Existence of the mild solution for nonlocal fractional differential equation of Sobolev type with iterated deviating arguments Alka Chadha∗, D. N. Pandey†, Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India February 19, 2015

Abstract This paper investigates a fractional nonlocal differential equation of Sobolev type with iterated deviating arguments in Banach space. The sufficient condition for providing the existence of mild solution to the nonlocal Sobolev type fractional differential equation with iterated deviating arguments is obtained via techniques of fixed point theorems with analytic semigroup. Finally, an example is provided for explaining the applicability of the abstract results developed. Keywords: Fractional calculus, Caputo derivative, fractional differential Equation, Nonlocal conditions. 2010 Mathematics Subject Classification: 26A33, 34K37, 34K40, 34K45, 35R11, 45J05, 45K05.

1

Introduction:

Recently, the investigation of fractional differential equation has been picking up much attention of researchers. This is due to the fact that fractional differential equations have various applications in engineering and scientific disciplines, for example, fluid dynamics, fractal theory, diffusion in porous media, fractional biological neurons, traffic flow, polymer rheology, neural network modeling, viscoelastic panel in super sonic gas flow, real system characterized by power laws, electrodynamics of complex medium, sandwich system identification, nonlinear oscillation of earthquake, models of population growth, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, nuclear reactors and theory of population dynamics. Also, the fractional differential equation is an important tool to describe the memory and hereditary properties of various materials and phenomena. The details on the theory and its applications may be found in books [1]-[4]and references therein. On the other hand, the nonlocal problem for abstract evolution equations has been studied by many authors. The existence of a solution for abstract Cauchy differential equation with nonlocal conditions in a Banach space has been considered first by Byszewski [5]. In physical science, the ∗ [email protected] † Email

(Alka Chadha) addresses: [email protected] (Dwijendra N Pandey)

1

nonlocal condition might be connected with better effect in applications than the classical initial condition since nonlocal conditions are normally more exact for physical estimations than the classical initial condition. For the study of nonlocal evolution equation, we refer to [5]-[6] and [9, 10, 11, 12, 15, 18] and references given therein. Our main purpose of this paper is to study the following nonlocal fractional differential equation of Sobolev type with iterated deviating arguments in illustrated by c

Dtβ [EBy(t)] y(0)

= Ly(t) + F(t, y(t), y(d1 (t, y(t)))),

t ∈ J = [0, T ],

= y0 + h(y), y0 ∈ X

(1) (2)

where d1 (t, y(t)) = b1 (t, u(b2 (t, · · · , y(bδ (t, y(t))) · · · ))), c Dtβ is the fractional derivative in Caputo derivative of order β, β ∈ (0, 1] and T ∈ (0, ∞). In (1), we assume that the operator L : D(L) ⊂ X → Z, B : D(B) ⊂ X → Y and E : D(E) ⊂ Y → Z are closed operators, where X, Y and Z are the Hilbert spaces such that Z is continuously and densely embedded in X, the state y(·) takes its values in X. Thus function F : [0, T ] × X × X → X is a appropriate function and h is a map from some space of functions satisfying some conditions to be stated later. For more study of Sobolev type differential equations, we refer to papers [8, 9, 10, 11, 12, 13, 14, 15] and reference cited therein. The article is organized as follows: Section 2 presents some basic definitions, lemmas and theorems. Section 3 focuses the existence result of mild to considered system by virtue of the theory of semigroup via fixed point technique. Section 4 considers an application for illustrating the discussed abstract results.

2

Preliminaries

In section, some essential facts about fractional calculus, semigroup theory, theorems and lemmas which will be required to obtain the our result are stated. β Definition 2.1. Let f ∈ L1 ([0, T ], R+ ). The fractional integral (J0,t ) of function f with order β is defined by Z t 1 β (t − s)β−1 f (s)ds, (3) J0,t f (t) = Γ(β) 0

where Γ denotes the classical gamma functions. We can also write J β f (t) = (gβ ∗ f )(t), here 1 β−1 , Γ(β) t

t > 0,

0,

t ≤ 0.

( gβ (t) :=

(4)

The notation ∗ stands the convolution of functions and limβ→0 gβ (t) = δ(t), δ means delta Dirac function. Definition 2.2. The Riemann-Liouville fractional derivative is given by Z t dn 1 RL β (t − s)n−β−1 f (s)ds, t > 0, β > 0, n − 1 < β < n, n ∈ N, D0,t f (t) = Γ(n − β) dtn 0 and f ∈ C n−1 ([0, a], X).

2

(5)

Definition 2.3. The Caputo fractional derivative is given by Z t 1 c β (t − s)n−β−1 f n (s)ds, n − 1 < α < n, D0,t f (t) = Γ(n − β) 0

(6)

where f ∈ C n−1 ((0, T ); X) ∩ L1 ((0, T ); X). Throughout the paper, we assume that (X, k·k, < ·, · >), (Y, k·k, < ·, · >) and (Z, k·k, < ·, · >) are separable Hilbert spaces, C([0, T ], X) represent the Banach space of continuous functions from [0, T ] into X with the norm kyk[0,T ] = sup{ky(t)k : t ∈ [0, T ]}. Let L(X) be the Banach space of bounded linear operators from X into X endowed with the norm kf kL(X) = sup{kf (y)k : kyk = 1}. Now, we impose following data on operators L and E and B: (C1) E : D(E) ⊂ Y → Z and B : D(B) ⊂ X → Y are linear operators and L : D(L) ⊂ X → Z is closed. (C2) D(B) ⊂ D(L), Im(B) ⊂ D(E) and E, B are bijective operators. (C3) The operators E−1 : Z → D(E) ⊂ Y and B−1 : Y → D(B) ⊂ X are assumed to be linear, bounded and compacts operators. By the hypothesis (C3), it follows that B−1 E−1 is closed and injective. Thus, its inverse is also closed i.e., EB is closed. By the hypothesis (C1) − (C3) and closed graph theorem, we conclude that the boundedness of the linear operator LB−1 E−1 . Therefore, LB−1 E−1 is the infinitesimal −1 −1 generator of a semigroup {S(t), t ≥ 0}, S(t) := e−LB E . Without loss of generality, we may assume that N0 := supt≥0 kS(t)k < ∞ and W1 = kE−1 k, W2 = kB−1 k. According to previous definitions, the system (1)-(2) is equivalent to the following integral equation Z t (t − s)β−1 [Ly(s) + F(s, y(s), y(h1 (s, y(s))))]ds, (7) [EB]y(t) = [EB]y(0) + Γ(β) 0 provided the integral in (7) exists for a.e. t ∈ [0, T ]. In this work, A = LB−1 E−1 : D(A) ⊂ Z → Z is the infinitesimal generator of a compact analytic semigroup of uniformly bounded linear operators S(·). Thus, it follows that there exists a positive constant N0 ≥ 1 such that kS(t)k ≤ N0 for each t ≥ 0. We assume that 0 ∈ ρ(A), ρ(A) means resolvent set of A. Therefore, we may define the fractional power Aα for α ∈ (0, 1] as closed linear operator with domain D(Aα ) with inverse A−α . Moreover, the subspace D(Aα ) is dense in X with the norm kykα = kAα yk for y ∈ D(Aα ). Thus, it is not difficult to show that D(Aα ) is a Banach space with supremum norm. Hence, we signify the space D(Aα ) by Xα endowed with the α-norm (k · kα ). We also have that for Xη ,→ Xα for 0 < α < η and therefore, the embedding mapping is continuous. Then, we may define X−α = (Xα )∗ for each α > 0, dual space of Xα , is a Banach space endowed with the norm kyk−α = kA−α yk for y ∈ X−α . For more details on the fractional powers of closed linear operators, we refer to book by Pazy [19]. Now, we present the following lemma follows from the results [16],[18] which will be used to establish the required result. Lemma 2.1. Let us assume that A is the infinitesimal generator of an analytic semigroup S(t), t ≥ 0 and 0 ∈ ρ(A). Then, (i) S(t) : X → D(Aα ) for every t > 0 and α ≥ 0. (ii) S(t)Aα y = Aα S(t)y for each y ∈ D(Aα ). (iii) For each t > 0, dj k j S(t)k ≤ Nj , j = 1, 2, (8) dt 3

where Nj , j = 1, 2 are some positive constants. (iv) The operator Aα S(t) is bounded and kAα S(t)k ≤ Nα t−α e−δt for each t > 0. (v) For each α ∈ (0, 1] and y ∈ D(Aα ), then kS(t)y − yk ≤ Cα tα kAα yk. Remark 2.1. [15] The operator A−α is a bounded linear operator in X such that D(Aα ) = Im(A−α ). We denote by C α = C([0, T ], Xα ) Banach space of all Xα -valued continuous functions on [0, T ] endowed with the supremum norm kykC α = supt∈[0,T ] ky(t)kα for each y ∈ C α . Now, we consider the space C α−1 = {y ∈ C α : ky(t) − y(s)k ≤ L|t − s|, ∀ t, s ∈ [0, T ]}.

(9)

Motivated by the paper [15], we present the following definition of mild solution of system (1)-(2). Definition 2.4. A continuous function y : [0, T ] → Xα is said to be a mild solution of system (1)-(2) if y(0) = y0 + h(y) and following integral equation Z y(t) = Uβ (t)[EB]h(y) +

t

(t − s)β−1 Vβ (t − s)F(s, y(s), y(d1 (s, y(s))))ds, t ∈ [0, T ],

(10)

0

is satisfied by y(·), where Z ∞ Uβ (t) = B−1 E−1 ξβ (ζ)S(tβ ζ)dζ, 0 Z ∞ Vβ (t) = βB−1 E−1 ζξβ (ζ)S(tβ ζ)dζ, 0

ξβ (ζ)

=

∞ −1 1X Γ(nβ + 1) 1 −1− β1 ζ (−1)n−1 ζ −βn−1 sin(nπβ), ζ ∈ (0, ∞), ψβ (ζ β ) ≥ 0, ψβ (ζ) = β π n=1 n!

and R ∞ ξβ (ζ) the probability density function defined on (0, ∞), i.e., ξβ (ζ) ≥ 0, ζ ∈ (0, ∞) with ξβ (ζ)dζ = 1. 0 Remark 2.2. [18]. For each ν ∈ [0, 1] Z ∞ Z ν ζ ξβ (ζ)dζ = 0

∞

ζ −βν ψβ (ζ)dζ =

0

Γ(1 + ν) . Γ(1 + βν)

(11)

Lemma 2.2. Let A be the infinitesimal generator of a semigroup of uniformly bounded linear operators S(t), t ≥ 0. Then, the operator Uβ (t) and Vβ (t), t ≥ 0 are bounded linear operator such that W2 N 0 (1) We have kUβ (t)yk ≤ W1 W2 N0 kyk and kVβ (t)yk ≤ W1Γ(β) kyk for each y ∈ X. (2) The families {Uβ (t), t ≥ 0} and {Vβ (t), t ≥ 0} are strongly continuous i.e., for 0 ≤ τ1 < τ2 ≤ T and y ∈ X, we have kUβ (τ2 )y − Uβ (τ1 )yk → 0 and kVβ (τ2 )y − Vβ (τ1 )yk → 0 as τ2 → τ1 . (3) If S(t), t ≥ 0 is compact, then Uβ (t) and Vβ (t), t ≥ 0 are compact operator. (4) For each y ∈ X, η ∈ (0, 1) and α ∈ (0, 1), we have AVβ (t)y = A1−η Vβ Aη y for t ∈ [0, T ]. We 1 W2 Nα Γ(2−α) −αβ also have kAα Vβ (t)k ≤ βWΓ(1+β(1−α)) t for each t ∈ (0, T ]. (5) For any y ∈ Xα and fixed t ≥ 0, we have kUβ (t)ykα ≤ W1 W2 N0 kykα and kVβ (t)ykα ≤ W1 W2 N 0 Γ(β) kykα .

4

Lemma 2.3. For each ϕ ∈ Lp ([0, T ], X) and 1 ≤ p < ∞, Z T kϕ(t + ) − ϕ(t)kp dt = 0, lim →0

(12)

0

where ϕ(s) = 0 for s ∈ / [0, T ]. Lemma 2.4. A measurable function ϕ : [0, T ] → X is Bochner integral if kϕk is Lebesgue integrable.

3

Main Result

In this segment, the sufficient condition for the existence of α-mild solution for system (1)-(2) is derived. To prove the result, we impose following assumptions on the data of the system (1)-(2). (J1) The function F : [0, T ] × Xα × Xα−1 → X is a H¨ older continuous and there exist constants LF > 0 and θ1 ∈ (0, 1] such that k F (t, z1 , w1 ) − F (s, z2 , w2 )k ≤ LF (|t − s|θ1 + k z1 − z2 kα + k w1 − w2 kα−1 ),

(13)

for each (t, z1 , w1 ), (s, z2 , w2 ) ∈ [0, T ] × Xα × Xα−1 . (J2) The functions bi : [0, ∞) × Xα−1 → [0, ∞), (i = 1, · · · , δ) are continuous functions and there are positive constants Lbi and 0 < θ2 ≤ 1 such that |bi (t, z) − bi (s, w)| ≤ Lbi (|t − s|θ2 + k z − wkα−1 ),

(14)

for all (t, z), (s, w) ∈ [0, T ] × Xα−1 . (J3) g : Xα → Xα is continuous function and there exists positive constant Lg such that kg(w1 ) − g(w2 )kα ≤ Lg kw1 − w2 kα ,

(15)

for each w1 , w2 ∈ Xα . Now, we consider the space S α = {y ∈ C α ∩ C α−1 : kykα ≤ R},

(16)

where R is a positive constant to be defined later. Theorem 3.1. If the assumptions (J1) − (J3) are fulfilled and K ∗ = W1 W2 N0 kEBkLh +

W1 W2 Nα Γ(2 − α) T β(1−α) LF (2 + LLb ) < 1. Γ(1 + β(1 − α)) β(1 − α)

(17)

Then, there exists at least one α-mild solution for system (1)-(2). Proof. Firstly, we consider the operator Υ : C α ∩ C α−1 → C α ∩ C α−1 defined by Z t (Υy)(t) = Uβ (t)[EB](y0 +h(y))+ (t−s)β−1 Vβ (t−s)F(s, y(s), y(d1 (s, y(s))))ds, t ∈ [0, T ]. (18) 0

Clearly, it is easy to show that Υ : C α → C α by using the fact that F and bi are continuous function. Now, it remains to show that Υy ∈ C α−1 . To this end, let τ1 , τ2 ∈ [0, T ] with τ1 < τ2 . 5

Then, we get k(Υy)(τ2 ) − (Υy)(τ1 )kα−1 Z τ2 k(Uβ (τ2 ) − Uβ (τ1 ))[EB](y0 + h(y))kα−1 + k (τ2 − s)β−1 Vβ (τ2 − s)F(s, y(s), y(d1 (s, y(s))))ds 0 Z τ1 (τ1 − s)β−1 Vβ (τ1 − s)F(s, y(s), y(d1 (s, y(s))))dskα−1 , − 0 Z τ2 ≤ k(Uβ (τ2 ) − Uβ (τ1 ))[EB]h(y)kα−1 + k (τ2 − s)β−1 Vβ (τ2 − s)F(s, y(s), y(d1 (s, y(s))))dskα−1 τ1 Z τ1 (τ2 − s)β−1 Vβ (τ2 − s)F(s, y(s), y(d1 (s, y(s))))ds +k 0 Z τ1 − (τ1 − s)β−1 Vβ (τ1 − s)F(s, y(s), y(d1 (s, y(s))))dskα−1 (19)

≤

0

From the first term of above inequality, we have Z ∞ B−1 E−1 ξβ (ζ)[S(τ2β ζ) − S(τ1β ζ)]Aα−1 ([EB]h(y))dζ. [Uβ (τ2 ) − Uβ (τ1 )]Aα−1 [EB](y0 + h(y)) = 0

Also, we have that for each x ∈ E [S(τ2β ζ)

−

S(τ1β ζ)]x

Z

τ2

= τ1

d S(sβ ζ)xds = ds

Z

τ2

βζsβ−1 AS(sβ ζ)xds.

τ1

Therefore, R ∞ −1 −1we estimateβ the first βterm asα−1 B E ξβ (ζ)kS(τ2 ζ) − S(τ1 ζ)kk A [EB](y0 + h(y))kdζ 0 Z

∞ −1

≤

ξβ (ζ)kB 0

Z

−1

E

Z

τ2

k

k[ τ1

d S(sβ ζ)kds]k[EB]kk y0 + h(y)kα−1 dζ, ds

∞

≤

ξβ (ζ)W1 W2 [N1 (τ2 − τ1 )]k[EB]kky0 + h(y)kα−1 dζ, Z ∞ ≤ K1 (τ2 − τ1 ) ξβ (ζ)dζ, 0

0

= K1 (τ2 − τ1 ),

(20)

where K1 = W1 W2 N1 k[EB]kky0 + h(y)kα−1 . The second term can be estimated Z τ1 k (τ1 − s)β−1 Vβ (τ1 − s) − (τ2 − s)β−1 Vβ (τ2 − s)kα−1 kF(s, y(s), y(d1 (s, y(s))))kds 0 Z τ1 Z ∞ d d −1 −1 ≤ kB E k ξβ (ζ)k [ S((ς − s)β ζ)|ς=τ2 − S((ς − s)β ζ)|ς=τ1 ]Aα−2 k dς dς 0 0 ×kF(s, y(s), y(d1 (s, y(s))))kdζds, Z τ1 Z ∞ Z τ2 d2 ≤ W1 W2 ξβ (ζ)[ k Aα−2 2 S((ς − s)β ζ)kdς]NF dζds, dς 0 0 τ1 Z τ1 Z ∞ ≤ W1 W2 ξβ (ζ)[k Aα−2 kN2 (t − τ )]NF dζds, 0

0

≤ K2 (τ2 − τ1 ),

(21)

6

where K2 = W1 W2 k Aα−2 kN2 NF T . The sixth integrals is estimated as Z τ2 k(τ2 − s)β−1 Vβ (τ2 − s)kα−1 kF(s, y(s), y(d1 (s, y(s))))kds τ1 Z τ2 Z ∞ ξβ (ζ)kB−1 E−1 kk [β(τ2 − s)β−1 ζAS((τ2 − s)β ζ)]Aα−2 k ≤ 0

τ1

×k F(s, y(s), y(d1 (s, y(s))))kdζds, Z τ2 Z ∞ d ξβ (ζ)k S((ς − s)β ζ)|ς=τ2 Aα−2 kNF dζds, ≤ W1 W2 dς 0 τ1 ≤ K3 (τ2 − τ1 ),

(22)

where K3 = N1 W1 W2 k Aα−2 kNF . Thus, from the inequality (19) to (22), we obtain that k (Υy)(τ2 ) − (Υy)(τ1 )kα−1 ≤ L(τ2 − τ1 ),

(23)

for a positive suitable constant L = min{Kl }, l = 1, 2, 3. Therefore, we conclude that (Υy) ∈ CTα−1 . Hence, we deduce that the operator Υ : CTα−1 → CTα−1 is well defined map. Next, we prove that Υ : BR → BR . For 0 ≤ t ≤ T and y ∈ BR , we get that k (Υy)(t)kα Z ≤

k Uβ (t)[EB](y0 + h(y))kα +

t

k (t − s)β−1 Vβ (t − s)F(s, y(s), y(d1 (s, y(s))))kα ds,

0

≤ W1 W2 N0 kEBk · ky0 + h(y)kα + ≤

W1 W2 Nα NF Γ(2 − α)T β(1−α) , (1 − α)Γ(1 + β(1 − α))

R.

(24)

Therefore, we conclude that Υ(BR ) ⊂ BR . Finally, we will show that Υ is a contraction map. For y ∗ , y ∗∗ ∈ BR and 0 ≤ t ≤ T , we get that k(Υy ∗ )(t) − (Υy ∗∗ )(t)kα ≤

∗

∗∗

Z

kUβ (t)[EB][h(y ) − h(y )]kα + k

t

(t − s)β−1 Vβ (t − s)[F(s, y ∗ (s), y ∗ (d1 (s, y ∗ (s))))

0

−F(s, y ∗∗ (s), y ∗∗ (d1 (s, y ∗∗ (s))))]dskα , Z βW1 W2 Nα Γ(2 − α) t (t − s)β(1−α)−1 Γ(1 + β(1 − α)) 0 ×kF(s, y ∗ (s), y ∗ (d1 (s, y ∗ (s)))) − F(s, y ∗∗ (s), y ∗∗ (d1 (s, y ∗∗ (s))))kds.

≤ W1 W2 N0 kEBkLh ky ∗ − y ∗∗ kα +

(25)

Now, we estimate k F(t, y ∗ (t), y ∗ (d1 (t, y ∗ (t)))) − F(t, y ∗∗ (t), y ∗∗ (d1 (t, y ∗∗ (t))))k ≤ LF [ky ∗ (t) − y ∗∗ (t)kα + ky ∗ (d1 (t, y ∗ (t))) − y ∗∗ (d1 (t, y ∗∗ (t)))kα−1 ] ≤ LF [ky ∗ (t) − y ∗∗ (t)kα + kA−1 k · ky ∗ (d1 (t, y ∗∗ (t))) − y ∗∗ (d1 (t, y ∗∗ (t)))kα +ky ∗ (d1 (t, y ∗ (t))) − y ∗ (d1 (t, y ∗∗ (t)))kα−1 ], Let dj (t, y(t)) = bj (t, y(bj+1 (t, · · · y(t, bm (t, y(t))) · · · ))), j = 1, 2, · · · , δ, y ∈ S α ,

7

(26)

with dm+1 (t, y(t)) = t [[21] p. 2183]. Thus, we obtain |d1 (t, y ∗ (t)) − d1 (t, y ∗∗ (t))| = |b1 (t, y ∗ (d2 (t, y ∗ (t)))) − b1 (t, y ∗∗ (d2 (t, y ∗∗ (t))))|, ≤ Lb1 ky ∗ (d2 (t, y ∗ (t))) − y ∗∗ (d2 (t, y ∗∗ (t)))kα−1 , ≤ Lb1 [ky ∗ (d2 (t, y ∗ (t))) − y ∗ (d2 (t, y ∗∗ (t)))kα−1 +ky ∗ (d2 (t, y ∗∗ (t))) − y ∗∗ (d2 (t, y ∗∗ (t)))kα−1 ], ≤ Lb1 [L|b2 (t, y ∗ (d3 (t, y ∗ (t))) − b2 (t, y ∗∗ (d3 (t, y ∗∗ (t)))| +kAk−1 ky ∗ − y ∗∗ kC α ], ··· ≤ [Lδ−1 Lb1 · · · Lbδ + Lδ−2 Lb1 · · · Lbδ−1 + · · · + LLb1 Lb2 +Lb1 ]kAk−1 ky ∗ − y ∗∗ kC α ,

(27)

Therefore, we get k F(t, y ∗ (t), y ∗ (d1 (t, y ∗ (t)))) − F(t, y ∗∗ (t), y ∗∗ (d1 (t, y ∗∗ (t))))k ≤ LF (2 + LLb kAk−1 )ky ∗ − y ∗∗ kC α , ≤ LF (2 + LLb )ky ∗ − y ∗∗ kC α .

(28)

where Lb = [Lδ−1 Lb1 · · · Lbδ + Lδ−2 Lb1 · · · Lbδ−1 + · · · + LLb1 Lb2 + Lb1 ] > 0. Thus, from the inequalities (25) and (28), we obtain k(Υy ∗ )(t) − (Υy ∗∗ )(t)kα ≤ [W1 W2 N0 kEBkLh + =

βW1 W2 Nα Γ(2 − α) T β(1−α) LF (2 + LLb ) ]ky ∗ − y ∗∗ kC α , Γ(1 + β(1 − α)) β(1 − α)

K ∗ ky ∗ − y ∗∗ kC α .

(29) β(1−α)

1 W2 Nα Γ(2−α) Since K ∗ = W1 W2 N0 kEBkLh + WΓ(1+β(1−α)) LF (2 + LLb ) T(1−α) < 1. It implies that Υ is a contraction mapping on S α with constant K ∗ < 1. Therefore, there exists a fixed point of the mapping Υ by Banach fixed point theorem which is just a mild solution for the problem (1)-(2).

4

Example

In this section, we present an example to show the effectiveness of the discussed abstract result. We consider the following nonlocal differential problem illustrated by c

Dtβ [wxx (t, x) − wxxxx (t, x)] +

w(0, x) =

n X

Cs w(ts , x),

∂ 2 w(t, x) e e x, w(t, x)), = H(x, w(t, x)) + G(t, ∂x2 x ∈ S, t ∈ [0, T ],

x ∈ [0, π],

(30) (31)

s=1

w(t, 0) = w(t, π) = 0,

0 < t ≤ 1,

(32)

where c Dtβ denotes the fractional derivative in Caputo sense of order β ∈ (0, 1], Cs > 0 are constants for s = 1, · · · , n.

8

Take X = Y = Z = L2 [0, π] and S = [0, π]. Let us consider the operator E, B, L on domains and ranges which contained in L2 [0, π] defined by Ey = y 00 ,

By = y − y 00 (EBy = y 00 − y 0000 ),

Ly = −y 00 ,

(33)

and M the identity with domain D(E), D(B), D(L) which are given by {y ∈ X : y, y 0 , y 00 , y 0000 are absolutely continuous, y 0000 ∈ X, y(0) = y(π) = 0}.

(34)

Thus, the operators have the following expression Ey

=

∞ p X

1 + n2 (y, yn )yn , By =

n=1

∞ p X 1 + n2 (y, yn )yn ,

(35)

n=1

p P∞ and Ly = n=1 (−n2 )(y, yn )yn with yn (t) = ( 2/π) sin(nt), n = 1, · · · , as the orthogonal set of eigenfunctions of L. Moreover, we have ∞ X

B−1 E−1 y

=

1 (y, yn )yn , 1 + n2 n=1

(36)

LB−1 E−1 y

=

∞ X −n2 (y, yn )yn , 1 + n2 n=1

(37)

S(t)x

=

∞ X

exp(

n=1

−n2 t )(y, yn )yn . 1 + n2

(38)

Clearly, the operator B−1 E−1 is bounded and compact such that kB−1 E−1 k ≤ 1. It is also well known that A = LB−1 E−1 is the infinitesimal generator of the strongly continuous semigroup S(t) on L2 [0, π] with kS(t)k e−t ≤ 1. P≤ n Let w(t) = w(t, ·), h(w) = s=1 Cs w(ts , x). Now, we define the function F : R+ × X × X → X as e e x, ξ), for x ∈ (0, π), F(t, ξ, ζ)(x) = H(x, ζ) + G(t,

(39)

e : [0, 1] × X → X is defined as where H Z

x

K(x, y)ζ(y)dy,

e H(x, ζ) =

(40)

0

e : R+ × [0, 1] × X → X satisfies following condition and G e x, ξ)k ≤ W (x, t)(1 + kξk1/2 ), kG(t,

(41)

where Q is continuous in t and Q(·, t) ∈ X. Now, from the definition of F and h, it can be easily shown that F and h satisfy the assumption (J1)-(J3). Applying the result of the Theorem 3.1, we can get that there exists a mild solution for the problem (30)-(32).

Acknowledgment The authors would like to thank the referee for valuable comments and suggestions. The work of the first author is supported by the University Grants Commission (UGC), Government of India, New Delhi and Indian Institute of Technology, Roorkee. 9

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