Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag-Leffler functions
aa r X i v : . [ m a t h . A P ] J un Solvability of Langevin equations with twoHadamard fractional derivatives viaMittag-Le ffl er functions Mohamed I. Abbas ∗ Department of Mathematics and Computer Science,Faculty of Science, Alexandria University, Alexandria 21511, Egypt
Maria Alessandra Ragusa † Dipartimento di Matematica e Informatica, Universit di CataniaViale Andrea Doria, 6, CATANIA - 95125, ITALYRUDN University, 6 Miklukho - Maklay St, Moscow, 117198, Russia
Abstract
In this paper we discuss the solvability of Langevin equations with twoHadamard fractional derivatives. The method of this discussion is to studythe solutions of the equivalent Volterra integral equation in terms of Mittag-Le ffl er functions. The existence and uniqueness results are establishedby using Schauder fixed point theorem and Banach fixed point theoremrespectively. An example is given to illustrate the main results. Mathematics Subject Classification:
Keywords:
Langevin equation, Mittag-Le ffl er functions, Hadamard fractionalderivative, Schauder fixed point theorem. In recent years, The fractional di ff erential equations have elicited as a richarea of research due to their applications in various fields of science and en-gineering such as aerodynamics, viscoelasticity, control theory, economics andblood flow phenomena, etc. For more details, we refer the reader to the textbooks [14, 16, 20, 23]. ∗ [email protected] † [email protected] ff ers from the precedingones in the sense that the kernel of the integral in its definition contains log-arithmic function of arbitrary exponent. A detailed description of Hadamardfractional derivative and integral can be found in [5–7, 14] and the referencestherein. For some recent results in Hadamard fractional di ff erential equations,we refer the reader to [1, 26, 27] and the references therein.The Langevin equation was able to attract the attention of researchers dueto its importance in mathematical physics that is used in modeling the phe-nomena occurring in fluctuating environment such as Brownian motion [8].In 1908, P. Langevin [17] derived the classical form of this equation in termsof ordinary derivatives. For the systems in complex media, classical Langevinequation does not provide the correct description of the dynamics. The gen-eralized Langevin equation was introduced in 1966 by Kubo [15], where africtional memory kernel was incorporated into the Langevin equation to de-scribe the fractal and memory properties. In the 1990s, Mainardi and collabo-rators [21, 22] introduced the fractional Langevin equation. Many interestingresults regarding the existence, uniqueness and stability results for fractionalorder Langevin equations have been studied by many researchers, see for ex-ample [2–4, 9, 10, 19, 24, 25] and references cited therein.As far as we know, there are no contributions associated with the solutionsof the equivalent Volterra integral equations of the fractional Langevin equa-tions in terms of Mittag-le ffl er functions.The objective of this paper is to investigate the existence and uniqueness ofsolutions for the following Langevin equations with two Hadamard fractionalderivatives: H D β , t ( H D α , t − λ ) x ( t ) = f ( t , x ( t )) , t ∈ [1 , e ] , λ > , < α, β ≤ , ( H D α , t − λ ) x ( e ) = , H I − α + x (1 + ) = c , c ∈ R , (1.1)where H D α , t , H D β , t denote Hadamard fractional derivatives of orders α, β (0 <α, β ≤
1) respectively , H I − α + denotes the left-sided Hadamard fractional integralof order 1 − α and f : [1 , e ] × R → R is given continuous function. Let C ([1 , e ] , R ) be the Banach space of all continuous functions from [1 , e ] to R endowed with the norm k x k C = sup t ∈ [1 , e ] | x ( t ) | .For 0 ≤ γ <
1, we define the weighted space of functions on [1 , e ] by C γ, ln ([1 , e ] , R ) = { : [1 , e ] → R | (ln t ) γ ( t ) ∈ C ([1 , e ] , R ) } . C γ, ln ([1 , e ] , R ) is the Banach space with the norm k k γ, ln = k (ln t ) γ ( t ) k C = sup t ∈ [1 , e ] | (ln t ) γ ( t ) | . The following definitions are devoted to the basic concepts of Hadamard frac-tional integrals and fractional derivatives.
Definition 2.1. (see [14] , p.110) The left-sided Hadamard fractional integral of order α ∈ R + of function ( t ) is defined by H I α + ( t ) = Γ ( α ) Z t (cid:18) ln ts (cid:19) α − ( s ) dss , < < t ≤ b < ∞ , where Γ ( · ) is the Gamma function. Definition 2.2. (see [14] , p.111) The left-sided Hadamard fractional derivative oforder α ∈ [ n − , n ) , n ∈ Z + of function ( t ) is defined by H D α + ( t ) = Γ ( n − α ) t ddt ! n Z t (cid:18) ln ts (cid:19) n − α + ( s ) dss , < < t ≤ b < ∞ , where Γ ( · ) is the Gamma function. Proposition 2.3. (see [14] , p.114,115) Let α > , β > and < < b < ∞ . Thenfor f ∈ C γ, ln ([1 , e ] , R ) , the following properties hold: H D α , t H I β + f = H I β − α + f and H D α , t H I α + f = f . Lemma 2.4. (see [14] , Theorem 2.3, p.116) Let β > , < < b < ∞ andf ∈ C γ, ln ([1 , e ] , R ) . Then H I β + H D β , t u ( t ) = u ( t ) − n X j = c j (ln t ) β − j , where c j ∈ R and n − < β < n. Lemma 2.5.
Given σ ∈ C γ, ln ([1 , e ] , R ) . Then the linear problem H D β , t ( H D α , t − λ ) x ( t ) = σ ( t ) , < t < e , ( H D α , t − λ ) x ( e ) = , H I − α + x (1 + ) = c , c ∈ R , (2.1) is equivalent to the integral equationx ( t ) = c (ln t ) α − E α,α ( λ (ln t ) α ) + Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) × " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − σ ( τ ) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − σ ( τ ) d ττ dss (2.2)3 roof. Firstly, we apply the Hadamard fractional integral of order β to bothsides of (2.1) and using the result of Lemma 2.4, we get( H D α , t − λ ) x ( t ) = H I β + σ ( t ) + c (ln t ) β − , (2.3)where c ∈ R . Using the boundary condition ( H D α , t − λ ) x ( e ) =
0, we get c = − Γ ( β ) Z e (1 − ln s ) β − σ ( s ) dss . Then we can rewrite equation (2.3) in the form( H D α , t − λ ) x ( t ) = Γ ( β ) Z t (cid:18) ln ts (cid:19) β − σ ( s ) dss − (ln t ) β − Γ ( β ) Z e (1 − ln s ) β − σ ( s ) dss . (2.4)Now, by ( [14], p. 234, (4.1.89)(4.1.95)), we conclude that equation (2.4) withthe boundary condition H I − α + x (1 + ) = c has a solution x ∈ C γ, ln ([1 , e ] , R ) thatsatisfies the volterra integral equation (2.2), which yields the required solution.By direct computation we can prove the converse. This ends the proof. (cid:3) Lemma 2.6. ( [18]) Let α, γ ∈ (0 , , < t < t ≤ e and α − + q > , γ < p . Then Z t (ln s ) − γ (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19) dss ≤ ln( t ) − p γ − p γ ! p (ln t ) q ( α − + − (ln t ) q ( α − + + (cid:16) ln t t (cid:17) q ( α − + q ( α − + , where p + q = , p , q > . Lemma 2.7. (see [11], 4.2.3 and 4.2.7) The Mittag-Le ffl er function E α,β satisfies thefollowing identities:(i) E α,β ( z ) = Γ ( β ) + z E α,β + α ( z ) ,(ii) E , ( z ) = P ∞ k = z k Γ ( k + = e z erfc( − z ) ,where erfc is complementary to the error function erf : erfc( z ) = √ π Z ∞ z e − u du = − erf( z ) , z ∈ C . Lemma 2.8. ( The mean value theorem for integrals ) If f is a continuous functionon the closed, bounded interval [ a , b ] , then there is at least one number ξ in ( a , b ) forwhich f ( ξ ) = b − a Z ba f ( t ) dt . In the following Lemma, we present some useful integrals which are usedfurther in this paper. 4 emma 2.9.
Using the famous beta function B ( m , n ) = R (1 − z ) m − z n − dz, we get ( i ) Z t (cid:18) ln ts (cid:19) β − (ln s ) − γ dss = (ln t ) β − γ B ( β, − γ ) . ( ii ) Z t (cid:18) ln ts (cid:19) β − dss = β (ln t ) β . ( iii ) Z e (1 − ln s ) β − (ln s ) − γ dss = B ( β, − γ ) . ( iv ) Z e (1 − ln s ) β − dss = β . To end this section, we present the Schauder fixed point theorem whichplays a main tool in the existence of the solution of (1.1).
Lemma 2.10. ( The Schauder fixed point theorem [12] ) Let E be a Banach space andQ be a nonempty bounded convex and closed subset of E, and N : Q → Q is a compactand continuous map. Then N has at least one fixed point in Q.
Let B r = n x ∈ C γ, ln ([1 , e ] , R ) : k x k γ, ln ≤ r o with r ≥ ω − L ω , where ω = | c | + L B ( α, + β ) Γ ( β + + L B ( α, β ) Γ ( β + ! E α,α ( λ ) , and ω = B ( β, − γ ) Γ ( β ) (cid:16) B ( α, + β − γ ) + B ( α, β ) (cid:17) E α,α ( λ ) . We define the operator ( H x )( t ) : B r → C γ, ln ([1 , e ] , R ) by( H x )( t ) = c (ln t ) α − E α,α ( λ (ln t ) α ) + Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) × " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ dss (3.1) Theorem 3.1.
Assume that: (H1) f : [1 , e ] × R → R is a continuous function, f ( · , x ( · )) ∈ C γ, ln ([1 , e ] , R ) and there exist constants ≤ L < ω − , L > such that | f ( t , x ) | ≤ L | x | + L , ∀ ( t , x ) ∈ [1 , e ] × R . Then (1.1) has at least one solution on [1 , e ] . roof. The proof will be through several steps.
Step 1.
We show that H ( B r ) ⊂ B r .For any x ∈ C γ, ln ([1 , e ] , R ) and Lemma 2.9, we have (cid:12)(cid:12)(cid:12) (ln t ) γ ( H x )( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) c (ln t ) γ + α − E α,α ( λ (ln t ) α ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (ln t ) γ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | c | (ln t ) γ + α − E α,α ( λ ) + (ln t ) γ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − | f ( τ, x ( τ )) | d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − | f ( τ, x ( τ )) | d ττ dss ≤ | c | E α,α ( λ ) + Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − ( L | x ( τ ) | + L ) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − ( L | x ( τ ) | + L ) d ττ dss | c | E α,α ( λ ) + Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) × " L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ k x k γ, ln d ττ + L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (ln τ ) − γ k x k γ, ln d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − d ττ dss = | c | E α,α ( λ ) + Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) " L k x k γ, ln Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L k x k γ, ln Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − dss ≤ | c | E α,α ( λ ) + L k x k γ, ln Γ ( β ) B ( β, − γ ) E α,α ( λ ) Z t (cid:18) ln ts (cid:19) α − (ln s ) β − γ dss + L Γ ( β + E α,α ( λ ) Z t (cid:18) ln ts (cid:19) α − (ln s ) β dss + L k x k γ, ln Γ ( β ) B ( β, − γ ) + L Γ ( β + ! E α,α ( λ ) Z t (cid:18) ln ts (cid:19) α − (ln s ) β − dss = | c | E α,α ( λ ) + L k x k γ, ln Γ ( β ) B ( β, − γ ) E α,α ( λ )(ln t ) α + β − γ B ( α, + β − γ ) + L Γ ( β + E α,α ( λ )(ln t ) α + β B ( α, + β ) + L k x k γ, ln Γ ( β ) B ( β, − γ ) + L Γ ( β + ! E α,α ( λ )(ln t ) α + β − B ( α, β ) ≤ ω + L ω k x k γ, ln ≤ ω + L ω r ≤ r , which implies that kH x k γ, ln ≤ r , therefore H ( B r ) ⊂ B r . Step 2.
We show that H is continuous. (cid:12)(cid:12)(cid:12) (ln t ) γ (( H x n )( t ) − ( H x )( t )) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (ln t ) γ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − ( f ( τ, x n ( τ )) − f ( τ, x ( τ ))) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − ( f ( τ, x n ( τ )) − f ( τ, x ( τ ))) d ττ dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E α,α ( λ ) Z t (cid:18) ln ts (cid:19) α − " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (cid:12)(cid:12)(cid:12) f ( τ, x n ( τ )) − f ( τ, x ( τ )) (cid:12)(cid:12)(cid:12) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (cid:12)(cid:12)(cid:12) f ( τ, x n ( τ )) − f ( τ, x ( τ )) (cid:12)(cid:12)(cid:12) d ττ dss E α,α ( λ ) Z t (cid:18) ln ts (cid:19) α − " Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ (cid:13)(cid:13)(cid:13) f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (ln τ ) − γ (cid:13)(cid:13)(cid:13) f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln d ττ dss = B ( β, − γ ) Γ ( β ) E α,α ( λ ) "Z t (cid:18) ln ts (cid:19) α − (ln s ) β − γ dss + Z t (cid:18) ln ts (cid:19) α − (ln s ) β − dss f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln = B ( β, − γ ) Γ ( β ) E α,α ( λ ) h (ln t ) α + β − γ B ( α, + β − γ ) + (ln t ) α + β − B ( α, β ) i (cid:13)(cid:13)(cid:13) f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln ≤ B ( β, − γ ) Γ ( β ) (cid:18) B ( α, + β − γ ) + B ( α, β ) (cid:19) E α,α ( λ ) (cid:13)(cid:13)(cid:13) f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln . Hence, we get (cid:13)(cid:13)(cid:13) H x n − H x (cid:13)(cid:13)(cid:13) γ, ln ≤ ω (cid:13)(cid:13)(cid:13) f ( · , x n ( · )) − f ( · , x ( · )) (cid:13)(cid:13)(cid:13) γ, ln , and the continuity of f implies that H is continuous. Step 3.
We show H is relatively compact on B r .According to Step 1 , we showed that H ( B r ) ⊂ B r . Thus H ( B r ) is uniformlybounded. It remains to show that H is equicontinuous.For 1 < t < t ≤ e and x ∈ B r , we have (cid:12)(cid:12)(cid:12) ( H x )( t ) − ( H x )( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) c (ln t ) α − E α,α (cid:16) λ (ln t ) α (cid:17) − c (ln t ) α − E α,α (cid:16) λ (ln t ) α (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) c (ln t ) α − E α,α (cid:16) λ (ln t ) α (cid:17) − c (ln t ) α − E α,α (cid:16) λ (ln t ) α (cid:17)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss − Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c E α,α ( λ (ln t ) α ) (cid:18) (ln t ) α − − (ln t ) α − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c (ln t ) α − (cid:18) E α,α (cid:16) λ (ln t ) α ) − E α,α ( λ (ln t ) α (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss − Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss − Z t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E α,α ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c (cid:18) (ln t ) α − − (ln t ) α − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c (ln t ) α − (cid:18) E α,α (cid:16) λ (ln t ) α ) − E α,α ( λ (ln t ) α (cid:17)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − f ( τ, x ( τ )) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − f ( τ, x ( τ )) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | c | E α,α ( λ ) (cid:18) ( α −
1) (ln ζ ) α − ζ | t − t | (cid:19) + | c | (cid:18) O| t − t | (cid:19) + Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − | f ( τ, x ( τ )) | d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − | f ( τ, x ( τ )) | d ττ (cid:21) dss + Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − | f ( τ, x ( τ )) | d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − | f ( τ, x ( τ )) | d ττ (cid:21) dss + Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − | f ( τ, x ( τ )) | d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − | f ( τ, x ( τ )) | d ττ (cid:21) dss ≤ | c | E α,α ( λ ) (cid:18) ( α −
1) (ln ζ ) α − ζ | t − t | (cid:19) + | c | (cid:18) O| t − t | (cid:19) + Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − ( L | x ( τ ) | + L ) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − ( L | x ( τ ) | + L ) d ττ (cid:21) dss + Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − ( L | x ( τ ) | + L ) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − ( L | x ( τ ) | + L ) d ττ (cid:21) dss + Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − ( L | x ( τ ) | + L ) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − ( L | x ( τ ) | + L ) d ττ (cid:21) dss | c | E α,α ( λ ) (cid:18) ( α −
1) (ln ζ ) α − ζ | t − t | (cid:19) + | c | (cid:18) O| t − t | (cid:19) + Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) × (cid:20) L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ k x k γ, ln d ττ + L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (ln τ ) − γ k x k γ, ln d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − d ττ (cid:21) dss + Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ k x k γ, ln d ττ + L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (ln τ ) − γ k x k γ, ln d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − d ττ (cid:21) dss + Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ k x k γ, ln d ττ + L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (ln τ ) − γ k x k γ, ln d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − d ττ (cid:21) dss = | c | E α,α ( λ ) (cid:18) ( α −
1) (ln ζ ) α − ζ | t − t | (cid:19) + | c | (cid:18) O| t − t | (cid:19) + Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss + Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss + Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss = | c | E α,α ( λ ) (cid:18) ( α −
1) (ln ζ ) α − ζ | t − t | (cid:19) + | c | (cid:18) O| t − t | (cid:19) + I + I + I , where ζ ∈ ( t , t ) and I = Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss , = Z t E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss , I = Z t t (cid:18) ln t s (cid:19) α − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss . For p + q = , p , q >
1, we have the following estimations. I = Z t (cid:18) ln t s (cid:19) α − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19)(cid:20) L r Γ ( β ) (ln s ) β − γ B ( β, − γ ) + L Γ ( β +
1) (ln s ) β + L r Γ ( β ) B ( β, − γ ) + L Γ ( β + ! (ln s ) β − (cid:21) dss = L r Γ ( β ) B ( β, − γ ) (cid:20) Z t (cid:18) ln t s (cid:19) α − (ln s ) β − γ (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) dss + Z t (cid:18) ln t s (cid:19) α − (ln s ) β − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) dss (cid:21) + L Γ ( β + (cid:20) Z t (cid:18) ln t s (cid:19) α − (ln s ) β (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) dss + Z t (cid:18) ln t s (cid:19) α − (ln s ) β − (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) dss (cid:21) ≤ L r Γ ( β ) B ( β, − γ ) (cid:20)(cid:18) Z t (cid:18) ln t s (cid:19) p ( α − (ln s ) p ( β − γ ) dss (cid:19) p (cid:18) Z t (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) q dss (cid:19) q + (cid:18) Z t (cid:18) ln t s (cid:19) p ( α − (ln s ) p ( β − dss (cid:19) p (cid:18) Z t (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) q dss (cid:19) q (cid:21) + L Γ ( β + (cid:20)(cid:18) Z t (cid:18) ln t s (cid:19) p ( α − (ln s ) p β dss (cid:19) p (cid:18) Z t (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) q dss (cid:19) q + (cid:18) Z t (cid:18) ln t s (cid:19) p ( α − (ln s ) p ( β − dss (cid:19) p (cid:18) Z t (cid:18) E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17) − E α,α (cid:16) λ (cid:18) ln t s (cid:19) α (cid:17)(cid:19) q dss (cid:19) q (cid:21) L r Γ ( β ) B ( β, − γ ) (cid:20)(cid:18) B ( p ( α − + , p ( β − γ ) + (cid:19) p (cid:18) Z t O ( | t − t | ) dss (cid:19) q + (cid:18) B ( p ( α − + , p ( β − + (cid:19) p (cid:18) Z t O ( | t − t | ) dss (cid:19) q (cid:21) + L Γ ( β + (cid:20)(cid:18) B ( p ( α − + , p β + (cid:19) p (cid:18) Z t O ( | t − t | ) dss (cid:19) q + (cid:18) B ( p ( α − + , p ( β − + (cid:19) p (cid:18) Z t O ( | t − t | ) dss (cid:19) q (cid:21) , which implies that I → t → t .For I , using Lemma 2.6, we have I ≤ L r Γ ( β ) B ( β, − γ ) E α,α ( λ ) (cid:20)(cid:18) Z t (ln s ) p ( β − γ ) dss (cid:19) p (cid:18) Z t (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19) q dss (cid:19) q + (cid:18) Z t (ln s ) p ( β − dss (cid:19) p (cid:18) Z t (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19) q dss (cid:19) q (cid:21) + L Γ ( β + E α,α ( λ ) (cid:20)(cid:18) Z t (ln s ) p β dss (cid:19) p (cid:18) Z t (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19) q dss (cid:19) q + (cid:18) Z t (ln s ) p ( β − dss (cid:19) p (cid:18) Z t (cid:18) (cid:18) ln t s (cid:19) α − − (cid:18) ln t s (cid:19) α − (cid:19) q dss (cid:19) q (cid:21) ≤ ( L r Γ ( β ) B ( β, − γ ) E α,α ( λ ) (cid:20)(cid:18) (ln t ) p ( β − γ ) + p ( β − γ ) + (cid:19) p + (cid:18) (ln t ) p ( β − + p ( β − + (cid:19) p (cid:21) + L Γ ( β + E α,α ( λ ) (cid:20)(cid:18) (ln t ) p β + p β + (cid:19) p + (cid:18) (ln t ) p ( β − + p ( β − + (cid:19) p (cid:21)) × (cid:18) q ( α − + (cid:18) (ln t ) q ( α − + − (ln t ) q ( α − + + (cid:18) ln t t (cid:19) q ( α − + (cid:19)(cid:19) , which implies that I → t → t .For I , using Lemma 2.8, we have I = L r Γ ( β ) B ( β, − γ ) E α,α ( λ ) (cid:20) (cid:18) ln t ξ (cid:19) α − (ln ξ ) β − γ ξ | t − t | + (cid:18) ln t ξ (cid:19) α − (ln ξ ) β − ξ | t − t | (cid:21) + L Γ ( β + E α,α ( λ ) (cid:20) (cid:18) ln t ξ (cid:19) α − (ln ξ ) β ξ | t − t | + (cid:18) ln t ξ (cid:19) α − (ln ξ ) β − ξ | t − t | (cid:21) , where ξ ∈ ( t , t ). Then I → t → t .Thus we get (cid:12)(cid:12)(cid:12) ( H x )( t ) − ( H x )( t ) (cid:12)(cid:12)(cid:12) → t → t which proves that theoperator H is equicontinuous operator. Hence, by Arzel`a-Ascoli theorem, we13onclude that H is relatively compact on B r . Therefore, the Schauder fixedpoint theorem shows that the operator H has a fixed point which correspondsto the solution of (1.1). This completes the proof. (cid:3) Next, we shall show that the following existence and uniqueness result viaBanach fixed point theorem.
Theorem 3.2.
Assume that: (H2)
Let f : [1 , e ] × R → R be a continuous function. There exists a constantL > such that | f ( t , x ) − f ( t , y ) | ≤ L | x − y | , for each t ∈ [1 , e ] and x , y ∈ R .Then (1.1) has a unique solution on [1 , e ] , provided that L ω < .Proof. As previously proven in
Step 1 of Theorem 3.1, the operator H : B k → B k defined in (3.1) is uniformly bounded. Then it remains to show that H is acontraction mapping.For x , y ∈ B k , where B k = { x ∈ C γ, ln ([1 , e ] , R ) : k x k γ, ln ≤ k } , we have (cid:12)(cid:12)(cid:12) (ln t ) γ (( H x )( t ) − ( H y )( t )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (ln t ) γ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) (cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (cid:16) f ( τ, x ( τ )) − f ( τ, y ( τ )) (cid:17) d ττ − (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (cid:16) f ( τ, x ( τ )) − f ( τ, y ( τ )) (cid:17) d ττ (cid:21) dss (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) (cid:20) Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − (cid:12)(cid:12)(cid:12) ( f ( τ, x ( τ )) − f ( τ, y ( τ )) (cid:12)(cid:12)(cid:12) d ττ + (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − (cid:12)(cid:12)(cid:12) ( f ( τ, x ( τ )) − f ( τ, y ( τ )) (cid:12)(cid:12)(cid:12) d ττ (cid:21) dss ≤ Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) (cid:20) L Γ ( β ) Z s (cid:18) ln s τ (cid:19) β − | x ( τ ) − y ( τ ) | d ττ + L (ln s ) β − Γ ( β ) Z e (1 − ln τ ) β − | x ( τ ) − y ( τ ) | d ττ (cid:21) dss ≤ L Γ ( β ) ( Z t (cid:18) ln ts (cid:19) α − E α,α (cid:18) λ (cid:18) ln ts (cid:19) α (cid:19) (cid:20) Z s (cid:18) ln s τ (cid:19) β − (ln τ ) − γ d ττ + (ln s ) β − Z e (1 − ln τ ) β − (ln τ ) − γ d ττ (cid:21) dss ) k x − y k γ, ln ≤ L B ( β, − γ ) E α,α ( λ ) Γ ( β ) (Z t (cid:18) ln ts (cid:19) α − (ln s ) β − γ dss + Z t (cid:18) ln ts (cid:19) α − (ln s ) β − dss ) k x − y k γ, ln ≤ L B ( β, − γ ) E α,α ( λ ) Γ ( β ) (cid:18) B ( α, + β − γ ) + B ( α, β ) (cid:19) k x − y k γ, ln , kH x − H y k γ, ln ≤ L ω k x − y k γ, ln . It follows that H is acontraction. As a consequence of Banach fixed point theorem, the operator H has a fixed point which corresponds to the unique solution of (1.1). Thiscompletes the proof. (cid:3) Consider the following Langevin equation with two Hadamard fractionalderivatives:
Example 4.1. H D , t ( H D , t − x ( t ) = f ( t , x ( t )) , t ∈ [1 , e ] , ( H D , t − x ( e ) = , H I + x (1 + ) = . (4.1) Here, α = , β = , λ = , γ = and c = . In order to illustrate Theorem 3.1, we take f ( t , x ) = sin | x | + + t ) for all t ∈ [1 , e ] .Clearly, | f ( t , x ) | ≤ | x | + . According to the assumption ( H , we get L = andL = .Thus,L ω = B (cid:16) , (cid:17) Γ (cid:16) (cid:17) (cid:20) B (cid:18) , (cid:19) + B (cid:18) , (cid:19)(cid:21) E , (1) = . E , (1) . Using Lemma 2.7, we get E , (1) = Γ (cid:16) (cid:17) + E , (1) = √ π + erfc(-1) × e = . . Therefore, L ω = . < , and according to Theorem 3.1, we conclude thatthe Langevin equation (4.1) with f ( t , x ) = sin | x | + + t ) has at least one solutionon [1 , e ] .For the illustration of Theorem 3.2, let us take f ( t , x ) = | x | (99 + t )(1 + | x | ) for all t ∈ [1 , e ] .Obviously, | f ( t , x ) − f ( t , y ) | ≤ | x − y | . Thus, the assumption ( H implies thatL = .The direct computations giveL ω = B (cid:16) , (cid:17) Γ (cid:16) (cid:17) (cid:20) B (cid:18) , (cid:19) + B (cid:18) , (cid:19)(cid:21) E , (1) = . E , (1) . Hence L ω = . < . According to Theorem 3.2, the Langevin equation(4.1) with f ( t , x ) = | x | (99 + t )(1 + | x | ) has a unique solution on [1 , e ] . eferences [1] M. I. Abbas, On the Hadamard and Riemann-Liouville fractional neutral func-tional integrodi ff erential equations with finite delay , J. Pseudo-Di ff er. Oper.Appl., (2), (2019) 1–10.[2] B. Ahmad, A. Alsaedi and S.K. Ntouyas, Nonlinear Langevin equations andinclusions involving mixed fractional order derivatives and variable coe ffi cientwith fractional nonlocal-terminal conditions , AIMS Mathematics, 4(3), (2019)626–647.[3] B. Ahmad, A. Alsaedi and S. Salem, On a nonlocal integral boundary valueproblem of nonlinear Langevin equation with di ff erent fractional orders , Adv.Di ff er. Eqns, (2019), 2019:57.[4] A. Berhail, N. Tabouche, M.M. Matar and J. Alzabut, On nonlo-cal integral and derivative boundary value problem of nonlinear HadamardLangevin equation with three di ff erent fractional orders , Bol. Soc. Mat. Mex.,doi.org / / s40590-019-00257-z (2019).[5] P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Compositions of Hadamard-type frac-tional integration operators and the semigroup property , J. Math. Anal. Appl., , (2002) 387–400.[6] P. L. Butzer, A. A. Kilbas, J. J. Trujillo,
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