ) g σ ( t ) + f ( t ) g ∆ ( t )= f ∆ ( t ) g ( t ) + f σ ( t ) g ∆ ( t ) , where we abbreviate here and throughout the text f ◦ σ by f σ . Similarly propertyholds for nabla derivatives (and we then use the notation f ρ = f ◦ ρ ).A function f : T → R is said to be a regulated function if its left-sided limitsexist at left-dense points, and its right-sided limits exist at right-dense points.A function f : T → R is called rd-continuous if it is continuous at right-dense points and if its left-sided limit exists at left-dense points. We denote theset of all rd-continuous functions by C rd and the set of all delta differentiablefunctions with rd-continuous derivative by C .Analogously, a function f : T → R is called ld-continuous, provided it iscontinuous at all left-dense points in T and its right-sided limits exist finite atall right-dense points in T .It is known that rd-continuous functions possess a delta antiderivative , i.e.,there exists a function F with F ∆ = f , and in this case the delta integral isdefined by R dc f ( t )∆ t = F ( c ) − F ( d ) for all c, d ∈ T . The delta integral has thefollowing property: Z σ ( t ) t f ( τ )∆ τ = µ ( t ) f ( t ) . A function G : T → R is called a nabla antiderivative of g : T → R , provided G ∇ ( t ) = g ( t ) holds for all t ∈ T k . Then, the nabla integral of g is defined by R ba g ( t ) ∇ t = G ( b ) − G ( a ).Let T be a time scale, and t , s ∈ T . Following [7], we define µ ts = σ ( t ) − s , η ts = ρ ( t ) − s , and f ⋄ α ( t ) to be the value, if one exists, such that for all ǫ > U of t such that for all s ∈ U | α [ f σ ( t ) − f ( s )] η ts + (1 − α ) [ f ρ ( t ) − f ( s )] µ ts − f ⋄ α ( t ) µ ts η ts | < ǫ | µ ts η ts | . A function f is said diamond- α differentiable on T kk provided f ⋄ α ( t ) exists forall t ∈ T kk . Let 0 ≤ α ≤
1. If f ( t ) is differentiable on t ∈ T kk both in thedelta and nabla senses, then f is diamond- α differentiable at t and the dynamicderivative f ⋄ α ( t ) is given by f ⋄ α ( t ) = αf ∆ ( t ) + (1 − α ) f ∇ ( t ) (2.1)(see [7, Theorem 3.2]). Equality (2.1) is given as definition of f ⋄ α ( t ) in [9]. Thediamond- α derivative reduces to the standard ∆ derivative for α = 1, or the3tandard ∇ derivative for α = 0. On the other hand, it represents a “weighteddynamic derivative” for α ∈ (0 , T when α = .Let f, g : T → R be diamond- α differentiable at t ∈ T kk . Then (cf. [9,Theorem 2.3]),(i) f + g : T → R is diamond- α differentiable at t ∈ T kk with( f + g ) ⋄ α ( t ) = f ⋄ α ( t ) + g ⋄ α ( t );(ii) For any constant c , cf : T → R is diamond- α differentiable at t ∈ T kk with( cf ) ⋄ α ( t ) = cf ⋄ α ( t );(iii) f g : T → R is diamond- α differentiable at t ∈ T kk with( f g ) ⋄ α ( t ) = f ⋄ α ( t ) g ( t ) + αf σ ( t ) g ∆ ( t ) + (1 − α ) f ρ ( t ) g ∇ ( t ) . Let a, b ∈ T , and h : T → R . The diamond- α integral of h from a to b is definedin [7, 9] by Z ba h ( τ ) ⋄ α τ = α Z ba h ( τ )∆ τ + (1 − α ) Z ba h ( τ ) ∇ τ, ≤ α ≤ , (3.1)provided that there exist delta and nabla integrals of h on T . In § α integral. We use a Darboux approachwithout the need to define previously delta and nabla integrals. Improper in-tegrals are introduced in § § α derivatives and in-tegrals. Moreover, a new notion of local extremum on time scales is proposed,which leads to a diamond-alpha first order optimality condition more similar tothe classical condition ( T = R ) than those of delta or nabla derivatives (cf. [4]). α integral Let T be a one-dimensional time scale, a, b ∈ T , a < b and [ a, b ] a closed,bounded interval in T . A partition of [ a, b ] is any finite ordered subset P = { t , t , . . . , t n } ⊂ [ a, b ] , where a = t < t < · · · < t n = b . The number n depends on the particularpartition, so we have n = n ( P ). We denote by P = P ( a, b ) the set of allpartitions of [ a, b ]. Let f be a real-valued bounded function on [ a, b ]. We set: M = sup { f ( t ) : t ∈ [ a, b ) } , m = inf { f ( t ) : t ∈ [ a, b ) } , = sup { f ( t ) : t ∈ ( a, b ] } , m = inf { f ( t ) : t ∈ ( a, b ] } , and for 1 ≤ i ≤ nM i = sup { f ( t ) : t ∈ [ t i − , t i ) } , m i = inf { f ( t ) : t ∈ [ t i − , t i ) } ,M i = sup { f ( t ) : t ∈ ( t i − , t i ] } , m i = inf { f ( t ) : t ∈ ( t i − , t i ] } . Let α ∈ [0 , ⊂ R . The upper Darboux ⋄ α -sum U ( f, P ) and the lower Darboux ⋄ α -sum L ( f, P ) of f with respect to P are defined respectively by U ( f, P ) = n X i =1 ( αM i + (1 − α ) M i )( t i − t i − ) ,L ( f, P ) = n X i =1 ( αm i + (1 − α ) m i )( t i − t i − ) . Note that U ( f, P ) ≤ n X i =1 ( αM + (1 − α ) M )( t i − t i − ) = ( αM + (1 − α ) M )( b − a )and L ( f, P ) ≥ n X i =1 ( αm + (1 − α ) m )( t i − t i − ) = ( αm + (1 − α ) m )( b − a ) . Thus, we have:( αm + (1 − α ) m )( b − a ) ≤ L ( f, P ) ≤ U ( f, P ) ≤ ( αM + (1 − α ) M )( b − a ) . (3.2)The upper Darboux ⋄ α -integral U ( f ) of f from a to b is defined by U ( f ) = inf { U ( f, P ) : P ∈ P ( a, b ) } and the lower Darboux ⋄ α -integral L ( f ) of f from a to b is defined by L ( f ) = sup { L ( f, P ) : P ∈ P ( a, b ) } . In view of (3.2), U ( f ) and L ( f ) are finite real numbers. Definition 3.1.
We say that f is ⋄ α -integrable from a to b (or on [ a, b ]) provided L ( f ) = U ( f ). In this case, we write R ba f ( t ) ⋄ α t for this common value. We callthis integral the Darboux ⋄ α -integral.Let U ( f ) and L ( f ) denote the upper and the lower Darboux ∆-integral of f from a to b , respectively; U ( f ) and L ( f ) denote the upper and the lowerDarboux ∇ -integral of f from a to b , respectively. Given the construction of U ( f ) and L ( f ), the equality (3.1) follows from the properties of supremum andinfimum. 5 orollary 3.2. If f is ∆ -integrable from a to b and ∇ -integrable from a to b ,then it is ⋄ α -integrable from a to b and Z ba f ( t ) ⋄ α t = α Z ba f ( t )∆ t + (1 − α ) Z ba f ( t ) ∇ t. Now, suppose that f is ⋄ α -integrable from a to b . Then L ( f ) = U ( f ) and αU ( f ) + (1 − α ) U ( f ) = αL ( f ) + (1 − α ) L ( f ) ,α ( U ( f ) − L ( f )) = (1 − α )( L ( f ) − U ( f )) . Since U ( f ) ≥ L ( f ) and U ( f ) ≥ L ( f ), we get the following result. Corollary 3.3.
Let f be ⋄ α -integrable from a to b .(i) If α = 1 , then f is ∆ -integrable from a to b .(ii) If α = 0 , then f is ∇ -integrable from a to b .(iii) If < α < , then f is ∆ -integrable and ∇ -integrable from a to b . Example 3.4.
Note that the strict inequalities in (iii) of the above corollary arenecessary. Consider the function f ( t ) = t and the time scale T = Z . We have R − f ( t )∆ t = − R f ( t ) ∇ t = 1. However, both R f ( t )∆ t and R − f ( t ) ∇ t do not exist.The following theorems may be showed in the same way as Theorem 5.5 andTheorem 5.6 in [2]. Theorem 3.5. If U ( f, P ) = L ( f, P ) for some P ∈ P ( a, b ) , then the function f is ⋄ α -integrable from a to b and R ba f ( t ) ⋄ α t = U ( f, P ) = L ( f, P ) . Theorem 3.6.
A bounded function f on [ a, b ] is ⋄ α -integrable if and only if foreach ε > there exists P ∈ P ( a, b ) such that U ( f, P ) − L ( f, P ) < ε . Lemma 3.7 ([2]) . For every δ > there exists some partition P ∈ P ( a, b ) given by a = t < t < · · · < t n = b such that for each i ∈ { , , . . . , n } either t i − t i − ≤ δ or t i − t i − > δ and ρ ( t i ) = t i − . We denote by P δ = P δ ( a, b ) the set of all P ∈ P ( a, b ) that possess theproperty indicated in Lemma 3.7. Theorem 3.8.
A bounded function f on [ a, b ] is ⋄ α -integrable if and only if foreach ε > there exists δ > such that P ∈ P δ ( a, b ) ⇒ U ( f, P ) − L ( f, P ) < ε . (3.3) Proof.
By Theorem 3.6 condition (3.3) implies integrability. Conversely, sup-pose that f is ⋄ α -integrable from a to b . If α = 1 or α = 0, then f is ∆-integrablefrom a to b or ∇ -integrable from a to b . Therefore condition (3.3) holds (see [2]).Now, let 0 < α <
1. By Corollary 3.3, f is ∆-integrable and ∇ -integrable from a to b . According to [2, Theorem 5.9], for each ε > δ > δ > P ∈ P δ ( a, b ) implies U ( f, P ) − L ( f, P ) < ε and P ∈ P δ ( a, b )implies U ( f, P ) − L ( f, P ) < ε . If P ∈ P δ ( a, b ) where δ = min { δ , δ } , then wehave U ( f, P ) − L ( f, P ) = αU ( f, P )+(1 − α ) U ( f, P ) − αL ( f, P ) − (1 − α ) L ( f, P ) <ε . 6et f be a bounded function on [ a, b ] and let P ∈ P ( a, b ) be given by a = t < t < · · · < t n = b . For 1 ≤ i ≤ n , choose arbitrary points ξ i in[ t i − , t i ), ξ i in ( t i − , t i ], and form the sum S = Σ ni =1 ( αf ( ξ i ) + (1 − α ) f ( ξ i ))( t i − t i − ) . We call S a Riemann ⋄ α -sum of f corresponding to P ∈ P ( a, b ). We say that f is Riemann ⋄ α -integrable from a to b if there exists a real number I with thefollowing property: for each ε > δ > P ∈ P δ ( a, b )implies | S − I | < ε independent of the choice of ξ i , ξ i for 1 ≤ i ≤ n . The number I is called the Riemann ⋄ α -integral of f from a to b . Theorem 3.9. If f is Riemann ∆ -integrable and Riemann ∇ -integrable from a to b , then it is Riemann ⋄ α -integrable from a to b and I = α R ba f ( t )∆ t + (1 − α ) R ba f ( t ) ∇ t .Proof. Assume that f is Riemann ∆-integrable and Riemann ∇ -integrable from a to b . Then, for each ε > δ > δ > P ∈ P δ ( a, b )implies | S − R ba f ( t )∆ t | < ε , and P ∈ P δ ( a, b ) implies | S − R ba f ( t ) ∇ t | < ε , where S is the Riemann ∆-sum of f corresponding to P , and S is the Riemann ∇ -sumof f corresponding to P . Now, if P ∈ P δ ( a, b ) with δ = min { δ , δ } , then wehave (cid:12)(cid:12)(cid:12) S − α Z ba f ( t )∆ t − (1 − α ) Z ba f ( t ) ∇ t (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αS + (1 − α ) S − α Z ba f ( t )∆ t − (1 − α ) Z ba f ( t ) ∇ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αS − α Z ba f ( t )∆ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1 − α ) S − (1 − α ) Z ba f ( t ) ∇ t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε . Thus, f is Riemann ⋄ α -integral from a to b and I = α R ba f ( t )∆ t +(1 − α ) R ba f ( t ) ∇ t .By construction of the ⋄ α -Riemann sum S , the following theorem may beproved in much the same way as [2, Theorem 5.11]. Theorem 3.10.
A bounded function f on [ a, b ] is Riemann ⋄ α -integrable if andonly if it is Darboux ⋄ α -integrable, in which case the values of the integrals areequal. We define Z aa f ( t ) ⋄ α t = 0and Z ba f ( t ) ⋄ α t = − Z ab f ( t ) ⋄ α t, a > b. orollary 3.11. Let f be Riemann ⋄ α -integrable from a to b .(i) If α = 1 , then f is Riemann ∆ -integrable from a to b .(ii) If α = 0 , then f is Riemann ∇ -integrable from a to b .(iii) If < α < , then f is is Riemann ∆ -integrable and Riemann ∇ -integrable from a to b . Theorem 3.12.
Let f : T → R and let t ∈ T . Then,(i) f is integrable from t to σ ( t ) and Z σ ( t ) t f ( s ) ⋄ α s = µ ( t )( αf ( t ) + (1 − α ) f σ ( t )) ; (3.4) (ii) f is integrable from ρ ( t ) to t and Z tρ ( t ) f ( s ) ⋄ α s = η ( t )( αf ρ ( t ) + (1 − α ) f ( t )) . Proof. (i) If σ ( t ) = t , then µ ( t ) = 0 and equality (3.4) is obvious. If σ ( t ) > t ,then P ( t, σ ( t )) contains only one element given by t = s < s = σ ( t ). Since[ s , s ) = t and ( s , s ] = σ ( t ), we have U ( f, P ) = αf ( t )( σ ( t ) − t ) + (1 − α ) f σ ( t )( σ ( t ) − t ) = L ( f, P ). By Theorem 3.5, f is ⋄ α -integrable from t to σ ( t )and (3.4) holds. Proof of (ii) is done in a similar way. Corollary 3.13.
Let a, b ∈ T and a < b . Then we have the following:(i) If T = R , then a bounded function f on [ a, b ] is ⋄ α -integrable from a to b if and only if is Riemann integrable on [ a, b ] in the classical sense, and in thiscase R ba f ( t ) ⋄ α t = R ba f ( t ) dt .(ii) If T = Z , then every function f defined on Z is ⋄ α -integrable from a to b , and R ba f ( t ) ⋄ α t = P b − t = a +1 f ( t ) + αf ( a ) + (1 − α ) f ( b ) .(iii) If T = h Z , then then every function f defined on h Z is ⋄ α -integrablefrom a to b , and R ba f ( t ) ⋄ α t = P bh − k = ah +1 f ( kh ) h + αf ( a ) h + (1 − α ) f ( b ) h . The following results are straightforward consequences of Theorems 3.9 and3.10 and properties of the Riemann delta (nabla) integral:1. Let a, b ∈ T and a < b . Every constant function f : T → R is ⋄ α -integrablefrom a to b and R ba f ( t ) ⋄ α t = c ( b − a ).2. Every monotone function f : T → R on [ a, b ] is ⋄ α -integrable from a to b .3. Every continuous function f : T → R on [ a, b ] is ⋄ α -integrable from a to b .4. Every bounded function f : T → R on [ a, b ] with only finitely manydiscontinuity points is ⋄ α -integrable from a to b .5. Every regulated function f : T → R on [ a, b ] is ⋄ α -integrable from a to b .6. Let f : T → R be a bounded function on [ a, b ] that is ⋄ α -integrable from a to b . Then, f is ⋄ α -integrable on every subinterval [ c, d ] of [ a, b ].8. Let f, g be ⋄ α -integrable from a to b and c ∈ R , d ∈ T , a < d < b . Then:(i) cf is ⋄ α -integrable from a to b and R ba ( cf )( t ) ⋄ α t = c R ba f ( t ) ⋄ α t ,(ii) f + g is ⋄ α -integrable from a to b and R ba ( f + g )( t ) ⋄ α t = R ba f ( t ) ⋄ α t + R ba g ( t ) ⋄ α t ,(iii) f g is ⋄ α -integrable from a to b ,(iv) R ba f ( t ) ⋄ α t = R da f ( t ) ⋄ α t + R bd f ( t ) ⋄ α t .8. If f, g are ⋄ α -integrable from a to b and if f ( t ) ≤ g ( t ) for all t ∈ [ a, b ], then R ba f ( t ) ⋄ α t ≤ R ba f ( t ) ⋄ α t .9. If f is ⋄ α -integrable from a to b , then so is | f | . Moreover, | R ba f ( t ) ⋄ α t | ≤ R ba | f ( t ) | ⋄ α t . Let T be a time scale, a ∈ T . Throughout this section we assume that thereexists a subset { t k : k ∈ N } ⊂ T , a = t < t < t < · · · , lim k →∞ t k = ∞ . Let us suppose that the real-valued function f is defined on [ a, ∞ ) and is Rie-mann ⋄ α -integrable from a to any point A ∈ T with A ≥ a . If the integral F ( A ) = Z Aa f ( t ) ⋄ α t approaches a finite limit as A → ∞ , we call that limit the improper diamond- α integral of first kind of f from a to ∞ and write Z ∞ a f ( t ) ⋄ α t = lim A →∞ Z Aa f ( t ) ⋄ α t ! . (3.5)In such case we say that the improper integral R ∞ a f ( t ) ⋄ α t exists or that it isconvergent. If the limit (3.5) does not exist, we say that the improper integraldoes not exist or is divergent. Note that Z ∞ a f ( t ) ⋄ α t = lim A →∞ α Z Aa f ( t ) △ t + (1 − α ) Z Aa f ( t ) ∇ t ! . Corollary 3.14.
If improper integrals R ∞ a f ( t ) △ t and R ∞ a f ( t ) ∇ t exist, then Z ∞ a f ( t ) ⋄ α t = α Z ∞ a f ( t ) △ t + (1 − α ) Z ∞ a f ( t ) ∇ t. On the other hand, the improper diamond- α integral may exist even if im-proper delta and nabla integrals do not exist.9 xample 3.15. Consider the function f ( t ) = ( t = 2 l − t = 2 l + 1 , l ∈ T , on the time scale T = N ∪ { } . We have Z ∞ f ( t ) △ t = ∞ X k =0 f ( k ) , Z ∞ f ( t ) ∇ t = ∞ X k =0 f ( k + 1) . Hence, the improper integrals R ∞ f ( t ) △ t and R ∞ f ( t ) ∇ t do not exist. On theother hand, for α = 1 / Z Aa f ( t ) ⋄ α t = 12 A − X k =0 f ( k ) + A − X k =0 f ( k + 1) ! = 12 A − X k =0 ( f ( k ) + f ( k + 1)) = 0and Z ∞ a f ( t ) ⋄ α t = lim A →∞ Z Aa f ( t ) ⋄ α t ! = 0 . Theorem 3.16 and Theorem 3.17 are exact analogies of mean value theorems fordelta (nabla) integrals and there are no differences in proofs of these theorems.However, the formulation of the mean value theorems 3.21 and 3.22 below, forthe diamond- α derivative, provide a generalization more similar to the classicalresults than the ones previously proved for the delta or nabla derivatives (cf.[3]). Theorem 3.16.
Let f and g be bounded and ⋄ α -integrable functions from a to b , and let g be nonnegative (or nonpositive) on [ a, b ] . Let m and M be theinfimum and supremum, respectively, of the function f on [ a, b ] . Then, thereexists a real number K satisfying the inequalities m ≤ K ≤ M such that Z ba f ( t ) g ( t ) ⋄ α t = K Z ba g ( t ) ⋄ α t. Theorem 3.17.
Let f be bounded and ⋄ α -integrable function on [ a, b ] . Let m and M be the infimum and supremum, respectively, of the function F ( s ) = R ta f ( s ) ⋄ α s on [ a, b ] . We have:(i) If a function g is non-increasing with g ( t ) ≥ on [ a, b ] , then there is anumber K such that m ≤ K ≤ M and Z ba f ( t ) g ( t ) ⋄ α t = Kg ( a ) . ii) If g is any monotonic function on [ a, b ] , then there is a number K such that m ≤ K ≤ M and Z ba f ( t ) g ( t ) ⋄ α t = [ g ( a ) − g ( b )] K + g ( b ) Z ba f ( t ) ⋄ α t. We now define a local maximum of a real function defined on a time scale T . The definition of a local minimum is done in a similar way. Definition 3.18.
We say that a function f : T → R assumes its local maximumat t ∈ T kk provided(i) if t is scattered, then f ( σ ( t )) ≤ f ( t ) and f ( ρ ( t )) ≤ f ( t );(ii) if t is dense, then there is a neighborhood U of t such that f ( t ) ≤ f ( t )for all t ∈ U ;(iii) if t is left-scattered and right-dense, then f ( ρ ( t )) ≤ f ( t ) and there is aneighborhood U of t such that f ( t ) ≤ f ( t ) for all t ∈ U with t > t ;(iv) if t is right-scattered and left-dense, then f ( σ ( t )) ≤ f ( t ) and there is aneighborhood U of t such that f ( t ) ≤ f ( t ) for all t ∈ U with t < t .Theorem 3.19 permits to introduce the notion of critical point in a similarway as done in classical calculus. We remark that equality f ⋄ α ( t ) = 0 inTheorem 3.19 does not always hold for the delta ( α = 1) or nabla ( α = 0) cases(cf. [3]). Theorem 3.19 (diamond-alpha Fermat’s theorem for stationary points) . Sup-pose f assumes its local extremum at t ∈ T kk and f is delta and nabla differen-tiable at t . Then, there exists α ∈ [0 , such that f ⋄ α ( t ) = 0 .Proof. Suppose that f assumes its local maximum at t ∈ T kk . Then, we have f △ ( t ) ≤ f ∇ ( t ) ≥
0. If f △ ( t ) = 0 ( f ∇ ( t ) = 0), then we put α = 1( α = 0). Therefore, we can assume that f △ ( t ) < f ∇ ( t ) >
0. Setting α = f ∇ ( t ) f ∇ ( t ) − f △ ( t ) , it is easy to see that 0 < α <
1, and we obtain the intended result.
Example 3.20.
Let T = {− , , , , , } , and f be defined by f ( −
1) = f (0) =5, f (1) = 0, f (2) = 1, and f (3) = f (4) = 3. The point 1 is a local minimizer,points 0 and 3 are local maximizers, and point 2 is neither a minimizer neithera maximizer (as well as -1 and 4, by definition). The delta-derivative is onlyzero at point 3 while the nabla-derivative is zero only at zero. According withTheorem 3.19, at all extremizers there exists an alpha such that the diamond-alpha derivative vanishes. Indeed, f ⋄ (0) = 0, f ⋄ / (1) = 0, and f ⋄ (3) = 0.Observe that f ⋄ α (2) = 0 for all α ∈ [0 , Theorem 3.21 (diamond-alpha Rolle’s mean value theorem) . Let f be a con-tinuous function on [ a, b ] that is delta and nabla differentiable on ( a, b ) with f ( a ) = f ( b ) . Then, there exists α ∈ [0 , and c ∈ ( a, b ) such that f ⋄ α ( c ) = 0 . roof. If f = const , then f ⋄ α ( c ) = 0 for all α ∈ [0 ,
1] and c ∈ ( a, b ). Hence,assume that f is not the constant function and f ( t ) ≥ f ( a ) for all t ∈ [ a, b ].Since function f is continuous on the compact set [ a, b ], f assumes its maximum M > f ( a ). Therefore, there exists c ∈ [ a, b ] such that M = f ( c ). As f ( a ) = f ( b ), a < c < b , clearly f assumes its local maximum at c and there exists α ∈ [0 , f ⋄ α ( c ) = 0.The following mean value theorem is a generalization of Theorem 3.21. Theorem 3.22 (diamond-alpha Lagrange’s mean value theorem) . Let f bea continuous function on [ a, b ] that is delta and nabla differentiable on ( a, b ) .Then, there exists α ∈ [0 , and c ∈ ( a, b ) such that f ⋄ α ( c ) = f ( b ) − f ( a ) b − a . Proof.
Consider the function g defined on [ a, b ] by g ( t ) = f ( a ) − f ( t ) + ( t − a ) f ( b ) − f ( a ) b − a . Clearly g is continuous on [ a, b ] and △ and ∇ differentiable on ( a, b ). Also g ( a ) = g ( b ) = 0. Hence, there exists α ∈ [0 ,
1] and c ∈ ( a, b ) such that g ⋄ α ( c ) =0. Since g ⋄ α ( t ) = αg △ ( t ) + (1 − α ) g ∇ ( t )= α (cid:18) − f △ ( t ) + f ( b ) − f ( a ) b − a (cid:19) + (1 − α ) (cid:18) − f ∇ ( t ) + f ( b ) − f ( a ) b − a (cid:19) , we conclude that0 = − αf △ ( c ) − (1 − α ) f ∇ ( c ) + f ( b ) − f ( a ) b − a ⇔ f ⋄ α ( c ) = f ( b ) − f ( a ) b − a . We end by proving a diamond-alpha Cauchy mean value theorem, which isthe more general form of the diamond-alpha mean value theorem.
Theorem 3.23 (diamond-alpha Cauchy’s mean value theorem) . Let f and g be continuous functions on [ a, b ] that are delta and nabla differentiable on ( a, b ) .Suppose that g ⋄ α ( t ) = 0 for all t ∈ ( a, b ) and all α ∈ [0 , . Then, there exists ¯ α ∈ [0 , and c ∈ ( a, b ) such that f ( b ) − f ( a ) g ( b ) − g ( a ) = f ⋄ ¯ α ( c ) g ⋄ ¯ α ( c ) . roof. Let us first observe that from the condition g ⋄ α ( t ) = 0 for all t ∈ ( a, b )and all α ∈ [0 ,
1] it follows from Theorem 3.21 that g ( b ) = g ( a ). Hence, we mayconsider the function F defined on [ a, b ] by F ( t ) = f ( t ) − f ( a ) − f ( b ) − f ( a ) g ( b ) − g ( a ) [ g ( t ) − g ( a )] . Clearly, F is continuous on [ a, b ] and delta and nabla differentiable on ( a, b ).Also, F ( a ) = F ( b ). Applying Theorem 3.21 to the function F and taking intoaccount that F ⋄ α ( t ) = αF △ ( t ) + (1 − α ) F ∇ ( t )= α (cid:18) f △ ( t ) − f ( b ) − f ( a ) g ( b ) − g ( a ) g △ ( t ) (cid:19) + (1 − α ) (cid:18) f ∇ ( t ) − f ( b ) − f ( a ) g ( b ) − g ( a ) g ∇ ( t ) (cid:19) , we conclude that there exists ¯ α ∈ [0 ,
1] and c ∈ ( a, b ) such that0 = f ⋄ ¯ α ( c ) − f ( b ) − f ( a ) g ( b ) − g ( a ) g ⋄ ¯ α ( c ) . Hence, f ⋄ ¯ α ( c ) = f ( b ) − f ( a ) g ( b ) − g ( a ) g ⋄ ¯ α ( c ) , and dividing by g ⋄ ¯ α ( c ) we complete the proof. Acknowledgments
Research partially supported by the
Centre for Research on Optimization andControl (CEOC) from the
Portuguese Foundation for Science and Technology (FCT), cofinanced by the European Community Fund FEDER/POCI 2010. Thefirst author was also supported by KBN under Bia lystok Technical UniversityGrant S/WI/1/08.
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