A new technique for solving Sobolev type fractional multi-order evolution equations
aa r X i v : . [ m a t h . A P ] F e b A new technique for solving Sobolev type fractional multi-orderevolution equations
Nazim I. Mahmudov ∗ , Arzu Ahmadova † , and Ismail T. Huseynov ‡ Eastern Mediterranean University, Department of Mathematics, Gazimagusa, 99628,TRNC, Mersin 10, Turkey
Abstract
A strong inspiration for studying Sobolev type fractional evolution equations comes from the fact thathave been verified to be useful tools in the modeling of many physical processes. We introduce a noveltechnique for solving Sobolev type fractional evolution equations with multi-orders in a Banach space.We propose a new Mittag-Leffler type function which is generated by linear bounded operators andinvestigate their properties which are productive for checking the candidate solutions for multi-termfractional differential equations. Furthermore, we propose an exact analytical representation of solutionsfor multi-dimensional fractional-order dynamical systems with nonpermutable and permutable matrices.
Mathematics Subject Classification:
Keywords:
Evolution equations, Caputo fractional differentiation operator, Mittag-Leffler type functions,Sobolev, nonpermutable linear operators
Fractional differential equations (FDEs) are a generalization of the classical ordinary and partial differentialequations, in which the order of differentiation is permitted to be any real (or even complex) number, notonly a natural number. FDEs containing not only one fractional derivative but also more than one fractionalderivative are intensively studied in many physical processes [20, 35, 38]. Many authors demonstrate twoessential mathematical ways to use this idea: multi-term equations [2, 6, 13, 23, 24] and multi-order systems[1, 14].Multi-term FDEs have been studied due to their applications in modelling, and solved using variousmathematical methods. Finding the solution to these equations is an interesting and challenging subject thatattracted many scientists in the last decades. Up to now, various analytical and computational techniqueshave been investigated to find the solution of multi-term FDEs, of which we mention a few as follows. Luchkoand several collaborators [13, 23, 24] used the method of operational calculus to solve multi-order FDEs withdifferent types of fractional derivatives. In the realm of ordinary differential equations, Mahmudov andother collaborators [2, 30] have derived an analytical representation of solutions for special cases of fractionaldifferential equations with multi-orders, namely: Langevin and Bagley-Torvik equations involving scalarcoefficients and permutable matrices by using Laplace transform method and fractional analogue of variationof constants formula, respectively, while the other authors [33] have solved multi-term differential equations inRiemann-Liouville’s sense with variable coefficients applying a new method to construct analytical solutions.Several results have been investigated on solving multi-dimensional time-delay deterministic and stochas-tic systems with permutable matrices [3, 16] in classical and fractional senses. In [8], Diblik et al. haveconsidered inhomogeneous system of linear differential equations of second order with multiple different de-lays & pairwise permutable matrices and represented a solution of corresponding initial value problem by ∗ Corresponding author. Email: [email protected] † Email: [email protected] ‡ Email: [email protected] AB = BA to solve the sequentialRiemann-Liouville type linear time-delay systems whilst Liang et al. [21] have also obtained an explicitsolution of the differential equation with pure delay and sequential Caputo type fractional derivative.However, there are a few papers involving non-permutable matrices which are recently studied frac-tional time-continuous [27] and discrete [28] systems with a constant delay using recursively defined matrix-equations, and also delayed linear difference equations [29] applying Z -transform technique by Mahmudov.Meanwhile, Sobolev type evolution equations and their fractional-order analogues have attracted a greatdeal of attention from applications’ point of view and studied by several authors [4, 5, 9, 41, 42] in recentdecades. In [4], Balachandran and Dauer have derived sufficient conditions for controllability of partialfunctional differential systems of Sobolev type in a Banach space by using compact semigroups and Schauder’sfixed point theorem. Moreover, Balachandran et al. [5] have considered existence results of solutions fornonlinear impulsive integrodifferential equations of Sobolev type with nonlocal conditions via Krasnoselkii’sfixed point technique. In terms of fractional differential equations, Wang et al. [41] have investigatedcontrollability results of Sobolev type fractional evolution equations in a seperable Banach space by usingthe theory of propagation family and contraction mapping principle. In addition, Feckan et al. [9] havepresented controllability of fractional functional evolution sytems of Sobolev type with the help of newcharacteristic solution operators and well-known Schauder’s fixed point approach. In addition, Mahmudov[25] have considered approximate controllability results for a class of fractional evolution equations of Sobolevtype by using fixed point approach. In [42], Wang and Li have discussed stability analysis of fractionalevolution equations of Sobolev type in Ulam-Hyers sense. In [7], Chang et al. have studied the asymptoticbehaviour of resolvent operators of Sobolev type and their applications to the existence and uniqueness ofmild solutions to fractional functional evolution equations in Banach spaces. Vijayakumar et al. [40] havepresented approximate controllability results for Sobolev type time-delay differential systems of fractional-order in Hilbert spaces.To the best of our knowledge, the fractional evolution equations of Sobolev type with non-permutableoperators and two independent fractional orders of differentiation α and β which are assumed to be in theinterval (1 ,
2] and (0 , < α ≤ < β ≤ J := [0 , T ]: ((cid:0) C D α Ey (cid:1) ( t ) − A (cid:16) C D β y (cid:17) ( t ) = B y ( t ) + g ( t ) , t > ,Ey (0) = η, Ey ′ (0) = ˜ η, (1.1)where C D α and C D β Caputo fractional differential operators of orders 1 < α ≤ < β ≤ E : D ( E ) ⊂ X → Y , A : D ( A ) ⊂ X → Y and B : D ( B ) ⊂ X → Y are linear, where X and Y are Banach spaces, y ( · ) is a X -valued function on J , i.e., y ( · ) : J → X and η, ˆ η ∈ Y . In addition, g ( · ) : J → Y is a continuous function. The domain D ( E ) of E becomes a Banach space with respect to k y k D ( E ) = k Ey k Y , y ∈ D ( E ).The main idea is that under the hypotheses ( H )-( H ) we transform Sobolev type fractional multi-termevolution equation with linear operators (1.1) to fractional-order evolution equation with multi-orders andlinear bounded operators (3.1). Secondly, we solve fractional evolution equation with nonpermutable lin-ear bounded operators by using Laplace transform technique which is used as a necessary tool for solvingand analyzing fractional-order differential equations and systems in [2],[16],[17],[37]. Then we propose exactanalytical representation of a mild solution of (3.1) and (1.1), respectively with the help of new definedMittag-Leffler function which is expressed via linear bounded operators by removing the exponential bound-edness of a forced term g ( · ) and (cid:16) C D β x (cid:17) ( · ) for β ∈ (0 ,
1] (or (cid:0) C D α x (cid:1) ( · ) for α ∈ (1 , A, B ∈ B ( Y ).2he structure of this paper contains important improvement in the theory of Sobolev type fractionalmulti-term evolution equations and is outlined as below. Section 2 is a preparatory section where we recallmain definitions and results from fractional calculus, special functions and fractional differential equations.In Section 3, we establish a new Mittag-Leffler type function which is generated by linear bounded operatorsvia a double infinity series and investigate some necessary properties of this function which are accuratetool for testing the candidate solutions of fractional-order dynamical equations. We also investigate that Q A,Bk,m with nonpermutable linear operators
A, B ∈ B ( Y ) is a generalization of well-known Pascal’s rulebinomial coefficients. Moreover, we introduce the sufficient conditions for exponential boundedness of (3.1) toguarantee the existence of Laplace integral transform of equation (3.1). Then we solve multi-order fractionalevolution equations (1.1) and (3.1) with the help of Laplace integral transform. Meanwhile, we tackle thisstrong condition and verify that the sufficient conditions can be omitted easily. Section 4 deals with ananalytical representation of a mild solution to Sobolev type evolution equations with commutative linearbounded operators. In addition, we propose exact solutions for multi-dimensional multi-term fractionaldynamical systems with commutative and noncommutative matrices. In Section 5 we discuss our maincontributions of this paper and future research work. We embark on this section by briefly presenting some notations and definition fractional calculus and frac-tional differential equations [20, 38, 35] which are used throughout the paper.Let C ( J , X ) := n y ∈ C ( J , X ) : y ′ , y ′′ ∈ C ( J , X ) o denote the Banach space of functions y ( t ) ∈ X for t ∈ J equipped with a norm k y k C ( J ,X ) = P i =0 sup t ∈ J k y ( i ) ( t ) k . The space of all bounded linear operators from X to Y is denoted by B ( X, Y ) and B ( Y, Y ) is written as B ( Y ). Definition 2.1. [20, 38, 35] The fractional integral of order α > g ∈ ([0 , ∞ ) , R ) is definedby ( I α + g )( t ) = 1Γ( α ) t Z ( t − s ) α − g ( s ) d s, t > , (2.1)where Γ( · ) is the well-known Euler’s gamma function. Definition 2.2. [20, 38, 35] The Riemann-Liouville fractional derivative of order n − < α ≤ n , n ∈ N fora function g ∈ ([0 , ∞ ) , R ) is defined by( RL D α + g )( t ) = 1Γ( n − α ) (cid:18) ddt (cid:19) n t Z ( t − s ) n − α − g ( s ) d s, t > , (2.2)where the function g ( · ) has absolutely continuous derivatives up to order n .The following theorem and its corollary is regarding fractional analogue of the eminent Leibniz integralrule for general order α ∈ ( n − , n ] , n ∈ N in Riemann-Liouville’s sense which is more productive tool for thetesting particular solution of inhomogeneous linear multi-order fractional differential equations with variableand constant coefficients is considered by Huseynov et al. [15]. Theorem 2.1.
Let the function K : J × J → R be such that the following assumptions are fulfilled:(a) For every fixed t ∈ J , the function ˆ K ( t, s ) = RL,t D α − s + K ( t, s ) is measurable on J and integrable on J with respect to some t ∗ ∈ J ;(b) The partial derivative RL,t D αs + K ( t, s ) exists for every interior point ( t, s ) ∈ ˆ J × ˆ J ;(c) There exists a non-negative integrable function g such that (cid:12)(cid:12) RL,t D αs + K ( t, s ) (cid:12)(cid:12) ≤ g ( s ) for every interiorpoint ( t, s ) ∈ ˆ J × ˆ J ;(d) The derivative d l − dt l − lim s → t − RL,t D α − ls + K ( t, s ) , l = 1 , , . . . , n exists for every interior point ( t, s ) ∈ ˆ J × ˆ J . hen, the following relation holds true for fractional derivative in Riemann-Liouville sense under Lebesgueintegration for any t ∈ ˆ J : RL D αt + t Z t K ( t, s )d s = n X l =1 d l − dt l − lim s → t − RL,t D α − ls + K ( t, s ) + t Z t RL,t D αs + K ( t, s )d s. (2.3) If we have K ( t, s ) = f ( t − s ) g ( s ) , t = 0 and assumptions of Theorem 2.1 are fulfilled, then followingequality holds true for convolution operator in Riemann-Liouville sense for any n ∈ N : RL D α t Z f ( t − s ) g ( s )d s = n X l =1 lim s → t − RL,t D α − ls + f ( t − s ) d l − dt l − lim s → t − g ( s )+ t Z RL,t D αs + f ( t − s ) g ( s )d s, t > . (2.4) where RL,t D γt + K ( t, s ) is a partial Riemann-Liouville fractional differentiation operator of order γ > [20] with respect to t of a function K ( t, s ) of two variables with lower terminal t and J = [ t , T ] , ˆ J = ( t , T ) . In the special cases, Riemann-Riouville type differentiation under integral sign holds for convolutionoperator [15]: • If α ∈ (0 , RL D α t Z f ( t − s ) g ( s )d s = lim s → t − RL,t D α − s + f ( t − s ) lim s → t − g ( s )+ t Z RL,t D αs + f ( t − s ) g ( s )d s, t > • If α ∈ (1 , RL D α t Z f ( t − s ) g ( s )d s = lim s → t − RL,t D α − s + f ( t − s ) lim s → t − g ( s )+ lim s → t − RL,t D α − s + f ( t − s ) lim s → t − g ( s ) + t Z RL,t D αs + f ( t − s ) g ( s )d s, t > . Definition 2.3. [20, 35] The Caputo fractional derivative of order, n − < α ≤ n , n ∈ N for a function g ∈ ([0 , ∞ ) , R ) is defined by( C D α + g )( t ) = 1Γ( n − α ) t Z ( t − s ) n − α − (cid:18) dds (cid:19) n g ( s ) d s, t > , (2.5)where the function g ( · ) has absolutely continuous derivatives up to order n . Definition 2.4. [20, 35] The relationship between Caputo and Riemann-Liouville fractional differentialoperators of order n − < α ≤ n , n ∈ N for a function g ∈ ([0 , ∞ ) , R ) is defined by( C D α + g )( t ) = RL D α + g ( t ) − n − X k =0 t k k ! g ( k ) (0) ! , t > , (2.6)where the function g ( · ) has absolutely continuous derivatives up to order n .4 emark . If g ( · ) is an abstract function with values in X , then the integrals which appear in Definition2.1, 2.2, 2.3 and 2.4 are taken in Bochner’s sense.The Laplace transform of the Caputo’s fractional differentiation operator [20] is defined by L (cid:8) ( C D α + g )( t ) (cid:9) ( s ) = s α G ( s ) − n X k =1 s α − k g ( k − (0) , n − < α ≤ n, n ∈ N , (2.7)where G ( s ) = L { g ( t ) } ( s ).In the particular cases, the Laplace integral transform of the Caputo fractional derivative is: • If α ∈ (0 , L (cid:8)(cid:0) C D α + x (cid:1) ( t ) (cid:9) ( s ) = s α X ( s ) − s α − x (0); • If α ∈ (1 , L (cid:8)(cid:0) C D α + x (cid:1) ( t ) (cid:9) ( s ) = s α X ( s ) − s α − x (0) − s α − x ′ (0) , where X ( s ) = L { x ( t ) } ( s ). Lemma 2.1 ([45]) . Suppose that A is linear bounded operator defined on the Banach space X and assumethat k A k < . Then, ( I − A ) − is linear bounded on X and ( I − A ) − = ∞ X k =0 A k . (2.8)The following well-known generalized Gronwall inequality which plays an important role in the qualitativeanalysis of the solutions to fractional differential equations is stated and proved in [12, 44] for β >
0. Inparticular case, if β = 1, then the following relations hold true: Theorem 2.2.
Suppose a ( t ) is a nonnegative function locally integrable on ≤ t < T (some T ≤ + ∞ ), b ( t ) is a nonnegative, nondecreasing continuous function defined on ≤ t < T , | b ( t ) | ≤ M , ( M is a positiveconstant) and suppose u ( t ) is a nonnegative and locally integrable on ≤ t < T with u ( t ) ≤ a ( t ) + b ( t ) t Z u ( s ) ds, on this interval; then u ( t ) ≤ a ( t ) + b ( t ) t Z exp ( b ( t )( t − s )) a ( s ) ds, ≤ t < T. Corollary 2.1.
Under the hypothesis of Theorem 2.2, let a ( t ) be a nondecreasing function on [0 , T ) . Then u ( t ) ≤ a ( t ) exp ( b ( t ) t ) , ≤ t < T. (2.9)The Mittag-Leffler function is a natural generalization of the exponential function, first proposed asa single parameter function of one variable by using an infinite series [32]. Extensions to two or threeparameters are well known and thoroughly studied in textbooks such as [11]. Extensions to two or severalvariables have been studied more recently [14, 1, 10, 39]. Definition 2.5 ([32]) . The classical Mittag-Leffler function is defined by E α ( t ) = ∞ X k =0 t k Γ( kα + 1) , α > , t ∈ R . (2.10)5he two-parameter Mttag-Leffler function [43] is given by E α,β ( t ) = ∞ X k =0 t k Γ( kα + β ) , α > , β ∈ R , t ∈ R . (2.11)The three-parameter Mittag-Leffler function [36] is determined by E γα,β ( t ) = ∞ X k =0 ( γ ) k Γ( kα + β ) t k k ! , α > , β, γ ∈ R , t ∈ R , (2.12)where ( γ ) k is the Pochhammer symbol denoting Γ( γ + k )Γ( γ ) . These series are convergent, locally uniformly in t ,provided the α > E α,β ( t ) = E α,β ( t ) , E α, ( t ) = E α ( t ) , E ( t ) = exp( t ) . Lemma 2.2 ([36]) . The Laplace transform of the three-parameter Mittag-Leffler function is given by L n t β − E γα,β ( λt α ) o ( s ) = s − β (cid:0) − λs − α (cid:1) − γ , (2.13) where α > , β, γ, λ ∈ R and Re ( s ) > . Definition 2.6. [10] A bivariate Mittag-Leffler type function which is a particular case of multivariateMittag-Leffler function [23] is defined by E α,β,γ ( λ t α , λ t β ) = ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) λ k λ m t kα + mβ Γ( kα + mβ + γ ) , α, β > , γ ∈ R . (2.14) (1.1) with non-permutablelinear operators In this section, we consider the Cauchy problem for fractional evolution equation of Sobolev type in a Banachspaces. Firstly, we introduce the following hypotheses on the linear operators A , B and E :( H ): A is a closed operator;( H ): B is a bounded operator;( H ): D ( E ) ⊂ D ( A ) and E is bijective;( H ): A linear operator E − : Y → D ( E ) ⊂ X is compact.It is important to stress out that ( H ) implies E − is bounded. Furthermore, ( H ) also implies that E is closed since the fact: E − is closed and injective, then its inverse is also closed. It comes from the closedgraph theorem, we acquire the boundedness of the linear operator A := A E − : Y → Y . Furthermore, B := B E − : Y → Y is a linear bounded operator since E − and B are bounded.Obviously, the substitution Ey ( t ) = x ( t ) is equivalent to y ( t ) = E − x ( t ). The central idea is thatapplying the substitution y ( t ) = E − x ( t ), under the hypotheses ( H ) − ( H ), we transform the Sobolev typefractional-order evolution system (1.1) to the following multi-term evolution system with linear boundedoperators A, B ∈ B ( Y ): ((cid:0) C D α x (cid:1) ( t ) − A (cid:16) C D β x (cid:17) ( t ) = Bx ( t ) + g ( t ) , t > ,x (0) = η, x ′ (0) = ˜ η, (3.1)where x ( · ) : J → Y and η, ˜ η ∈ Y .This signifies that a mild solution of the Cauchy problem for Sobolev type multi-term fractional evolutionequation (1.1) is the multiplication of E − ∈ B ( Y ) and the solution of an initial value problem for fractionalevolution equation with multi-orders and linear bounded operators (3.1).6 emark . Alternatively, we can modify the assumptions which are given above in a similar way:( H ′ ): A is a bounded operator;( H ′ ): B is a closed operator;( H ′ ): D ( E ) ⊂ D ( B ) and E is bijective;( H ′ ): E − : Y → D ( E ) ⊂ X is compact.It follows from the closed graph theorem B := B E − : Y → Y is a linear bounded operator. Furthermore, A := A E − : Y → Y is also a linear bounded operator since A and E − are bounded. In conclusion, underthe assumptions ( H ′ ) − ( H ′ ), the Sobolev type fractional multi-term evolution equation with initial conditions(1.1) is converted to the fractional evolution system with linear bounded operators (3.1) by using the sametransformation y ( t ) = E − x ( t ).To get an analytical representation of the mild solution of (3.1), first, we need to show that exponentiallyboundedness of x ( · ) and its Caputo derivatives (cid:0) C D α x (cid:1) ( · ) , (cid:16) C D β x (cid:17) ( · ) for 1 < α ≤ < β ≤
1, respectively. To do this, we need to assume exponential boundedness for one of the given fractionaldifferentiation operators and a forced term with the aid of following Theorem 3.1.
Theorem 3.1.
Assume (3.1) has a unique continuous solution x ( t ) , if g ( t ) is continuous & exponentiallybounded and (cid:16) C D β + x (cid:17) ( t ) for < β ≤ is exponentially bounded on [0 , ∞ ) , then x ( t ) and its Caputoderivative (cid:0) C D α + x (cid:1) ( t ) is exponentially bounded for < α ≤ on [0 , ∞ ) and, thus, their Laplace transformsexist.Proof. Since g ( t ) and (cid:16) C D β + x (cid:17) ( t ) for 0 < β ≤ L, P, δ and sufficient large T such that k g ( t ) k ≤ L exp( δt ) and k (cid:16) C D β + x (cid:17) ( t ) k ≤ P exp( δt ) for any t ≥ T . It isclear that the system (3.1) is equivalent to the following Volterra fractional integral equation of second-kind: x ( t ) = (cid:18) − At α − β Γ( α − β + 1) (cid:19) η + t ˆ η + A Γ( α − β ) t Z ( t − r ) α − β − x ( r ) dr + 1Γ( α ) t Z ( t − r ) α − [ Bx ( r ) + g ( r )] dr. (3.2)This means that every solution of (3.2) is also a solution of (3.1) and vice versa. For t ≥ T , (3.2) can beexpressed as x ( t ) = (cid:18) − At α − β Γ( α − β + 1) (cid:19) η + t ˆ η + A Γ( α − β ) T Z ( t − r ) α − β − x ( r ) dr + 1Γ( α ) T Z ( t − r ) α − [ Bx ( r ) + g ( r )] dr + A Γ( α − β ) t Z T ( t − r ) α − β − x ( r ) dr + 1Γ( α ) t Z T ( t − r ) α − [ Bx ( r ) + g ( r )] dr. In view of hypotheses of Theorem 3.1, the solution x ( t ) , ( x (0) = η, x ′ (0) = ˆ η ) is unique and continuous on[0 , ∞ ), then Ax ( t ) and Bx ( t ) + g ( t ) are bounded on [0 , T ], namely: ∃ M > k Ax ( t ) k ≤ M, ∀ t ∈ [0 , T ] , and ∃ N > k Bx ( t ) + g ( t ) k ≤ N, ∀ t ∈ [0 , T ] .
7e have k x ( t ) k ≤ (cid:18) k A k t α − β Γ( α − β + 1) (cid:19) k η k + t k ˆ η k + M Γ( α − β ) T Z ( t − r ) α − β − dr + N Γ( α ) T Z ( t − r ) α − dr + k A k Γ( α − β ) t Z T ( t − r ) α − β − k x ( r ) k dr + k B k Γ( α ) t Z T ( t − r ) α − k x ( r ) k dr + 1Γ( α ) t Z T ( t − r ) α − k g ( r ) k dr. Multiplying last inequality by exp( − δt ) and note thatexp( − δt ) ≤ exp( − δr ) , r ∈ [ T, t ] and exp( − δt ) ≤ exp( − δT ) , k g ( t ) k ≤ L exp( δt ) , t ≥ T. Using the aforementioned inequalities, we attain k x ( t ) k exp( − δt ) ≤ k η k exp( − δt ) + k A k t α − β Γ( α − β + 1) k η k exp( − δt )+ t k η k exp( − δt ) + M exp( − δt )Γ( α − β ) T Z ( t − r ) α − β − dr + N exp( − δt )Γ( α ) T Z ( t − r ) α − dr + k A k exp( − δt )Γ( α − β ) t Z T ( t − r ) α − β − k x ( r ) k dr + k B k exp( − δt )Γ( α ) t Z T ( t − r ) α − k x ( r ) k dr + exp( − δt )Γ( α ) t Z T ( t − r ) α − k g ( r ) k dr ≤ k η k exp( − δT ) + k A k t α − β Γ( α − β + 1) k η k exp( − δT ) + t k η k exp( − δT )+ M exp( − θT )Γ( α − β + 1) ( t α − β − ( t − T ) α − β ) + N exp( − δT )Γ( α + 1) ( t α − ( t − T ) α )+ k A k Γ( α − β ) t Z T ( t − r ) α − β − k x ( r ) k exp( − δr ) dr + k B k Γ( α ) t Z T ( t − r ) α − k x ( r ) k exp( − δr ) dr + L Γ( α ) t Z T ( t − r ) α − exp( δ ( r − t )) dr ≤ k η k exp( − δT ) + k A k t α − β Γ( α − β + 1) k η k exp( − δT )+ t k η k exp( − δT ) + M exp( − δT )Γ( α − β + 1) T α − β + N exp( − δT )Γ( α + 1) T α + t Z (cid:18) k A k ( t − r ) α − β − Γ( α − β ) + k B k ( t − r ) α − Γ( α ) (cid:19) k x ( r ) k exp( − δr ) dr L Γ( α ) t Z ( t − r ) α − exp( − δ ( t − r )) dr ≤ k η k exp( − δT ) + k A k t α − β Γ( α − β + 1) k η k exp( − δT )+ t k η k exp( − δT ) + M exp( − δT )Γ( α − β + 1) T α − β + N exp( − δT )Γ( α + 1) T α + (cid:18) k A k t α − β − Γ( α − β ) + k B k t α − Γ( α ) (cid:19) t Z k x ( r ) k exp( − δr ) dr + Lδ α , t ≥ T. Denote a ( t ) = k A k t α − β Γ( α − β +1) k η k exp( − δT ) + t k η k exp( − δT ) + k η k exp( − δT )+ M exp( − δT )Γ( α − β +1) T α − β + N exp( − δT )Γ( α +1) T α + Lδ α ,b ( t ) = k A k t α − β − Γ( α − β ) + k B k t α − Γ( α ) ,v ( t ) = k x ( t ) k exp( − δt ) . Thus, we attain v ( t ) ≤ a ( t ) + b ( t ) t Z v ( s ) ds, t ≥ T. (3.3)According to the Gronwall’s inequality (2.9), we have v ( t ) ≤ a ( t ) exp( tb ( t )) ≤ exp( a ( t ) + tb ( t )) . (3.4)Then, it yields from (3.4) that k x ( t ) k ≤ exp( a ( t ) + tb ( t ) + δt ) , t ≥ T. Since g ( t ) and (cid:16) C D β + x (cid:17) ( t ) for β ∈ (0 ,
1] are exponentially bounded on [0 , ∞ ), from equation (3.1), weacquire k (cid:0) C D α + x (cid:1) ( t ) k ≤ k A kk (cid:16) C D β + x (cid:17) ( t ) k + k B kk x ( t ) k + k g ( t ) k≤ k A k P exp( δt ) + k B k exp( a ( t ) + tb ( t ) + δt ) + L exp( δt ) ≤ ( k A k P + k B k + L ) exp( a ( t ) + tb ( t ) + δt ) , t ≥ T. In other words, (cid:0) C D α + x (cid:1) ( t ) is also exponentially bounded, the Laplace integral transforms of x ( t ) and itsCaputo derivatives (cid:0) C D α + x (cid:1) ( t ), (cid:16) C D β + x (cid:17) ( t ) exist for α ∈ (1 ,
2] and β ∈ (0 , x ( · )and its derivatives (cid:0) C D α + x (cid:1) ( · ) , (cid:16) C D β + x (cid:17) ( · ) of order 1 < α ≤ < β ≤
1, respectively in Caputo’ssense on [0 , ∞ ). Theorem 3.2.
Assume (3.1) has a unique continuous solution x ( t ) , if g ( t ) is continuous & exponentiallybounded and (cid:0) C D α + x (cid:1) ( t ) for < α ≤ is exponentially bounded on [0 , ∞ ) , then x ( t ) and its Caputoderivative (cid:16) C D β + x (cid:17) ( t ) is exponentially bounded for < β ≤ on [0 , ∞ ) and, thus, their Laplace transformsexist.Proof. This proof is similar to the proof of Theorem 3.1. So, we omit it here.9 efinition 3.1.
We define a new Mittag-Leffler function E A,Bα,β,γ ( · ) : R → Y generated by nonpermutablelinear bounded operators A, B ∈ B ( Y ) as follows: E A,Bα,β,γ ( t ) := ∞ X k =0 ∞ X m =0 Q A,Bk,m t kα + mβ Γ( kα + mβ + γ ) , α, β > , γ ∈ R , (3.5)where Q A,Bk,m ∈ B ( Y ), k, m ∈ N is given by Q A,Bk,m := k X l =0 A k − l BQ A,Bl,m − , k, m ∈ N , Q A,Bk, := A k , k ∈ N , Q A,B ,m := B m , m ∈ N . (3.6)A linear bounded operator Q A,Bk,m can be represented explicitly in Table 1.Table 1: Explicit representation of Q A,Bk,m for r, s ∈ N Q A,Bk,m k=0 k=1 k=2 . . . k=r m = 0 I A A . . . A r m = 1 B AB + BA A B + ABA + BA . . . A r B + . . . + BA r m = 2 B AB + BAB + B A A B + ABAB + AB A + BA B + BABA + B A . . . A r B + . . . + B A r . . . . . . . . . . . . . . . . . . m = s B s AB s + . . . + B s A A B s + . . . + B s A . . . A r B s + . . . + B s A r From the above table, it can be easily seen that, in the case of commutativity AB = BA , we have Q A,Bk,m := (cid:0) k + mm (cid:1) A k B m , k, m ∈ N . Theorem 3.3.
A linear operator Q A,Bk,m ∈ B ( Y ) for k, m ∈ N has the following properties: ( i ) Q A,Bk,m , k, m ∈ N generalizes classical Pascal’s rule for linear operators A, B ∈ B ( Y ) as follows: Q A,Bk,m = AQ A,Bk − ,m + BQ A,Bk,m − , k, m ∈ N ; (3.7)( ii ) If AB = BA , then we have Q A,Bk,m = (cid:18) k + mm (cid:19) A k B m , k, m ∈ N . (3.8) Proof. ( i ) By making use of the mathematical induction principle, we can prove (3.7) is true for all k ∈ N .It is obvious that the relation (3.7) is true for k = 1. With the help of (3.6) we obtain: Q A,B ,m = X l =0 A − l BQ A,Bl,m − = ABQ
A,B ,m − + BQ A,B ,m − = AQ A,B ,m + BQ A,B ,m − , where Q A,B ,m = BQ A,B ,m − .Suppose that the formula (3.7) is true for ( k − ∈ N . Then, by applying definition (3.6) for ( k − k ∈ N as below: Q A,Bk,m = k X l =0 A k − l BQ A,Bl,m − = k − X l =0 A k − l BQ A,Bl,m − + BQ A,Bk,m − = A k − X l =0 A k − l − BQ A,Bl,m − + BQ A,Bk,m − = AQ A,Bk − ,m + BQ A,Bk,m − .
10o show ( ii ) we will use proof by induction with regard to m ∈ N via the definition of Q A,Bk,m (3.6).Obviously, for m = 0 ,
1, we have Q A,Bk, := A k , Q A,Bk, = k X l =0 A k − l BQ A,Bl, = k X l =0 A k − l BA l = ( k + 1) A k B = (cid:18) k + 11 (cid:19) A k B. Suppose that it is true for m = n ∈ N : Q A,Bk,n = (cid:18) k + nn (cid:19) A k B n . Let us prove it for m = n + 1: Q A,Bk,n +1 = k X l =0 A k − l BQ A,Bl,n = k X l =0 A k − l B (cid:18) l + nn (cid:19) A l B n = A k B n +1 k X l =0 (cid:18) l + nn (cid:19) = (cid:18) k + n + 1 n + 1 (cid:19) A k B n +1 . The proof is complete.According to the above theorem, a linear operator Q A,Bk,m for k, m ∈ N satisfies the following Pascal’s rulefor permutable linear operators A, B ∈ B ( Y ) as follows: (cid:18) k + mm (cid:19) A k B m = (cid:18) k + m − m (cid:19) A k − B m + (cid:18) k + m − m − (cid:19) A k B m − , k, m ∈ N . (3.9)By using the property of Q A,Bk,m (3.8) we define the following bivariate Mittag-Leffler function via permutablelinear bounded operators which is similar to (2.14).
Definition 3.2.
We define a Mittag-Leffler function E α,β,γ ( A ( · ) α , B ( · ) β ) : R → Y generated by permutablelinear bounded operators A, B ∈ B ( Y ) as follows: E α,β,γ ( At α , Bt β ) := ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m t kα + mβ Γ( kα + mβ + γ ) , α, β > , γ ∈ R . (3.10)In the special case, bivariate Mittag-Leffler function (3.10) via commutative linear bounded operatorsconverts to the product of classical exponential functions as follows. Lemma 3.1. If α = β = γ = 1 , then we get the double exponential function: E , , ( At, Bt ) = exp( At ) exp( Bt ) = exp(( A + B ) t ) , t ∈ R . Proof.
Applying the formula (3.10), we attain E , , ( At, Bt ) = ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m t k + m ( k + m )!= ∞ X k =0 A k t k k ! ∞ X m =0 B m t m m ! = exp( At ) exp( Bt ) = exp(( A + B ) t ) , t ∈ R . The following lemma plays a crucial role for solving the given Cauchy problem (3.1) with linear boundedoperators. In general case, it holds true whenever α > α > β , γ ∈ R .11 emma 3.2. For
A, B ∈ B ( Y ) which are satisfying AB = BA , we have: L − (cid:26) s γ s ( m +1) β (cid:2) ( s α − β I − A ) − B (cid:3) m ( s α − β I − A ) − (cid:27) ( t )= ∞ X k =0 Q A,Bk,m Γ( k ( α − β ) + mα + α − γ ) t k ( α − β )+ mα + α − γ − , m ∈ N := N ∪ { } . (3.11) Proof.
To prove, we will use a mathematical induction principle with regard to m ∈ N . Obviously, accordingto the relation (2.13), (3.11) is true for m = 0, which establishes the basis for induction: L − (cid:8) s γ − β ( s α − β I − A ) − (cid:9) ( t ) = t α − γ − E α − β,α − γ ( At α − β )= t α − γ − E α − β,α − γ ( At α − β ) = ∞ X k =0 A k t k ( α − β )+ α − γ − Γ( k ( α − β ) + α − γ )= ∞ X k =0 Q A,Bk, t k ( α − β )+ α − γ − Γ( k ( α − β ) + α − γ ) , where Q A,Bk, := A k , k ∈ N . (3.12)For m = 1, we use the convolution property of Laplace integral transform and formula (3.12): L − (cid:8) s γ − β ( s α − β I − A ) − B ( s α − β I − A ) − (cid:9) ( t )= L − (cid:8) s − β ( s α − β I − A ) − B (cid:9) ( t ) ∗ L − (cid:8) s γ − β ( s α − β I − A ) − (cid:9) ( t )= t α − E α − β,α ( At α − β ) B ∗ t α − γ − E α − β,α − γ ( At α − β )= t Z ( t − s ) α − E α − β,α ( A ( t − s ) α − β ) Bs α − γ − E α − β,α − γ ( As α − β )d s. (3.13)Then interchanging the order of integration and summation in (3.13) which is permissible in accordance withthe uniform convergence of the series (2.11), we attain: L − (cid:8) s γ − β ( s α − β I − A ) − B ( s α − β I − A ) − (cid:9) ( t )= ∞ X k =0 ∞ X l =0 A k BA l Γ( k ( α − β ) + α )Γ( l ( α − β ) + α − γ ) t Z ( t − s ) k ( α − β )+ α − s l ( α − β )+ α − γ − d s = ∞ X k =0 ∞ X l =0 A k BA l Γ( k ( α − β ) + α )Γ( l ( α − β ) + α − γ ) t ( k + l )( α − β )+2 α − γ − B ( k ( α − β ) + α, l ( α − β ) + α − γ )= ∞ X k =0 ∞ X l =0 A k BA l Γ(( k + l )( α − β ) + 2 α − γ ) t ( k + l )( α − β )+2 α − γ − , (3.14)where B ( · , · ) is a well-known beta function.Applying Cauchy product formula to the double infinity series in (3.14), we get: L − (cid:8) s γ − β ( s α − β I − A ) − B ( s α − β I − A ) − (cid:9) ( t )= ∞ X k =0 k X l =0 A k − l BA l Γ( k ( α − β ) + 2 α − γ ) t k ( α − β )+2 α − γ − = ∞ X k =0 k X l =0 A k − l BQ A,Bl, Γ( k ( α − β ) + 2 α − γ ) t k ( α − β )+2 α − γ − = ∞ X k =0 Q A,Bk, Γ( k ( α − β ) + α + α − γ ) t k ( α − β )+ α + α − γ − , where Q A,Bk, := k X l =0 A k − l BQ A,Bl, , k ∈ N . (3.15)12o verify the induction step, we assume that (3.11) holds true for m = n where n ∈ N : L − n s γ − ( n +1) β (cid:2) ( s α − β I − A ) − B (cid:3) n ( s α − β I − A ) − o ( t )= ∞ X k =0 k X l =0 A k − l BQ A,Bl,n − Γ( k ( α − β ) + ( n + 1) α − γ ) t k ( α − β )+( n +1) α − γ − = ∞ X k =0 Q A,Bk,n Γ( k ( α − β ) + nα + α − γ ) t k ( α − β )+ nα + α − γ − , where Q A,Bk,n := k X l =0 A k − l BQ A,Bl,n − , k ∈ N . (3.16)Then it yields that for m = n + 1 as follows: L − n s γ − ( n +2) β (cid:2) ( s α − β I − A ) − B (cid:3) n +1 ( s α − β I − A ) − o ( t )= L − (cid:8) s − β ( s α − β I − A ) − B (cid:9) ( t ) ∗ L − n s γ − ( n +1) β (cid:2) ( s α − β I − A ) − B (cid:3) n ( s α − β I − A ) − o ( t )= t α − E α − β,α ( At α − β ) B ∗ ∞ X l =0 Q A,Bl,n Γ( l ( α − β ) + ( n + 1) α − γ ) t l ( α − β )+( n +1) α − γ − = t Z ( t − s ) α − E α − β,α ( A ( t − s ) α − β ) B ∞ X l =0 Q A,Bl,n Γ( l ( α − β ) + ( n + 1) α − γ ) s l ( α − β )+( n +1) α − γ − d s = ∞ X k =0 ∞ X l =0 A k BQ A,Bl,n Γ( k ( α − β ) + α )Γ( l ( α − β ) + ( n + 1) α − γ ) t Z ( t − s ) k ( α − β )+ α − s l ( α − β )+( n +1) α − γ − d s = ∞ X k =0 ∞ X l =0 A k BQ A,Bl,n Γ( k ( α − β ) + α )Γ( l ( α − β ) + ( n + 1) α − γ ) t ( k + l )( α − β )+( n +1) α + α − γ − ×B ( k ( α − β ) + α, l ( α − β ) + ( n + 1) α − γ )= ∞ X k =0 ∞ X l =0 A k BQ A,Bl,n
Γ(( k + l )( α − β ) + ( n + 1) α + α − γ ) t ( k + l )( α − β )+( n +1) α + α − γ − = ∞ X k =0 k X l =0 A k − l BQ A,Bl,n Γ( k ( α − β ) + ( n + 1) α + α − γ ) t k ( α − β )+( n +1) α + α − γ − = ∞ X k =0 Q A,Bk,n +1 Γ( k ( α − β ) + ( n + 1) α + α − γ ) t k ( α − β )+( n +1) α + α − γ − , where Q A,Bk,n +1 := k X l =0 A k − l BQ A,Bl,n , k ∈ N . (3.17)Thus, (3.17) holds true whenever (3.16) is true, and by the principle of mathematical induction, we concludethat the formula (3.11) holds true for all m ∈ N . Theorem 3.4.
Let
A, B ∈ B ( Y ) with non-zero commutator, i.e., [ A, B ] := AB − BA = 0 . Assume that g ( · ) : J → Y and (cid:16) C D β x (cid:17) ( t ) where < β ≤ are exponentially bounded. A mild solution x ( · ) ∈ C ( J , Y ) of the Cauchy problem (3.1) can be represented as x ( t ) = I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) B (cid:17) η + t E A,Bα − β,α, ( t )ˆ η + t Z ( t − s ) α − E A,Bα − β,α,α ( t − s ) g ( s )d s, t > , (3.18) where I ∈ B ( Y ) is an identity operator.Proof. We recall that the existence of Laplace transform of x ( · ) and its Caputo derivatives C D α + x ( · ) and C D β + x ( · ) for 1 < α ≤ < β ≤
1, respectively, is guaranteed by Theorem 3.1. Thus, to find the mildsolution x ( t ) of (3.1) satisfying the initial conditions x (0) = η , x ′ (0) = ˆ η , we can use the Laplace integraltransform. By assuming T = ∞ , taking the Laplace transform on both sides of equation (3.1) and using thefollowing facts that L n C D α + x ( t ) o ( s ) = s α X ( s ) − s α − η − s α − ˆ η, L n C D β + x ( t ) o ( s ) = s β X ( s ) − s β − η, which implies that (cid:0) s α I − As β − B (cid:1) X ( s ) = s α − η + s α − ˆ η − s β − Aη + G ( s ) , where X ( s ) and G ( s ) represent the Laplace integral transforms of x ( t ) and g ( t ), respectively.Thus, after solving the above equation with respect to the X ( s ), we get X ( s ) = s α − (cid:0) s α I − As β − B (cid:1) − η + s α − (cid:0) s α I − As β − B (cid:1) − ˆ η − s β − (cid:0) s α I − As β − B (cid:1) − Aη + (cid:0) s α − I − As β − B (cid:1) − G ( s )= s − η + s − (cid:0) s α I − As β − B (cid:1) − Bη + s α − (cid:0) s α I − As β − B (cid:1) − ˆ η + (cid:0) s α − I − As β − B (cid:1) − G ( s ) . On the other hand, in accordance with the relation (2.8), for sufficiently large s , such that k ( s α − β I − A ) − Bs − β k < . Thus, for nonpermutable linear operators
A, B ∈ B ( Y ) and sufficiently large s , we have (cid:0) s α I − As β − B (cid:1) − = (cid:0) s β (cid:2) s α − β I − A − Bs − β (cid:3)(cid:1) − = (cid:0) s β ( s α − β I − A ) (cid:2) I − ( s α − β I − A ) − Bs − β (cid:3)(cid:1) − = (cid:16) s β h I − (cid:0) s α − β I − A (cid:1) − Bs − β i(cid:17) − (cid:0) s α − β I − A (cid:1) − = (cid:2) I − ( s α − β I − A ) − Bs − β (cid:3) − s − β (cid:0) s α − β I − A (cid:1) − = ∞ X m =0 s βm h(cid:0) s α − β I − A (cid:1) − B i m s − β (cid:0) s α − β I − A (cid:1) − = ∞ X m =0 s ( m +1) β h(cid:0) s α − β I − A (cid:1) − B i m (cid:0) s α − β I − A (cid:1) − Then, by taking inverse Laplace transform, we have x ( t ) = L − (cid:8) s − (cid:9) ( t ) η + L − ( ∞ X m =0 s − s ( m +1) β h(cid:0) s α − β I − A (cid:1) − B i m (cid:0) s α − β I − A (cid:1) − ) ( t ) Bη + L − ( ∞ X m =0 s α − s ( m +1) β h(cid:0) s α − β I − A (cid:1) − B i m (cid:0) s α − β I − A (cid:1) − ) ( t )ˆ η L − ( ∞ X m =0 s ( m +1) β h(cid:0) s α − β I − A (cid:1) − B i m (cid:0) s α − β I − A (cid:1) − G ( s ) ) ( t ) . (3.19)Therefore, in accordance with Lemma 3.2, we acquire x ( t ) = I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s := (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) B (cid:17) η + t E A,Bα − β,α, ( t )ˆ η + t Z ( t − s ) α − E A,Bα − β,α,α ( t − s ) g ( s )d s, t > . (3.20)It should stressed out that the assumption on the exponential boundedness of the function g ( · ) and (cid:16) C D β x (cid:17) ( · ) where 0 < β ≤ (cid:0) C D α x (cid:1) ( · ) for 1 < α ≤
2) can be omitted. As is shown below,the statement of the above theorem holds for a more general function g ( · ) ∈ C ( J , Y ). Theorem 3.5.
Let
A, B ∈ B ( Y ) with non-zero commutator, i.e., [ A, B ] := AB − BA = 0 . A mild solution x ( · ) ∈ C ( J , Y ) of the Cauchy problem (3.1) can be represented as x ( t ) = I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s := (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) B (cid:17) η + t E A,Bα − β,α, ( t )ˆ η + t Z ( t − s ) α − E A,Bα − β,α,α ( t − s ) g ( s )d s, t > . (3.21) Proof.
For making use of verification by substitution, we apply superposition principle for the initial valueproblem of linear inhomogeneous multi-order fractional evolution equation (3.1). For this, firstly let usconsider the following homogeneous system with inhomogeneous initial conditions: ((cid:0) C D α x (cid:1) ( t ) − A (cid:16) C D β x (cid:17) ( t ) − Bx ( t ) = 0 , t > x (0) = η, x ′ (0) = ˜ η, (3.22)has a mild solution x ( t ) = I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η = (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) B (cid:17) η + t E A,Bα − β,α, ( t )˜ η. (3.23)With the help of verification by substitution and the property of Q A,Bk,m (3.7), we confirm that (3.23) is a mildsolution of linear homogeneous fractional evolution equation (3.22): (cid:16) C D α x (cid:17) ( t ) = C D α (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) B (cid:17) η + C D α (cid:16) t E A,Bα − β,α, ( t ) (cid:17) ˆ η = C D α I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η C D α ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ! ˆ η. In this case, we first apply the property of Q A,Bk,m (3.7) before Caputo differentiation the first and secondterms above, in accordance with the following formula [35]: C D ν (cid:18) t η Γ( η + 1) (cid:19) = t η − ν Γ( η − ν +1) , η > ⌊ ν ⌋ , , η = 0 , , , . . . , ⌊ ν ⌋ , undefined , otherwise . (3.24)Then, we have (cid:16) C D α x (cid:17) ( t ) = C D α h B t α Γ( α ) + ∞ X k =1 ∞ X m =0 AQ A,Bk − ,m B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1)+ ∞ X k =0 ∞ X m =1 BQ A,Bk,m − B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) i η + C D α h tI + ∞ X k =1 ∞ X m =0 AQ A,Bk − ,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2)+ ∞ X k =0 ∞ X m =1 BQ A,Bk,m − t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) i ˆ η = Bη + ∞ X k =1 ∞ X m =0 AQ A,Bk − ,m B t k ( α − β )+ mα Γ( k ( α − β ) + mα + 1) η + ∞ X k =0 ∞ X m =1 BQ A,Bk,m − B t k ( α − β )+ mα Γ( k ( α − β ) + mα + 1) η + ∞ X k =1 ∞ X m =0 AQ A,Bk − ,m t k ( α − β )+ mα +1 − α Γ( k ( α − β ) + mα + 2 − α ) ˆ η + ∞ X k =0 ∞ X m =1 BQ A,Bk,m − t k ( α − β )+ mα +1 − α Γ( k ( α − β ) + mα + 2 − α ) ˆ η. Next, we can attain that (cid:16) C D α x (cid:17) ( t ) = Bη + ∞ X k =0 ∞ X m =0 AQ A,Bk,m
B t k ( α − β )+ mα + α − β Γ( k ( α − β ) + mα + α − β + 1) η + ∞ X k =0 ∞ X m =0 BQ A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) η + ∞ X k =0 ∞ X m =0 AQ A,Bk,m t k ( α − β )+ mα +1 − β Γ( k ( α − β ) + mα + 2 − β ) ˆ η + ∞ X k =0 ∞ X m =0 BQ A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η. Then, the Caputo fractional differentiation of x ( t ) (3.23) of order 0 < β ≤ (cid:16) C D β x (cid:17) ( t ) = C D β (cid:16) I + t α E A,Bα − β,α,α +1 ( t ) (cid:17) η + C D β (cid:16) t E A,Bα − β,α, ( t ) (cid:17) ˆ η = C D β I + ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mβ + α Γ( k ( α − β ) + mβ + α + 1) ! η C D β ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mβ +1 Γ( k ( α − β ) + mβ + 2) ! ˆ η = ∞ X k =0 ∞ X m =0 Q A,Bk,m
B t k ( α − β )+ mβ + α − β Γ( k ( α − β ) + mβ + α − β + 1) η + ∞ X k =0 ∞ X m =0 Q A,Bk,m t k ( α − β )+ mβ +1 − β Γ( k ( α − β ) + mβ + 2 − β ) ˆ η. Finally, taking a linear combination of above results, we acquire the desired result: (cid:16) C D α x (cid:17) ( t ) − A (cid:16) C D β x (cid:17) ( t ) = Bη + ∞ X k =0 ∞ X m =0 BQ A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) η + ∞ X k =0 ∞ X m =0 BQ A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η := Bx ( t ) . Next, we consider the following linear inhomogeneous fractional evolution equation: (cid:16) C D α x (cid:17) ( t ) − A (cid:16) C D β x (cid:17) ( t ) − Bx ( t ) = g ( t ) , (3.25)with zero initial conditions: x (0) = x ′ (0) = 0 , has an integral representation of a mild solution which is a particular solution of (3.1):¯ x ( t ) = t Z ( t − s ) α − E A,Bα − β,α,α ( t − s ) g ( s )d s = t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . In accordance with fractional analogue of variation of constants formula any particular mild solution ofinhomogeneous differential equation of fractional-order (3.25) should be looked for in the form of¯ x ( t ) = t Z ( t − s ) α − E A,Bα − β,α,α ( t − s ) f ( s )d s = t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s, t > , (3.26)where f ( s ) is unknown function for s ∈ [0 , t ] with ¯ x (0) = ¯ x ′ (0) = 0.Because of this homogeneous initial values ¯ x (0) = ¯ x ′ (0) = 0, it follows that in this case, for any givenorder either in (1 ,
2] and (0 , Q A,Bk,m (3.7) and having Caputo differentiation of order1 < α ≤ x ( t ), we obtain: (cid:16) C D α ¯ x (cid:17) ( t ) = (cid:16) RL D α ¯ x (cid:17) ( t )= RL D α h t Z ( t − s ) α − Γ( α ) f ( s )d s + t Z ∞ X k =1 ∞ X m =0 AQ A,Bk − ,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s + t Z ∞ X k =0 ∞ X m =1 BQ A,Bk,m − ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s i (cid:16) RL D α ( I α f ) (cid:17) ( t ) + RL D α h t Z ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα +2 α − β − Γ( k ( α − β ) + mα + 2 α − β ) f ( s )d s i + RL D α h t Z ∞ X k =0 ∞ X m =0 BQ A,Bk,m ( t − s ) k ( α − β )+ mα +2 α − Γ( k ( α − β ) + mα + 2 α ) f ( s )d s i . By making use of the fractional Leibniz integral rules (2.4) in Riemann-Liouville’s sense for the secondand third terms of the above expression, we get (cid:16) C D α ¯ x (cid:17) ( t ) = (cid:16) RL D α ¯ x (cid:17) ( t )= f ( t ) + lim s → t − RL,t D α − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα +2 α − β − Γ( k ( α − β ) + mα + 2 α − β ) ! lim s → t − f ( s )+ lim s → t − RL,t D α − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα +2 α − β − Γ( k ( α − β ) + mα + 2 α − β ) ! ddt lim s → t − f ( s )+ t Z RL,t D α ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα +2 α − β − Γ( k ( α − β ) + mα + 2 α − β ) f ( s )d s + lim s → t − RL,t D α − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα +2 α − Γ( k ( α − β ) + mα + 2 α ) ! lim s → t − f ( s )+ lim s → t − RL,t D α − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα +2 α − Γ( k ( α − β ) + mα + 2 α ) ! ddt lim s → t − f ( s )+ t Z RL,t D α ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα +2 α − Γ( k ( α − β ) + mα + 2 α ) f ( s )d s = f ( t ) + lim s → t − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα + α − β Γ( k ( α − β ) + mα + α − β + 1) lim s → t − f ( s )+ lim s → t − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα + α − β +1 Γ( k ( α − β ) + mα + α − β + 2) ddt lim s → t − f ( s )+ t Z ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα + α − β − Γ( k ( α − β ) + mα + α − β ) f ( s )d s + lim s → t − RL,t D α − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! lim s → t − f ( s )+ lim s → t − ∞ X k =0 ∞ X m =0 AQ A,Bk,m lim s → t − ( t − s ) k ( α − β )+ mα + α +1 Γ( k ( α − β ) + mα + α + 2) ddt lim s → t − f ( s )+ t Z ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s = f ( t ) + Z t ∞ X k =0 ∞ X m =0 AQ A,Bk,m ( t − s ) k ( α − β )+ mα + α − β − Γ( k ( α − β ) + mα + α − β ) f ( s )d s + Z t ∞ X k =0 ∞ X m =0 BQ A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s x ( t ) of order of 0 < β ≤ (cid:16) C D β ¯ x (cid:17) ( t ) = (cid:16) RL D β ¯ x (cid:17) ( t )= RL D β h t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s i = lim s → t − RL,t D β − ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) ! lim s → t − f ( s )+ t Z RL D β ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s = lim s → t − ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − β Γ( k ( α − β ) + mα + α − β + 1) lim s → t − f ( s )+ t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − β − Γ( k ( α − β ) + mα + α − β ) f ( s )d s = t Z ∞ X k =0 ∞ X m =0 Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − β − Γ( k ( α − β ) + mα + α − β ) f ( s )d s. Thus, linear combinations of above results yield that (cid:16) C D α ¯ x (cid:17) ( t ) − A (cid:16) C D β ¯ x (cid:17) ( t )= f ( t ) + t Z ∞ X k =0 ∞ X m =0 BQ A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) f ( s )d s = f ( t ) + B ¯ x ( t ) = g ( t ) + B ¯ x ( t ) . Therefore, f ( t ) = g ( t ), t > y ( t ) = E − x ( t ), we can acquire a mild solution of (1.1) asbelow. Theorem 3.6.
Let
A, B ∈ B ( Y ) with non-zero commutator, i.e., [ A, B ] := AB − BA = 0 . A mild solution y ( · ) ∈ C ( J , X ) of the Cauchy problem (1.1) can be represented as y ( t ) = E − + ∞ X k =0 ∞ X m =0 E − Q A,Bk,m
B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 E − Q A,Bk,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 E − Q A,Bk,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s := (cid:16) E − + t α E − E A,Bα − β,α,α +1 ( t ) B (cid:17) η + tE − E A,Bα − β,α, ( t )ˆ η + t Z ( t − s ) α − E − E A,Bα − β,α,α ( t − s ) g ( s )d s, t > . (3.27)19 emark . Let A = Θ be a zero operator. Then, a mild solution y ( · ) ∈ C ( J , X ) of the Cauchy problem(3.28) ((cid:0) C D α Ey (cid:1) ( t ) − B y ( t ) = g ( t ) , t > , < α ≤ ,y (0) = η, y ′ (0) = ˜ η, (3.28)can be determined by means of two parameter Mittag-Leffler functions as follows y ( t ) = ∞ X m =0 E − B m t mα Γ( mα + 1) η + ∞ X m =0 E − B m t mα +1 Γ( mα + 2) ˆ η + t Z ∞ X m =0 E − B m ( t − s ) mα + α − Γ( mα + α ) g ( s )d s := E − E α, ( Bt α ) η + tE − E α, ( Bt α )ˆ η + t Z ( t − s ) α − E − E α,α ( B ( t − s ) α ) g ( s )d s, t > . (3.29)Similar problem to (3.28) has been considered in [9] for Sobolev type functional evolution equations withfractional-order as follows: ( C D qt ( Ex ( t )) + Ax ( t ) = f ( t, x t ) , t ∈ J := [0 , a ] ,x ( t ) = φ ( t ) , − r ≤ t ≤ , (3.30)where C D qt is the Caputo fractional derivative of order 0 < q < A : D ( A ) ⊂ X → Y and E : D ( E ) ⊂ X → Y , where X, Y are Banach spaces. Moreover, f ( · , · ) : J × C → Y with C := C ([ − r, , X ). x ( · ) : J ∗ := [ − r, a ] → X is continuous, x t is the element of C defined by x t ( s ) := x ( t + s ), − r ≤ s ≤
0. The domain D ( E ) of E becomes a Banach space with norm k x k D ( E ) := k Ex k Y , x ∈ D ( E ) and φ ∈ C ( E ) := C ([ − r, , D ( E )).Feckan et al. [9] have introduced the following assumptions on the operators A and E :( ˆ H ): A and E are linear operators and A is closed;( ˆ H ): D ( E ) ⊂ D ( A ) and E is bijective;( ˆ H ): Linear operator E − : Y → D ( E ) ⊂ X is compact.By making use of the substitution x ( t ) = E − y ( t ), under the hypotheses ( ˆ H )-( ˆ H ), we transform theSobolev type fractional-order functional evolution system (3.30) to the following evolution system with alinear bounded operator ˆ A := − AE − : Y → Y : ( C D αt y ( t ) − ˆ Ay ( t ) = f ( t, E − y t ) , t ∈ J,y ( t ) = ϕ ( t ) , − r ≤ t ≤ , (3.31)where y ( · ) : J ∗ → Y , ϕ ( t ) = Eφ ( t ) , t ∈ [ − r,
0] and y t ( s ) = y ( t + s ) , s ∈ [ − r, y ( t ) = E q, ( ˆ At q ) ϕ (0) + t Z ( t − s ) q − E q,q ( ˆ A ( t − s ) q ) f ( s, E − y s )d s, t > . (3.32)Thus, the mild solution of Sobolev type functional evolution equation of fractional-order should be repre-sented by x ( t ) = E − E q, ( ˆ At q ) Eφ (0) + t Z ( t − s ) q − E − E q,q ( ˆ A ( t − s ) q ) f ( s, x s )d s, t > . (3.33)However, the mild solution of (3.30) was represented via characteristic solution operators (see Lemma 3.1 in[9]) instead of Mittag-Leffler functions generated by a linear operator ˆ A := − AE − ∈ B ( Y ). It should bestressed out that if E = I , then a mild solution of (3.30) can be expressed with the help of characteristicsolution operators (see Remark 3.1 in [9]), otherwise, under hypotheses ( ˆ H ) − ( ˆ H ) it should be determinedby classical Mittag-Leffler functions with two parameters which are compact linear operators in Y as (3.33).20 emark . In particular case, we consider the following initial value problem for multi-dimensional multi-term fractional differential equation with noncommutative matrices ((cid:0) C D α y (cid:1) ( t ) − A (cid:16) C D β y (cid:17) ( t ) − B y ( t ) = g ( t ) , t > ,y (0) = η, y ′ (0) = ˜ η, (3.34)where C D α and C D β Caputo fractional derivatives of orders 1 < α ≤ < β ≤
1, respectively, withthe lower limit zero. E = I ∈ R n × n is an identity matrix, the matrices A , B ∈ R n × n are nonpermutablei.e., AB = BA , y ( t ) ∈ R n is a vector-valued function on J , i.e., y ( · ) : J → R n and η, ˆ η ∈ R n . In addition, aforced term g ( · ) : J → R n is a continuous function.The exact analytical representation of solution y ( · ) ∈ C ( J , R n ) of (3.34) can be expressed by y ( t ) := (cid:16) t α E A ,B α − β,α,α +1 ( t ) B (cid:17) η + t E A ,B α − β,α, ( t )ˆ η + t Z ( t − s ) α − E A ,B α − β,α,α ( t − s ) g ( s )d s = ∞ X k =0 ∞ X m =0 Q A ,B k,m B t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 Q A ,B k,m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 Q A ,B k,m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . (3.35) (1.1) with permutable linearoperators To get an analytical representation of a mild solution of (3.1) with permutable linear operators i.e., AB = BA ,first, we need to prove auxiliary lemma for making use of Laplace integral transform according to the Theorem3.1. Moreover, the scalar analogue of following theorem has been considered by Ahmadova and Mahmudovfor fractional Langevin equations with constant coefficients in [2]. In general, the following theorem is truefor α > α > β and γ ∈ R . Theorem 4.1.
Let m ∈ N and Re ( s ) > . For A, B ∈ B ( Y ) with [ A, B ] = AB − BA = 0 , we have: L − n s γ B m ( s α I − As β ) m +1 o ( t ) = t mα + α + γ − ∞ X k =0 (cid:18) k + mm (cid:19) A k B m t k ( α − β ) Γ( k ( α − β ) + mα + α − γ )= t mα + α − γ − E m +1 α − β,mα + α − γ ( At α − β ) B m . Proof.
By using the Taylor series representation of − t ) m +1 , m ∈ N of the form1(1 − t ) m +1 = ∞ X k =0 (cid:18) k + mm (cid:19) t k , | t | < , we achieve that s γ B m ( s α I − As β ) m +1 = s γ B m ( s α I ) m +1 − As α − β ) m +1 = s γ B m s ( m +1) α ∞ X k =0 (cid:18) k + mm (cid:19)(cid:16) As α − β (cid:17) k = ∞ X k =0 (cid:18) k + mm (cid:19) A k B m s ( m +1) α + k ( α − β ) − γ .
21y using the inverse Laplace integral formula for the above function, we get the desired result: L − n s γ B m ( s α I − As β ) m +1 o ( t ) = ∞ X k =0 A k B m (cid:18) k + mm (cid:19) L − n s k ( α − β )+( m +1) α − γ o ( t )= ∞ X k =0 A k B m (cid:18) k + mm (cid:19) t k ( α − β )+ mα + α − γ − Γ( k ( α − β ) + mα + α − γ )= t mα + α − γ − E m +1 α − β,mα + α − γ ( At α − β ) B m . We have required an extra condition on s such that s α − β > k A k , for proper convergence of the series. But, this condition can be removed at the end of calculation sinceanalytic continuation of both sides, to give the desired result for all s ∈ C which is satisfying Re ( s ) > Theorem 4.2.
Let
A, B ∈ B ( Y ) with zero commutator, i.e., [ A, B ] := AB − BA = 0 . Assume that g ( · ) : J → X and (cid:16) C D β + x (cid:17) ( · ) for < β ≤ are exponentially bounded. A mild solution x ( · ) ∈ C ( J , Y ) of the Cauchy problem (3.1) can be represented by means of bivariate Mittag-Leffler type functions (2.14) asfollows x ( t ) = (cid:0) I + t α BE α − β,α,α +1 ( At α − β , Bt α ) (cid:1) η + tE α − β,α, ( At α − β , Bt α )ˆ η + t α − E α − β,α,α ( At α − β , Bt α ) ∗ g ( t )= I + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m +1 t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . (4.1) Proof.
We recall that the existence of Laplace transform of x ( · ) and its Caputo derivatives (cid:0) C D α + x (cid:1) ( · ) and (cid:16) C D β + x (cid:17) ( · ) for 1 < α ≤ < β ≤
1, respectively, is guaranteed by Theorem 3.1. Thus, to find the mildsolution x ( t ) of (1.1) with permutable linear operators, i.e., AB = BA , we can use the Laplace transformtechnique. By assuming T = ∞ , applying the Laplace transform technique on both sides of equation (3.1)and solving the equation with respect to the X ( s ), we get X ( s ) = s − η + s − (cid:0) s α I − As β − B (cid:1) − Bη + s α − (cid:0) s α I − As β − B (cid:1) − ˆ η + (cid:0) s α − I − As β − B (cid:1) − G ( s ) . On the other hand, since (2.8) for sufficiently large s , we have k (cid:0) s α I − As β (cid:1) − B k < . Then, for permutable linear operators
A, B ∈ B ( Y ) and sufficiently large s , one can attain (cid:0) s α I − As β − B (cid:1) − = (cid:0) s α I − As β (cid:1) − (cid:16) I − (cid:0) s α I − As β (cid:1) − B (cid:17) − = (cid:0) s α I − As β (cid:1) − ∞ X m =0 (cid:0) s α I − As β (cid:1) − m B m = ∞ X m =0 B m ( s α I − As β ) ( m +1) . x ( t ) = L − (cid:8) s − (cid:9) ( t ) η + L − ( ∞ X m =0 s − B m +1 ( s α I − As β ) ( m +1) ) ( t ) Bη + L − ( ∞ X m =0 s α − B m ( s α I − As β ) ( m +1) ) ( t )ˆ η + L − ( ∞ X m =0 B m ( s α I − As β ) ( m +1) G ( s ) ) ( t ) . (4.2)Therefore, in accordance with Theorem (4.1), we acquire x ( t ) = ( I + ∞ X k =0 ∞ X m =0 (cid:18) k + mk (cid:19) A k B m +1 t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ) η + ∞ X k =0 ∞ X m =0 (cid:18) k + mk (cid:19) A k B m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 (cid:18) k + mk (cid:19) A k B m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s := (cid:0) I + t α BE α − β,α,α +1 ( At α − β , Bt α ) (cid:1) η + tE α − β,α, ( At α − β , Bt α )ˆ η + t Z ( t − s ) α − E α − β,α,α ( A ( t − s ) α − β , B ( t − s ) α ) g ( s )d s, t > . (4.3) Remark . The analytical mild solution for the initial value problem for (3.1) can be attained from theproperty of Q A,Bk,m (3.8) for linear bounded operators
A, B ∈ B ( Y ) satisfying AB = BA where Q A,Bk,m = (cid:18) k + mm (cid:19) A k B m , k, m ∈ N . It should be emphasized that the assumption on the exponential boundedness of the function g ( · ) and (cid:16) C D β x (cid:17) ( · ) for 0 < β ≤ (cid:0) C D α x (cid:1) ( · ) for 1 < α ≤ Theorem 4.3.
Let
A, B ∈ B ( Y ) with zero commutator, i.e., [ A, B ] := AB − BA = 0 . A mild solution x ( · ) ∈ C ( J , Y ) of the Cauchy problem (3.1) can be expressed as x ( t ) = I + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m +1 t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . (4.4) Proof.
For linear homogeneous and inhomogeneous cases, by using the following Pascal identity for binomialcoefficients: (cid:18) k + mm (cid:19) = (cid:18) k + m − m (cid:19) + (cid:18) k + m − m − (cid:19) , k, m ∈ N , Theorem 4.4.
Let
A, B ∈ B ( Y ) with zero commutator, i.e., [ A, B ] := AB − BA = 0 . A mild solution y ( · ) ∈ C ( J , X ) of the Cauchy problem (1.1) can be determined as below y ( t ) = (cid:0) E − + t α E − BE α − β,α,α +1 ( At α − β , Bt α ) (cid:1) η + tE − E α − β,α, ( At α − β , Bt α )ˆ η + t α − E − E α − β,α,α ( At α − β , Bt α ) ∗ g ( t )= E − + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) E − A k B m +1 t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) E − A k B m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) E − A k B m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . (4.5) Remark . In a special case, the exact analytical representation of solution y ( · ) ∈ C ( J , R n ) of Cauchyproblem for multi-dimensional fractional differential equation with multi-orders and permutable matrices A , B ∈ R n × n i.e., A B = B A (3.34) can be represented by, y ( t ) = ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m +10 t k ( α − β )+ mα + α Γ( k ( α − β ) + mα + α + 1) ! η + ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m t k ( α − β )+ mα +1 Γ( k ( α − β ) + mα + 2) ˆ η + t Z ∞ X k =0 ∞ X m =0 (cid:18) k + mm (cid:19) A k B m ( t − s ) k ( α − β )+ mα + α − Γ( k ( α − β ) + mα + α ) g ( s )d s, t > . (4.6) In this research work, first, we convert Sobolev type fractional evolution equation with multi-orders (1.1)to multi-term fractional evolution equation with linear bounded operators (3.1). Secondly, we give thesufficient conditions to guarantee the rationality of solving multi-term fractional differential equations withlinear bounded operators by the Laplace transform method. Then we solve linear inhomogeneous fractionalevolution equation with nonpermutable & permutable linear bounded operators
A, B ∈ B ( Y ) by making useof Laplace integral transform. Next we propose exact analytical representation of a mild solution of (3.1)and (1.1), respectively with the help of newly defined Mittag-Leffler type function which is generated bylinear bounded operators by removing the strong condition which is an exponential boundedness of a forcedterm and one of fractional orders with the help of analytical methods, namely: verification by substitutionand fractional analogue of variation of constants formula.The main contributions of this paper are as follows: • we introduce a new Mittag-Leffler type function which is generated by linear bounded operators A, B ∈ B ( Y ) via a double infinity series ; • we derive new properties of Mittag-Leffler type function which are useful tool for checking the candidatesolutions of multi-term fractional differential equations; • we propose the property of Q A,Bk,m with nonpermutable linear operators
A, B ∈ B ( Y ) which is a gener-alization of well-known Pascal’s rule binomial coefficients.24 we acquire the analytical representation of a mild solution for linear Sobolev type fractional multi-termevolution equations with nonpermutable and permutable linear operators; • we derive the exact analytical representation of multi-dimensional fractional differential equations withtwo independent orders and nonpermutable & permutable matrices.The possible directions for future work in which to extend the results of this paper is looking at Sobolevtype fractional functional evolution equations with multi-orders (5.1). Furthermore, one can expect theresults of this paper to hold for a class of problems such as Sobolev type functional evolution system governedby ((cid:0) C D α Ey (cid:1) ( t ) − A (cid:16) C D β y (cid:17) ( t ) = B y ( t − τ ) + g ( t ) , ≥ α > β > ,Ey ( τ ) = Eφ ( t ) , − τ ≤ t ≤ . 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