Open Access

Convex Solutions of a Nonlinear Integral Equation of Urysohn Type

Fixed Point Theory and Applications20092009:917614

DOI: 10.1155/2009/917614

Received: 4 August 2009

Accepted: 25 September 2009

Published: 5 November 2009

Abstract

We study the solvability of a nonlinear integral equation of Urysohn type. Using the technique of measures of noncompactness we prove that under certain assumptions this equation possesses solutions that are convex of order for each , with being a given integer. A concrete application of the results obtained is presented.

1. Introduction

Existence of solutions of differential and integral equations is subject of numerous investigations (see, e.g., the monographs [13] or [4]). Moreover, a lot of work in this domain is devoted to the existence of solutions in certain special classes of functions (e.g., positive functions or monotone functions). We merely mention here the result obtained by Caballero et al. [5] concerning the existence of nondecreasing solutions to the integral equation of Urysohn type
(1.1)

where is a positive constant. In the special case when (or even ), the authors proved in [5] that if is positive and nondecreasing, is positive and nondecreasing in the first variable (when the other two variables are kept fixed), and they satisfy some additional assumptions, then there exists at least one positive nondecreasing solution to (1.1). A similar existence result, but involving a Volterra type integral equation, has been obtained by Banaś and Martinon [6].

It should be noted that both existence results were proved with the help of a measure of noncompactness related to monotonicity introduced by Banaś and Olszowy [7]. The reader is referred also to the paper by Banaś et al. [8], in which another measure of noncompactness is used to prove the solvability of an integral equation of Urysohn type on an unbounded interval.

The main purpose of the present paper is twofold. First, we generalize the result from the paper [5] to the framework of higher-order convexity. Namely, we show that given an integer , if and are convex of order for each , then (1.1) possesses at least one solution which is also convex of order for each . Second, we simplify the proof given in [5] by showing that it is not necessary to make use of the measure of noncompactness related to monotonicity introduced by Banaś and Olszowy [7].

2. Measures of Noncompactness

Measures of noncompactness are frequently used in nonlinear analysis, in branches such as the theory of differential and integral equations, the operator theory, or the approximation theory. There are several axiomatic approaches to the concept of a measure of noncompactness (see, e.g., [911] or [12]). In the present paper the definition of a measure of noncompactness given in the book by Banaś and Goebel [12] is adopted.

Let be a real Banach space, let be the family consisting of all nonempty bounded subsets of , and let be the subfamily of consisting of all relatively compact sets. Given any subset of , we denote by and the closure and the convex hull of , respectively.

Definition 2.1 (see [12]).

A function is said to be a measure of noncompactness in if it satisfies the following conditions.

(1)The family ker (called the kernel of ) is nonempty and it satisfies .

(2) whenever satisfy .

(3) for all .

(4) for all and all .

(5)If is a sequence of closed sets from such that for each positive integer and if , then the set is nonempty.

An important and very convenient measure of noncompactness is the so-called Hausdorff measure of noncompactness , defined by
(2.1)
The importance of this measure of noncompactness is given by the fact that in certain Banach spaces it can be expressed by means of handy formulas. For instance, consider the Banach space consisting of all continuous functions , endowed with the standard maximum norm
(2.2)
Given , , and , let
(2.3)
be the usual modulus of continuity of . Further, let
(2.4)
and . Then it can be proved (see Banaś and Goebel [12, Theorem  7.1.2]) that
(2.5)

For further facts concerning measures of noncompactness and their properties the reader is referred to the monographs [9, 11] or [12]. We merely recall here the following fixed point theorem.

Theorem 2.2 (see [12, Theorem  5.1]).

Let be a real Banach space, let be a measure of noncompactness in , and let be a nonempty bounded closed convex subset of . Further, let be a continuous operator such that for each subset of , where is a constant. Then has at least one fixed point in .

3. Convex Functions of Higher Orders

Let be a nondegenerate interval. Given an integer , a function is said to be convex of order or -convex if
(3.1)
for any system of points in , where
(3.2)
is called the divided difference of at the points . With the help of the polynomial function defined by
(3.3)
the previous divided difference can be written as
(3.4)
An alternative way to define the divided difference is to set
(3.5)
whenever . Finally, we mention a representation of the divided difference by means of two determinants. It can be proved that
(3.6)
where
(3.7)

Note that a convex function of order is a nonnegative function, a convex function of order is a nondecreasing function, while a convex function of order is an ordinary convex function.

Let be a nondegenerate interval, let be an arbitrary function, and let . The difference operator with the span is defined by
(3.8)
for all for which . The iterates of are defined recursively by
(3.9)
It can be proved (see, e.g., [13, page 368, Corollary  3]) that
(3.10)
for every for which . On the other hand, the equality
(3.11)

holds for every nonnegative integer and every for which .

Let be a nondegenerate interval. Given an integer , a function is called Jensen convex of order or Jensen -convex if
(3.12)

for all and all such that . Due to (3.11), it is clear that every convex function of order is also Jensen convex of order . In general, the converse does not hold. However, under the additional assumption that is continuous, the two notions turn out to be equivalent.

Theorem 3.1 (see [13, page 387, Theorem  1]).

Let be a nondegenerate interval, let be an integer, and let be a continuous function. Then is convex of order if and only if it is Jensen convex of order .

Finally, we mention the following result concerning the difference of order of a product of two functions:

Lemma 3.2.

Let be a nondegenerate interval, and let be a nonnegative integer. Given two functions , the equality
(3.13)

holds for every such that .

4. Main Results

Throughout this section is a positive real number. In the space , consisting of all continuous functions , we consider the usual maximum norm
(4.1)

Our first main result concerns the integral equation of Urysohn type (1.1) in which , , and are given functions, while is the unknown function. We assume that the functions , , and satisfy the following conditions:

is a given integer number;

is a continuous function which is convex of order for each ;

is a continuous function such that for all and the function
(4.2)

is convex of order for each whenever is convex of order for each ;

there exists a continuous function which is nondecreasing in each variable and satisfies
(4.3)

for all and all ;

is a continuous function such that the function is convex of order for each whenever and ;

there exists a continuous nondecreasing function such that
(4.4)
there exists such that
(4.5)

Theorem 4.1.

If the conditions ( )–( ) are satisfied, then (1.1) possesses at least one solution which is convex of order for each .

Proof.

Consider the operator , defined on by
(4.6)

Then whenever (see [5, the proof of Theorem  3.2]).

We claim that is continuous on . To this end we fix any in and prove that is continuous at . Let , and let
(4.7)
Further, let . The uniform continuity of on as well as that of on ensures the existence of a real number such that
(4.8)
for all and all satisfying . Then for every such that and every we have
(4.9)

Therefore, the inequality holds for every in satisfying . This proves the continuity of at .

Next, let be the positive real number whose existence is assured by ( ), and let be the subset of , consisting of all functions such that and is convex of order for each . Obviously, is a nonempty bounded closed convex subset of . We claim that maps into itself. To prove this, let be arbitrarily chosen. For every we have
(4.10)
Since is convex of order (i.e., nonnegative), according to ( ) and ( ) we also have
(4.11)
This inequality and ( ) yield
(4.12)
Taking into account that , by ( ), ( ), and ( ) we conclude that
(4.13)
On the other hand, for every we have
(4.14)
where are the functions defined by
(4.15)
respectively. According to Lemma 3.2, we have
(4.16)
for every and every such that . But
(4.17)
where . By virtue of ( ) we have , whence
(4.18)

This inequality together with (4.16), ( ), and ( ) ensures that the function is Jensen convex of order for each . Since is continuous on , by Theorem 3.1 it follows that is convex of order for each . Taking into account (4.13), we conclude that maps into itself, as claimed.

Finally, we prove that the operator satisfies the Darbo condition with respect to the Hausdorff measure of noncompactness . To this end let be an arbitrary nonempty subset of and let . Further, let and let be such that . We have
(4.19)
Letting
(4.20)
we get
(4.21)
Thus
(4.22)
whence
(4.23)
Taking into account that is uniformly continuous on , is uniformly continuous on and is uniformly continuous on , we have that , and as . So letting we obtain , that is,
(4.24)

by virtue of (2.5).

By ( ) and Theorem 2.2 we conclude the existence of at least one fixed point of in . This fixed point is obviously a solution of (1.1) which (in view of the definition of ) is convex of order for each .

Theorem 4.1 can be further generalized as follows. Given an integer number and a sequence , we denote by the set consisting of all functions with the property that for each the function is convex of order . For instance, if and , then consists of all functions in that are nonnegative, nonincreasing, and convex on .

Recall (see, e.g., Roberts and Varberg [14, pages 233-234]) that a function is called absolutely monotonic (resp., completely monotonic) if it possesses derivatives of all orders on and
(4.25)

for each and each integer . By [13, Theorem  6, page 392] it follows that if is an absolutely monotonic (resp., a completely monotonic) function, then belongs to every set with and (resp., ) for each .

Instead of the conditions ( ), ( ), ( ), and ( ) we consider the following conditions.

is a given integer number and is a sequence such that either
(4.26)
or
(4.27)

belongs to .

is a continuous function such that for all and the function
(4.28)

belongs to whenever

is a continuous function such that the function belongs to whenever and .

Theorem 4.2.

If the conditions ( )–( ), ( ), ( ), and ( )-( ) are satisfied, then (1.1) possesses at least one solution .

Proof.

Consider the operator , defined on , as in the proof of Theorem 4.1. As we have already seen in the proof of Theorem 4.1 we have whenever and is continuous on .

Instead of the set , considered in the proof of Theorem 4.1, we take now to be the subset of consisting of all functions such that . Then is a nonempty bounded closed convex subset of . We claim that maps into itself. Indeed, according to (4.13) we have whenever satisfies . On the other hand, admits the representation (4.14), where are defined by (4.15). Given any , note that
(4.29)
whence
(4.30)
for every such that . By proceeding as in the proof of Theorem 4.1 one can show that
(4.31)

Therefore .

The rest of the proof is similar to the corresponding part in the proof of Theorem 4.1 and we omit it.

5. An Application

As an application of the results established in the previous section, in what follows we study the solvability of the integral equation
(5.1)

in which is a given positive integer and is a positive real parameter. Note that (5.1) is similar to the Chandrasekhar equation, arising in the theory of radiative transfer (see, e.g., Chandrasekhar [15] or Banaś et al. [16], and the references therein).

We are going to prove that if , then (5.1) possesses at least one continuous nonnegative solution, which is nonincreasing and convex. To this end, we apply Theorem 4.2 for and . Take , , and . It is immediately seen that all the conditions ( )–( ), ( ), ( ), and ( ) are satisfied if the functions and are defined by
(5.2)
respectively. It remains to show that ( ) is satisfied, too. Taking into account the expressions of and , condition ( ) is equivalent to the following statement. If , then there exists an such that
(5.3)
Clearly, such an must satisfy . Let be the functions defined by
(5.4)
respectively. Since
(5.5)
one can see that attains a maximum at , the maximum value being . On the other hand, we have
(5.6)

If , then for all . If and , then , while if and , then . Note that .

Assume now that . Then we can select an sufficiently close to such that . Obviously, satisfies (5.3).

Authors’ Affiliations

(1)
Faculty of Mathematics and Computer Science, Babeş-Bolyai University

References

  1. Corduneanu C: Integral Equations and Applications. Cambridge University Press, Cambridge, UK; 1991:x+366.View ArticleMATHGoogle Scholar
  2. Guo D, Lakshmikantham V, Liu X: Nonlinear Integral Equations in Abstract Spaces, Mathematics and Its Applications. Volume 373. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:viii+341.View ArticleMATHGoogle Scholar
  3. O'Regan D, Meehan M: Existence Theory for Nonlinear Integral and Integrodifferential Equations, Mathematics and Its Applications. Volume 445. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:viii+218.View ArticleMATHGoogle Scholar
  4. Precup R: Methods in Nonlinear Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2002:xiv+218.View ArticleMATHGoogle Scholar
  5. Caballero J, O'Regan D, Sadarangani K: On nondecreasing solutions of cubic integral equations of Urysohn type. Commentationes Mathematicae. Prace Matematyczne 2004,44(1):39–53.MathSciNetMATHGoogle Scholar
  6. Banaś J, Martinon A: Monotonic solutions of a quadratic integral equation of Volterra type. Computers & Mathematics with Applications 2004,47(2–3):271–279. 10.1016/S0898-1221(04)90024-7MathSciNetView ArticleMATHGoogle Scholar
  7. Banaś J, Olszowy L: Measures of noncompactness related to monotonicity. Commentationes Mathematicae. Prace Matematyczne 2001, 41: 13–23.MathSciNetMATHGoogle Scholar
  8. Banaś J, Rocha Martin J, Sadarangani K: On solutions of a quadratic integral equation of Hammerstein type. Mathematical and Computer Modelling 2006,43(1–2):97–104. 10.1016/j.mcm.2005.04.017MathSciNetView ArticleMATHGoogle Scholar
  9. Akhmerov RR, Kamenskiĭ MI, Potapov AS, Rodkina AE, Sadovskiĭ BN: Measures of Noncompactness and Condensing Operators, Operator Theory: Advances and Applications. Volume 55. Birkhäuser, Basel, Switzerland; 1992:viii+249.View ArticleGoogle Scholar
  10. Appell J, Erzakova NA, Falcon Santana S, Väth M: On some Banach space constants arising in nonlinear fixed point and eigenvalue theory. Fixed Point Theory and Applications 2004,2004(4):317–336. 10.1155/S1687182004406068View ArticleMathSciNetMATHGoogle Scholar
  11. Ayerbe Toledano JM, Dominguez Benavides I, Lopez Acedo G: Noncompactness in Metric Fixed Point Theory. Birkhäuser, Basel, Switzerland; 1997.View ArticleMATHGoogle Scholar
  12. Banaś J, Goebel K: Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics. Volume 60. Marcel Dekker, New York, NY, USA; 1980:vi+97.Google Scholar
  13. Kuczma M: An Introduction to the Theory of Functional Equations and Inequalities: Cauchy's Equation and Jensen's Inequality, Prace Naukowe Uniwersytetu Ślaskiego w Katowicach. Volume 489. Uniwersytet Ślaski, Katowice, Poland; 1985:523.Google Scholar
  14. Roberts AW, Varberg DE: Convex Functions, Pure and Applied Mathematics, vol. 5. Academic Press, New York, NY, USA; 1973:xx+300.Google Scholar
  15. Chandrasekhar S: Radiative Transfer. Oxford University Press, London, UK; 1950:xiv+393.MATHGoogle Scholar
  16. Banaś J, Lecko M, El-Sayed WG: Existence theorems for some quadratic integral equations. Journal of Mathematical Analysis and Applications 1998,222(1):276–285. 10.1006/jmaa.1998.5941MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Tiberiu Trif. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.