J. Fixed Point Theory Appl. 9:02)81( https://doi.org/10.1007/s11784-018-0574-x c Springer International Publishing AG,  part of Springer Nature 2018

Journal of Fixed Point Theory and Applications

Existence of solutions for some classes of integro-differential equations in the Sobolev space W n,p (R+) Mohsen Hossein Zadeh Moghaddam, Reza Allahyari, Mohsen Erfanian Omidvar and Ali Shole Haghighi Abstract. In this paper, first, we introduce a new measure of noncompactness in the Sobolev space W n,p (R+ ) and then, as an application, we study the existence of solutions for a class of the functional integral– differential equations using Darbo’s fixed point theorem associated with this new measure of noncompactness. Mathematics Subject Classification. Primary 47H08; Secondary 45J05, 47H10. Keywords. Measures of noncompactness, Darbo’s fixed point theorem, integro-differential equations, Sobolev spaces, Carath´eodory condition.

1. Introduction Sobolev spaces [12], i.e., the class of functions with derivatives in Lp , play an outstanding role in the modern analysis. In the last decades, there has been increasing attempts to study of these spaces. Their importance comes from the fact that solutions of partial differential equations are naturally found in Sobolev spaces. They also highlighted in approximation theory, calculus of variation, differential geometry, spectral theory, etc. On the other hand, integral–differential equations (IDE) have a great deal of applications in some branches of sciences. It arises especially in a variety of models from applied mathematics, biological science, physics and another phenomenon, such as the theory of electrodynamics, electromagnetic, fluid dynamics, heat and oscillating magnetic, etc. [6,9,11,13,19,20,22,23,26]. There have appeared recently a number of interesting papers [2,10,25,29] on the solvability of various integral equations with help of measures of noncompactness. The first such a measure was defined by Kuratowski [27]. Next, Bana´s et al. [8] proposed a generalization of this notion which is more convenient in the applications. The technique of measures of noncompactness is frequently

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applicable in several branches of nonlinear analysis, in particular the technique turns out to be very useful tool in the existence theory for several types of integral and integral–differential equations. Furthermore, they are often used in the functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory [1,3,7,14,16–18,28,30,31]. The most important application of measures of noncompactness in the fixed point theory is contained in the Darbo’s fixed point theorem [4,5]. Now, in this paper, we introduce a new measure of noncompactness in the Sobolev space W n,p (R+ ) as a more effective approach. Then, we study the problem of existence of solutions of the functional integral–differential equation:    ∞ k(t, s)x(s)ds (1.1) x(n+1) (t) = f t, x(t), x (t), . . . , x(n) (t), 0

in the Sobolev space W n,p (R+ ).

2. Preliminaries In this section, we recall some basic facts concerning measures of noncompactness, which are defined axiomatically in terms of some natural conditions. Denote by R the set of real numbers and put R+ = [0, +∞). Let (E,  · ) be a real Banach space with zero element 0. Let B(x, r) denote the closed ball centered at x with radius r. The symbol B r stands for the ball B(0, r). For X, a nonempty subset of E, we denote by X and ConvX the closure and the closed convex hull of X, respectively. Moreover, let us denote by ME the family of nonempty bounded subsets of E and by NE its subfamily consisting of all relatively compact subsets of E. Definition 2.1 [8]. A mapping μ : ME −→ R+ is said to be a measure of noncompactness in E if it satisfies the following conditions: 1◦ 2◦ 3◦ 4◦ 5◦ 6◦

The family ker μ = {X ∈ ME : μ(X) = 0} is nonempty and ker μ ⊂ NE . X ⊂ Y =⇒ μ(X) ≤ μ(Y ). μ(X) = μ(X). μ(ConvX) = μ(X). μ(λX + (1 − λ)Y ) ≤ λμ(X) + (1 − λ)μ(Y ) for λ ∈ [0, 1]. If {Xn } is a sequence of closed sets from ME , such that Xn+1 ⊂ Xn for n = 1, 2, . . ., and if limn→∞ μ(Xn ) = 0, then X∞ = ∩∞ n=1 Xn = ∅.

In what follows, we recall the well known fixed point theorem of Darbo type [14]. Theorem 2.1. Let Ω be a nonempty, bounded, closed and convex subset of a space E and let F : Ω −→ Ω be a continuous mapping, such that there exists a constant k ∈ [0, 1) with the property μ(F X) ≤ kμ(X)

(2.1)

for any nonempty subset X of Ω. Then, F has a fixed point in the set Ω.

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Lemma 2.2 [24]. Let (Ei , .i ), for i = 1, 2 be Banach spaces and let L : E1 −→ E2 be a one-to-one, continuous linear operator of E1 onto E2 . If μ2 is a measure of noncompactness on E2 , define, for X ∈ ME1 , μ 2 (X) := μ2 (LX). Then μ 2 is a measure of noncompactness on E1 .

3. Construction of measures of noncompactness on Sobolev spaces Let Lp (R+ ) denote the space of Lebesgue integrable functions on R+ with the standard norm:  ∞  p1 |u(t)|p dt for 1 ≤ p < ∞. up = 0

Recall that the Sobolev space W n,p (R+ ) is defined to consist of those measurable functions u which, together with all their distributional derivatives u(k) of order k ≤ n, belong to Lp (R+ ) with the standard norm: un,p = max u(k) p , 0≤k≤n

(0)

where f = f . Before introducing the new measures of noncompactness on the spaces W n,p (R+ ), we need to characterize the compact subsets of W n,p (R+ ). Theorem 3.1 [12,21]. Let F be a bounded set in Lp (RN ) and 1 ≤ p < ∞. Then F is relatively compact if and only if the following conditions are satisfied: (i) limh−→0 τh f −f p = 0 uniformly with respect to f ∈ F, where τh f (x) = f (x + h) for x, h ∈ RN . (ii) For ε > 0 there exists a bounded and measurable subset Ω ⊂ RN , such that f Lp (RN \Ω) < ε for f ∈ F. Obviously, we have the following Theorem. Theorem 3.2. Let Fbe a bounded set in W n,p (R+ ) and (1 ≤ p < ∞). Then, F is relatively compact if and only if the following conditions are satisfied: (i) limh−→0 τh f (k) − f (k) p = 0 uniformly with respect to f ∈ F and 0 ≤ k ≤ n for x, h ∈ R. (ii) For ε > 0, there exists a bounded and measurable subset Ω ⊂ R+ , such that f (k) Lp (R+ \Ω) < ε for f ∈ F and 0 ≤ k ≤ n.

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Let X be a bounded subset of the space W n,p (R+ ). For u ∈ X, and ε > 0. Let us denote   p1 ∞ ω(u, ε) = sup |u(k) (t + h) − u(k) (t)|p dt : |h| < ε, 0 ≤ k ≤ n , 0

ω(X, ε) = sup{ω(u, ε) : u ∈ X}, ω(X) = lim ω(X, ε) ε→0

and



∞

dT (X) = sup

|u

(k)

 p1 (s)| ds : u ∈ X, 0 ≤ k ≤ n , p

T

d(X) = lim dT (X), T →∞

ω0 (X) = ω(X) + d(X). We have the following fact. Theorem 3.3. The function ω0 , where ω0 : MW n,p (R+ ) −→ R+ is a measure of noncompactness on W n,p (R+ ), and moreover, ker ω0 = NW n,p (R+ ) . Proof. First, we show that ω0 satisfies condition 1◦ . Suppose that X ∈ MW n,p (R+ ) is such that ω0 (X) = 0. Since ω0 (X) = 0 then limε→0 ω(X, ε) = 0 and limT →∞ dT (X) = 0. Therefore, for any η > 0, there exist δ > 0, such that  ∞  p1 |u(k) (t + h) − u(k) (t)|p dt <η 0

for all u ∈ X, 0 ≤ k ≤ n and h ∈ R, such that |h| < δ. Since η > 0 was arbitrary, we get  ∞  p1 |u(k) (t + h) − u(k) (t)|p dt =0 lim h→0

0

uniformly with respect to u ∈ X. Using again the fact that ω0 (X) = 0, we have lim dT (X) = 0

T →∞

and therefore, for ε > 0, there exists T > 0, such that  p1  ∞ (k) p |u (s)| ds <ε T

for all u ∈ X and 0 ≤ k ≤ n. Thus, by Theorem 3.2, we infer that X is relatively compact and ker ω0 ⊆ NW n,p (R+ ). The fact that ω0 satisfies condition 2◦ is obvious. Now, we check that condition 3◦ holds. For this purpose, suppose that X ∈ MW n,p (R+ ) and u ∈ X. Therefore, there exists a sequence un in X, such that um → u ∈ X in W n,p (R+ ). By the definition of ω(X, ε) and d(X), we have  ∞  p1 (k) p |u(k) (t + h) − u (t)| dt ≤ ω(X, ε) m m 0

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9

for any n ∈ N and |h| < ε, and  ∞  p1 p |u(k) ≤ dT (X) m (t)| dt T

for any n ∈ N and T > 0. Letting m → ∞, we get  ∞  p1 |u(k) (t + h) − u(k) (t)|p dt ≤ ω(X, ε) 0

for any |h| < ε, and



∞

|u(k) (t)|p dt

 p1

≤ dT (X)

T

for any T > 0. Hence lim ω(X, ε) ≤ lim ω(X, ε)

ε→0

→0

and lim dT (X) ≤ lim dT (X).

T →∞

T →∞

Consequently, we have ω(X) ≤ ω(X)

and d(X) ≤ d(X),

(3.1)

and by 2◦ we get ω0 (X) = ω0 (X). Hence, ω0 satisfies condition 3◦ of Definition 2.1. The proof of conditions 4◦ and 5◦ can be carried out similarly using the inequality:   p1   p1 1  p p p p |λx(t) + (1 − λ)y(t)| ≤λ |x(t)| dt + (1 − λ) |y(t)| dt Ω

Ω

Ω

for any Ω ⊆ R+ . Furthermore, observe that the function ω0 satisfies the following property, which is called the maximum property (cf. [8]): ω0 (X ∪ Y ) = max{ω0 (X), ω0 (Y )}. We omit the standard proof. Now, we can utilize the lemma contained in paper [10] which asserts that since ω0 satisfies conditions 1◦ , 2◦ and has the maximum property, then ω0 satisfies condition 6◦ . Thus, the proof is complete. Finally, we verify ker ω0 = NW n,p (R+ ) . By condition 1◦ , we know that ker ω0 ⊆ NW n,p (R+ ) . Now to prove NW n,p (R+ ) ⊆ ker ω0 . For this purpose, suppose that X ∈ NW n,p (R+ ) . Thus, X is relatively compact, and for any ε > 0 there exists T > 0, such that  p1  ∞ |u(k) (t)|p dt <ε (3.2) T

for all u ∈ X and 0 ≤ k ≤ n. In addition,  ∞  p1 lim |u(k) (t + h) − u(k) (t)|p dt =0 h−→0

0

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uniformly with respect to u ∈ X and 0 ≤ k ≤ n. Therefore, there exists δ > 0, such that  p1  ∞ (k) (k) p |u (t + h) − u (t)| dt <ε 0

for any |h| < δ, and for all u ∈ X   p1 ∞ (k) (k) p ω(u, δ) = sup |u (t + h) − u (t)| dt : |h| < δ ≤ ε. 0

Thus ω(X, δ) = sup{ω(x, δ) : x ∈ X} ≤ ε. This proves that lim ω(X, δ) = 0.

(3.3)

δ→0

On the other hand, using (3.2), we can conclude that lim dT (X) = 0.

(3.4)

T →∞

Now, from (3.3) and (3.4), we have ω0 (X) = 0, which means NW n,p (R+ ) ⊆  ker ω0 . Hence, ker ω0 = NW n,p (R+ ) holds.

4. Existence of solutions for some classes of integro-differential equations In this section, we study the existence of solutions for Eq. (1.1). Let E 0,p = {et u(t) : u ∈ Lp (R+ )} and E n,p = {et u(t) : u ∈ W n,p (R+ )} for all n ∈ N, with the norm:  ∞  p1 e−pt |u(t)|p dt uE 0,p = 0

and uE n,p = max u(k) E 0,p = max 0≤k≤n



0≤k≤n

∞

e−pt |u(k) (t)|p dt

 p1

0

for all n ∈ N. In addition, E is isomorphic by Lp (R+ ) with continuous 0,p p linear operator I0 : E −→ L (R+ ) defined by 0,p

I0 u(t) = e−t u(t) and E n,p is isomorphic by W n,p (R+ ) with continuous linear operator In : E n,p −→ W n,p (R+ ) defined by In u(t) = e−t u(t) for all n ∈ N. Moreover, W n,p (R+ ) ⊂ E n,p . Consider, for X ∈ ME n,p , μ(X) := ω0 (In X). By using Lemma 2.2, μ is a measure of noncompactness on E n,p .

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Definition 4.1. A function f : R+ ×Rn −→ R is said to have the Carath´ eodory property if (i) For all x ∈ Rn the function t → f (t, x) is measurable on R+ . (ii) For almost all t ∈ R+ the function x → f (t, x) is continuous on Rn . Lemma 4.1 [15]. Let Ω be a Lebesgue measurable subset of Rn and 1 ≤ p ≤ ∞. If {fn } is convergent to f ∈ Lp (Ω) in the Lp -norm, then there is a subsequence {fnk } which converges to f a.e., and there is g ∈ Lp (Ω), g ≥ 0, such that |fnk (x)| ≤ g(x),

a.e. x ∈ Ω.

Theorem 4.2 (Minkowki’s Inequality for Integrals) [6]. Suppose that (X, M, μ) and (Y, N , ν) are σ-finite measure spaces and let f is an (M⊗N )-measurable function on X × Y . If f ≥ 0 and 1 ≤ p < ∞, then      p1  p1 p ≤ f (x, y)p dμ(x) dν(y). f (x, y)dν(y) dμ(x) To investigate the following theorem, we need definition of the Gamma function with some its basic properties. Here, we are only referring to the definition of the Gamma function ( for more details see calculus books). The Gamma function can be defined as a definite integral for any real number z > 0 (Euler’s integral form) by  ∞ tz−1 e−t dt. Γ(z) = 0

We will consider Eq. (1.1) under the following assumptions: (i) u0 , u1 , . . . , un ∈ R eodory conditions. Moreover, (ii) f : R+ × Rn+2 −→ R satisfies the Carath´ there exist functions a, b ∈ Lq (R+ ), such that b(t) = et a(t), p1 + 1q = 1 and |f (t, x0 , x1 , . . . , xn+1 )| ≤ a(t) max |xi |. 0≤i≤n+1

(4.1)

(iii) k : R+ × R+ −→ R+ is a measurable function, such that the integral operator K generated by k, that is  ∞ k(t, s)x(s)ds (Kx)(t) = 0

is a continuous map from E into itself and K ≤ 1. (iv) The following inequality is satisfied 0

(Γ(p(n − k) + 1)) p1  aq < 1. D := 2 max 0≤k≤n pn−k (n − k)! n

Theorem 4.3. Under assumptions (i)–(iv), Eq. (1.1) has at least a solution in the space E n+1,p . Proof. The integral–differential equation (1.1) has at least one solution in the space E n+1,p if and only if the nonlinear integral equation:

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u(t) = p(t) +

1 n!

M. H. Z. Moghaddam et al.



t

(t − s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))ds

0

(4.2) has at least one solution in the space E n,p , where p(t) =

n uk k=0

and K on E 0,p is defined by



(Ku)(t) =

k!

tn

∞

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

Let us define the operator F on E n,p by  1 t F u(t) = p(t) + (t − s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), n! 0 Ku(s))ds. (4.3) First, by considering the Carath´ eodory conditions, we infer that F u is measurable for any u ∈ E n,p . In addition, for any t ∈ R+ , 1 ≤ k ≤ n, we have  t dk (F u) 1 (k) (t) = p (t)+ (t−s)n−k f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), dtk (n−k)! 0 Ku(s))ds, and F u has measurable derivative of order k (1 ≤ k ≤ n). Using conditions (i)–(iv), for arbitrarily fixed t ∈ R+ , we have 

∞

0

1 e−pt |F u(t)|p dt p

≤ pE n,p  t  ∞ 1 1 + e−pt | (t−s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))ds|p dt p n! 0 0 ≤ pE n,p  ∞  ∞ 1 + e−pt | χ[0,t] (s)(t − s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), n! 0 0 1 Ku(s))ds|p dt p ≤ pE n,p  ∞  ∞ 1 1 + (t−s)pn e−pt|f (s, u(ξ(s)), u (ξ(s)), . . . , u(n)(ξ(s)), Ku(s))|p dt p ds n! 0 s ≤ pE n,p 1  ∞ (Γ(pn + 1)) p + e−s |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))|ds pn n! 0 ≤ pE n,p 1  ∞ (Γ(pn + 1)) p + e−s |a(s)| max{|u(ξ(s))| . . . , |u(n) (ξ(s))|, |Ku(s))|}ds n p n! 0 ≤ pE n,p

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(Γ(pn + 1)) p aq max uE 0,p , u E 0,p , . . . , u(n) E 0,p , KuE 0,p , n p n! 1

+

and similarly  ∞  p1 dk (F u) p e−pt | (t)| dt dtk 0

(Γ(p(n − k) + 1)) p aq max uE 0,p , u E 0,p , . . . , n p (n − k)!  (n) u E 0,p , KuE 0,p . 1

≤ pE n,p +

Hence

(Γ(p(n − k) + 1)) p1  F uE n,p ≤ pE n,p + max aq uE n,p . (4.4) 0≤k≤n pn (n − k)! ¯r into itself, where r0 = From inequality (4.4), F transforms the ball B 0 pE n,p . Next, we show that the map F is continuous. Let {um } be an (1 − D) arbitrary sequence in E n,p which converges to u ∈ E n,p in the E n,p -norm. Since the integral operator K maps (continuously) the space E 0,p into itself, so Kum converge to Ku. Using Lemma 4.1, there is a subsequence {umk } (k) which converges to u a.e., {umk } converges to u(k) a.e., {Kumk } converges to Ku a.e. for all 1 ≤ k ≤ n and there is h ∈ E 0,p , h ≥ 0, such that (n) (ξ(t))|, |Kumk (t)|} ≤ h(t), max{|umk (ξ(t))|, |umk (ξ(t))|, |umk (ξ(t))|, . . . , |um k

a.e. on R+ .

(4.5)

eodory Since umk → u almost everywhere in R+ and f satisfies the Carath´ conditions, so (n) (ξ(s)), Kumk (s)) f (s, umk (ξ(s)), . . . , um k

−→ f (s, u(ξ(s)), . . . , u(n) (ξ(s)), Ku(s))

(4.6)

for almost all t ∈ R+ . From inequalities (4.1) and (4.5), we infer that (n) (ξ(s)), Kumk (s))| ≤ a(s)h(s), |f (s, umk (ξ(s)), . . . , um k

a.e. on R+ .

(4.7)

As a consequence of the Lebesgue’s Dominated Convergence Theorem, (4.6) and (4.7), we get  t (n) (t − s)n f (s, umk (ξ(s)), . . . , um (ξ(s)), Kumk (s))ds k 0  t (t − s)n f (s, u(ξ(s)), . . . , u(n) (ξ(s)), Ku(s))ds −→ 0

for almost all t ∈ R+ . Inequality (4.7) implies that

 t

1

n (n)

(t − s) f (s, umk (s), . . . , umk (s), Kumk (s))ds

|F (umk )(t)| ≤ n! 0



1

t n

≤ (4.8) (t − s) a(s)h(s)ds

n! 0

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for almost all t ∈ R+ . From assumptions on a, we obtain 

∞

0

p  1

 t 

p e−pt

(t − s)n a(s)h(s)ds

dt ≤ 0

∞

0

a(s)h(s) 1 p

 

∞

s ∞

1 e−pt (t − s)pn dt p ds

≤

(Γ(pn + 1)) pn n!

≤

(Γ(pn + 1)) p aq hE 0,p . pn n!

0

e−s a(s)h(s)ds

1

(4.9)

Inequalities (4.8), (4.9) and the Lebesgues dominated convergence theorem imply F umk − F uE 0,p −→ 0. Since any sequence {um } converging to u in E 0,p has a subsequence {umk }, such that F umk −→ F u in E 0,p , we can conclude that F : E 0,p −→ E 0,p is a continuous operator. In addition, by a similar argument, we can conclude k (F u) that d dt : E 0,p −→ E 0,p is a continuous operator. Thus, F : E n,p −→ E n,p k is a continuous operator. To finish the proof we have to verify that condition (2.1) is satisfied. Let ¯r , and without loss of generality, X be a nonempty and bounded subset of B 0 assume that 0 < ε ≤ 1 is an arbitrary constant. Let t, h ∈ R+ be such that |h| ≤ ε and u ∈ X. We obtain  ∞  p1 |(In F u)(t + h) − (In F u)(t)|p dt 0  ∞  p1 ≤ |e−t p(t) − e−t−h p(t + h)|p dt 0 

 t+h ∞ 1

+ e−pt (t − s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) n! 0 t  p1

p

(ξ(s)), Ku(s))ds dt 1 + n!



−t

∞



0

t+h

[e−t−h (t + h − s)n

0 

−e (t − s) ]f (s, u(ξ(s)), u (ξ(s)), . . . , u n

(n)

p

(ξ(s)), Ku(s))ds dt

 p1 .

(4.10) Now, by condition (ii) and Theorem 4.2, we have

 

t+h ∞ 1 −pt e (t − s)n f (s, u(ξ(s)), u (ξ(s)), . . . , u(n)

t n! 0

p  p1

(ξ(s)), Ku(s))ds dt

Existence of solutions for some classes of integro-differential

1 ≤ n!



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9

∞

|a(s)| max{|u(ξ(s))|, |u (ξ(s))|, . . . , |u(n) (ξ(s))|,  p1  s (t − s)np e−pt dt ds |Ku(s))|} s−h  hn ∞ ≤ |a(s)| max{|u(ξ(s))|, |u (ξ(s))|, . . . , |u(n) (ξ(s))|, n! 0  e−p(s−h) − e−ps  p1 |Ku(s))|} ds p 1  hn |eph − 1| p ∞ −s ≤ e |a(s)| max{|u(ξ(s))|, |u (ξ(s))|, . . . , |u(n) (ξ(s))|, √ n! p p 0 |Ku(s))|}ds 0

1

hn |eph − 1| p ≤ aq uE n,p √ n! p p 1

≤

hn |eph − 1| p aq r0 √ n! p p

and 1 n!



∞



0

t+h

[e−t−h (t + h − s)n − e−t (t − s)n ]f (s, u(ξ(s)),

0



(n)

u (ξ(s)), . . . , u ≤

≤

(4.11)

1 n! 



∞

0 ∞

 p1

e−s |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))|

|e−(t+h−s) (t + h − s)n − e−(t−s) (t − s)n |p dt

s−h  ∞

1 n! 

p

(ξ(s)), Ku(s))ds dt

 p1

ds

e−s |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))|

0 ∞

 p1 |e−(t+h−s) (t + h − s)n − e−(t−s) (t − s)n |p dt ds s−1  M (ε) ∞ −s ≤ e |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) (ξ(s)), Ku(s))|ds n! 0 M (ε) aq r0 , ≤ (4.12) n! where M (ε) = sup

 

∞ s−1

|e−(t+h−s) (t + h − s)k − e−(t−s) (t − s)k |p dt : |h| ≤ ε, 1 ≤ k ≤ n .

Now, by (4.10), (4.11) and (4.12) we have

 p1

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 p1 |(In F u)(t + h) − (In F u)(t)|p dt 0  ∞  p1 ≤ |e−t p(t) − e−(t+h) p(t + h)|p dt ∞

0 1

+

hn |eph − 1| p M (ε) aq r0 + aq r0 . √ n! p p n!

(4.13)

Similarly, for all 1 ≤ k ≤ n, we have 

∞

0

≤

|(In F u)(k) (t + h) − (In F u)k (t)|p dt

k k   i=0

+

i

≤

|e−t p(i) (t) − e−(t+h) p(i) (t + h)|p dt

 p1

0

k k  n−i

h

i=0

+

∞

 p1

i

1

|eph − 1| p √ aq r0 (n − i)! p p

k k  M (ε) aq r0 . i (n − i)! i=0

(4.14)

Therefore, by 4.13 and 4.14 we have ω(In F u, ε) ≤ 2n ω(In p, ε)  k  k  aq r0  εn−i |epε − 1| p1 + M (ε) . + max √ p p 0≤k≤n i (n − i)! i=0

(4.15)

Since u was an arbitrary element of X in (4.15), we obtain ω(In F (X), ε) ≤ 2n ω(In p, ε) 1 k k 

 aq r0  εn−i |epε − 1| p + M (ε) . + max √ p p 0≤k≤n i (n − i)! i=0 Since {In p} is a compact set, so we have ω(In p, ε) → 0 as ε −→ 0. Then, we obtain ω(In F (X)) = 0.

(4.16)

Next, let us fix an arbitrary number T > 0. Then, taking into account our assumptions, for an arbitrary function u ∈ X, we have 

∞

 p1



∞

|In F u(t)| dt ≤ e−pt |F u(t)|p dt T T   p1 1  ∞ −pt ≤ e |p(t)|p dt n! T p

 p1

Existence of solutions for some classes of integro-differential

1 + n!



∞



T

∞

Page 13 of 16

9

e−pt (t − s)np |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n)

s

 p1 (ξ(s)), Ku(s))|p dt ds   1 T  ∞ −pt e (t − s)np |f (s, u(ξ(s)), u (ξ(s)), . . . , u(n) + n! 0 T  p1 (ξ(s)), Ku(s))|p dt ds   ∞  p1 1  ∞ −pt np  p1 −pt p ≤ e |p(t)| dt + e t dt bq uE n,p n! T T 1  ∞ (Γ(pn + 1)) p a + e−ps max{|u(ξ(s))|p , q pn n! T |u (ξ(s))|p , . . . , |u(n) (ξ(s))|p ,  p1 |Ku(s))|p }ds , and by similar argument, for any 1 ≤ k ≤ n, we have  ∞  p1 |(In F u)(k) (t)|p dt T

≤

k k   i=0

i

∞

e−pt |p(i) (t)|p dt

k k 

e−pt tnp dt

 p1

i

1 (n − i)!

1 k k  (Γ(p(n − i) + 1)) p

i

∞

T

i=0

i=0

 +

T

aq uE n,p +

 p1

pn−i (n − i)!

aq



 p1 . . . , |u(n) (ξ(s))|p , |Ku(s))|p }ds .

∞

e−ps max{|u(ξ(s))|p , |u (ξ(s))|p ,

T

Since {In p} is a compact set, so we get  ∞  p1 e−pt |p(t)|p dt −→ 0, T k k   ∞ i

i=0

and



∞

 p1

−→ 0

T

−pt np

e

e−pt |p(i) (t)|p dt

t dt

 p1

−→ 0

as

T → ∞.

T

Then, we deduce that d(In F X) ≤ D d(In X).

(4.17)

Furthermore, combining (4.16) and (4.17), we get μ(F X) ≤ D μ(X).

(4.18)

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M. H. Z. Moghaddam et al.

Since D < 1, from (4.18) and applying Theorem 2.1 we obtain that the ¯r . Hence, Eq. (1.1) has at least a solution operator F has a fixed point x in B 0 n+1,p .  in E

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Mohsen Hossein Zadeh Moghaddam, Reza Allahyari and Mohsen Erfanian Omidvar Department of Mathematics, Mashhad Branch Islamic Azad University Mashhad Iran e-mail: [email protected]; [email protected]; [email protected]

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Ali Shole Haghighi Department of Mathematics, Sari Branch Islamic Azad University Sari Iran e-mail: [email protected]