Long time behavior for a nonlinear fractional model

J. Math. Anal. Appl. 332 (2007) 441–454 www.elsevier.com/locate/jmaa

Long time behavior for a nonlinear fractional model ✩ Khaled M. Furati, Nasser-eddine Tatar ∗ King Fahd University of Petroleum and Minerals, Department of Mathematical Sciences, Dhahran 31261, Saudi Arabia Received 15 November 2005 Available online 15 November 2006 Submitted by I. Podlubny

Abstract The asymptotic behavior for solutions of a weighted Cauchy-type nonlinear fractional problem is investigated. We find bounds for solutions on infinite time intervals and also provide sufficient conditions assuring decay to zero. This work improves earlier results by the same authors. © 2006 Elsevier Inc. All rights reserved. Keywords: Fractional differential equation; Riemann–Liouville integral; Singular kernel; Weighted Cauchy-type problem

1. Introduction We shall be concerned by the following fractional differential problem ⎧ t ⎪ ⎪   ⎪ α ⎨ D u(t) = f (t, u) + g t, s, u(s) ds, t > 0, ⎪ ⎪ ⎪ ⎩

t 1−α u(t) t=0 = b,

0

(1)

where 0 < α < 1, b ∈ R, f and g are continuous functions. D α denotes the fractional derivative of order α in the sense of Riemann–Liouville. This problem has been studied by the present authors in [4,6]. In [4] an existence result was proved and in [6] bounds for solutions were found on arbitrarily large intervals assuring global existence. Existence and uniqueness results ✩

The authors are very grateful for the financial support provided by King Fahd University of Petroleum and Minerals.

* Corresponding author.

E-mail addresses: [email protected] (K.M. Furati), [email protected] (N.-e. Tatar). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.10.027

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K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454

have been proved in the special case g ≡ 0 by several authors (see references in [10,12]) in the Lipschitzian case and the non-Lipschitzian case of f (t, u). See also [9] for references concerning the problem with the initial condition (I 1−α u)(0) = b (where the operator I is defined below) and the linear case. For a long time researchers have been expressing the concepts in Newtonian terms just for convenience. But, in the last three decades, it has been shown that in many fields, phenomena with estrange kinetics cannot be described within the framework of classical theory using integer order derivatives. For a high accuracy, fractional derivatives are then used to describe the dynamics of some structures. It is also recognized that they have some other attractive features. They are also used to describe many anomalous phenomena. More precisely, experimental research has shown that a better agreement of the experimental data could be achieved with fractional models than the usual models. Nowadays, we can find applications and models involving fractional derivatives in Probability, Physics, Astrophysics, Chemical physics, Anomalous diffusion, Seismic analysis, Finance, Optic and signal processing, Robust control, Electromagnetism, Biology, Viscoelasticity, Acoustics, . . . (see for instance references in [11,15,20,22,24,26]). In our case, problem (1) may describe a grey Brownian motion (see [11,12,21] and references therein). As a result we are witnessing a growing interest in developing the theory of fractional calculus. The integral term in (1) accounts for the memory of the system. It arises in mathematical models which reflect the “after-effect” of a system. It is a nonlocal (in time) term usually called “Volterra” operator (or “causal” operator in the engineering literature). Nonlocal terms stem from the observation that they are more realistic than local terms. Many phenomena cannot be found by ordinary differential equations and are observed in specific examples of Volterra type. Indeed, it is clear that many phenomena (in Physics, Mechanics, Biology, Medicine, Economics, Control theory, Information theory, etc.) exhibit an hereditary behavior (see [7,15,23]). The evolution of the process depends on the whole past history the system has undergone and not only on the present state. This memory term can also be regarded as a distributed delay. One can find many books and many research papers on delays where their importance in applications (Heat conduction in materials with memory, Electrodynamics, Biology, Ecology, Physiology, Epidemiology, . . .) is well established (see for instance [15]). Therefore, we are lead to an integrodifferential equation whenever we would like to take into account an hereditary phenomena. This integrodifferential equation (with α = 1, in our case) is said to be of Volterra type. Although the theory of Volterra integrodifferential equations has undergone rapid developments during the last four decades, it remains wide open for further progress. It is important to note that, in general, Volterra integrodifferential equations cannot be reduced to finite-dimensional ordinary differential equations and their properties could be very different from those in ordinary differential equations. Moreover, many of the classical methods (as the Lyapunov method) fail to give satisfactory results. Most of the existing results on well-posedness are concerned about the linear case and for smooth and bounded kernels. The existence part is proved usually only locally or on a finite time. Indeed, linear Volterra integrodifferential equations (f and g linear, or f (t, u) ≡ f (t)) have been extensively studied during the last four decades (see [2,3,7,8,15,23], to cite but a few). Existence and uniqueness of classical solutions and absolutely continuous solutions are proved for regular kernels. Very often, when the asymptotic behavior is investigated, the rate of decay is not determined. We can, however, cite the problem treated in [1, Chapter 5] where an exponential decay result has been proved in the linear case when the kernel itself is exponentially decaying to zero.

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Although the linear case is an acceptable first approximation, for the sake of accuracy one is lead to consider nonlinear terms in the dynamic of the process. In case f is nonlinear, g is linear and the linear part, in some sense, dominates the nonlinear part then the nonlinear term is considered as a small perturbation (see [7, Chapter 11]). Conversely, sometimes the nonlinear memory term is considered as a perturbation of the (nonlinear) remaining part of the equation. Well-posedness, stability and convergence to zero, for problem (1) with the classical initial data u(0) = u0 , have been discussed under certain growth conditions (such as the Nagumo type condition) on the nonlinearity but for regular kernels and without any explicit decay rate (see [15, Chapters 4–6]) (see also [2]). The global existence and uniqueness for the problem





t

u (t) = h t, u,

k(t, s)u(s) ds 0

under a generalized Lipschitz-type condition on h and for a continuous kernel k(t, s) on Δ = {(t, s); 0  s  t < T  ∞} is proved in [3, Chapter 3]. Well-posedness has been proved in [8, Chapter 3.1] for a similar problem using Darbo’s fixed point theorem for a smooth kernel and under some growth condition on the nonlinearity involving the Kuratowski measure of noncompactness. The global existence of classical solution is established under the additional assumption that the kernel be summable over (0, ∞) and its L1 -norm be small enough (see [8, Chapter 3.3]). We mention here that problem (1) with f (t, u) ≡ f (t) and a classical initial data has been considered in [7, p. 360] and a local existence result was proved there for an L1loc kernel. In fact p a similar result exists in the literature for Lloc kernels. In the present problem our kernel is not necessarily smooth. It could be singular at 0 and it is not summable (take, for instance, equality in (3) and (4)). It is worth noting here, as it is mentioned in literature, that it was precisely these singular kernels that were the first in order of time to arise (even in the integral equation case). We can cite the generalization of the tautochronous curve considered by Abel (see [26]). It becomes important to have a better insight and understanding of the solutions of problem (1). Consequently, there is every reason to analyse carefully problems of Volterra type and to pay attention to the specific peculiarities of the memory effect. Finally, we point out that the question of geometric and physical interpretation of the fractional integration and differentiation remained open for more than 300 years (since 1695). We refer the reader to Podlubny [25] for a suggestion in this sense. In this work we determine sufficient conditions assuring global existence of solutions and also prove some results on decay of power type for the solutions. This improves earlier results by the authors (see [6]). To this end we combine some well-known inequalities with a technique of desingularization due to Medved [17,18], Kirane and Tatar [13,14], Tatar [27] and Mazouzi and Tatar [16]. The rest of the paper is divided into two sections. In the first section we prepare some material needed to establish our results. The second section contains the statements and proofs of our main results concerning boundedness and asymptotic behavior.

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2. Preliminaries In this section we present some definitions, lemmas and notation which will be used in our theorems. Definition 1. The Riemann–Liouville fractional integral of order α > 0 of a Lebesguemeasurable function f : R+ → R is defined by (the Abel-integral operator) 1 I f (t) = Γ (α)

t (t − s)α−1 f (s) ds

α

0

provided that the integral exists. Definition 2. The fractional derivative (in the sense of Riemann–Liouville) of order 0 < α < 1 of a continuous function f : R+ → R is defined as the left inverse of the fractional integral of f d  1−α  I f (t). D α f (t) = dt That is t d 1 α D f (t) = (t − s)−α f (s) ds, Γ (1 − α) dt 0

provided that the right-hand side exists. See [20,22,24,26] for more on fractional integrals and fractional derivatives. For h > 0, we define the space

    Cr0 [0, h] := v ∈ C 0 (0, h] : lim t r v(t) exists and is finite . t→0+

Here is the usual space of continuous functions on [0, h]. The space Cr0 ([0, h]) endowed with the norm vr := max t r v(t) C 0 ((0, h])

0th

is a Banach space. Then, we define the space        α 0 0 C1−α [0, h] : = v ∈ C1−α [0, h] : there exist c ∈ R and v ∗ ∈ C1−α [0, h]  such that v(t) = ct α−1 + I α v ∗ (t) . α ([0, h]) endowed with the norm v If α > 1/2, then the space C1−α 1−α,α := v1−α + α D v1−α is a Banach space. To problem (1) we associate the integral equation 

t s     1 α−1 α−1 + (t − s) (2) u(t) = bt f s, u(s) + g s, z, u(z) dz ds. Γ (α) 0

0

The differential problem (1) and its associated integral equation (3) are equivalent in the space α ([0, h]) (see [5,6] for the proof). C1−α We will also need the following lemmas.

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445

Lemma 1. For all α > 0 and μ > −1, we have t (t − s)α−1 s μ ds = Dt α+μ ,

t  0,

0

where D = D(α − 1, μ) =

Γ (α)Γ (μ+1) Γ (α+μ+1) .

Lemma 2. If λ, ν, ω > 0, then for any t > 0 we have t t

1−ν

(t − s)ν−1 s λ−1 e−ωs ds  C,

0

where C = C(ν − 1, λ − 1, ω) is a positive constant independent of t. In fact,   C = max 1, 21−ν Γ (λ)(1 + λ/ν)ω−λ . For the proof of this lemma we refer to [16] or [19]. Lemma 3. For a, b, c  0 and γ > 0, we have   (a + b + c)γ  3γ −1 a γ + bγ + cγ . We will assume the following hypotheses on the nonlinearities f and g. 0 (R+ ) and (F) t 1−α f (t, u) is continuous on R+ × C1−α f (t, u)  t μ ϕ(t)|u|m1 , μ  0, m1 > 1.

(3)

0 (R+ ) where (G) s 1−α g(t, s, u(s)) is continuous on DR+ × C1−α   DR+ = (t, s) ∈ R+ × R+ , 0  s  t ,

and

  g t, s, u(s)  (t − s)β−1 s σ ψ(s)|u|m2 ,

0 < β < 1, σ  0, m2 > 1,

(4)

where ϕ(t) and ψ(s) are such that: (Φ) ϕ(t) is continuous and t μ−(1−α)m1 ϕ(t) is continuous in case μ − (1 − α)m1 < 0. (Ψ ) ψ(t) is continuous and t σ −(1−α)m2 ψ(t) is continuous in case σ − (1 − α)m2 < 0. Under these assumptions there exists at least one local solution to problem (1) in the space α ([0, h]). C1−α Theorem 1. (See [4].) Assume that the above hypotheses (F), (G), (Φ) and (Ψ ) on f , g, ϕ and ψ hold on [0, h], h > 0. Suppose further that 1 + μ − (1 − α)m1 > 0 and 1 + σ − (1 − α)m2 > 0. ˜ with h˜ < h. Then problem (1) admits at least one solution on a sufficiently small interval [0, h]

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3. Global existence and polynomial decay In this section we state and prove our main results on the global existence and behavior of solutions to problem (1) as time goes to infinity. In particular we show that solutions decay to zero at the rate of a polynomial. Let q = α + 1, p = β + 1 and q and p their conjugate exponents, respectively. For some 0 < δ < 1 − α and a > 0, we denote by 1  D1 := D q q(α − 1), q(μ − δm1 ) /Γ (α),

1  D2 := D p p(β − 1), p(σ − δm2 ) ,

D3 := D2 /Γ (α),

D4 := D3 D(α − 1, β + σ − δm2 − 1/p ),

D5 := |b|a δ+α−1 ,

D6 := D1 a δ+α+μ−δm1 −1/q ,

and D7 := D4 a δ+α+β+σ −δm2 −1/p .

We will need the assumptions



(ΦΨ ) ϕ ∈ Lq , ψ ∈ Lp and H := 3

q −1

 K :=

 1 mm 1 D9 ϕ

q m1 p

 q m 1 p

q q



p m2 + q

 p m 2 q

q

p D10 ψp

 < mm−1 em(1−w0 ) ,

p

ˆ ˆ 2 0) D12 ϕq + mm ˆ m−1 em(1−y 2 D13 ψp < m

where m := min(m1 , p m2 /q ), m ˆ := q m/p , w0 := D8 := 3q p

p

D11 := 3p −1 (D5 + D6 ), D12 := 3p (H) The parameters satisfy

−1

p

D6 and D13 := 3p

−1

−1

and (α + β)m2 + σ +

1 < m2 . p

Observe that the last assumption requires that α + β < 1. (I) The intersection of the intervals   1 + q(μ + α − 1) 1 + qμ I1 := , q(m1 − 1) qm1 and

 I2 :=

1 + p(σ + β − 1) 1 + pσ , p(m2 − 1) pm2

p

D7 .

1 + qμ < m1 q(1 − α)



is not empty. Observe that the assumption (H) ensures that I1 and I2 are not empty.

q

q

(D5 + D7 ), y0 :=

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447

Theorem 2. Assume that the hypotheses (F), (G), (H), (I), (Φ), (Ψ ) and (ΦΨ ) hold. Then the solutions of problem (1) decay polynomially as t −δ for some δ > 0 away from zero. In fact, u(t)  Ct −δ , t  a > 0, (5) for some positive constant C. Proof. Let us multiply the integral equation (2) by t δ with 0 < δ < 1 − α in I1 ∩ I2 and use the assumptions (3) and (4) on f and g, we find t u(t)  |b|t δ+α−1 +

δ

t

tδ Γ (α)

m (t − s)α−1 s μ ϕ(s) u(s) 1 ds

0

tδ + Γ (α)

s

t (t − s)α−1 0

Put

v(t) = t δ |u(t)|, v(t)  |b|t

(6)

0

then (6) can be rewritten as

δ+α−1

t

tδ + Γ (α)

(t − s)α−1 s μ−δm1 ϕ(s)v m1 (s) ds 0

+

 m (s − z)β−1 zσ ψ(z) u(s) 2 dz ds.

(t − s) 0

β−1 σ −δm2

(s − z)

α−1

Γ (α)



s

t

tδ

z

ψ(z)v

m2

(z) dz ds.

(7)

0

For the second term in the right-hand side of (7), using the Hölder inequality with q = α + 1 and q = α+1 α , we get tδ Γ (α)

t (t − s)α−1 s μ−δm1 ϕ(s)v m1 (s) ds 0

tδ  Γ (α)

 1 t

t



q

(t − s)q(α−1) s q(μ−δm1 ) ds

q

ϕ (s)v

0

q m

1

(s) ds

1 q

.

0

Since 1 + q(μ − δm1 ) > 0 (because δ ∈ I1 ) and q(α − 1) = α 2 − 1 > −1 we can apply Lemma 1. We obtain tδ Γ (α)

t (t − s)α−1 s μ−δm1 ϕ(s)v m1 (s) ds 0

 D1 t δ+α+μ−δm1

−1/q

t

 q

ϕ (s)v

q m

1

(s) ds

1 q

,

0

where D1 is the constant which arises from the application of Lemma 1 1  D1 := D q q(α − 1), q(μ − δm1 ) /Γ (α).

(8)

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K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454

Similarly, since 1 + p(σ − δm2 ) > 0 (because δ ∈ I2 ) and p(β − 1) = β 2 − 1 > −1, it follows that  1

s s (s − z)β−1 zσ −δm2 ψ(z)v m2 (z) dz  D2 s β+σ −δm2 −1/p



p

ψ p (z)v m2 p (z) dz

0

0 1 p

where D2 := D (p(β − 1), p[σ − δm2 ]). Therefore, 1 Γ (α)



s

t (t − s)

β−1 σ −δm2

(s − z)

α−1

0

z

0

t  D3

α−1 β+σ −δm2 −1/p

(t − s)

ψ(z)v

0

(z) dz ds 

s

s

m2

p

ψ (z)v

p m2

1 p

(z) dz

ds

0

with D3 := D2 /Γ (α). Applying Lemma 1 again, we get 1 Γ (α)



s

t (t − s) 0

β−1 σ −δm2

(s − z)

α−1 0

 D4 t

α+β+σ −δm2 −1/p

t

z

ψ(z)v

m2

(z) dz ds

 p

ψ (z)v

p m2

1 p

(9)

(z) dz

0

where D4 := D3 D(α − 1, β + σ − δm2 − 1/p ). Using the estimations (8) and (9) in (7) we find v(t)  |b|t

δ+α−1

+ D4 t

+ D1 t

δ+α+μ−δm1 −1/q

δ+α+β+σ −δm2 −1/p



t

q

ϕ (s)v

q m1

0

t

(s) ds



p

ψ (z)v

p m2

1 q

1 p

(z) dz

.

0

Since the exponents of t are non-positive, we entail that  1

t

t q

v(t)  D5 + D6

ϕ (s)v

q m1

q

(s) ds

+ D7

 p

ψ (z)v

0

p m2

(z) dz

1 p

(10)

0

for all t  a > 0, with D6 := D1 a δ+α+μ−δm1 −1/q ,

(1) If q

q p

D7 := D4 a δ+α+β+σ −δm2 −1/p .

D5 := |b|a δ+α−1 ,

 1 i.e. α  β, then by Lemma 3 t

v (t)  D8 + D9

q

ϕ (s)v 0

q m1

t (s) ds + D10

ψ p (z)v p 0

m

2

(z) dz

(11)

K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454 q

q

with D8 := 3q −1 (D5 + D7 ), D9 := 3q side of (11) by w(t), we have



v q (t)  w(t),

−1

q

D6 and D10 := 3q

−1

449

q

D7 . Denoting the right-hand

t  0 and w(0) = D8 .

(12)

By differentiation and from (12), we infer that w  (t) = D9 ϕ q (t)v q

m

1

(t) + D10 ψ p (t)v p



 D9 ϕ q (t)w m1 (t) + D10 ψ p (t)v

m

p m2 q

2

(t) (13)

(t).

Let m := min(m1 , p m2 /q ). Multiplying both sides of (13) by e−mw and using the inequality x a e−bx  (a/eb)a ,

a, b, x > 0,

we obtain w  (t)e−mw  h(t)/(em)m

(14)

where q 1 h(t) := mm 1 D9 ϕ (t) +



p m2 q

 p m 2 q

D10 ψ p (t).

Next, we integrate (14) from 0 to t to arrive at  1 1  −mw(0) e − e−mw  m (em)m

t h(s) ds 

H (em)m

(15)

0 p m  +∞ q p p m2 q 2 1 where H := 0 h(s) ds = mm D7 ψp . By the hypotheses on ϕ and 1 D6 ϕq + ( q ) ψ (H < mm−1 em(1−w0 ) ), we entail from (15) that   (em)m 1 w(t)  ln =: W. (16) m (me1−w0 )m − mH

From (16) and (12) and the fact that v(t) = t δ |u(t)|, we deduce that 1 u(t)  W q t −δ , t  a > 0.

(2) If pq  1 i.e. α  β, then raising to the power p both sides of (10) and applying Lemma 3 to (10) we find t

p

v (t)  D11 + D12

q

ϕ (s)v

q m1

t (s) ds + D13

0 p

p −1

p −1

p

D6 and D13 := 3p

t  0 and y(0) = D11 .

A differentiation of y(t), together with (18) yields y  (t)  D12 ϕ q (t)y

m

2

(z) dz

(17)

0 p

with D11 := 3 (D5 + D6 ), D12 := 3 hand side of (17) by y(t), we may write v p (t)  y(t),

ψ p (z)v p

q m1 p

(t) + D13 ψ p (t)y m2 (t).

−1

p

D7 . Denoting the right(18)

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K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454

Using a similar argument as in the first case, we arrive at   (em) ˆ mˆ 1 =: Y y(t)  ln m ˆ (me ˆ 1−y0 )mˆ − mK ˆ where m ˆ := q m1 p

q m1 p

q m p ,

k(t) := ( q pm 1 )

q m1 p

q q



2 p D12 ϕ q (t) + mm 2 D13 ψ (t) and K :=

 +∞ 0

k(s) ds =

p p

2 ˆ ˆ 0 ) . Finally, the hypotheses set on ϕ and ( ) D12 ϕ + mm ˆ m−1 em(1−y 2 D13 ψ < m ψ and (18) allow us to conclude (5). In fact, 1 u(t)  Y p t −δ , t  a > 0. 2

Example. As an example we can take f (t, u) as in (3) and g as in (4) with an equality in both relations and ϕ and ψ satisfying the conditions (Φ) and (Ψ ), respectively. One can pick polynomials of order bigger than μ − (1 − α)m1 and σ − (1 − α)m2 , respectively. Remark 1. Note that the power type decay is a result of the particular behavior of the solution nearby 0 and also of the power type decay of the kernel. This is the best decay we can get even if we assume that the kernel is exponentially decaying to zero. That is, we cannot get exponential decay in case the kernel is exponentially decaying to zero. The next result shows that we can dispense of most of the hypotheses on the parameters at

the expense of requiring the functions ϕ and ψ satisfy eε1 t ϕ ∈ Lq and eε2 t ψ ∈ Lp for some ε1 , ε2 > 0. Theorem 3. If β < 1 − α, then we have decay of order t δ with δ + β  1 − α provided that

1 + q(μ − δm1 ) > 0, 1 + p(σ − δm2 ), ϕ and ψ are such that eε1 t ϕ ∈ Lq and eε2 t ψ ∈ Lp for

q p some ε1 , ε2 > 0 and satisfy a certain smallness condition (ϕq + ψp < M). Proof. From the relation (7) and by multiplying by e−εs eεs (with ε = min(ε1 , ε2 )), we get v(t)  |b|t

δ+α−1

t

tδ + Γ (α)

(t − s)α−1 s μ−δm1 e−εs eεs ϕ(s)v m1 (s) ds

0

tδ + Γ (α)



s

t (t − s) 0

β−1 σ −δm2 −εz εz

(s − z)

α−1

z

e

e ψ(z)v

m2

(z) dz ds.

0

Next using Lemmas 1 and 2, we infer that v(t)  |b|t

δ+α−1

t δ+α−1 + C1 Γ (α)



t e

q εs q

0

tδ + C2 Γ (α)

t (t − s)

ϕ (s)v

0

s

e 0

1 q

(s) ds 

s

α−1 β−1

q m1

p εz

p

ψ (z)v

p m2

(z) dz

1 p

K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454

t δ+α−1  |b|t δ+α−1 + C1 Γ (α) t δ+α+β−1 + C3 Γ (α)



t e

q εs

q

ϕ (s)v

q m

1

0

t e

p εs

ψ (s)v

p m2

1 q

(s) ds

 p

451

1 p

(s) ds

0

where the constants come from the application of the lemmas 1  C1 := C q q(α − 1), q(μ − δm1 ), qε , 1  C2 := C p p(β − 1), p(σ − δm2 ), pε and C3 := C2 D(α − 1, β − 1). For t  a > 0, we can write

t v(t)  D11 + D12

 e

q εs q

ϕ (s)v

q m1

(s) ds



t

1 q

+ D13

0

e

q εs

p

ψ (s)v

q m2

1 q

(s) ds

,

0 a aδ+α−1

with D11 := |b|a δ+α−1 , D12 := C1 Γ (α) , D13 := C3 to the one of Theorem 2. We omit the details. 2

a δ+α+β−1 Γ (α)

. The rest of the proof is similar

The next theorem considers the case where we have a weak singular kernel of the form (t − s)β−1 e−γ (t−s) with γ > 0 in (G). Here we do not need the condition β < 1 − α. Theorem 4. Assume the first condition in (H), 1 + p(σ − δm2 ) > 0 for some δ  1 − α, ϕ ∈ Lq

and eμt ψ ∈ Lp for some μ  γ . If (G) is replaced by

(G) |g(t, s, u(s))|  (t − s)β−1 e−γ (t−s) s σ ψ(s)|u|m2 , γ > 0, then the conclusion of Theorem 2 remains true provided that ϕ and ψ satisfy some smallness condition. Proof. Because of the first condition in (H), the estimate (8) in the proof of Theorem 2 is valid. Suppose first that μ = γ . We have 

s t (s − z)β−1 e−γ (s−z) zσ −δm2 ψ(z)v m2 (z) dz ds

(t − s)α−1 0

0

t

α−1 −γ s

(t − s)

 0

e 0

p(β−1) p(σ −δm2 )

(s − z)

e

0

s

×

1

s

γp z

p

ψ (z)v

z

 p m2

(z) dz

p

dz

1 p

ds

(19)

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and by Lemma 1, relation (19) gives

s t (t − s) 0



β−1 −γ (s−z) σ −δm2

(s − z)

α−1

e

z

0

t  A1

(t − s)α−1 s

β−1+σ −δm2 + p1

e−γ s

ψ(z)v

(z) dz ds 

s e

0

m2

γ q z

q

ψ (z)v

q m

2

(z) dz

1 q

ds

0

1 p

with A1 := D (p(β − 1)p(σ − δm2 )). Now since β + σ − δm2 + p1 > 0, we can apply Lemma 2 to get 

s t (s − z)β−1 e−γ (s−z) zσ −δm2 ψ(z)v m2 (z) dz ds

(t − s)α−1 0

0



t

 A1 C1 t

α−1

e

γ q s

q

ψ (s)v

q m2

1 q

(s) ds

0

with C1 := C(α − 1, β − 1 + σ − δm2 + p1 , γ ). The rest of the proof is similar to that of Theorem 2. 2 If μ > γ , we multiply by e−μz .eμz inside the inner integral so that when we apply Hölder inequality we find t

s

α−1 −γ s

(t − s) 0

e

0

t

α−1 −γ s

(t − s)



(s − z)β−1 eγ z e−μz .eμz zσ −δm2 ψ(z)v m2 (z) dz

0

p(β−1) −p(μ−γ )z p(σ −δm2 )

(s − z)

e

e

0

s

×

1

s

e

p μz

p

ψ (z)v

 p m2

z

p

dz

1 p

(z) dz

ds.

0

By applying Lemma 2 twice, we see that t

α−1 −γ s

(t − s) 0

s

e

0

t  C2

(s − z)β−1 eγ z e−μz .eμz zσ −δm2 ψ(z)v m2 (z) dz

α−1 −γ s β−1

(t − s) 0

e

 C2 C3 t

ds

 e

p μs

e

p μs

p

ψ (s)v

p m2

p

ψ (s)v

0

t α−1

s

t



p m2

1 p

(s) ds

1 p

(s) ds

0 1 p

where C2 := C (p(β − 1), p(σ − δm2 ), p(μ − γ )) and C3 := C(α − 1, β − 1, γ ).

K.M. Furati, N.-e. Tatar / J. Math. Anal. Appl. 332 (2007) 441–454

453

Remark 2. Using generalizations of the Gronwall inequality (such as the Gronwall–Bihari inequality) one can extend our results easily to nonlinearities of the form h(u) (non-decreasing) other than polynomials (of the form |u|m ). Acknowledgment The authors are very grateful to the anonymous referee for the comments and suggestions which have helped improving the paper.

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