Study of a nonlinear fractional boundary value problem via the dichotomy-type technique
aa r X i v : . [ m a t h . C A ] J a n STUDY OF A NONLINEAR FRACTIONAL BOUNDARYVALUE PROBLEM VIA THE DICHOTOMY–TYPETECHNIQUEKateryna Marynets Abstract
We present a new view onto the successive approximations’ approachin study of the two–point nonlinear fractional boundary value problems. Inorder to reduce the original problem and further construct its approximatesolution we use the co–called ’freezing’ technique and the dichotomy–typeapproach. These lead to improvement of the sufficient conditions for appli-cation of the aforementioned method and sharpen the obtained estimates.
MSC 2010 : Primary 34A08; 34K07; Secondary 34K28.
Key Words and Phrases : nonlinear fractional boundary value problem,successive approximations, dichotomy–type approach, ’freezing’ technique,bifurcation equations.
1. Introduction
The fractional differential equations have been waking a high interestduring the last decades. The variety of their applications in biology, physics,engineering and economics lead to development of the proper and precisetechniques to study behavior of solutions of the aforementioned equationsand their systems.Particular attention is paid to the class of nonlinear fractional boundaryvalue problems (FBVPs), since construction of their exact solutions maybe impossible or one may face computational difficulties trying to find theiranalytical representation. However, the high precise constructive methodsof approximation of solutions may help to simplify and even solve this task.Is the current paper we give a new view on the successive approxi-mations approach, recently used in study of the FBVPs for periodic andc (cid:13)
Year Diogenes Co., Sofiapp. xxx–xxx K. MarynetsCauchy–Nicoletti type boundary conditions (see [2]– [5]). An original ’freez-ing’ technique, initially suggested for the nonlinear systems of ordinarydifferential equations (see discussions [8], [7]), and a dichotomy–type ap-proach (see [9], [10]) lead to investigation of solutions of two ’model’–typeFBVPs, containing some artificially introduced parameters. The approx-imate solutions of these problems are constructed analytically, while thenumerical values of parameters are determined as solutions of the so–called’bifurcation’ equations.It should be emphasized that the suggested in this paper techniquefor study of the FBVPs allows us to improve the applicability conditionsof the successive approximations approach and to essentially sharpen theestimates, obtained in the earlier papers (see [2]– [5]).
2. Problem setting
Consider a two–point nonlinear boundary–value problem for a systemof fractional differential equations (FDEs) Ca D pt x = f ( t, x ( t )) , t ∈ [ a, b ] , x, f ∈ R n (2.1)for some p ∈ (0 , Ca D pt is the generalized Caputo fractional de-rivative with lower limit at a (see [12, Definition 1.8], [11, Definition 2.3]) f : G f → R n is a continuous vector–function and G f := [ a, b ] × D , D ⊂ R n is a closed and bounded domain, subjected to the two–point boundaryconstraint g ( x ( a ) , x ( b )) = 0 , (2.2)where g : D × D → R n is a continuous function.Together with the FBVP (2.1), (2.2) we study two ’model’–type FBVPswith separated two–point linear boundary conditions Ca D pt u = f ( t, u ( t )) , t ∈ (cid:20) a, a + b (cid:21) , u, f ∈ R n , (2.3) u ( a ) = z, u (cid:18) a + b (cid:19) = λ (2.4)and C a + b D pt v = f ( t, v ( t )) , t ∈ (cid:20) a + b , b (cid:21) , v, f ∈ R n , (2.5) v (cid:18) a + b (cid:19) = λ, v ( b ) = η, (2.6)where z, λ, η ∈ R n are considered as parameters. The problem is to find a continuous function x : [ a, b ] → D satisfyingthe system of FDEs (2.1) and the nonlinear boundary conditions (2.2) . TUDY OF A NONLINEAR FBVP... . . . 3
Remark . Note that in the FBVPs (2.3), (2.4) and (2.5), (2.6) thelength of the interval of definition of the independent variable t is equal to I := b − a , that is a half of the interval in the original problem (2.1), (2.2).The results, presented in the upcoming sections show, that this approachenables us to reduce some values in the qualitative analysis of the givenFBVP and to essentially improve the estimates of the constructed iterationschemes.
3. Reduction of the original FBVP and some subsidiarystatements
Let D a , D a + b , D b ⊂ R n be some convex domains containing boundaryvalues of the continuous solution x of the FBVP (2.1), (2.2): x ( a ) ∈ D a , x (cid:18) a + b (cid:19) ∈ D a + b , x ( b ) ∈ D b , (3.1)and let us introduce a set D a, a + b := (1 − θ ) z + θλ, z ∈ D a , λ ∈ D a + b , θ ∈ [0 ,
1] (3.2)with its ρ u neighborhood of the form: D u := B ( D a, a + b , ρ u ) . Here B (Ω , r ) := ∪ y ∈ Ω B ( y, r )is a componentwise r –neighborhood of a bounded connected set Ω ⊂ R n , where under B ( y, r ) := { ξ ∈ R n : | ξ − y | ≤ r } we understand the componentwise ρ –neighborhood of a point y ∈ R n with r to be some non–negative real vector (see [10], Definition 1).Similarly, based on the sets D a + b , D b we introduce a set D a + b ,b := (1 − θ ) λ + θη, λ ∈ D a + b , η ∈ D b , θ ∈ [0 ,
1] (3.3)and its ρ v neighborhood D v := B ( D a + b ,b , ρ v ) . Using the aforementioned ’freezing’ technique (see discussions [8], [7],[10]) we introduce the following vector parameters z = col ( z , z , . . . , z n ) , λ = col ( λ , λ , . . . , λ n ) , η = col ( η , η , . . . , η n ) K. Marynetsassigning them values of solution x of the FBVP (2.1), (2.2) at the points t = a , t = a + b , t = b : z := x ( a ) , λ := x (cid:18) a + b (cid:19) , η := x ( b ) . (3.4)The parametrization (3.4) reduces study of the original FBVP (2.1), (2.2)with nonlinear boundary conditions at the full interval [ a, b ] to investiga-tion of solutions of two ’model–type’ problems (2.3), (2.4) and (2.5), (2.6)with parametrized linear boundary conditions, defined on the half intervals (cid:2) a, a + b (cid:3) and (cid:2) a + b , b (cid:3) respectively. It is worth to emphasize that the set of solutions of the FBVP (2.1) , (2.2) coincides with the set of solutions of the modified problems (2.3) , (2.4) and (2.5) , (2.6) under additional conditions (3.4) . The following lemmas hold.
Lemma . Let f ( t ) be a continuous function for t ∈ [ a, b ] .Then for all t ∈ [ a, b ] the following estimate is true: p ) (cid:12)(cid:12)(cid:12)(cid:12)Z tτ ( t − s ) p − f ( s ) ds − (cid:18) t − τ I (cid:19) p Z τ + I τ ( τ + I − s ) p − f ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ α ( t, τ, I ) max t ∈ [ τ,τ + I ] | f ( t ) | , (3.5) where α ( t, τ, I ) = ( t − τ ) p Γ( p + 1) (cid:20) − I p − + 2 (cid:18) − t − τ I (cid:19) p (cid:21) (3.6) and Γ( · ) is the Gamma–function.Proof. The direct calculations show that1Γ( p ) (cid:12)(cid:12)(cid:12)(cid:12)Z tτ ( t − s ) p − f ( s ) ds − (cid:18) t − τ I (cid:19) p Z τ + I τ ( τ + T − s ) p − f ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) | f ( s ) | ds + (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − | f ( s ) | ds (cid:21) ≤ p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) ds TUDY OF A NONLINEAR FBVP... . . . 5+ (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − ds (cid:21) max t ∈ [ τ,τ + I ] | f ( t ) | = α ( t, τ, I ) max t ∈ [ τ,τ + I ] | f ( t ) | . Note, that we could omit the absolute value for some terms under theintegrals, since ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − = ( t − s ) p − " − (cid:18) t − τ I (cid:19) p (cid:18) t − sτ + I − s (cid:19) p − ≥ ( t − s ) p − " − (cid:18) t − τ I (cid:19) p (cid:18) t − τ I (cid:19) p − = ( t − s ) p − (cid:20) − (cid:18) t − τ I (cid:19) p (cid:21) = ( t − s ) p − τ + I − t I ≥ , t ∈ [ τ, τ − I ] . ✷ Lemma . Let { α m ( · , τ, I ) } m ∈ N be a sequence of continuous func-tions at the interval [ a, b ] given by α m ( t, τ, I ) :=:= 1Γ( p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) α m − ( s, τ, I ) ds + (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − α m − ( s, τ, I ) ds (cid:21) , m ∈ N , (3.7) where α ( · , τ, I ) = 1 and α ( · , τ, I ) defined by formula (3.6) .Then the following estimate holds: α m ( t, τ, I ) ≤ I ( m − p ( m − p − Γ m − ( p + 1) α ( t, τ, I ) ≤ I mp m (2 p − Γ m ( p + 1) , (3.8) for all m ∈ N . K. Marynets
Proof.
Let us first estimate α ( t, τ, I ). Indeed, using its explicit form (3.6)we get: α ( t, τ, I ) = ( t − τ ) p Γ( p + 1) (cid:18) − t − τ I (cid:19) p ≤ t − τ ) p Γ( p + 1) (cid:18) − t − τ I (cid:19) p ≤ I p p − Γ( p + 1) . For m = 2 from the recurrent formula (3.8) we obtain: α ( t, τ, I )= 1Γ( p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) α ( s, τ, I ) ds + (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − α ( s, τ, I ) ds (cid:21) ≤ p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) ds + (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − ds (cid:21) I p p − Γ( p + 1) ≤I p p − Γ ( p + 1) . Suppose that for ( m −
1) the estimate α m − ( t, τ, I ) ≤ I ( m − p ( m − p − Γ m − ( p + 1) α ( t, τ, I ) ≤ I ( m − p ( m − p − Γ m − ( p + 1)is true, and let us prove it in the case of m . The direct calculations showthat α m ( t, τ, I )= 1Γ( p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) α m − ( s, τ, I ) ds + (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − α m − ( s, τ, I ) ds (cid:21) ≤ p ) (cid:20)Z tτ (cid:18) ( t − s ) p − − (cid:18) t − τ I (cid:19) p ( τ + I − s ) p − (cid:19) ds TUDY OF A NONLINEAR FBVP... . . . 7+ (cid:18) t − τ I (cid:19) p Z τ + I t ( τ + I − s ) p − ds (cid:21) I ( m − p ( m − p − Γ m − ( p + 1) ≤ I mp m (2 p − Γ m ( p + 1) . The last inequality proves lemma. ✷
4. Successive approximation techniques on the half intervals
Suppose that f ∈ Lip ( K u , D u ) with ρ u satisfying an inequality ρ u ≥ ( b − a ) p M u p − Γ( p + 1) . (3.1)Let us connect with the first ’model’–type FBVP (2.3), (2.4) the fol-lowing sequence of functions u m ( t, z, λ ) := z + 1Γ( p ) Z ta ( t − s ) p − f ( s, u m − ( s, z, λ )) ds − p ) (cid:18) t − a ) b − a (cid:19) p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u m − ( s, z, λ )) ds + (cid:18) t − a ) b − a (cid:19) p [ λ − z ] , (3.2)with the zero–approximation to the exact solution given by u ( t, z, λ ) := (cid:20) − (cid:18) t − a ) b − a (cid:19) p (cid:21) z + (cid:18) t − a ) b − a (cid:19) p λ, (3.3)for all m ∈ N and ( t, z, λ ) ∈ G u , G u := [ a, a + b ] × D a × D a + b .Similarly to the FBVP (2.3), (2.4) we assume that for the problem(2.5), (2.6) function f ∈ Lip ( K v , D v ) with ρ v satisfying an inequality ρ v ≥ ( b − a ) p M v p − Γ( p + 1) (3.4)and construct an appropriate sequence of functions v m ( t, λ, η ) := λ + 1Γ( p ) Z t a + b ( t − s ) p − f ( s, v m − ( s, λ, η )) ds − p ) (cid:18) t − b ) b − a + 1 (cid:19) p Z b a + b ( b − s ) p − f ( s, v m − ( s, λ, η )) ds + (cid:18) t − b ) b − a + 1 (cid:19) p [ η − λ ] , (3.5) K. Marynetswhere the zero–approximation to the exact solution is v ( t, λ, η ) := (cid:20) − (cid:18) t − b ) b − a + 1 (cid:19) p (cid:21) λ + (cid:18) t − b ) b − a + 1 (cid:19) p η, (3.6)for all m ∈ N and ( t, λ, η ) ∈ G v , G v := [ a + b , b ] × D a + b × D b . Theorem . Let for a parametrized FBVP (2.3) , (2.4) there existsa non–negative vector ρ u satisfying an inequality (3.1) such that f ∈ Lip ( K u , D u ) on an interval t ∈ (cid:2) a, a + b (cid:3) and for the matrix Q u := ( b − a ) K u p − Γ( p + 1) (3.7) an inequality holds r ( Q u ) < . (3.8) Then for arbitrary pair of vector parameters ( z, λ ) ∈ D a × D a + b : (1) All functions of the sequence are continuous on the interval (cid:2) a, a + b (cid:3) andsatisfy the linear boundary conditions (2.4) . (2) The sequence of functions (3.2) for t ∈ (cid:2) a, a + b (cid:3) converges uniformly as m → ∞ to its limit function u ∞ ( t, z, λ ) = lim m →∞ u m ( t, z, λ ) . (3.9)(3) The limit function (3.9) satisfies boundary conditions u ∞ ( a, z, λ ) = z, u ∞ (cid:18) a + b , z, λ (cid:19) = λ (3.10) and is a unique solution of an integral equation u ( t ) := z + 1Γ( p ) Z ta ( t − s ) p − f ( s, u ( s )) ds − p ) (cid:18) t − a ) b − a (cid:19) p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u ( s )) ds + (cid:18) t − a ) b − a (cid:19) p [ λ − z ] , t ∈ (cid:20) a, a + b (cid:21) (3.11) in the domain D u , i.e. it is a solution of the corresponding Cauchy problemfor a perturbed system of FDEs: Ca D pt u = f ( t, u ( t )) + (cid:18) b − a (cid:19) p ∆( z, λ ) , t ∈ (cid:20) a, a + b (cid:21) , (3.12) u ( a ) = z, (3.13)TUDY OF A NONLINEAR FBVP... . . . 9 where ∆ : D a × D a + b → R n is a mapping given by formula: ∆( z, λ ) := Γ( p + 1)[ λ − z ] − p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u ( s )) ds. (3.14)(4) The following error estimation holds: | u ∞ ( t, z, λ ) − u m ( t, z, λ ) | ≤ ( b − a ) p p − Γ( p + 1) Q mu ( I n − Q u ) − M u , (3.15) where I n is the n –dimensional unit matrix.Proof. Simple calculations show that the first statement of the theoremholds, i.e. all functions of the sequence (3.2) are continuous and satisfy theparametrized boundary restrictions (2.4).Now we prove that for all m ∈ N functions u m of the sequence (3.2)will remain in its domain of definition, i.e. the iteration process can lastinfinitely long. For this purpose let us estimate the differences d m ( t, z, λ ) := | u m ( t, z, λ ) − u ( t, z, λ ) | , m ∈ N , where functions u m ( · , z, λ )and u ( · , z, λ ) are defined by formulas (3.2), (3.3). We get d m ( t, z, λ ) = (cid:12)(cid:12)(cid:12)(cid:12) p ) Z ta ( t − s ) p − f ( s, u m − ( s, z, λ )) ds − p ) (cid:18) t − a ) b − a (cid:19) p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u m − ( s, z, λ )) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ p ) "Z ta ( ( t − s ) p − − (cid:18) t − a ) b − a (cid:19) p (cid:18) a + b − s (cid:19) p − ) ds + (cid:18) t − a ) b − a (cid:19) p Z a + b t (cid:18) a + b − s (cid:19) p − ds × max ( t,z,λ ) ∈ G u | f ( t, u m − ( t, z, λ )) | = M u α (cid:18) t, a, b − a (cid:19) , (3.16)where α (cid:0) t, a, b − a (cid:1) is defined by (3.6) and M u := max ( t,z,λ ) ∈ G u | f ( t, u m − ( t, z, λ )) | , m ∈ N . Let us now analyse the difference d mm +1 ( t, z, λ ) := | u m +1 ( t, z, λ ) − u m ( t, z, λ ) | , ∀ m ∈ N , where u m ( · , z, λ ) are functions of the sequence (3.2).From the inequality (3.16) for m = 0 we already obtained an estimate d ( t, z, λ ) ≤ M u α (cid:18) t, a, b − a (cid:19) . m one gets d mm +1 ( t, z, λ )= 1Γ( p ) (cid:12)(cid:12)(cid:12)(cid:12)Z ta ( t − s ) p − [ f ( s, u m ( s, z, λ )) − f ( s, u m − ( s, z, λ ))] ds − (cid:18) t − a ) b − a (cid:19) p Z a + b a (cid:18) a + b − s (cid:19) p − [ f ( s, u m ( s, z, λ )) − f ( s, u m − ( s, z, λ ))] ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ K u Γ( p ) "Z ta ( ( t − s ) p − − (cid:18) t − a ) b − a (cid:19) p (cid:18) a + b − s (cid:19) p − ) | u m ( s, z, λ ) − u m − ( s, z, λ ) | ds + (cid:18) t − a ) b − a (cid:19) p Z a + b t (cid:18) a + b − s (cid:19) p − | u m ( s, z, λ ) − u m − ( s, z, λ ) | ds K u M u Γ( p ) "Z ta ( ( t − s ) p − − (cid:18) t − a ) b − a (cid:19) p (cid:18) a + b − s (cid:19) p − ) α m (cid:18) t, a, b − a (cid:19) ds + (cid:18) t − a ) b − a (cid:19) p Z a + b t (cid:18) a + b − s (cid:19) p − α m (cid:18) t, a, b − a (cid:19) ds ≤ (cid:18) I p K u p − Γ( p + 1) (cid:19) m M u α (cid:18) t, a, b − a (cid:19) = Q m M u α (cid:18) t, a, b − a (cid:19) ≤ ( b − a ) p p − Γ( p + 1) Q m M. (3.17)In view of the inequality (3.17) d mm + j ( t, z, λ ) ≤ j X k =1 d m + k − m + k ( t, z, λ ) ≤ j X k =1 K m + k − u M u α m + k ( t ) ≤ j X k =1 K m + k − u ( b − a ) ( m + k − p ( m + k − p − Γ m + k − ( p + 1) M u α (cid:18) t, a, b − a (cid:19) = j − X k =0 Q m + ku M u α (cid:18) t, a, b − a (cid:19) = Q mu j − X k =0 Q ku M u α (cid:18) t, a, b − a (cid:19) ≤ ( b − a ) p p − Γ( p + 1) Q mu j − X k =0 Q ku M u . (3.18)Due to the condition (3.8) the spectrum radius of the matrix Q u doesnot exceed 1.TUDY OF A NONLINEAR FBVP... . . . 11This means that j − X k =0 Q ku ≤ ( I n − Q u ) − , lim m →∞ Q mu = O n , where O n is the zero n –dimension matrix.Passing in (3.18) to the limit for j → ∞ , we get the estimate (3.15).Moreover, according to the Cauchy criteria the sequence of functions { u m ( · , z, λ ) } , defined by the iterative formula (3.2), is uniformly conver-gent in the domain G u to the limit function u ∞ ( · , z, λ ).Since all functions of the sequence (3.15) satisfy two–point parametrizedboundary conditions (2.4), the limit function u ∞ ( · , z, λ ) also satisfied them.Analogically to Theorem 1 in [2] it is easy to show, that letting m → ∞ in the relation (3.2), the limit function (3.9) is the solution of the integralequation (3.11), i.e. it is a unique solution of the Cauchy problem (3.12),(3.11) with the perturbation term ∆( z, λ ) defined by (3.14). ✷ Under similar to Theorem conditions one can prove convergence ofthe sequence of functions v m ( · , λ, η ), i.e. theorem holds. Theorem . Let for a parametrized FBVP (2.5) , (2.6) there existsa non–negative vector ρ v satisfying an inequality (3.4) such that f ∈ Lip ( K v , D v ) , ∀ t ∈ (cid:2) a + b , b (cid:3) and for the matrix Q v := ( b − a ) p K v p − Γ( p + 1) (3.19) an inequality holds r ( Q v ) < . (3.20) Then for arbitrary pair of vector parameters ( λ, η ) ∈ D a + b × D b : (1) All functions of the sequence are continuous on the interval (cid:2) a + b , b (cid:3) andsatisfy the separated boundary conditions (2.6) . (2) The sequence of functions (3.5) for t ∈ (cid:2) a + b , b (cid:3) converges uniformly as m → ∞ to its limit function v ∞ ( t, λ, η ) = lim m →∞ v m ( t, λ, η ) . (3.21)(3) The limit function (3.21) satisfies boundary conditions v ∞ (cid:18) a + b , λ, η (cid:19) = λ, u ∞ ( b, z, λ ) = η (3.22)2 K. Marynets and is a unique solution of an integral equation v ( t ) := λ + 1Γ( p ) Z t a + b ( t − s ) p − f ( s, v ( s )) ds − p ) (cid:18) t − b ) b − a + 1 (cid:19) p Z b a + b ( b − s ) p − f ( s, v ( s ) ds + (cid:18) t − b ) b − a + 1 (cid:19) p [ η − λ ] (3.23) in the domain D v , i.e. it is a solution of the corresponding Cauchy problemfor a perturbed system of FDEs: C a + b D pt v = f ( t, v ( t )) + (cid:18) b − a (cid:19) p Θ( λ, η ) , t ∈ (cid:20) a + b , b (cid:21) , (3.24) v (cid:18) a + b (cid:19) = λ, (3.25) where Θ : D a + b × D b → R n is a mapping, given by formula: Θ( λ, η ) := Γ( p + 1)[ η − λ ] − p Z b a + b ( b − s ) p − f ( s, v ( s )) ds. (3.26)(4) The following error estimation holds: | v ∞ ( t, z, λ ) − v m ( t, z, λ ) | ≤ ( b − a ) p p − Γ( p + 1) Q mv ( I n − Q v ) − M v . (3.27) Proof.
The proof is similar to the aforementioned Theorem . ✷ Remark . Theorem and guarantee that under assumedconditions functions u ∞ ( t, z, λ ) : (cid:20) a, a + b (cid:21) × D a × D a + b → R n ,v ∞ ( t, λ, η ) : (cid:20) a + b , b (cid:21) × D a + b × D b → R n (3.28)are well defined for all pairs of artificially introduced parameters( z, λ ) ∈ × D a × D a + b and ( λ, η ) ∈ D a + b × D b .Then by putting x ∞ ( t, z, λ, η ) := u ∞ ( t, z, λ ) , t ∈ (cid:2) a, a + b (cid:3) ,v ∞ ( t, λ, η ) , t ∈ (cid:2) a + b , b (cid:3) (3.29)TUDY OF A NONLINEAR FBVP... . . . 13we obtain a well defined continuous function x ∞ ( · , z, λ, η ), which at thepoint t = a + b attains the value x ∞ (cid:18) a + b , z, λ, η (cid:19) = u ∞ (cid:18) a + b , z, λ (cid:19) = v ∞ (cid:18) a + b , z, η (cid:19) = λ. (3.30)
4. Main result
Let us now study two fractional initial value problems (FIVP) withsome constant perturbation vector terms: Ca D pt u = f ( t, u ( t )) + (cid:18) b − a (cid:19) p µ u , t ∈ (cid:20) a, a + b (cid:21) , (4.1) u ( a ) = z (4.2)and C a + b D pt v = f ( t, v ( t )) + (cid:18) b − a (cid:19) p µ v , t ∈ (cid:20) a + b , b (cid:21) , (4.3) v (cid:18) a + b (cid:19) = λ, (4.4)where µ u = col ( µ u , µ u , . . . , µ un ), µ v = col ( µ v , µ v , . . . , µ vn ) ∈ R n we will call’control parameters’. Theorem . Let z ∈ D a , λ ∈ D a + b and η ∈ D b are fixed values ofparameters. Assume that conditions of Theorem , Theorem hold.Then the solutions u ( · , z, λ ) and v ( · , λ, η ) of the FIVPs (4.1) , (4.2) and (4.3) , (4.4) respectively will satisfy conditions u (cid:18) a + b , z, λ (cid:19) = λ, (4.5) and v ( b, λ, η ) = η, (4.6) i.e. they will be solutions of the ’model–type‘ FBVPs with separated two–point parametrized boundary conditions if and only if the control parameters µ u , µ v in (4.1) , (4.3) have the form: µ u = Γ( p + 1)[ λ − z ] − p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u ∞ ( s, z, λ )) ds (4.7) and µ v = Γ( p + 1)[ η − λ ] − p Z b a + b ( b − s ) p − f ( s, v ∞ ( s, λ, η )) ds (4.8)4 K. Marynets respectively, where u ∞ ( · , z, λ ) , v ∞ ( · , λ, η ) are the limit functions (3.9) , (3.21) .Proof. The proof can be carried out using a similar approach described inTheorem 2 (see discussion [2]). ✷ Theorem . Assume that conditions of Theorem and Theo-rem are true. Then (1)
Function x ∞ ( · , z, λ, η ) : [ a, b ] × D a × D a + b × D b → R n is a continuoussolution of the original nonlinear FBVP (2.1) , (2.2) if and only if the triplet ( z, λ, η ) satisfies the system of determining equations ∆( z, λ ) = 0 , (4.9)Θ( λ, η ) = 0 , (4.10)Ξ( z, λ, η ) = 0 , (4.11) where ∆ and Θ are the mappings defined by formulas (3.14) , (3.26) respec-tively, and Ξ : D u × D v → R n , given by Ξ( z, λ, η ) := g (cid:18) u ∞ ( a, z, λ ) , u ∞ (cid:18) a + b , z, λ (cid:19)(cid:19) + g (cid:18) v ∞ (cid:18) a + b , λ, η (cid:19) , v ∞ ( b, λ, η ) (cid:19) . (2) For every function X ( · ) of the FBVP (2.1) , (2.2) with values (cid:0) X ( a ) , X (cid:0) a + b (cid:1) , X ( b ) (cid:1) ∈ D a × D a + b × D b , there exists a triplet ( z , λ , η ) such that X ( · ) = x ∞ ( t, z, λ, η ) , where function x ∞ is defined by (3.29) .Proof. We refer to proofs of Theorem 3 (see discussion in [2]) and Theorem 3(see [8]) and note that the equations (3.30), (3.12), (3.24), (4.9), (4.10)lead straightforward to the continuity of function x ∞ ( · , z, λ, η ) at the point t = a + b . Moreover, according to the definition (3.29) of the aforementionedfunction, its continuity at all other points of the interval [ a, b ] holds as well. ✷
5. Some remarks
In practice it is more reasonable to consider an approximate determin-ing system ∆ m ( z, λ ) := Γ( p + 1)[ λ − z ] − p Z a + b a (cid:18) a + b − s (cid:19) p − f ( s, u m ( s, z, λ )) ds = 0 , (5.1)TUDY OF A NONLINEAR FBVP... . . . 15Θ m ( λ, η ) := Γ( p +1)[ η − λ ] − p Z b a + b ( b − s ) p − f ( s, v m ( s, λ, η )) ds = 0 , (5.2)Ξ m ( z, λ, η ) := g (cid:18) u m ( a, z, λ ) , u m (cid:18) a + b , z, λ (cid:19)(cid:19) + g (cid:18) v m ( a + b , λ, η ) , v m ( b, λ, η ) (cid:19) = 0 (5.3)instead of the exact one. Here ∆ m : D a × D a + b → R n , Θ m : D a + b × D b → R n and Ξ m : D u × D v → R n are continuous mappings.Using our conclusions about function x ∞ ( · , z, λ, η ) given by (3.29), it isnatural that its m –th approximation will be defined as x m ( t, z, λ, η ) := u m ( t, z, λ ) , t ∈ (cid:2) a, a + b (cid:3) ,v m ( t, λ, η ) , t ∈ (cid:2) a + b , b (cid:3) , (5.4)where the sequences of function u m ( · , z, λ ), v m ( · , λ, η ) have the form (3.2),(3.5) accordingly. Theorem . If the values of parameters z, λ, η satisfy the m –approximate system of determining equations (5.1) – (5.3) , then the function x m ( · , z, λ, η ) in (5.4) is continuous on [ a, b ] .Proof. Since the functions u m ( · , z, λ ) and v m ( · , λ, η ), defined by the succes-sive approximations 3.2, 3.5, satisfy the consistency condition u m (cid:18) a + b , z, λ (cid:19) = v m (cid:18) a + b , λ, η (cid:19) = λ, (5.5)it follows that Ca D pt u m (cid:18) a + b , z, λ (cid:19) = f (cid:18) a + b , u m (cid:18) a + b , z, λ (cid:19)(cid:19) − (cid:18) b − a (cid:19) p p Z a + b a (cid:18) a + b − s (cid:19) p − f (cid:18) a + b , u m (cid:18) a + b , z, λ (cid:19)(cid:19) ds + (cid:18) b − a (cid:19) p Γ( p + 1)[ λ − z ] (5.6)6 K. Marynetsand C a + b D pt v m (cid:18) a + b , λ, η (cid:19) = f (cid:18) a + b , v m (cid:18) a + b , λ, η (cid:19)(cid:19) − (cid:18) b − a (cid:19) p p Z b a + b ( b − s ) p − f (cid:18) a + b , v m (cid:18) a + b , λ, η (cid:19)(cid:19) ds + (cid:18) b − a (cid:19) p Γ( p + 1)[ η − λ ] . (5.7)Due to assumptions of the theorem, parameters z, λ, η satisfy the so–called ’bifurcation equations’ (5.1), (5.2). This means that (5.6), (5.7) maybe simplified to the form Ca D pt u m (cid:18) a + b , z, λ (cid:19) = f (cid:18) a + b , u m (cid:18) a + b , z, λ (cid:19)(cid:19) (5.8)and C a + b D pt v m (cid:18) a + b , λ, η (cid:19) = f (cid:18) a + b , v m (cid:18) a + b , λ, η (cid:19)(cid:19) (5.9)respectively.Since (5.5) holds, from (5.8), (5.9) we come to the conclusion that Ca D pt u m (cid:18) a + b , z, λ (cid:19) = C a + b D pt v m (cid:18) a + b , λ, η (cid:19) , which under relation (5.4) prove the continuity of function x m ( · , z, λ, η ) atthe point t = a + b . The fact, that this function is also continuous at all theother points follows straightforward from its definition. ✷ References [1] F. Batelli, M. Feˇckan,
Handbook of Differential Equations: ordinary differential equa-tions Vol. IV. , 1. Ed., Elsevier/North-Holland, Amsterdam (2008).[2] M. Feˇckan, K. Marynets, Approximation approach to periodic BVP for fractional dif-ferential systems,
European Physical Journal:
Special Topics 226, 3681–3692 (2017).[3] M. Feˇckan, K. Marynets, Approximation approach to periodic BVP for mixed frac-tional differential systems,
Journal of Computational and Applied Mathematics
Mathematical Methods in the Applied Sciences (2019), 42, 3616–3632. https://doi.org/10.1002/mma.5601[5] K. Marynets, Solvability analysis of a special type fractional differential system,
Computational and Applied Mathematics (2019), doi: 10.1007/s40314-019-0981-7.[6] K. Marynets, On the Cauchy–Nicoletti type two–point boundary–value problemfor fractional differential systems,
Differential Equations and Dynamical Systems (2019), (submitted).
TUDY OF A NONLINEAR FBVP... . . . 17 [7] M. I. Ronto, K.V. Marynets’, On the parametrization of boundary-value problemswith two-point nonlinear boundary conditions,
Nonlinear Oscillations (2012), 14(3), 379-413. doi: 10.1007/s11072-012-0165-5.[8] K. Marynets, On the parametrization of nonlinear boundary value problems withnonlinear boundary conditions,
Miskolc Mathematical Notes (2011), 12 (2), 209-223.https://doi.org/10.18514/MMN.2011.403.[9] A. Ront´o, M. Ront´o, N. Shchobak, Boundary Value Problems (2014) 2014: 164. doi:10.1186/s13661-014-0164-9[10] M. Ront´o, Y. Varha, Successive approximations and interval halving for integralboundary value problems,
Miskolc Mathematical Notes (2014), 16 (2), 1129–1152.https://doi.org/10.18514/MMN.2015.1708.[11] J. Wang, M. Feˇckan, Y. Zhou,
A survey on impulsive fractional differential equations ,Fract. Calc. Appl. Anal. 19, 806-831 (2016).[12] Y. Zhou,