Acta Mathematica Sinica, English Series

Nov., 2006, Vol. 22, No. 6, pp. 1745–1750

Published online: Jun. 27, 2006

DOI: 10.1007/s10114-005-0782-3

Http://www.ActaMath.com

Positive and Nontrivial Solutions for the Urysohn Integral Equation

Daniel FRANCO1)

Departamento de Matematica Aplicada, Universidad Nacional de Educacion a Distancia,

Apartado de Correos 60149, Madrid, 28080, Spain

E-mail: dfranco@ind.uned.es

Gennaro INFANTEDipartimento di Matematica, Universita della Calabria,

87036 Arcavacata di Rende, Cosenza, Italy

E-mail: g.infante@unical.it

Donal O’REGANDepartment of Mathematics, National University of Ireland, Galway, Ireland

E-mail: donal.oregan@nuigalway.ie

Abstract We establish new criteria for the existence of either positive or nonzero solutions of the

Urysohn integral equation. We also discuss the existence of an interval of positive eigenvalues and

sufficient conditions for the existence of at least a positive eigenvalue with a nonzero or positive eigen-

function for the Urysohn integral operator. Among others, we employ techniques based on fixed point

index theory for compact maps, which are new for this type of equation.

Keywords Urysohn integral equation, positive solutions, eigenvalues

MR(2000) Subject Classification 45G10; 47H10, 47H30

1 IntroductionIn this paper we study the problem of the existence of nontrivial solutions for the Urysohnequation

u(t) =∫ 1

0

g(t, s, u(s)) ds, (1)

or equivalently the existence of a nontrivial fixed point for the integral Urysohn operatorS : C[0, 1] → C[0, 1] defined by

[Su](t) =∫ 1

0

g(t, s, u(s)) ds. (2)

Here, as usual, we denote by C[0, 1] the Banach space of continuous functions u : [0, 1] → R

with norm |u|0 = sup{|u(x)| : x ∈ [0, 1]}.Equation (1) was discussed by Urysohn in 1924 in [1]. Equations of this type appear in many

applications. For example, it occurs in solving problems arising in economics, engineering, andphysics (see [2] and references therein).

Recently several authors have considered the problem of establishing the existence of solu-tions for (1) using different techniques. For example, in [3, 4] the authors discuss the Newton–Kantorovich method and present existence and uniqueness results for (1). The applicability of

Received December 14, 2004, Accepted July 18, 2005The first and third authors have been supported in part by Ministerio de Ciencia y Tecnologıa (Spain) MTM2004-

06652-C03-031) Corresponding author

1746 Franco D., et al.

the upper and lower solutions method is considered in [5]. In [6] existence theorems are obtainedas a consequence of the Darbo fixed point principle and its Leray–Schauder-like analogue. TheSchauder Tychonoff fixed point theorem is employed in [7], whereas a fixed point theorem forcontraction maps is used in [8]. Degree theory, theory of monotone operators, variational meth-ods and topological transversality are used in [9–12], respectively. Finally, numeric approachesare considered, for example, in [13], where the validity of the Galerkin and collocation methodsfor (1) are discussed.

Here, we shall employ fixed point theorems in conical shells (see [14, 15] or the recent papersin this journal [16, 17] for more details). This type of technique relies on fixed point index theoryfor completely continuous maps. The approach we are going to consider here has been usedbefore in the study of Hammerstein integral equations in [18, 19], where the authors dealt withthe existence of nontrivial and positive solutions, respectively.

In the present paper we give new results on the existence of positive and nontrivial solutionsof (1). We also give new results on the existence of an interval of positive eigenvalues and of atleast one eigenvalue with a positive or nonzero eigenfunction for the operator (2) (see [20–23]).

Several remarks are presented in the paper about possible extensions of our results.

2 PreliminariesIn this section we shall introduce some notation and background material used throughout thispaper.

Let (E, ‖ · ‖) be a Banach space. We say that K ⊂ E is a cone if it is closed, convex,nonempty, K �= {0} and whenever x ∈ K and λ ≥ 0, then λx ∈ K. If D is a subset of E wewrite DK = D∩K, and we denote by ∂KD and DK the boundary and the closure of D ∩K inK. Recall that a completely continuous map means a continuous map which transforms everybounded set into a relatively compact set.

The following lemma is a consequence of fixed point index:Lemma 1 Assume Ω1, Ω2 are open bounded sets with Ω1

K �= ∅, Ω1K ⊂ Ω2

K , and let S : Ω2K →

K be a completely continuous map with u �= Su for u ∈ (∂KΩ1) ∪ (∂KΩ2) and either :• There exists e ∈ K \ {0} such that u �= Su+ λe for all u ∈ ∂KΩ1 and all λ > 0;• ‖Su‖ ≤ ‖u‖ for u ∈ ∂KΩ2;

or,• There exists e ∈ K \ {0} such that u �= Su+ λe for all u ∈ ∂KΩ2 and all λ > 0;• ‖Su‖ ≤ ‖u‖ for u ∈ ∂KΩ1.Then S has a fixed point in Ω2

K \ Ω1K .

We shall make the following assumptions on g in (1):(C1) g is a Caratheodory function, i.e., for each u ∈ R, (t, s) → g(t, s, u) is measurable,

and for almost every (t, s) ∈ [0, 1] × [0, 1], u→ g(t, s, u) is continuous.(C2) For each r > 0 there exist measurable functions βr : [0, 1] × [0, 1] → [0,∞) and

γr : [0, 1] × [0, 1] × [0, 1] → [0,∞) such that|g(t, s, u)| ≤ βr(t, s) and |g(t, s, u) − g(τ, s, u)| ≤ γr(t, τ, s),

for all t, s, τ ∈ [0, 1] and u ∈ R with |u| ≤ r.(C3)

limt→τ

∫ 1

0

γr(t, τ, s) ds = 0 uniformly for τ ∈ [0, 1].

(C4) There is a number Mr <∞ such that∫ 1

0

βr(t, s) ds ≤Mr, for all t ∈ I.

(C5) There exist a closed interval [a, b] ⊂ [0, 1], a constant c ∈ (0, 1], and a measurablefunction Φ: [0, 1] × R → [0,∞) such that

|g(t, s, u)| ≤ Φ(s, u) for t ∈ [0, 1], u ∈ R, and almost every s ∈ [0, 1],

Urysohn Integral Equation 1747

cΦ(s, u) ≤ g(t, s, u) for t ∈ [a, b], u ∈ R, and almost every s ∈ [0, 1].

(C6) For each r > 0 there is Nr < ∞ such that∫ 1

0Φ(s, u) ds ≤ Nr, for all u ∈ R with

|u| ≤ r.The interval [a, b] in condition (C5) could be changed to an arbitrary measurable subset

G ⊂ [0, 1] with positive measure, but we choose this presentation for clarity.Lemma 2 Assume (C1)–(C6) hold. Then S is a completely continuous operator which mapsC[0, 1] into the cone K = {u ∈ C[0, 1] : mint∈[a,b] u(t) ≥ c|u|0 }, where c ∈ (0, 1] was introducedin (C5).Proof By Proposition 3.1 in Martin’s book [24, p. 164] conditions (C1)–(C4) guarantee thatS is completely continuous.

Next, to show that S maps C[0, 1] into K, let u ∈ C[0, 1] and t ∈ [0, 1]. We have from(C5)–(C6) that |Su(t)| ≤ ∫ 1

0|g(t, s, u(s))| ds ≤ ∫ 1

0Φ(s, u(s)) ds.

Therefore

|Su|0 ≤∫ 1

0

Φ(s, u(s)) ds. (3)

On the other hand, using (C5) and (3) we have, for each u ∈ C[0, 1] and t ∈ [a, b], that

Su(t) =∫ 1

0

g(t, s, u(s)) ds ≥ c

∫ 1

0

Φ(s, u(s)) ds ≥ c|Su|0.Thus mint∈[a,b] Su(t) ≥ c|Su|0, and S maps C[0, 1] into the cone and particularly the cone

back into itself.

3 Nontrivial SolutionsAs we have announced in the previous section we are going to employ Lemma 1 to prove asufficient condition for the existence of a nontrivial fixed point for (2). In order to do that weneed to select adequate open bounded sets. Instead of using two open balls as is common inthe literature, we are going to change one of the balls for a different set which permits us toget a more general result. Following Lan [19], for each, r > 0 we write

Ωr = {u ∈ C[0, 1] : mint∈[a,b]

u(t) < cr }, and Kr = {u ∈ C[0, 1] : |u|0 < r} .Lemma 2.5 in [19] gives some properties of these sets which for completeness we list here.

Lemma 3 The sets defined above have the following properties :(a) Ωr

K and KrK are open relative to K;

(b) KcrK ⊂ Ωr

K ⊂ KrK ;

(c) u ∈ ∂KΩr iff u ∈ K and mint∈[a,b] u(t) = cr;(d) If u ∈ ∂KΩr, then cr ≤ u(t) ≤ r for each t ∈ [a, b].Before presenting the main result in this section, we write down the main hypotheses:(D1) There exists α > 0 and φ : [a, b] × [a, b] → (0,∞) measurable such that

g(t, s, u) ≥ cαφ(t, s) for all cα ≤ u ≤ α, for all t ∈ [a, b] and a.e. s ∈ [a, b]

and inft∈[a,b]

∫ b

aφ(t, s) ds ≥ 1.

(D2) There exists β > 0 and ψ : [0, 1] × [0, 1] → (0,∞) measurable such that|g(t, s, u)| ≤ βψ(t, s) for all t ∈ [0, 1], a.e. s ∈ [0, 1] and all u ∈ [−β, β]

and supt∈[0,1]

∫ 1

0ψ(t, s) ds ≤ 1.

Theorem 1 Assume (C1)–(C6) hold. Also assume that there exist α, β > 0 such that (D1)–(D2) hold with either (A) β < cα or (B) α < β. Then (1) has a nontrivial solution u ∈ K

with u ∈ ΩαK \Kβ

K if β < cα or u ∈ KβK \ Ωα

K if α < β.Proof We claim that

(I) There exists e ∈ K \ {0} such that u �= Su+ λe for u ∈ ∂KΩα and λ > 0.(II) |Su|0 ≤ |u|0 for u ∈ ∂KK

β .

1748 Franco D., et al.

We start with (I). Let e(t) = 1 for t ∈ [0, 1]. Then e ∈ K \ {0}. Next, suppose thatthere exist u ∈ ∂KΩα and λ > 0 such that u = Su + λe. Then from Lemma 3(d) we havecα ≤ u(t) ≤ α for t ∈ [a, b]. From (C5) and (D1) we have, for each t ∈ [a, b],

u(t) = Su(t) + λ =∫ 1

0

g(t, s, u(s)) ds+ λ

≥∫ b

a

g(t, s, u(s)) ds+ λ ≥ cα

∫ b

a

φ(t, s) ds+ λ ≥ cα+ λ.

(4)

Thus, we have mint∈[a,b] u(t) ≥ cα+ λ > cα, contradicting Lemma 3(c), and so (I) holds.Now we consider (II). Let u ∈ ∂KK

β . Then −β ≤ u(t) ≤ β for t ∈ [0, 1] and, using (D2),we have

|Su(t)| ≤∫ 1

0

|g(t, s, u(s))| ds ≤∫ 1

0

βψ(t, s) ds ≤ β supt∈[0,1]

∫ 1

0

ψ(t, s) ds ≤ β.

Hence, |Su|0 ≤ |u|0 for all u ∈ ∂KKβ , and so (II) holds.

Lemma 2 guarantees that the restrictions S : ΩαK → K and S : Kβ

K → K are well-definedcompact maps.

Assume that (A) holds. By (b) of Lemma 3, we have KβK ⊂ Kcα

K ⊂ ΩαK since β < cα.

Then either S has a fixed point on ∂KΩα ∪ ∂KKβ or it follows from Lemma 1 that S has

a fixed point in ΩαK \ Kβ

K . To justify this (note Ωα is unbounded) we need to show thatΩα

K = (Ωα ∩Kδ)K with δ > α and ΩαK = (Ωα ∩Kδ)K . The first equality follows immediately

from Lemma 3(b). Next if u ∈ ΩαK then from Lemma 3(c) we have that c|u|0 ≤ mint∈[a,b] u(t) ≤

cα < cδ so u ∈ (Ωα∩Kδ)∩K. Now, since Ωα and Kδ are open sets, we have Ωα∩Kδ ⊆ Ωα ∩Kδ

and so (Ωα ∩Kδ)K ⊂ (Ωα ∩Kδ)K . Thus u ∈ (Ωα ∩Kδ)K so ΩαK ⊆ (Ωα ∩Kδ)K . The reverse

inclusion is trivial.The other assertion is proved similarly using Ωα

K ⊂ ΩβK ⊂ Kβ

K since α < β.Remark 1 If [a, b] = [0, 1] in (C5) and (D1) we obtain a sufficient condition for the existenceof positive solutions for (1).Remark 2 Using Theorem 1 repeatedly (and Remark 1) one can establish the existence ofmultiple nontrivial (positive) solutions for (1).

We present an example to show the applicability of our last result.Example 1 The Urysohn integral equation

u(t) =∫ 1

0

12 + t+ s+ sinu(s)

ds, t ∈ [0, 1]

has at least a positive solution u.Define g(t, s, u) = 1

2+t+s+sin u . Since g is continuous we see immediately that (C1)–(C4) aresatisfied. Next, let [a, b] = [0, 1], c = 1

5 and Φ ≡ 1, and we see that (C5)–(C6) hold.Finally, since limu→0

g(t,s,u)u = ∞ and limu→∞

g(t,s,u)u = 0 uniformly on t, s ∈ [0, 1], there

exist α, β such that (D1) and (D2) hold. Indeed, let for example α = 12 , β = 1 and φ ≡ ψ ≡ 1.

Then for u ∈ [ 110 ,

12 ] and t, s ∈ [0, 1] we have 5

αg(t, s, u) ≥ 5α

13+t+s ≥ 1

α = 2 ≥ 1. Also, foru ∈ [−1, 1] and t, s ∈ [0, 1] we obtain 1

β g(t, s, u) ≤ 1β

11+t+s ≤ 1

β = 1.Therefore, the existence of the solution follows from Theorem 1.

4 Eigenvalue ResultsIn this section we shall discuss the existence of eigenvalues for the Urysohn integral operator.Recall that we say that λ is an eigenvalue for the Urysohn operator (2) if there exists uλ suchthat

λuλ(t) =∫ 1

0

g(t, s, uλ(s)) ds := Suλ(t). (5)

Our first result is based on the Schauder fixed point theorem.

Urysohn Integral Equation 1749

Theorem 2 Assume (C1)–(C6) hold. Then for each r > 0 there exists cr > 0 such that λ isan eigenvalue of S for each λ ∈ (cr,∞).Proof From Lemma 2 we know that S : Kr

K → K is completely continuous; here Kr is as inSection 3 and note that Kr

K == {u ∈ K : |u|0 ≤ r}. Now let us define

cr =1rNr, (6)

where Nr was introduced in (C6).We have, for u ∈ Kr

K and t ∈ [0, 1],

|Su(t)| ≤∫ 1

0

|g(t, s, u(s))|ds ≤∫ 1

0

Φ(s, u(s))ds ≤ Nr = cr r.

Fix λ ∈ [cr,∞). Then 1λ |Su(t)| ≤ cr r

λ ≤ r and so | 1λSu|0 ≤ r for each u ∈ KrK . Thus

1λS maps Kr

K into KrK . Now Schauder’s fixed point theorem guarantees that 1

λS has a fixedpoint in Kr

K since 1λS is completely continuous. Evidently, this fixed point is a solution of (5)

and therefore λ is an eigenvalue of (2).Notice that in Theorem 2 the eigenfunction uλ could be 0. In the following result we present

a condition to guarantee that there exists at least one eigenvalue λ of S such that uλ �= 0. Theexistence of such an eigenvalue will follow as a direct consequence of the following well-knownresult (see, for example, Lemma 1.1, Chapter 5 of [25]).Lemma 4 Let T : Kr

K → K be completely continuous and suppose that infx∈∂KKr ‖Tx‖ > 0.Then there exist λ0 > 0 and x0 ∈ ∂KK

r such that λ0x0 = Tx0.Theorem 3 Assume that (C1)–(C6) hold and that

(E1) There exists α > 0 and ϕα : [a, b] × [a, b] → [0,∞) measurable such thatg(t, s, u) ≥ ϕα(t, s) for all cα ≤ u ≤ α, all t ∈ [a, b] and a.e. s ∈ [a, b];

and,(E2) τ : = supt∈[a,b]

∫ b

aϕα(t, s) ds > 0.

Then there exist λ0 and u0 ∈ ∂KKα such that λ0u0 = Su0.

Proof Since S satisfies the hypotheses of Lemma 2, S : KrK → K is completely continuous.

Suppose that (E1) holds and take u ∈ ∂KKα. Then we have, for every t ∈ [a, b], cα ≤ u(t) ≤ α.

For t ∈ [a, b], |Su(t)| ≥ ∫ b

ag(t, s, u(s)) ds ≥ ∫ b

aϕα(t, s) ds. Thus |Su|0 ≥ supt∈[0,1] |Su(t)| ≥

supt∈[a,b] |Su(t)| ≥ τ, so infu∈∂KKα |Su|0 > 0. By Lemma 4 we obtain the existence of aneigenvalue λ0 > 0.Remark 3 If [a, b] = [0, 1] in the last result, then the eigenfunction is positive.

For brevity and clarity, we have presented our results in a simple case. However, severalextensions can be done without difficulty.

We have considered only the existence of nontrivial solutions for positive nonlinearities g,but this technique could be employed to prove the existence of nontrivial solutions when g hasa constant sign behavior.

Using similar ideas to those in [20, 26] the results in this paper can also be extended to asystem

ui(t) =∫ 1

0

gi(t, s, u(s)) ds, t ∈ [0, 1] , i = 1, 2, . . . , n.

Acknowledgments The authors thank the anonymous referee for her/his comments whichhelp to improve the paper.

This research was partially done during a visit by Daniel Franco to the National Universityof Ireland at Galway and the University of Glasgow. That visit was supported by Secretarıa deEstado de Universidades e Investigacion (Spain).

1750 Franco D., et al.

References[1] Urysohn, P. S.: On a type of nonlinear integral equation. Mat. Sb., 31 236–355 (1924)

[2] Zabreıko, P. P., Koshelev, A. I., et al.: Integral Equations: A Reference Text, Noordhoff Int. Publ.,

Leyden, 1975

[3] Appell, J., De Pascale, E., Zabreıko, P. P.: On the application of the Newton-Kantorovich method to

nonlinear integral equations of Uryson type. Numer. Funct. Anal. Optim., 12, 271–283 (1991)

[4] Appell, J., De Pascale, E., Lysenko, J. V., Zabreıko, P.P.: New results on Newton-Kantorovich approxima-

tions with applications to nonlinear integral equations. Numer. Funct. Anal. Optim., 18, 1–17 (1997)

[5] Pachpatte, B. G.: Method of upper and lower solutions for nonlinear integral equations of Urysohn type.

J. Integral Equations, 6, 119–125 (1984)

[6] O’Regan, D.: Volterra and Urysohn integral equations in Banach spaces. J. Appl. Math. Stochastic Anal.,

11(4), 449–464 (1998)

[7] Darwish, M. A.: On integral equations of Urysohn-Volterra type. Appl. Math. Comput., 136, 93–98 (2003)

[8] El-Sayed, W. G., El-Bary, A. A., Darwish, M. A.: Solvability of Urysohn integral equation. Appl. Math.

Comput., 145, 487–493 (2003)

[9] Georgescu, C.: Existence results for Urysohn integral equation, Preprint

[10] Joshi, M.: Existence theorems for Urysohn’s integral equation. Proc. Amer. Math. Soc., 49, 387–392

(1975)

[11] Kolomy, J.: The solvability of nonlinear integral equations. Comment. Math. Univ. Carolinae, 8, 273–289

(1967)

[12] O’Regan, D., Meehan, M.: Existence theory for nonlinear integral and integrodifferential equations, Kluwer

Academic Publishers, Dordrecht, 1998

[13] Atkinson, K. E., Rotra, F. A.: Projection and iterated projection methods for nonlinear integral equations.

SIAM J. Numer. Anal., 24, 1352–1373 (1987)

[14] Krasnosel’skii, M. A.: Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964

[15] Guo, D., Lakshmikantham, V.: Nonlinear problems in abstract cones, Academic Press, Boston, 1988

[16] Ma, R. Y., Ma, Q. Z.: Positive solutions for semipositone m-point boundary-value problems. Acta Mathe-

matica Sinica, English Series, 20(2), 273–282 (2004)

[17] Yao, Q. L.: Existence, Multiplicity and Infinite Solvability of Positive Solutions for One-Dimensional p-

Laplacian. Acta Mathematica Sinica, English Series, 21(2), 691–698 (2005)

[18] Infante, G., Webb, J. R. L.: Nonzero solutions of Hammerstein integral equations with discontinuous kernels.

J. Math. Anal. Appl., 272, 30–42 (2002)

[19] Lan, K. Q.: Multiple positive solutions of semilinear differential equations with singularities. J. London

Math. Soc., 63, 690–704 (2001)

[20] Agarwal, R. P., O’Regan, D., Wong, P. J. Y.: Eigenvalues of a system of Fredholm integral equations. Math.

Comput. Modelling, 39, 1113–1150 (2004)

[21] Infante, G.: Eigenvalues of some non-local boundary-value problems. Proc. Edinb. Math. Soc. (2), 46(1),

75–86 (2003)

[22] Lan, K. Q.: Multiple positive eigenvalues of conjugate boundary value problems with singularities. Discrete

Contin. Dyn. Syst., Suppl., 501–506 (2003)

[23] Lan, K. Q.: Eigenvalues of second order differential equations with singularities. Discrete Contin. Dyn.

Syst., Added Volume, 241–247 (2001)

[24] Martin, R. H.: Nonlinear operators and differential equations in Banach spaces, Jonh Wiley and Sons, New

York, 1976

[25] Krasnosel’skii, M. A.: Topological methods in the theory of nonlinear integral equations, Translated by A.

H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, Macmillan, New York, 1964

[26] Franco, D., Infante, G., O’Regan, D.: Nontrivial solutions in abstract cones for Hammerstein integral

systems. Dynam. Contin. Discrete Impuls. Systems, to appear