Sharp estimates for the unique solution of the Hadamard-type two-point fractional boundary value problems
aa r X i v : . [ m a t h . G M ] J u l Sharp estimates for the unique solution of the Hadamard-typetwo-point fractional boundary value problems ∗ Zaid Laadjal § , Adjeroud Nacer ‡ Submitted on 24 April 2020; accepted for publication in “Applied Mathematics E-Notes”
Abstract
In this short note, we present the sharp estimate for the existence of a unique solution for a Hadamard-type fractional differential equations with two-point boundary value conditions. The method of analysisis obtained by using the integral of the Green’s function and the Banach contraction principle. Further,we will also obtain a sharper lower bound of the eigenvalues for an eigenvalue problem. Two examplesare presented to clarify the applicability of the essential results.
In the book [1] Kelley and Peterson considered the following classical two-point boundary value problems: (cid:26) u ′′ ( x ) = F ( x, u ( x )) , a < x < b,u ( a ) = A, u ( b ) = B, A, B ∈ R , (1)where a, b ∈ R , and they included the following result: Theorem 1 ([1], Theorem 7.7)
Let F : [ a, b ] × R → R be a continuous function satisfying the assumption: ( H ) There exists
K > such that |F ( x, ω ) − F ( x, ̟ ) | ≤ K | ω − ̟ | for all ( x, ω ) , ( x, ̟ ) ∈ [ a, b ] × R . Then the boundary value problem (1) has a unique solution on [ a, b ] if b − a < p /K. Ferreira in 2016 [2] discussed the existence and uniqueness of solutions for the following fractional bound-ary value problems with Reimman-Liouville fractional derivative: (cid:26) R D σa u ( x ) = −F ( x, u ( x )) , a < x < b, < σ ≤ ,u ( a ) = 0 , u ( b ) = B, B ∈ R , (2)so he included this result given by: Theorem 2 ([2])
Let F : [ a, b ] × R → R be a continuous function satisfying the assumption ( H ) . Then theboundary value problem (2) has a unique solution on [ a, b ] if b − a < ( σ ( σ +1) /σ Γ /σ ( σ )) / ( K /σ ( σ − ( σ − /σ ) . In 2019, Ferreira [3] corrected a recent uniqueness result [4] for a two-point fractional boundary valueproblem with Caputo derivative: (cid:26) C D σa u ( x ) = −F ( x, u ( x )) , a < x < b, < σ ≤ ,u ( a ) = A, u ( b ) = B, A, B ∈ R , (3)and he came to the following theorem: ∗ Mathematics Subject Classifications: 34A08, 34A40, 26A33. § Department of Mathematics and Computer Sciences, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 40004,Algeria. Email: [email protected] ‡ Department of Mathematics and Computer Sciences, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 40004,Algeria. Email: adjeroud [email protected] . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud
Let F : [ a, b ] × R → R be a continuous function satisfying the assumption ( H ) . Then theboundary value problem (2) has a unique solution on [ a, b ] if b − a < ( σ ( σ +1) /σ Γ /σ ( σ )) / ( K /σ ( σ − ( σ − /σ ) . In 2019, Ferreira [3] corrected a recent uniqueness result [4] for a two-point fractional boundary valueproblem with Caputo derivative: (cid:26) C D σa u ( x ) = −F ( x, u ( x )) , a < x < b, < σ ≤ ,u ( a ) = A, u ( b ) = B, A, B ∈ R , (3)and he came to the following theorem: ∗ Mathematics Subject Classifications: 34A08, 34A40, 26A33. § Department of Mathematics and Computer Sciences, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 40004,Algeria. Email: [email protected] ‡ Department of Mathematics and Computer Sciences, ICOSI Laboratory, Abbes Laghrour University, Khenchela, 40004,Algeria. Email: adjeroud [email protected] . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud Theorem 3 ([3])
Let F : [ a, b ] × R → R be a continuous function satisfying the assumption ( H ) . Then theboundary value problem (3) has a unique solution on [ a, b ] if M ( σ, a, b ) < /K, where M ( σ, a, b ) = 1Γ( σ + 1) max x ∈ [ a,b ] (cid:16) − x − ϕ ( x )) σ + 2 ( x − a )( b − ϕ ( x )) σ b − a + ( x − a ) σ − ( x − a )( b − a ) σ − (cid:17) , with ϕ ( x ) = (cid:16) ( x − ab − a ) σ − b − (cid:17) / (cid:16) ( x − ab − a ) σ − − (cid:17) . See [2, 3] and references therein for more details.In the last few decades, the differential equations involving a fractional order have witnessed a wide atten-tion from the researchers, because they were extensively implemented in daily life and in various scientific andtechnological fields and in many branches including physics, biology, chemistry, economics, astronomy, con-trol theory, viscoelastic materials, robotics, signal processing, electromagnetism, electrodynamics of complexmedium, anomalous diffusion and fractured media, electromagnetism, potential theory and electro statistics,polymer rheology, and aerodynamics, ... etc, we refer the interested reader to paper [7], and the referencescontained therein.It is well known that the existence of solution plays a main important role in the theory and applicationsof fractional differential equations with boundary conditions. Recently, many researchers are interested instudying the Hadamard-type fractional boundary value problems, where there are several results about theexistence of solutions for the differential equations with Hadamard derivative, we refer the reader to thebook [8] that contains the most important works that have been published in this domain. In addition,some researchers are interested in studying the stability of solutions to fractional differential equations,including Laypunov stability, exponential stability, Mittag-Leffler stability, and HyersUlam stability, havebeen introduced. Among these concepts, HyersUlam stability analysis was recognized as a simple method ofinvestigation. We refer the readers to [10, 11, 12, 13, 14, 15], and the references contained therein.Motivated by the above mentioned works and the papers [5, 6], in this paper, we investigated the sharpestimate for the unique solution of the following fractional differential equation with Hadamard derivative: (cid:26) H D σa u ( x ) = −F ( x, u ( x )) , < a < x < b, < σ ≤ ,u ( a ) = 0 , u ( b ) = B, B ∈ R , (4)where F is a given function, H D σa denotes the Hadamard fractional derivative of order σ , and B is realconstant. Further, we will also obtain a sharp estimate for lower bound for the eigenvalues for the follwingeigenvalue problem (cid:26) H D σa u ( x ) = λu ( x ) , < a < x < b, < σ ≤ ,u ( a ) = 0 = u ( b ) . (5)We start now to present some fundamental definitions and lemmas which will be used in this work. Definition 4 ([8, 9])
Let < a ≤ b and σ ∈ R + where n − < σ ≤ n with n ∈ N . The Hadamard fractionalintegral of ordre σ for a function g is defined by: H I a g ( x ) = g ( x ) and H I σa g ( x ) = 1Γ( σ ) Z xa (cid:16) ln xτ (cid:17) σ − g ( τ ) dττ for σ > . (6) Definition 5 ([8, 9])
Let < a < b ; δ = x ddx and let AC [ a, b ] be the space of functions g which areabsolutely continuous on [ a, b ] , and AC nδ [ a, b ] = { g : [ a, b ] × R → R s.t. δ n − [ g ( x )] ∈ AC [ a, b ] } . TheHadamard fractional derivative of order σ ≥ for a function g ∈ AC nδ [ a, b ] is defined by: H D a g ( x ) = g ( x ) , and H D σa g ( x ) = 1Γ( n − σ ) (cid:18) x ddx (cid:19) n Z xa (cid:16) ln xτ (cid:17) n − σ − g ( τ ) dττ for σ > , (7) where n − < σ ≤ n, n ∈ N . . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud
Let < a < b ; δ = x ddx and let AC [ a, b ] be the space of functions g which areabsolutely continuous on [ a, b ] , and AC nδ [ a, b ] = { g : [ a, b ] × R → R s.t. δ n − [ g ( x )] ∈ AC [ a, b ] } . TheHadamard fractional derivative of order σ ≥ for a function g ∈ AC nδ [ a, b ] is defined by: H D a g ( x ) = g ( x ) , and H D σa g ( x ) = 1Γ( n − σ ) (cid:18) x ddx (cid:19) n Z xa (cid:16) ln xτ (cid:17) n − σ − g ( τ ) dττ for σ > , (7) where n − < σ ≤ n, n ∈ N . . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud Lemma 6 ([8, 9])
Let < a ≤ b , and σ > where n − < σ ≤ n , n ∈ N . The differential equation H D σa u ( x ) = 0 has the general solution: u ( x ) = i = n X i =1 c i (cid:16) ln xa (cid:17) σ − i , x ∈ [ a, b ] , (8) where c i ∈ R ( i = 1 , ..., n ) are arbitrary constants. And moreover H I σa H D σa u ( x ) = u ( x ) + i = n X i =1 c i (cid:16) ln xa (cid:17) σ − i . (9) Lemma 7
Let y ∈ C ([ a, b ] , R ) ∩ L ([ a, b ] , R ) , the solution of following linear fractional boundary valueproblem (cid:26) H D σa u ( x ) = − y ( x ) , < a < x < b, < σ ≤ ,u ( a ) = 0 , u ( b ) = B, B ∈ R , (10) is given by u ( x ) = Z ba G ( x, τ ) y ( τ ) dτ + B ln xa ln ba ! σ − , where G ( x, τ ) = 1Γ( σ ) (cid:16) ln xa ln ba (cid:17) σ − (cid:0) ln bτ (cid:1) σ − τ − (cid:0) ln xτ (cid:1) σ − τ , a ≤ τ ≤ x ≤ b, (cid:16) ln xa ln ba (cid:17) σ − (cid:0) ln bτ (cid:1) σ − τ , a ≤ x ≤ τ ≤ b. (11) Proof.
Applying the operator H I σa on the equation H D σa u ( x ) = − y ( x ), we get u ( x ) = − σ ) Z xa (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + c (cid:16) ln xa (cid:17) σ − + c (cid:16) ln xa (cid:17) σ − , (12)where c , c ∈ R .Using the boundary conditions u ( a ) = 0 and u ( b ) = B, we get c = 0 and c = 1Γ( σ ) (cid:18) ln ba (cid:19) − σ Z ba (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ + B (cid:18) ln ba (cid:19) − σ . Substituting the values of c and c in (12), we obtain u ( x ) = 1Γ( σ ) ln xa ln ba ! σ − Z ba (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ − σ ) Z xa (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + B ln xa ln ba ! σ − = 1Γ( σ ) Z xa ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − − (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + 1Γ( σ ) Z bx ln xa ln ba ! σ − × (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ + B ln xa ln ba ! σ − = Z ba G ( x, τ ) dτ + B ln xa ln ba ! σ − . Hence, the proof is completed. . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud
Applying the operator H I σa on the equation H D σa u ( x ) = − y ( x ), we get u ( x ) = − σ ) Z xa (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + c (cid:16) ln xa (cid:17) σ − + c (cid:16) ln xa (cid:17) σ − , (12)where c , c ∈ R .Using the boundary conditions u ( a ) = 0 and u ( b ) = B, we get c = 0 and c = 1Γ( σ ) (cid:18) ln ba (cid:19) − σ Z ba (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ + B (cid:18) ln ba (cid:19) − σ . Substituting the values of c and c in (12), we obtain u ( x ) = 1Γ( σ ) ln xa ln ba ! σ − Z ba (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ − σ ) Z xa (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + B ln xa ln ba ! σ − = 1Γ( σ ) Z xa ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − − (cid:16) ln xτ (cid:17) σ − y ( τ ) dττ + 1Γ( σ ) Z bx ln xa ln ba ! σ − × (cid:18) ln bτ (cid:19) σ − y ( τ ) dττ + B ln xa ln ba ! σ − = Z ba G ( x, τ ) dτ + B ln xa ln ba ! σ − . Hence, the proof is completed. . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud This section is devoted to prove the main results of the problem (4), and present a lower bound of theeigenvalues for the eigenvalue problem (5).
Lemma 8
The Green’s function G defined in Lemma 7 has the following property: max x ∈ [ a,b ] Z ba | G ( x, τ ) | dτ = ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ +1 Γ( σ ) . (13) Proof.
From Lemma 4 of [5], we have G ( x, τ ) ≥ x, τ ) ∈ [ a, b ] × [ a, b ] . Therefore,Γ( σ ) Z ba | G ( x, τ ) | dτ = Z xa ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − − (cid:16) ln xτ (cid:17) σ − dττ + Z bx ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − dττ = ln xa ln ba ! σ − Z ba (cid:18) ln bτ (cid:19) σ − dττ − Z xa (cid:16) ln xτ (cid:17) σ − dττ = 1 σ (cid:18) ln ba (cid:19) − σ (cid:16) ln xa (cid:17) σ − (cid:18) ln ba (cid:19) σ − σ (cid:16) ln xa (cid:17) σ , which yields Γ( σ + 1) Z ba G ( x, τ ) dτ = (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − (cid:16) ln xa (cid:17) σ . (14)It follows that we need to get the maximum value of the function g ( x ) = (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − (cid:16) ln xa (cid:17) σ , x ∈ [ a, b ] . (15)Observe that g ( x ) ≥ x ∈ [ a, b ] , and g ( a ) = g ( b ) = 0 . Now we differentiate g ( x ) on ( a, b ) to get g ′ ( x ) = ( σ − x (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − σx (cid:16) ln xa (cid:17) σ − , from which follows that g ′ ( x ∗ ) = 0 has a unique zero, attained at the point x ∗ = a (cid:18) ba (cid:19) ( σ − /σ . It is easily seen that x ∗ ∈ ( a, b ) . Because g ( x ) is continuous function and x ∗ ∈ ( a, b ), we conclude thatmax x ∈ [ a,b ] g ( x ) = g ( x ∗ )= (cid:18) ln ba (cid:19) ln (cid:18) ba (cid:19) ( σ − /σ ! σ − − ln (cid:18) ba (cid:19) ( σ − /σ ! σ = 1 σ − (cid:18) σ − σ ln ba (cid:19) σ = ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ . (16)By (14), (15) and (16) we get the formula (13). The proof is completed. . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud
From Lemma 4 of [5], we have G ( x, τ ) ≥ x, τ ) ∈ [ a, b ] × [ a, b ] . Therefore,Γ( σ ) Z ba | G ( x, τ ) | dτ = Z xa ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − − (cid:16) ln xτ (cid:17) σ − dττ + Z bx ln xa ln ba ! σ − (cid:18) ln bτ (cid:19) σ − dττ = ln xa ln ba ! σ − Z ba (cid:18) ln bτ (cid:19) σ − dττ − Z xa (cid:16) ln xτ (cid:17) σ − dττ = 1 σ (cid:18) ln ba (cid:19) − σ (cid:16) ln xa (cid:17) σ − (cid:18) ln ba (cid:19) σ − σ (cid:16) ln xa (cid:17) σ , which yields Γ( σ + 1) Z ba G ( x, τ ) dτ = (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − (cid:16) ln xa (cid:17) σ . (14)It follows that we need to get the maximum value of the function g ( x ) = (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − (cid:16) ln xa (cid:17) σ , x ∈ [ a, b ] . (15)Observe that g ( x ) ≥ x ∈ [ a, b ] , and g ( a ) = g ( b ) = 0 . Now we differentiate g ( x ) on ( a, b ) to get g ′ ( x ) = ( σ − x (cid:18) ln ba (cid:19) (cid:16) ln xa (cid:17) σ − − σx (cid:16) ln xa (cid:17) σ − , from which follows that g ′ ( x ∗ ) = 0 has a unique zero, attained at the point x ∗ = a (cid:18) ba (cid:19) ( σ − /σ . It is easily seen that x ∗ ∈ ( a, b ) . Because g ( x ) is continuous function and x ∗ ∈ ( a, b ), we conclude thatmax x ∈ [ a,b ] g ( x ) = g ( x ∗ )= (cid:18) ln ba (cid:19) ln (cid:18) ba (cid:19) ( σ − /σ ! σ − − ln (cid:18) ba (cid:19) ( σ − /σ ! σ = 1 σ − (cid:18) σ − σ ln ba (cid:19) σ = ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ . (16)By (14), (15) and (16) we get the formula (13). The proof is completed. . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud Theorem 9
Let F : [ a, b ] × R → R be a continuous function satisfying the assumption ( H ) . If ba < exp σ ( σ +1) /σ Γ /σ ( σ )( σ − ( σ +1) /σ K /σ ! . (17) Then the Fractional boundary value problem (4) has a unique solution on [ a, b ] . Proof.
Let E = C ([ a, b ] , R ) be the Banach space endowed with the norm k u k = sup x ∈ [ a,b ] | u ( x ) | (seeProposition 2.18 in [16]), and we define the operator R : E → E by R u ( x ) = Z ba G ( x, τ ) u ( τ ) dτ + B ln xa ln ba ! σ − , where the function G is given by (11).Notice that the prolem (4) has a solution u if only if u is fixed point of the operator R .For all ( x, u ) , ( x, v ) ∈ [ a, b ] × E, we have | R u ( x ) − R v ( x ) | ≤ Z ba G ( x, τ ) |F ( τ, u ( τ )) − F ( τ, v ( τ )) | dτ ≤ Z ba KG ( x, τ ) | u ( τ ) − v ( τ ) | dτ ≤ K Z ba G ( x, τ ) dτ k u − v k , using the formula (13) yields k R u − R v k ≤ K ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ +1 Γ( σ ) k u − v k . Can be easily check that the assumption (17) leads to principle of contraction mapping. Hence, the operator R is contraction mapping, we conclude that the problem (4) has a unique solution. Now we present a lower bound of the eigenvalues for the eigenvalue problem (5).
Theorem 10
If the eigenvalue problem (5) has a non-trivial continuous solution, then | λ | ≥ σ σ +1 Γ( σ )( σ − σ − (cid:0) ln ba (cid:1) σ , (18) Proof.
From Lemma 7, the solution of the problem (5) can be written as follows u ( x ) = Z ba λG ( x, τ ) u ( τ ) dτ. which yields k u k ≤ | λ | k u k max x ∈ [ a,b ] Z ba | G ( x, τ ) | dτ Since u is non-trivial, then k u k 6 = 0 . So, using now to the formula of the Green function G proved in Lemma8, we get 1 ≤ | λ | max x ∈ [ a,b ] Z ba | G ( x, τ ) | dτ = | λ | ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ +1 Γ( σ ) , from which the inequality (18) follows. The proof is completed.We have the following result about the nonexistence for solutions of the boundary value problem (5). . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud
From Lemma 7, the solution of the problem (5) can be written as follows u ( x ) = Z ba λG ( x, τ ) u ( τ ) dτ. which yields k u k ≤ | λ | k u k max x ∈ [ a,b ] Z ba | G ( x, τ ) | dτ Since u is non-trivial, then k u k 6 = 0 . So, using now to the formula of the Green function G proved in Lemma8, we get 1 ≤ | λ | max x ∈ [ a,b ] Z ba | G ( x, τ ) | dτ = | λ | ( σ − σ − (cid:0) ln ba (cid:1) σ σ σ +1 Γ( σ ) , from which the inequality (18) follows. The proof is completed.We have the following result about the nonexistence for solutions of the boundary value problem (5). . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud Corollary 11 If | λ | < σ σ +1 Γ( σ )( σ − σ − (cid:0) ln ba (cid:1) σ , (19) Then the boundary value problem (5) has no non-trivial solution.
Proof.
Assume the contrary. Then the boundary value problem (5) has a non-trivial solution u. By Theorem10, inequality (18) holds. This contradicts assumption (19). The proof is completed.
Example 12
We consider the following Hadamard frational boundary value problem (cid:26) H D / a u ( x ) = ( x − + p x − u ( x ) , < x < e,u (1) = 0 , u ( e ) = 1 , (20) where e is an irrational number and it’s defined by the infinite series e = P + ∞ k =0 1 k ! and approximately equalto . . Here σ = and F ( x, u ) = ( x − + √ x − u . For all ( x, u ) ∈ (1 , e ] × R , we have | ∂ u F ( x, u ) | = | u |√ x − u ≤ . Choose K = 1 . So, by using the given values we get exp σ ( σ +1) /σ Γ /σ ( σ )( σ − ( σ +1) /σ K /σ ! = exp (cid:18)
34 (9 π ) / (cid:19) > e. Then the inequality (17) is satisfied. Hence, by Theorem 9, we conclude that the Hadamard fractionalboundary value problem (20) has a unique solution on the interval [1 , e ] . Example 13
Consider the following eigenvalue problem (cid:26) H D / a u ( x ) = λu ( x ) , < x < e,u (1) = 0 = u ( e ) , (21) Here σ = , and [ a, b ] = [1 , e ] . So, we obtain σ σ +1 Γ( σ )( σ − σ − (cid:0) ln ba (cid:1) σ = 9 √ π , (22) By Theorem 10, we conclude that: If λ is an eigenvalue of the problem (21), we must have | λ | ≥ √ π/ . Acknowledgment.
The authors would like to thank the anonymous referees for their useful remarksthat led that improved the paper.
References [1] W. C. Kelley, A. C. Peterson, The theory of differential equations. Second edition, Universitext, Springer,New York, 2010.[2] A. R. C. Ferreira, Existence and uniqueness of solutions for two-point fractional boundary value prob-lems, Electron. J. Diff. Equat., 2016, Paper No. 202, (2016), 5 pp. . Laadjal and N. Adjeroud. Laadjal and N. Adjeroud