Integrable solutions of a nonlinear functional integral equation on an unbounded interval

Nonlinear Analysis 71 (2009) 4131–4136

Contents lists available at ScienceDirect

Nonlinear Analysis journal homepage: www.elsevier.com/locate/na

Integrable solutions of a nonlinear functional integral equation on an unbounded interval Mohamed Aziz Taoudi ∗ Department of Mathematics and Computer Science, Gabes University, Cité Erriadh, Zrig, 6072, Gabes, Tunisia

article

abstract

info

Article history: Received 22 December 2008 Accepted 18 February 2009

In this paper we prove the existence of integrable solutions of a generalized functionalintegral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included. © 2009 Elsevier Ltd. All rights reserved.

MSC: 47H10 47H30 Keywords: Fixed point theorem Measure of weak noncompactness Functional integral equation Superposition operator

1. Introduction Integral equations have many useful applications in describing numerous problems of the real world. They have arisen in many branches of science [1,2], such as in the theory of optimal control, economics etc. The purpose of this paper is to consider the existence of integrable solutions for the following functional-integral equation x(t ) = g (t , x(t )) + f1

 Z t  t, k(t , s)f2 (s, x(s))ds ,

t ∈ R+ ,

(1.1)

0

where f1 , f2 , k and g are given measurable functions while x ∈ L1 (R+ ) is an unknown function. This equation includes many key integral and functional equations that arise in nonlinear analysis and its applications such as the classical Volterra or Hammerstein–Volterra integral equations. Recently, Banaś and Chlebowicz [3] study the solvability of Eq. (1.1) in the special case where g ≡ 0. The methods used in [3], based on the conjunction of the technique of measures of weak noncompactness with the Schauder fixed point principle, do not apply directly in our more general context. This is due mainly to the lack of compactness related to the presence of the perturbation term g. To overcome this difficulty we establish a new fixed point theorem of Krasnosl’skii type, which is an improvement of a deep result recently proved in [4]. This result applies directly to solve the functional-integral equation (1.1) (see Theorem 4.1). The paper is organized in four sections, including the introduction. Some preliminaries, notations and auxiliary facts are presented in Section 2. In Section 3, a new fixed point theorem of Krasnoselskii type is proved. An existence theory for the functional-integral equation (1.1) is the topic of Section 4. An example is given at the end of the last section to illustrate the applicability of Theorem 4.1 and to show that our result is more general than the corresponding one of [3].

∗

Tel.: +216 21033803. E-mail addresses: [email protected], [email protected].

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.02.072

4132

M. Aziz Taoudi / Nonlinear Analysis 71 (2009) 4131–4136

2. Notations and auxiliary facts Let R be the field of real numbers, R+ be the interval [0, ∞[ and L1 (I ) be the space of Lebesgue integrable functions on a measurable subset I of R, endowed with the standard norm

Z kxkL1 (I ) =

|x(t )|dt .

I

Following [3], The space L1 (R+ ) will be shortly denoted by L1 . Now, we make a short note about the so-called superposition operator, which is one of the simplest and most important operator that is investigated in nonlinear functional analysis (see [5]). Let I be an interval of R bounded or not. Consider a function f (t , x) = f : I × R → R. We say that f satisfies Carathéodory conditions if it is measurable in t for any x ∈ R and continuous in x for almost all t ∈ I. Then to every function x(t ) being measurable on I we may assign the function (Nf x)(t ) = f (t , x(t )), t ∈ I. The operator Nf defined in such a way is called the superposition operator generated by the function f . The theory concerning superposition operators is presented in [5]. We recall the following result due to Krasnosel’skii [6], which states a basic fact for these operators in L1 -spaces. Theorem 2.1. The superposition operator F maps continuously the space L1 (I ) into itself if and only if

|f (t , x)| ≤ a(t ) + b|x|, for all t ∈ I and x ∈ R, where a(t ) ∈ L1 (I ) and b ≥ 0. Also, we recall the following criterion for weak noncompactness due to Dieudonné [7,8] which is of fundamental importance in our subsequent analysis. Theorem 2.2. A bounded set S is relatively weakly compact in L1 if Rand only if the following two conditions are satisfied: (ı) for any ε > 0 there exists δ > 0 such that if meas(D) ≤ δ then D |x(t )|dt ≤ ε for all x ∈ X .

(ıı) for any ε > 0 there is τ > 0 such that

R∞ τ

|x(t )|dt ≤ ε for any x ∈ X .

3. Fixed point theorems Let X be a Banach space, let B (X ) denote the collection of all nonempty bounded subsets of X and W (X ) the subset of B (X ) consisting of all relatively weakly compact subsets of X . Finally, let Br denote the closed ball centered at 0 with radius r . Definition 3.1 ([9]). A function µ : B (X ) → R+ is said to be a measure of weak noncompactness if it satisfies the following conditions: 1. The family ker(µ) = {M ∈ B (X ) : µ(M ) = 0} is nonempty and ker(µ) ⊂ W (X ). 2. M1 ⊂ M2 ⇒ µ(M1 ) ⊂ µ(M2 ). 3. µ(co(M )) = µ(M ), where co(M ) is the convex hull of M . 4. µ(λM1 + (1 − λ)M2 ) ≤ λµ(M1 ) + (1 − λ)µ(M2 ) for λ ∈ [0, 1]. 5. If (Mn )n≥1 is a sequence of nonempty,Tweakly closed subsets of X with M1 bounded and M1 ⊇ M2 ⊇ · · · ⊇ Mn ⊇ · · · such ∞ that limn→∞ µ(Mn ) = 0, then M∞ := n=1 Mn is nonempty. The first important example of measure of weak noncompactness has been defined by De Blasi (see [10]). Generally, it is rather difficult to express the De Blasi measure of weak noncompactness with a convenient formula in a concrete Banach space. Such a formula is only known in the case of the space L1 (I ) where I is a bounded subinterval of R (see [11]). Based on a criterion of weak noncompactness due to Dieudonné [7], Banas and Knap [12] constructed a useful measure of weak noncompactness in L1 = L1 (R+ ) as follows: For a nonempty and bounded subset M of the space L1 we put

µ(M ) = c (M ) + d(M ),

(3.1)

where





c (M ) = lim sup sup →0

Z

x∈X

|x(t )|dt : D ⊂ R+ , meas(D) ≤ 



,

(3.2)

D

and d(M ) = lim

τ →∞



Z sup τ

∞

|x(t )|dt : x ∈ X



.

It was shown [12] that the function µ is a measure of weak noncompactness in the space L1 . For further purposes, let us recall the following definitions and results.

(3.3)

M. Aziz Taoudi / Nonlinear Analysis 71 (2009) 4131–4136

4133

Definition 3.2 ([13]). Let (X , d) be a metric space. We say that B : X → X is a large contraction if d(Bx, By) < d(x, y) for x, y ∈ X with x 6= y and if ∀ > 0 ∃δ < 1 such that [x, y ∈ X , d(x, y) ≥ ] ⇒ d(Bx, By) ≤ δ d(x, y). Definition 3.3 ([14]). Let (X , d) be a metric space. We say that B : X → X is a separate contraction if there exist two functions ϕ, ψ : R+ → R+ satisfying (1) ψ(0) = 0, ψ is strictly increasing, (2) d(Bx, By) ≤ ϕ(d(x, y)), (3) ψ(r ) + ϕ(r ) ≤ r for r > 0. Remark 1. (1) It is easy to see that every contraction is a separate contraction. (2) In [14], it was shown that every large contraction is a separate contraction. The converse is not true. Some examples of separate contractions that are not large contractions can be found in [15]. In what follows, the following lemmas will play a crucial role. Lemma 3.4 ([14, Lemma 1.2]). Let X be a normed space. If S ⊂ X and if B is a separate contraction mapping, then (I − B) is a homeomorphism of S onto (I − B)S . Lemma 3.5 ([14, Theorem 2.1]). Let (X , d) be a complete metric space and B : X → X a separate contraction. Then B has a unique fixed point in X . Before we state the main result of this section, we recall the following definition. Definition 3.6 ([16]). Let M be a subset of a Banach space X . A continuous map A : M → X is said to be (ws)- compact if for any weakly convergent sequence (xn )n∈N in M the sequence (Axn )n∈N has a strongly convergent subsequence in X . Remark 2. (1) Some concrete examples of (ws)-operators can be found in [17,4]. (2) A map A is (ws)-compact if and only if it maps relatively weakly compact sets into relatively compact ones (it follows immediately from Definition 3.6). Now we are ready to state and prove the following result which is an improvement of [4, Theorem 2.1.]. Theorem 3.7. Let M be a nonempty bounded closed convex subset of a Banach space X . Suppose that A : M −→ X and B : M −→ X such that: (i) A is (ws)-compact; (ii) there exists γ ∈ [0, 1[ such that µ(AS + BS ) ≤ γ µ(S ) for all S ⊆ M ; Here µ is an arbitrary measure of weak noncompactness on X ; (iii) B is a separate contraction; (iv) AM + BM ⊆ M . Then there is x ∈ M such that Ax + Bx = x. Proof. Let y be fixed in M, the map which assigns to each x ∈ M the value Bx + Ay defines a separate contraction from M into M. So, using Lemma 3.5 together with Lemma 3.4 and the assumption (iii), the equation x = Bx + Ay has a unique solution x = (I − B)−1 Ay in M . Therefore (I − B)−1 A(M ) ⊂ M . The rest of the proof runs as in [4, Theorem 2.1.].  Remark 3. (1) Theorem 3.7 remains valid if we replace the condition AM + BM ⊆ M by the weaker one (x = Bx + Ay, y ∈ M) ⇒ x ∈ M introduced in [18]. (2) Unlike [4, Theorem 2.1.], the result of Theorem 3.7 holds for an arbitrary measure of weak noncompactness and demands only that B belongs to the class of separate contractions which contains strictly the class of strict contractions. (3) In case B = 0, Theorem 3.7 corresponds to an improvement of [17, Theorem 2.2.]. 4. Existence theory In this section, we are concerned with the solvability of the functional integral equation (1.1). This equation will be studied under the following assumptions: (a) The function g : R+ × R → R satisfies Carathéodory conditions and there are a constant α > 0 and a function β ∈ L1 such that |g (t , x)| ≤ β(t ) + α|x|, for t ∈ R+ and for x ∈ R. (b) The function g is a separate contraction with respect to the second variable. (c) The functions fi : R+ × R → R satisfy Carathéodory conditions and there exist a constant bi > 0 and a function ai ∈ L1 such that |fi (t , x)| ≤ ai (t ) + bi |x|, for t ∈ R+ and for x ∈ R (i = 1, 2).

4134

M. Aziz Taoudi / Nonlinear Analysis 71 (2009) 4131–4136

(d) The function κ : R+ × R+ → R satisfies Carathéodory conditions and the linear Volterra integral operator K defined by Z t κ(t , s)x(s)ds (Kx)(t ) = 0

transform the space L1 into itself (and then K is continuous [19]). (e) α + b1 b2 kK k < 1 (the constant kK k denotes the norm of the operator K ). Our existence result for Eq. (1.1) is as follows Theorem 4.1. Let the assumptions (a) –(e) be satisfied. Then Eq. (1.1) has at least one solution x ∈ L1 . Proof. First of all, observe that the problem (1.1) may be written in the form x = Ax + Bx where B := Ng is the superposition operator associated to the function g (., .) and A = Nf1 KNf2 where Nfi is the superposition operator associated to fi , i = 1, 2 and K is the linear integral operator defined in assumption (d). Also, note that for a given x ∈ L1 the function Ax + Bx belongs to L1 which is a consequence of the assumptions (a), (c) and (d). Our strategy is to apply Theorem 3.7 in order to find a fixed point for the operator A + B in L1 . The proof will be given in several steps. Step 1: A is (ws)-compact. The continuity of A follows from assumptions (a), (c), (d) and Theorem 2.1. Arguing exactly in the same way as in [3, p. 6] we can prove that A is (ws)-compact. Step 2: There exists γ ∈ [0, 1[ such that µ(AS + BS ) ≤ γ µ(S ) for all bounded subset S of L1 , where µ is the measure of weak noncompactness defined by (3.1). Take an arbitrary number ε > 0 and a nonempty subset D of R+ such that D is measurable and meas(D) ≤ ε . Then for any x ∈ S we have:

Z

|(Ax)(t )|dt ≤ D

= ≤ = ≤ ≤

Z

Z Z Z β(t )dt + α |x(t )|dt + a1 (t )dt + b1 (KNf2 x)(t ) dt D D ZD Z D β(t )dt + a1 (t )dt + αkxkL1 (D) + b1 kKNf2 xkL1 (D) ZD ZD β(t )dt + a1 (t )dt + αkxkL1 (D) + b1 kK kkNf2 xkL1 (D) ZD ZD Z β(t )dt + a1 (t )dt + αkxkL1 (D) + b1 kK k |f2 (t , x(t ))|dt ZD ZD ZD β(t )dt + a1 (t )dt + αkxkL1 (D) + b1 kK k (a2 (t ) + b2 |x(t )|) dt D ZD ZD Z Z β(t )dt + a1 (t )dt + b1 kK k a2 (t )dt + (α + b1 b2 kK k) |x(t )|dt . D

D

D

D

Taking into account the fact that the set consisting of one element is weakly compact, the use of (3.2) and Theorem 2.2 leads to c (AS ) ≤ (α + b1 b2 kK k)c (S ).

(4.1)

Next, let τ > 0 be a fixed. Then for any x ∈ S we have: ∞

Z τ

|(Ax)(t )|dt ≤ = ≤ = ≤

≤

Z ∞ Z ∞ Z ∞ (KNf x)(t ) dt β(t )dt + α |x(t )|dt + a1 (t )dt + b1 2 τ τ Zτ ∞ Z ∞τ β(t )dt + a1 (t )dt + αkxkL1 ([τ ,∞)) + b1 kKNf2 xkL1 ([τ ,∞)) Zτ ∞ Zτ ∞ β(t )dt + a1 (t )dt + αkxkL1 ([τ ,∞)) + b1 kK kkNf2 xkL1 ([τ ,∞)) Zτ ∞ Zτ ∞ Z ∞ β(t )dt + a1 (t )dt + αkxkL1 ([τ ,∞)) + b1 kK k |f2 (t , x(t ))|dt τ Zτ ∞ Zτ ∞ β(t )dt + a1 (t )dt + αkxkL1 ([τ ,∞)) τ Z ∞ τ + b1 kK k (a2 (t ) + b2 |x(t )|) dt τ Z ∞ Z ∞ Z ∞ Z ∞ β(t )dt + a1 (t )dt + b1 kK k a2 (t )dt + (α + b1 b2 kK k) |x(t )|dt . Z

τ

∞

τ

τ

τ

M. Aziz Taoudi / Nonlinear Analysis 71 (2009) 4131–4136

4135

In view of (3.3) and Theorem 2.2 we obtain d(AS ) ≤ (α + b1 b2 kK k)d(S ).

(4.2)

Combining (4.1) and (4.2) we arrive at

µ(AS ) ≤ γ µ(S ),

(4.3)

where γ := α + b1 b2 kK k. By hypothesis we have γ ∈ [0, 1[. Step 3: There is a positive number r > 0 such that A(Br ) + B(Br ) ⊆ Br . Let x and y be an arbitrary functions in L1 . In view of our assumptions we get ∞

  Z t dt g (t , y(t )) + f1 t , κ( t , s ) f ( s , x ( s )) ds 2 0 0  Z t Z ∞ β(t ) + α|y(t )| + a1 (t ) + b1 κ(t , s)f2 (s, x(s))ds dt ≤ Z

kAx + Byk =

0

0

= kβk + αkyk + ka1 k + b1 kKNf2 xk Z ∞ ≤ kβk + αkyk + ka1 k + b1 kK k |f2 (t , x(t ))|dt Z0 ∞ ≤ kβk + αkyk + ka1 k + b1 kK k (a2 (t ) + b2 kx(t )k) dt 0

≤ kβk + αkyk + ka1 k + b1 kK kka2 k + b1 b2 kK kkxk = αkyk + b1 b2 kK kkxk + kβk + ka1 k + b1 kK kka2 k. Let r be the real defined by r =

kβk + ka1 k + b1 kK kka2 k . 1 − α − b1 b2 kK k

The hypothesis (e) ensures that r > 0. Clearly the last estimate guarantees that A(Br ) + B(Br ) ⊆ Br . Next, from hypothesis (b) it follows that B is a separate contraction. Thus the hypotheses of Theorem 3.7 are fulfilled. This gives a fixed point for A + B and hence an integrable solution to Eq. (1.1).  If we take g ≡ 0 in Theorem 4.1, we get the following result which was proved in [3]. Corollary 4.2. Let the assumptions (c) –(e) be satisfied. Then the equation

 Z t  x(t ) = f1 t , k(t , s)f2 (s, x(s))ds ,

t ∈ R+

0

has at least one solution x ∈ L1 . Remark 4. In [12], it was shown that if ess sups≥0

R∞ s

|κ(t , s)|dt < ∞ then the linear Volterra integral operator K defined in R∞ |κ(t , s)|dt . s

Assumption (d) maps L1 into itself and the norm kK k of this operator is majorized by the number ess sups≥0 Example 4.3. Consider that following Volterra integral equation x(t ) =

1

1

+ x( t ) +

t2 + 1

4

t

Z

e−(t +s) sin2 (t )



1 s2 + 1

0

+

s 2s + 1

where t ∈ R+ . It is easily seen that Eq. (4.4) is a particular case of Eq. (1.1), where g (t , x) =

1 t2 + 1

1

+ x, 4

f1 (t , x) = x,

κ(t , s) = e−(t +s) sin2 (t ) and f2 (t , x) =

1 t2 + 1

+

Easy computations show that +∞

Z

|κ(t , s)|dt = s

1 5

sin2 (s) + sin(2s) + 2 e−2s




and therefore +∞

Z

|κ(t , s)|dt =

kK k ≤ sup s ≥0

s

2 5

.

t 2t + 1



x(s) sin(x(s)) ds,

x sin x.

(4.4)

4136

M. Aziz Taoudi / Nonlinear Analysis 71 (2009) 4131–4136

Besides, the functions g (t , x), f1 (t , x) and f2 (t , x) satisfy the assumptions of Theorem 4.1, with a1 (t ) = 1, a2 (t ) = 1 , β(t ) = t 21+1 , b1 = 1, b2 = 21 and α = 41 . Obviously, t 2 +1

α + b1 b2 kK k ≤

1 4

+

2 5

< 1.

Now, applying Theorem 4.1 we infer that Eq. (4.4) has at least one solution in the space L1 . References [1] C. Corduneanu, Integral Equations and Applications, Cambridge Univ. Press, Cambridge, 1991. [2] K. Deimling, Nonlinear Functional Analysis, Springer Verlag, Berlin, 1985. [3] J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020. [4] K. Latrach, M.A. Taoudi, Existence results for a generalized nonlinear Hammerstein equation on L1 spaces, Nonlinear Anal. 66 (2007) 2325–2333. [5] J. Appell, Zabrejko, Nonlinear Superposition Operators, in: Cambridge Tracts in Mathematics, vol. 95, Cambridge University Press, 1990. [6] M.A. Krasnosel’skii, On the continuity of the operator Fu(x) = f (x, u(x)), Dokl. Akad. Nauk SSSR 77 (1951) 185–188 (in Russian). [7] J. Dieudonné, Sur les espaces de Köthe, J. Anal. Math. 1 (1951) 81–115. [8] N. Dunford, J.T. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York, 1958. [9] J. Banas, J. Rivero, On measures of weak noncompactness, Ann. Math. Pure Appl. 151 (1988) 213–224. [10] F.S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. Roumanie 21 (1977) 259–262. [11] J. Appell, E. De Pascale, Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. Unicone Mat. Ital. B(6) 3 (1984) 497–515. [12] J. Banas, Z. Knap, Measure of weak noncompactness and nonlinear integral equations of convolution type, J. Math. Anal. Appl. 146 (1990) 353–362. [13] T.A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124 (1996) 2383–2390. [14] Y.C. Liu, Z.X. Li, Schafer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl. 316 (1) (2006) 237–255. [15] Y.C. Liu, Z.X. Li, Krasnoselskii Type fixed point theorems and applications, Proc. Amer. Math. Soc. 136 (4) (2008) 1213–1220. [16] J. Jachymski, On Isac’s fixed point theorem for selfmaps of a Galerkin cone, Ann. Sci. Math. Québec 18 (2) (1994) 169–171. [17] K. Latrach, M.A. Taoudi, A. Zeghal, Some fixed point theorems of the Schauder and Krasnosel’skii type and application to nonlinear transport equations, J. Differential Equations 221 (1) (2006) 256–271. [18] T.A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998) 85–88. [19] P.P. Zabrejko, A.I. Koshelev, M.A. Krasnosel’skii, S.G. Mikhlin, L.S. Rakovshchik, V.J. Stecenko, Integral Equations, Noordhoff, Leyden, 1975.