Existence of solutions for some classes of integro-differential ... - EMIS

Electronic Journal of Qualitative Theory of Differential Equations 2015, No. 41, 1–18; doi: 10.14232/ejqtde.2015.1.41

http://www.math.u-szeged.hu/ejqtde/

Existence of solutions for some classes of integro-differential equations via measure of noncompactness Reza Allahyari1 , Reza Arab B 2 and Ali Shole Haghighi3 1,3 Department

of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran

2 Department

Received 27 February 2015, appeared 14 July 2015 Communicated by Hans-Otto Walther Abstract. In this present paper, we introduce a new measure of noncompactness on the space consisting of all real functions which are n times bounded and continuously differentiable on R+ . As an application, we investigate the problem of the existence of solutions for some classes of the functional integral-differential equations which enables us to study the existence of solutions of nonlinear integro-differential equations. In our considerations we apply the technique of measures of noncompactness in conjunction with Darbo’s fixed point theorem. Finally, we give some illustrative examples to verify the effectiveness and applicability of our results. Keywords: measures of noncompactness, Darbo’s fixed point theorem, integrodifferential equations. 2010 Mathematics Subject Classification: 45J05, 47H08, 47H10.

1

Introduction

Integro-differential equation (IDE) have a great deal of application in different branches of sciences and engineering. It arises naturally in a variety of models from biological science, applied mathematics, physics, and other disciplines, such as the theory of elasticity, biomechanics, electromagnetic, electrodynamics, fluid dynamics, heat and mass transfer, oscillating magnetic field, etc, see [14, 18, 21, 23]. Many papers have been devoted to the study of (IDE) and its versions by using different techniques, for example, some of the existing numerical methods can be found in [6, 7, 12, 13, 19, 22, 27, 30–32] and the references therein. The tools utilized in these papers are: the tau method, direct methods, collocation methods, Runge–Kutta methods, wavelet methods and spline approximation. On the other hand, measures of noncompactness are very useful tools in the theory of operator equations in Banach spaces. They are frequently used in the theory of functional B Corresponding

author. Email: [email protected]

2

R. Allahyari, R. Arab and A. Shole Haghighi

equations, including ordinary differential equations, equations with partial derivatives, integral and integro-differential equations, optimal control theory, etc. In particular, the fixed point theorems derived from them have many applications. There exists an enormous amount of considerable literature devoted to this subject (see for example [1, 8–11, 15, 16, 20, 21, 28, 29]). The first measure of noncompactness was introduced by Kuratowski [24] in the following way.   n [ Si for some Si with diam(Si ) ≤ δ for 1 ≤ i ≤ n ≤ ∞ . α(S) := inf δ > 0 S = i =1

Here diam( T ) denotes the diameter of a set T ⊂ X, namely diam( T ) := sup{d( x, y) | x, y ∈ T }. Another important measure is the so-called Hausdorff (or ball) measure of noncompactness is defined as follows χ( X ) = inf{ε : X has a finite ε-net in E}. (1.1) These measures share several useful properties [5, 9]. These measures seem to be nice, but they are rather rarely applied in practice. Hence, in order to resolve this problem, Bana´s presented some measures of noncompactness being the most frequently utilized in applications (see [10, 11]). The principal application of measures of noncompactness in the fixed point theory is contained in Darbo’s fixed point theorem [9]. The technique of measures of noncompactness in conjunction with it turned into a tool to investigate the existence and behavior of solutions of many classes of integral equations such as Volterra, Fredholm and Urysohn type integral equations (see [2–4, 9, 11, 16, 17, 25]). Now, in this paper, as a more effective approach, similar to the measures of noncompactness considered in [10, 11], in the first place we introduce a new measure of noncompactness on the space consisting of all real functions which are n times bounded and continuously differentiable on R+ . Then we study the problem of existence of solutions of the functional integral-differential equation x (t) = p(t) + q(t) x (t) +

Z t 0

g(t, s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)), Tx (s)) ds

on this space. As a special case of (1.2) we can refer to the integro-differential equation  (n) x (t) = f 1 t, x (ξ (t)), x 0 (ξ (t)), . . . , x (n−1) (ξ (t)),  Z ∞ 0 ( n −1) k (t, s) f 2 (s, x (s), x (s), . . . , x (s) ds ,

(1.2)

(1.3)

0

x (0 ) = x 0 , x 0 (0 ) = x 1 , . . . , x ( n −1) (0 ) = x n −1 , and the integro-differential equation  (n) x (t) = f 1 t, x (ξ (t)), x 0 (ξ (t)), . . . , x (n−1) (ξ (t)), Z t 0

0

k (t, s) f 2 (s, x (s), x (s), . . . , x

( n −1)



(1.4)

(s) ds ,

x (0 ) = x 0 , x 0 (0 ) = x 1 , . . . , x ( n −1) (0 ) = x n −1 , which will be investigated in this paper. In our considerations, we apply Darbo’s fixed point theorem associated with this new measure of noncompactness. Finally, some examples are presented to verify the effectiveness and applicability of our results.

Existence of solutions for some classes of integro-differential equations

2

3

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, k · k) 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 Br stands for the ball B(0, r ). For X, a nonempty subset of E, we denote by X and Conv X 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 ([9]). A mapping µ : ME −→ R+ is said to be a measure of noncompactness in E if it satisfies the following conditions: 1◦ the family ker µ = { X ∈ ME : µ( X ) = 0} is nonempty and ker µ ⊂ NE ; 2◦ X ⊂ Y =⇒ µ( X ) ≤ µ(Y ); 3◦ µ ( X ) = µ ( X ); 4◦ µ(Conv X ) = µ( X ); 5◦ µ(λX + (1 − λ)Y ) ≤ λµ( X ) + (1 − λ)µ(Y ) for λ ∈ [0, 1]; 6◦ 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 6 = ∅. In what follows, we recall the well known fixed point theorem of Darbo type [9]. Theorem 2.2. 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( FX ) ≤ kµ( X )

(2.1)

for any nonempty subset X of Ω. Then F has a fixed point in the set Ω. Here BC (R+ ) is the Banach space of all bounded and continuous function on R+ equipped with the standard norm k x ku = sup {| x (t)| : t ≥ 0} . For any nonempty bounded subset X of BC (R+ ), x ∈ X, T > 0 and ε ≥ 0 let ω T ( x, ε) = sup{| x (t) − x (s)| : t, s ∈ [0, T ] , |t − s| ≤ ε}. n o ω T ( X, ε) = sup ω T ( x, ε) : x ∈ X , ω0T ( X ) = lim ω T ( X, ε), ε →0

ω0 ( X ) = lim ω0T ( X ), T →∞

X (t) = { x (t) : x ∈ X } , and µ1 ( X ) = ω0 ( X ) + lim sup diam X (t). t→∞

It was demonstrated in [9] that the function µ is a measure of noncompactness in the space BC (R+ ) .

R. Allahyari, R. Arab and A. Shole Haghighi

4

3

Main results

In this section, we introduce a measure of noncompactness on BC n (R+ ). Let BC n (R+ ) = { f ∈ C n (R+ ) : e−t | f (i) (t)| is bounded for all t ≥ 0, i = 1, 2, . . . , n}, where ( 0 ) f = f . It is easy to see that BC n (R+ ) is a Banach space with norm

k f k BCn (R+ ) = max kh f (k) ku , 0≤ k ≤ n

where h(t) = e−t for all t ∈ R+ . Theorem 3.1. Suppose 1 ≤ n < ∞ and X be a bounded subset of BC n (R+ ). Then µ : MBCn (R+ ) −→ R+ given by µ( X ) = max µ1 ( X (k) ) (3.1) 0≤ k ≤ n

defines a measure of noncompactness on BC n (R+ ), where X (k) = { hx (k) : x ∈ X }. The proof relies on the following useful observations. Lemma 3.2 ([5]). Suppose µ1 , µ2 , . . . , µn are measures of noncompactness in Banach spaces E1 , E2 , . . . , En , respectively. Moreover assume that the function F : Rn+ −→ R+ is convex and F ( x1 , . . . , xn ) = 0 if and only if xi = 0 for i = 1, 2, . . . , n. Then µ( X ) = F (µ1 ( X1 ), µ2 ( X2 ), . . . , µn ( Xn )), defines a measure of noncompactness in E1 × E2 × · · · × En where Xi denotes the natural projection of X into Ei , for i = 1, 2, . . . , n. Lemma 3.3 ([26]). Let ( Ei , k · ki ), 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 , e2 ( X ) := µ2 ( LX ). µ e2 is a measure of noncompactness on E1 . Then µ Proof of Theorem 3.1. First, consider E = ( BC (R+ ))n+1 equipped with the norm

k( x1 , . . . , xn , xn+1 )k = max k xi ku . 1≤ i ≤ n +1

Also, F ( x1 , . . . , xn+1 ) = max1≤i≤n+1 xi for any ( x1 , . . . , xn+1 ) ∈ Rn++1 , therefore all the conditions of Lemma 3.2 are satisfied. Then µ2 ( X ) := max µ1 ( Xi ), 1≤ i ≤ n +1

defines a measure of noncompactness in the space E where Xi denotes the natural projection of X, for i = 1, 2, . . . , n + 1. Now, we define the operator L : BC n (R+ ) −→ E by the formula L( x ) = (hx, hx 0 , hx 00 , hx (3) , . . . , hx (n) ). Obviously, L is a one-to-one and continuous linear operator. We show that L( BC n (R+ )) is closed in E. To do this, let us choose { xn } ⊂ BC n (R+ ) such that L( xn ) is a Cauchy sequence in E. Thus, for any ε > 0 there exists N ∈ N such that for any k, m > N we have

k L( xk − xm )k < ε.

Existence of solutions for some classes of integro-differential equations

5

So, we deduce (i )

(i )

k xk − xm k BCn (R+ ) = max kh( xk − xm )ku 0≤ i ≤ n

(n)

(n)

0 = k(h( xk − xm ), h( xk0 − xm ), . . . , h( xk − xm ))k

= k L( xk − xm )k < ε. Therefore, { xn } is a Cauchy sequence of BC n (R+ ), and there exists x ∈ BC n (R+ ) such that xn −→ x. Since L is continuous, so we have L( xn ) −→ L( x ). This implies that Y = L( BC n (R+ )) is closed. Thus, the operator L : BC n (R+ ) −→ Y be a one-to-one and continuous linear operator of BC n (R+ ) onto Y. Since Y is a closed subspace of X, so µ2 is a measure of noncompactness on Y. Hence, for X ∈ MBCn (R+ ) , µe2 ( X ) = µ2 ( LX ) = max µ1 ( X (k) ) = µ( X ). 0≤ k ≤ n

Now using Lemma 3.3, the proof is complete. Corollary 3.4. Let F be a bounded set in BC n (R+ ) with 1 ≤ n < ∞. Also, assume that for every ε > 0 and T > 0, there exist δ > 0 such that for all 0 ≤ k ≤ n, t, s ∈ [0, T ] with |t − s| < δ and f ∈F | f (k) (t) − f (k) (s)| < ε, and lim |e−t f (k) (t) − e−t g(k) (t)| = 0

t−→∞

(3.2)

uniformly with respect to f , g ∈ F , for all 0 ≤ k ≤ n. Then F will be a totally bounded subset of BC n (R+ ). Proof. It is enough to show that µ(F ) = 0. Take an arbitrary ε > 0 and T > 0, there exists δ > 0 such that for all 0 ≤ k ≤ n, t, s ∈ [0, T ] with |t − s| < δ and f ∈ F , we have

| f (k) (t) − f (k) (s)| < ε. Thus, we obtain n o max ω T (h f (k) , δ) = max sup e−t | f (k) (t) − f (k) (s)| : t, s ∈ [0, T ], |t − s| ≤ δ ≤ ε

0≤ k ≤ n

0≤ k ≤ n

for all f ∈ F , and we deduce max ω T (F (k) , δ) ≤ ε.

0≤ k ≤ n

Therefore, we obtain ω0 T (F (k) ) = 0 for all 0 ≤ k ≤ n, and finally max ω0 (F (k) ) = 0.

0≤ k ≤ n

On the other hand, take ε > 0, by 3.2, there exists T > 0 such that e−t | f (k) (t) − g(k) (t)| < ε for all t > T and f , g ∈ F . Thus, we have n o max diam F (k) (t) ≤ max sup e−t | f (k) (t) − g(k) (t)| : f , g ∈ F ≤ ε

0≤ k ≤ n

0≤ k ≤ n

(3.3)

R. Allahyari, R. Arab and A. Shole Haghighi

6

for all t > T, so we deduce max lim sup diam F (k) (t) = 0. t→∞

0≤ k ≤ n

(3.4)

Further, combining (3.3) and (3.4), we get µ(F ) = max µ1 (F (k) ) = max 0≤ k ≤ n

0≤ k ≤ n



 ω0 (F (k) ) + lim sup diam F (k) (t) = 0, t→∞

(3.5)

and consequently F will be a totally bounded subset of BC n (R+ ).

4

Existence of solutions for some classes of integro-differential equations

In this section we study the existence of solutions for (1.2)–(1.4). Further, we present some illustrative examples to verify the effectiveness and applicability of our results. We will consider the Equation (1.2) under the following assumptions:

(i) ξ : R+ −→ R+ is a continuous function; (ii) p, q ∈ BC n (R+ ) and ( λ := sup

i =k 

∑

i =0

)  k kq(k−i) ku : 0 ≤ k ≤ n < 1; i

(4.1)

(iii) g : R+ × R+ × Rn+2 −→ R is continuous and has a continuous derivative of order n with respect to the first argument such that g(t, t, x0 , x1 , . . . , xn+1 ) = 0

for t ∈ R+ , x0 , x1 , . . . , xn+1 ∈ R,

(4.2)

and there exists a nondecreasing and continuous function ψ : R+ −→ R+ such that   Z t k
 ∂ g  −t : t ∈ R+ ,  t, s, x ( ξ ( s )) , x ( ξ ( s )) , . . . , x ( s ) ds sup e 0 1 n +1  0 ∂tk       k xi ku ≤ r, 1 ≤ k ≤ n ≤ ψ(r ),      Z t  
  − t  g t, s, x0 (ξ (s)), x1 (ξ (s)), . . . , xn+1 (s) ds : t ∈ R+ , k xi ku ≤ r ≤ ψ(r ) sup e 0

(4.3) for any r ∈ R+ . Moreover, for any r ∈ R+ and 1 ≤ k ≤ n  Z t 

  −t   g t, s, x0 (ξ (s)), . . . , xn+1 (s) − g t, s, y0 (ξ (s)), . . . , yn+1 (s) ds = 0, e tlim →∞ 0 Z t  k 
∂k g
 ∂ g  − t  lim e t, s, x0 (ξ (s)), . . . , xn+1 (s) − k t, s, y0 (ξ (s)), . . . , yn+1 (s) ds = 0  k t→∞ ∂t ∂t 0 (4.4) ¯ uniformly with respect to xi , yi ∈ Br ;

Existence of solutions for some classes of integro-differential equations

7

(iv) T : BC n (R+ ) −→ BC0 (R+ ) is a continuous operator such that for any x ∈ BC n (R+ ) we have k T ( x )k BC0 (R+ ) ≤ k x k BCn (R+ ) ; (4.5) (v) there exists a positive solution r0 to the inequality k pk BCn (R+ ) + λr + ψ(r ) ≤ r. Theorem 4.1. Under assumptions (i)–(v), the equation (1.2) has at least a solution in the space BC n (R+ ). Proof. First of all we define the operator F : BC n (R+ ) −→ BC n (R+ ) by Fx (t) = p(t) + q(t) x (t) +

Z t 0


g t, s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)), Tx (s) ds.

(4.6)

First, notice that the continuity of Fx (t) for any x ∈ BC n (R+ ) is obvious. Also, for any t ∈ R+ , 1 ≤ k ≤ n and by (4.2), we have i =k   dk ( Fx ) k (i ) (k) (t) = p (t) + ∑ x ( t ) q ( k −i ) ( t ) k i dt i =0

+

Z t k ∂ g 0 ∂tk


t, s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)), Tx (s) ds,

and Fx has continuous derivative of order k (1 ≤ k ≤ n). Using conditions (i)–(iv), for arbitrarily fixed t ∈ R+ , we have e−t | Fx (t)| ≤ e−t | p(t)| + e−t |q(t)|| x (t)| Z t
−t 0 (n) + e g t, s, x (ξ (s)), x (ξ (s)), . . . , x (ξ (s)), Tx (s) ds 0

≤ k pk BCn (R+ ) + kqku k x k BCn (R+ ) + ψ(k x k BCn (R+ ) ), and similarly e

−t

k i =k   d ( Fx ) k − t (i ) − t ( k ) ( k −i ) (t)| dtk (t) ≤ e | p (t)| + ∑ i e | x (t)||q i =0 Z t k   ∂ g −t 0 (n) +e t, s, x (ξ (s)), x (ξ (s)), . . . , x (ξ (s)), Tx (s) ds 0 ∂tk i =k   k (k) ≤ k p ku + ∑ kq(k−i) ku k x k BCn (R+ ) + ψ(k x k BCn (R+ ) ). i i =0

Hence

k Fx k BCn (R+ ) ≤ k pk BCn (R+ ) + λk x k BCn (R+ ) + ψ(k x k BCn (R+ ) ).

(4.7)

Due to Inequality (4.7) and using (v), the function F maps B¯ r0 into B¯ r0 . We also show that the map F is continuous. For this, take x ∈ BC n (R+ ), ε > 0 arbitrarily and consider y ∈ BC n (R+ )

R. Allahyari, R. Arab and A. Shole Haghighi

8

with k x − yk BCn (R+ ) < ε, then we obtain e−t | Fx (t) − Fy(t)| ≤ e−t |q(t)|| x (t) − y(t)| Z t −t +e g(t, s, x (ξ (s)), . . . , x (n) (ξ (s)), Tx (s)) ds 0

−

Z t 0

g(t, s, y(ξ (s)), . . . , y

(n)

(ξ (s)), Ty(s)) ds

(4.8)

≤ kqku k x − yk BCn (R+ ) Z t + e−t g(t, s, x (ξ (s)), . . . , x (n) (ξ (s)), Tx (s)) 0

− g(t, s, y(ξ (s)), . . . , y(n) (ξ (s)), Ty(s)) ds, and similarly k dk ( Fy) −t d ( Fx ) (t) − (t) ≤ e dtk dtk

i =k 

 k ∑ i kq(k−i) kkx − ykBCn (R+ ) i =0 Z t k −t d g (t, s, x (ξ (s)), . . . , x (n) (ξ (s)), Tx (s)) +e 0 dtk dk g (n) − k (t, s, y(ξ (s)), . . . , y (ξ (s)), Ty(s)) ds. dt

Furthermore, considering condition (iii), there exists T > 0 such that for t > T, we have e−t | Fx (t) − Fy(t)| ≤ kqku k x − yku + ε, and similarly e

−t

k   d ( Fx ) dk ( Fy) i=k k ( k −i ) k k x − yk BCn (R+ ) + ε. dtk (t) − dtk (t) ≤ ∑ i kq i =0

Also, if t ∈ [0, T ], then from (4.8) and (4.9), it follows that e−t | Fx (t) − Fy(t)| ≤ kqku k x − yku + Tθ T (ε), and similarly k dk ( Fy) −t d ( Fx ) e (t) − (t) ≤ dtk dtk

i =k 

∑

i =0

!  k ( k −i ) kq k k x − yk BCn (R+ ) + TϑT (ε), i

where 

θ T (ε) = sup e−t g(t, s, x0 , x1 . . . , xn+1 ) − g(t, s, y0 , y1 , . . . , yn+1 ) : t, s ∈ [0, T ],  xi , yi ∈ [−b, b], | xi − yi | ≤ ε , k kg d g d ϑT (ε) = sup e k (t, s, x0 , x1 . . . , xn+1 ) − k (t, s, y0 , y1 , . . . , yn+1 ) : t, s ∈ [0, T ], dt dt  xi , yi ∈ [−b, b], | xi − yi | ≤ ε , 

−t

b = e T (k x k + ε).

(4.9)

Existence of solutions for some classes of integro-differential equations

9

dk g

By using the continuity of g and dtk on [0, T ] × [0, T ] × [−b, b]n+2 , we have θ T (ε) −→ 0 and ϑT (ε) −→ 0 as ε −→ 0.Thus F is a continuous operator on BC n (R+ ) into BC n (R+ ). Now, let X be a nonempty and bounded subset of B¯ r0 , and assume that T > 0 and ε > 0 are arbitrary constants. Let t1 , t2 ∈ [0, T ], with |t2 − t1 | ≤ ε and x ∈ X. We obtain e−t | Fx (t1 ) − Fx (t2 )| Z t1 −t = e p ( t1 ) + q ( t1 ) x ( t1 ) + g(t1 , s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)))ds 0   Z t2 g(t2 , s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)))ds − p ( t2 ) + q ( t2 ) x ( t2 ) + 0

−t

−t

≤ e | p(t1 ) − p(t2 )| + e |q(t1 ) − q(t2 )|| x (t1 )| + |q(t2 )|| x (t1 ) − x (t2 )| Z t 1 −t +e g(t1 , s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n) (ξ (s)))ds 0 Z t2 0 (n) − g(t2 , s, x (ξ (s)), x (ξ (s)), . . . , x (ξ (s)))ds

(4.10)

0

−t

≤ e | p(t1 ) − p(t2 )| + e−t |q(t1 ) − q(t2 )|| x (t1 )| + e−t |q(t2 )|| x (t1 ) − x (t2 )| Z t 2 −t 0 (n) +e g(t1 , s, x (ξ (s)), x (ξ (s)), . . . , x (ξ (s)))ds t1 Z t 2   −t (n) (n) g(t1 , s, x (ξ (s)), . . . , x (ξ (s))) − g(t2 , s, x (ξ (s)), . . . , x (ξ (s))) ds +e 0

T

≤ ω ( p, ε) + r0 ω T (q, ε) + λω T (hx, ε) + Ur0 ε + TωrT0 ( g, ε), and similarly k dk ( Fx ) −t d ( Fx ) ( t1 ) − (t2 ) ≤ ω T ( p(k) , ε) + r0 ω T (q(k) , ε) + λ max ω T (hx (i) , ε) e k k dt dt 0≤ i ≤ k  k  d g + Wr0 ε + TωrT0 ,ε dtk

(4.11)

where 

−t



Ur0 = sup e | g(t, s, x0 , x1 . . . , xn+1 )| : t, s ∈ [0, T ], | xi | ≤ r0 , k   −t d g Wr0 = sup e k (t, s, x0 , x1 . . . , xn+1 ) : t, s ∈ [0, T ], 1 ≤ k ≤ n, | xi | ≤ r0 , dt  ωrT0 ( g, ε) = sup e−t | g(t1 , s, x0 , x1 . . . , xn+1 ) − g(t2 , s, x0 , x1 . . . , xn+1 )| : s, t1 , t2 ∈ [0, T ],  | x i | ≤ r0 , | t1 − t2 | ≤ ε , k  k   d g dk g T −t d g : ωr0 , ε = sup e ( t , s, x , x . . . , x ) − ( t , s, x , x . . . , x ) 0 2 0 1 1 n + 1 1 n + 1 dtk dtk dtk  s, t1 , t2 ∈ [0, T ], 1 ≤ k ≤ n, | xi | ≤ r0 , |t1 − t2 | ≤ ε . Since x was arbitrary element of X in (4.10) and (4.11), we obtain ω T ([ F ( X )](0) , ε) ≤ ω T ( p, ε) + r0 ω T (q, ε) + λω T ( X (0) , ε) + Ur0 ε + TωrT0 ( g, ε),

R. Allahyari, R. Arab and A. Shole Haghighi

10

and similarly T

ω ([ F ( X )]

(k)

T

, ε) ≤ ω ( p

(k)

T

, ε ) + r0 ω ( q

(k)

T

(i )

, ε) + λ max ω ( X , ε) + Wr0 ε + 0≤ i ≤ k

TωrT0



dk g ,ε dtk



for all 1 ≤ k ≤ n. Thus, by the uniform continuity of p(i) and q(i) on the compact set [0, T ] for all 0 ≤ i ≤ n, and g,

∂k g ∂tk

on the compact sets [0, T ] × [0, T ] × [−b, b]n+2 , we have ω ( p(i) , ε) −→ 0, ∂k g

ω (q(i) , ε) −→ 0, ω ( g, ε) −→ 0 and ω ( ∂tk , ε) −→ 0 as ε −→ 0. Therefore, we obtain ω0T ([ F ( X )](k) ) ≤ λ max ω0T ( X (i) ), 0≤ i ≤ k

for any 0 ≤ k ≤ n, and finally max ω0 ([ F ( X )](k) ) ≤ λ max ω0 ( X (k) ).

0≤ k ≤ n

0≤ k ≤ n

(4.12)

On the other hand, for all x, y ∈ X and t ∈ R+ we get e−t | Fx (t) − Fy(t)|

≤ e−t |q(t)|| x (t) − y(t)| Z t −t 0 (n) 0 (n) +e [ g(t, s, x (s), x (s), . . . , x (s)) − g(t, s, y(s), y (s), . . . , y (s))]ds 0 ≤ kqku diam( X (0) ) + ζ 0 (t), and similarly e

−t

k d ( Fx ) dk ( Fy) diam( X (i) ) + ζ k (t), dtk (t) − dtk (t) ≤ λ 0max ≤i ≤ k

where Z t   ζ 0 (t) = sup e g(t, s, x0 (s), . . . , xn+1 (s)) − g(t, s, y0 (s), . . . , yn+1 (s)) ds : 0  xi , yi ∈ BC (R+ ) , Z t  k   dk g d g −t : ( t, s, x ( s ) , . . . , x ( s )) − ( t, s, y ( s ) , . . . , y ( s )) ds ζ k (t) = sup e 0 0 n + 1 n + 1 dtk dtk 0  xi , yi ∈ BC (R+ ), 1 ≤ k ≤ n . 

−t

Thus diam([ FX ](k) ) ≤ λ max diam( X (i) ) + ζ k (t). 0≤ i ≤ k

(4.13)

Taking t −→ ∞ in the inequality (4.13), then using (iii) we arrive at max lim sup diam([ FX ](k) ) ≤ λ max lim sup diam( X (k) ). 0≤ k ≤ n

t−→∞

0≤ k ≤ n

t−→∞

Further, combining (4.12) and (4.14) we get n o max ω0 ([ F ( X )](k) ) + lim sup diam([ FX ](k) ) 0≤ k ≤ n t−→∞ n o (k) ≤ λ max ω0 ( X ) + lim sup diam( X (k) ) , 0≤ k ≤ n

t−→∞

(4.14)

(4.15)

Existence of solutions for some classes of integro-differential equations

11

or, equivalently µ( FX ) ≤ λµ( X ), where λ ∈ [0, 1). From Theorem 2.2 we obtain that the operator F has a fixed point x in B¯ r0 and thus the functional integral-differential equation (1.2) has at least a solution in BC n (R+ ). Corollary 4.2. Assume that the following conditions are satisfied:

(i) k : R+ × R+ −→ R and ξ : R+ −→ R+ are continuous functions such that Z ∞   −t s sup e k (t, s)e ds : t ∈ R+ ≤ 1; 0

and lim e

t−→∞

−t

Z ∞ s 0 k (t, s)e ds = 0.

Moreover, x0 , x1 , . . . , xn−1 ∈ R+ .

(ii) f 1 : R+ × Rn+1 −→ R is continuous and there exist continuous functions a, b : R+ −→ R+ such that | f 1 (t, x0 , x1 , . . . , xn )| ≤ a(t)b( max | xi |). (4.16) 0≤ i ≤ n

Moreover, there exists a positive constant D such that Z t   −t k sup e a(s)(t − s) ds : t ∈ R+ , 0 ≤ k ≤ n − 1 ≤ D, 0

and

 lim sup e

−t

t−→∞

Z t  a(s)(t − s)k ds : 0 ≤ k ≤ n − 1 = 0. 0

(iii) f 2 : R+ × Rn −→ R is continuous such that | f 2 (t, x0 , x1 , . . . , xn−1 )| ≤ max | xi |. 0≤ i ≤ n −1

(4.17)

(iv) There exists a positive solution r0 to the inequality Db(r ) ≤ r. Then the functional integro-differential equation (1.3) has at least a solution in the space BC n (R+ ). Proof. It is easy to see that Eq. (1.3) has at least one solution in the space BC n (R+ ) if and only if equation x n −1 n −1 t ( n − 1) ! ( t − s ) n −1  f s, x (ξ (s)), x 0 (ξ (s)), . . . , x (n−1) (ξ (s)), ( n − 1) ! 1

x ( t ) = x0 + x1 t + · · · +

+

Z t 0

Z ∞ 0

 k (s, v) f 2 (v, x (v), x 0 (v), . . . , x (n−1) (v))dv ds

(4.18)

R. Allahyari, R. Arab and A. Shole Haghighi

12

has at least a solution in the space BC n−1 (R+ ). Eq. (4.18) is a special case of Eq. (1.2) where g(t, s, x0 , x1 , x2 , . . . , xn ) = Tx (t) =

( t − s ) n −1 f (s, x0 , x1 , x2 , . . . , xn ), ( n − 1) ! 1 Z ∞ 0

k (t, s) f 2 (s, x (s), x 0 (s), . . . , x (n−1) (s)) ds.

p ( t ) = x0 + x1 t + . . . +

x n −1 n −1 t , ( n − 1) !

q(t) = 0 From the definitions of p, q and ξ, hypotheses (i) and (ii) of Theorem 4.1 obviously are satisfied with λ = 0. Also we have   ( t − t ) n −1 sup{| g(t, t, x0 , . . . , xn )| : t ∈ R+ , xi ∈ R} = sup f (t, x0 , . . . , xn ) : t ∈ R+ , xi ∈ R ( n − 1) ! 1

= 0, and similarly   k ∂ g sup k (t, t, x0 , x1 , . . . , xn ) : t ∈ R+ , xi ∈ R, 1 ≤ k ≤ n = 0. ∂t Now, assume that r > 0 is an arbitrary constant. Let xi ∈ R+ with k xi ku ≤ r, ψ(r ) = Db(r ), we obtain Z t Z t  
( t − s ) n −1 a(s)b max {| xi (ξ (s))|} ds e−t g t, s, x0 (ξ (s)), . . . , xn (ξ (s)) ds ≤ e−t 0≤ i ≤ n 0 0 ( n − 1) ! D b (r ) ≤ ψ (r ), ≤ ( n − 1) ! and similarly Z t k
∂ g D e t, s, x0 (ξ (s)), . . . , xn (ξ (s)) ds ≤ b (r ) ≤ ψ (r ). k ( n − k − 1) ! 0 ∂t −t

Thus, we conclude that the function g satisfies condition (4.3). Moreover, for any r ∈ R+ Z t 

 −t lim e g t, s, x0 (ξ (s)), . . . , xn (ξ (s)) − g t, s, y0 (ξ (s)), . . . , yn (ξ (s)) ds t→∞

0

≤ lim e−t t→∞

≤ lim e

Z t  0



 g t, s, x0 (ξ (s)), . . . , xn (ξ (s)) + g t, s, y0 (ξ (s)), . . . , yn (ξ (s)) ds

Z t ( t − s ) n −1 −t

h    i a(s) b max {| xi (ξ (s))|} + b max {|yi (ξ (s))|} ds

( n − 1) ! 0≤ i ≤ n Z t n − 1 (t − s) ≤ lim 2b(r )e−t a(s) ds t→∞ 0 ( n − 1) ! = 0, t→∞

0

0≤ i ≤ n

and similarly Z t  k 
∂k g
∂ g =0 lim e−t t, s, x ( ξ ( s )) , . . . , x ( ξ ( s )) − t, s, y ( ξ ( s )) , . . . , y ( ξ ( s )) ds n n 0 0 t→∞ ∂tk ∂tk 0

Existence of solutions for some classes of integro-differential equations

13

uniformly with respect to xi , yi ∈ B¯ r . Hence, g satisfies the condition (4.4) and hypothesis (iii) of Theorem 4.1. Next, hypothesis (iv) implies that condition (v) of Theorem 4.1 holds. To finish the proof we only need to verify that T is continuous. For this, take x ∈ BC n (R+ ) and ε > 0 arbitrarily, and consider y ∈ BC n (R+ ) with k x − yk BCn (R+ ) < ε. Then, considering condition (i), there exists T > 0 such that for t > T we obtain e−t | Tx (t) − Ty(t)| Z ∞

k (t, s) f 2 (s, x (s), x 0 (s), . . . , x (n−1) (s)) − f 2 (s, y(s), y0 (s), . . . , y(n−1) (s)) ds Z0 ∞ h −t ≤e k (t, s) | f 2 (s, x (s), x 0 (s), . . . , x (n−1) (s))| 0 (4.19) i + | f 2 (s, y(s), y0 (s), . . . , y(n−1) (s))| ds

≤e

−t

≤ 2(k x k BCn−1 (R+ ) + ε)e−t

Z ∞

k (t, s)es ds

0

≤ 2(k x k BCn−1 (R+ ) + ε)ε Also if t ∈ [0, T ], then the first inequality in (4.19) follows that e−t | Tx (t) − Ty(t)| ≤ ϑ (ε),

(4.20)

where n ϑ (ε) = | f 2 (t, x0 , x1 , . . . , xn−1 ) − f 2 (t, y0 , y1 , . . . , yn−1 )| : t ∈ [0, T ], o xi , yi ∈ [−q x , q x ], | xi − yi | ≤ ε , with q x = e T (k x k + ε). By using the continuity of g on the compact set [0, T ] × [−q x , q x ]n , we have ϑ (ε) −→ 0 as ε −→ 0. Thus from (4.20) we infer that T is a continuous function. Moreover by hypothesis (i) we easily obtain that

k T ( x )k BC0 (R+ ) ≤ k x k BCn (R+ ) , and complete the proof. Corollary 4.3. Assume that the following conditions are satisfied:

(i) k : R+ × R+ −→ R+ and ξ : R+ −→ R+ are continuous functions such that Z t   −t s sup e k (t, s)e ds : t ∈ R+ ≤ 1; 0

and lim e

t−→∞

−t

Z ∞ s 0 k (t, s)e ds = 0.

Moreover, x0 , x1 , . . . , xn−1 ∈ R+

(ii) f 1 : R+ × Rn+1 −→ R is continuous and there exist continuous functions a, b : R+ −→ R+ such that   | f 1 (t, x0 , x1 , . . . , xn )| ≤ a(t)b max | xi | . 0≤ i ≤ n

R. Allahyari, R. Arab and A. Shole Haghighi

14

Moreover, there exists a positive constant D such that Z t   −t k sup e a(s)(t − s) ds : t ∈ R+ , 0 ≤ k ≤ n − 1 ≤ D 0

and

 lim

t−→∞

Z t  e−t a(s)(t − s)k ds : 0 ≤ k ≤ n − 1 = 0. 0

(iii) f 2 : R+ × Rn −→ R is continuous such that | f 2 (t, x0 , x1 , . . . , xn−1 )| ≤ max | xi |. 0≤ i ≤ n −1

(iv) There exists a positive solution r0 to the inequality Db(r ) ≤ r. Then the functional integro-differential equation (1.4) has at least a solution in the space BC n (R+ ). Proof. Since the proof of this Corollary can be completed essentially on the line of the proof of Corollary 4.2, hence details are omitted. Example 4.4. Consider the following functional integral-differential equation x ( t ) = e − t −3 x ( t )

+

√ √ √ Z t (t − s)3 ln(| x ( s)| + 1) + sin((t − s)7 ) tanh( x 0 ( s)) + (t − s)3 sin( x 00 ( s)) √ 2 + sin( x (3) ( s))

0

ds.

(4.21) Eq. (4.21) is a special case of Eq. (1.2) with q ( t ) = e − t −3 ,

p(t) = 0, g(t, s, x0 , x1 , x2 , x3 , x4 ) =

(t

− s)3 ln(| x

ξ (t) =

0 | + 1) + sin(( t

√

t,

T = 0,

− s)7 ) tanh( x

1) +

2 + sin( x3 )

(t − s)3 sin( x2 )

.

It is easy to see that p, q ∈ BC3 (R+ ) and λ = e83 . Also, g is continuous and has a continuous derivative of order 3 with respect to the first argument such that condition (4.2) is satified. By simple calculation we obtain that e−t | g(t, s, x0 , x1 , x2 , x3 , x4 )| −t ∂g e (t, s, x0 , x1 , x2 , x3 , x4 ) ∂t 2 ∂ g e−t 2 (t, s, x0 , x1 , x2 , x3 , x4 ) ∂t 3 −t ∂ g e 3 (t, s, x0 , x1 , x2 , x3 , x4 ) ∂t

≤ e−t (2t3 + 1)φ(max{ xi }), ≤ e−t (6t2 + 7t6 )φ(max{ xi }), ≤ e−t (12t + 42t5 + 49t12 )φ(max{ xi }), ≤ e−t (12 + 210t4 + 294t12 + 343t18 + 588t11 )φ(max{ xi }),

where φ(t) = max{ln(t + 1), sin t, tanh t}. If we define ψ(t) = 1447φ(t). Then, conditions (4.3) and (4.4) hold. On the other hand, since T = 0, then condition (iv) of Theorem 4.1 is correct.

Existence of solutions for some classes of integro-differential equations

15

Also, it is easy to see that each number r ≥ 25000 satisfies the inequality in condition (v) of Theorem 4.1, i.e., 8 k pk + λr + ψ(r ) ≤ 0 + 3 r + ψ(r ) ≤ r. e Thus, as the number r0 we can take r0 = 25000. Consequently, all the conditions of Theorem 4.1 are satisfied. Hence the functional integral-differential equation (4.21) has at least a solution which belongs to the space BC3 (R+ ) . Example 4.5. Consider the following functional integro-differential equation x (5) ( t ) =

1 −2t 2 1 4 e x (t ) + ∑ e−(k+1)t x (k) (t2 ) 2 2 k =1   Z ∞ −s 2 4 e ( t + 1) 1 x (i ) + e−2t x ( s ) tanh ( s ) + ∑ si + 4 ds, 10 s2 + 4 0 i =1

(4.22)

u(0) = 0, u0 (0) = 1, u00 (0) = 2, u(3) (0) = 3, u(4) (0) = 4. Eq. (4.22) is a special case of Eq. (1.3) with 1 4 1 1 −2t e x0 + ∑ e−(k+1)t xk + e−2t x5 , 2 2 k =1 2 " # 4 1 xi f 2 (t, x0 , x1 , x2 , . . . , x4 ) = x0 tanh(t) + ∑ i , 5 t +1 i =1 f 1 (t, x0 , x1 , x2 , . . . , x5 ) =

k(t, s) =

e − s ( t2 + 1) , s2 + 4

It is easy to see that k and ξ are continuous, Z ∞ Z    −t s −t sup e k (t, s)e ds : t ∈ R+ ≤ sup e 0

and lim e

t−→∞

−t

ξ ( t ) = t2 .

∞ t2 0

 π + 1 ds : t ∈ R+ ≤ ≤1 2 s +4 4

Z ∞ Z ∞ 2 t + 1 s −t lim e ds = 0. 0 k (t, s)e ds ≤ t−→ 2 ∞ 0 s +4

Then, condition (i) of Corollary 4.2 is valid. Also, f 1 is continuous and satisfying the inequality (4.3) of Corollary 4.2, where a(t) = e−2t and b(t) = 12 t. Moreover, we deduce Z t   −t k sup e a(s)(t − s) ds : t ∈ R+ , 0 ≤ k ≤ 4 0

≤ sup{e−t (t4 + t3 + t2 + t + 1 + e−2t ) : t ∈ R+ } ≤2 and  lim sup e

t−→∞

−t

Z t  a(s)(t − s)k ds : 0 ≤ k ≤ 4 = lim e−t (t4 + t3 + t2 + t + 1 + e−2t ) = 0. 0 t−→∞

Now with choosing D = 2 the condition (ii) of Corollary 4.2 is satisfied. Next, f 2 is continuous and " # 4 1 xi f 2 (t, x0 , x1 , x2 , . . . , x4 ) = x0 tanh(t) + ∑ i ≤ max | xi |. 5 t +1 0≤ i ≤4 i =1

16

R. Allahyari, R. Arab and A. Shole Haghighi

Hence, the condition (iii) of Corollary 4.2 holds. It is easy to see that each number r ≥ 1 satisfies the inequality in condition (iv) of Corollary 4.2, i.e., 1 Db(r ) = 2 r ≤ r. 2 Thus, as the number r0 we can take r0 = 1. Consequently, all the conditions of Corollary 4.2 are satisfied. This shows that the functional integro-differential equation (1.3) has a solution which belongs to the space BC5 (R+ ).

References [1] R. P. Agarwal, M. Benchohra, D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55(2009), No. 3–4, 221–230. MR2571191 [2] R. Agarwal, M. Meehan, D. O’Regan, Fixed point theory and applications, Cambridge Univ. Press, 2004. url [3] A. Aghajani, J. Bana´s, Y. Jalilian, Existence of solutions for a class of nonlinear Volterra singular integralm equations, Comput. Math. Appl. 62(2011), 1215–1227. MR2824709 [4] A. Aghajani, Y. Jalilian, Existence and global attractivity of solutions of a nonlinear functional integral equation, Commun. Nonlinear Sci. numer. Simul. 15(2010), 3306–3312. MR2646160 [5] R. R. Akmerov, M. I. Kamenski, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii, Measures of noncompactness and condensing operators, Birkhäuser Verlag, Basel, 1992. MR1153247 [6] A. Ayad, Spline approximation for first order Fredholm delay integro-differential equations, Int. J. Comput. Math. 70(1999), No. 3, 467–476. MR1712663 [7] A. Ayad, Spline approximation for first order Fredholm integro-differential equations, Studia Univ. Babes, –Bolyai Math. 41(1996), No. 3, 1–8. MR1644426 [8] J. Bana´s, Measures of noncompactness in the study of solutions of nonlinear differential and integral equations, Cent. Eur. J. Math. 10(2012), No. 6, 2003–2011. MR2983142 [9] J. Bana´s, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Dekker, New York, 1980. MR0591679 [10] J. Bana´s, D. O’Regan, K. Sadarangani, On solutions of a quadratic Hammerstein integral equation on an unbounded interval, Dynam. Systems Appl. 18(2009), 251–264. MR2543230 [11] J. Bana´s, R. Rzepka, An application of a measure of noncompactness in the study of asymptotic stability, Appl. Math. Lett. 16(2003), No. 1, 1–6. MR1938185 [12] S. H. Behiry, H. Hashish, Wavelet methods for the numerical solution of Fredholm integro-differential equations, Int. J. Appl. Math. 11(2002), No. 1, 27–35. MR1956354

Existence of solutions for some classes of integro-differential equations

17

ˇ , , S. Mures, an, Application of a trapezoid inequality to neutral [13] A. M. Bica, V. A. Caus Fredholm integro-differential equations in Banach spaces, JIPAM. J. Inequal. Pure Appl. Math. 7(2006), Article 173. MR2268628 [14] F. Bloom, Asymptotic bounds for solutions to a system of damped integro-differential equations of electromagnetic theory, J. Math. Anal. Appl. 73(1980), No. 2, 524–542. MR0564002 [15] M. A. Darwish, On monotonic solutions of a singular quadratic integral equation with supremum, Dynam. Systems Appl. 17(2008), 539–549. MR2569518 [16] M. A. Darwish, J. Henderson, D. O’Regan, Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument, Bull. Korean Math. Soc. 48(2011), 539–553. MR2827764 [17] B. C. Dhage, S. S. Bellale, Local asymptotic stability for nonlinear quadratic functional integral equations, Electron. J. Qual. Theory Differ. Equ. 2008, No. 10, 1–13. MR2385414 [18] L. K. Forbes, S. Crozier, D. M. Doddrell, Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math. 57(1997), No. 2, 401–425. MR1438760 [19] L. A. Garey, C. J. Gladwin, Direct numerical methods for first order Fredholm integrodifferential equations, Int. J. Comput. Math. 34(1990), No. 3–4, 237–246. url [20] D. Guo, Existence of solutions for nth-order integro-differential equations in Banach spaces, Comput. Math. Appl. 41(2001), No. 5–6, 597–606. MR1822586 [21] K. Holmaker, Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal. 24(1993), No. 1, 116–128. MR1199530 [22] S. M. Hosseini, S. Shahmorad, Tau numerical solution of Fredholm integro-differential equations with arbitrary polynomial base, Appl. Math. Modelling 27(2003), No. 2, 145–154. url [23] R. P. Kanwal, Linear integral differential equations theory and technique, Academic Press, New York, 1971. MR2986176 [24] K. Kuratowski, Sur les espaces complets (in French), Fund. Math. 15(1930), 301–309. url [25] L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. 309(2005), No. 2, 638–649. MR2154141 [26] J. Mallet-Paret, R. D. Nussbaum, Inequivalent measures of noncompactness and the radius of the essential spectrum, Proc. Amer. Math. Soc. 139(2010), 917–930. MR2745644 [27] G. Micula, G. Fairweather, Direct numerical spline methods for first order Fredholm integro-differential equations, Rev. Anal. Numér. Théor. Approx. 22(1993), No. 1, 59–66. MR1621548

18

R. Allahyari, R. Arab and A. Shole Haghighi

[28] M. Mursaleen, S. A. Mohiuddine, Applications of noncompactness to the infinite system of differential equations in ` p spaces, Nonlinear Anal. 75(2012), No. 4, 2111–2115. url [29] L. Olszowy, Solvability of infinite systems of singular integral equations in Fréchet space of continuous functions, Comp. Math. Appl. 59(2010), 2794–2801. MR2607984 [30] B. G. Pachpatte, On Fredholm type integro-differential equation, Tamkang J. Math. 39(2008), 85–94. MR2416183 [31] J. Pour Mahmoud, M. Y. Rahimi-Ardabili, S. Shahmorad, Numerical solution of the system of Fredholm integro-differential equations by the tau method, Appl. Math. Comput. 168(2005), No. 1, 465–478. MR2170845 [32] H. Qiya, Interpolation correction for collocation solutions of Fredholm integro-differential equations, Math. Comput. 67(1998), No. 223, 987–999. MR1464145