On Monotonic Integrable Solutions for Quadratic Functional Integral Equations

We study the solvability of functional quadratic integral equations in the space of integrable functions on the interval I = [0, 1]. We concentrate on a.e. monotonic solutions for considered problems. The existence result is obtained under the assumption that the functions involved in the investigated equation satisfy Carathéodory conditions. As a solution space we consider both L1(I) and Lp(I) spaces for p > 1.


Introduction
Linear and nonlinear integral equations are considered as a branch of the applications of functional analysis. This branch is a great importance not only for the specialist in this field but also for those whose interest lies in other branch of mathematics with especial reference to mathematical physics, engineering and biology.
Our problem, as well as, the particular cases was investigated mainly in cases when the solutions are elements of the space of continuous functions. Thus the proofs are based on very special properties of this space (the compactness criterion, in particular), cf. [20,40].
On the other hand, by the practical interest it is worthwhile to consider discontinuous solutions. Here we are looking for integrable solutions. Thus the operators F 1 , F 2 and U should take their values in the space L 1 (I). Let us recall that we are interested in finding monotonic solutions (a.e. monotonic in the case of integrable solutions). In such a case discontinuous solutions are expected even in a simplest case i.e. when An interesting example of discontinuous solutions for integral equations is taken from [39,Example 3.5]: In the paper [24], we study the particular case of the above problem on R + when f 1 (t, x) = g(t) and f 2 (t, x) = x. Here we extend the earlier result by considering functional integral equation in a more general form. Moreover, we prove the existence of solutions in some subspaces of L 1 (0, 1). Let us add a few comments about functional dependence, i.e. functions ψ 1 and ψ 2 . Our set of assumptions is based on the paper [14]. Functions of the form ψ i (t) = t α (α > 0) or ψ i (t) = t − τ (t) with some set of assumptions for τ are most important cases covered in our paper. Let us note that functional equations with state dependent delay are very useful in many mathematical models including the population dynamics, the position control or the cell biology. A very interesting survey about such a theory and their applications can be found in [33].
The last aspect of our results is to investigate the monotonicity property of solutions. This is important property and there are many papers devoted to its study. Let us note some recent ones [16,17,24,30], for instance.
The results obtained in the current paper create some extensions for several known ones i.e. in addition to those mentioned previously also for the results from earlier papers or books ( [3,9,21,27,34,43,44,48], for example).

Notation and Auxiliary Facts
Let R be the field of real numbers, R + be the interval [0, ∞) and L 1 (I) be the space of Lebesgue integrable functions (equivalence classes of functions) on a measurable subset I of R, with the standard norm Recall that by L p we will denote the space of (equivalences classes of) functions x satisfying 1 0 |x(s)| p ds < ∞. In this paper we will denotes by I an interval [0, 1]. By · p we will denote the norm in L p (I).
One of the most important operator studied in nonlinear functional analysis is the so-called superposition operator [3]. Definition 2.1. Assume that a function f : I × R → R satisfies the Carathéodory conditions i.e. 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 The operator F f defined in such a way is called the superposition (Nemytskii) operator generated by the function f .
Furthermore, for every f ∈ L 1 and every φ : I → I we define the superposition operator generated by the functions f and φ, F φ,f : In L p (I) we have the "automatic" continuity of the Nemytskii operator ( [3,36] for all t ∈ I and x ∈ R, where a ∈ L q (I) and b ≥ 0.
It should be also noted that the superposition operator F takes its values in L ∞ (I) iff the generating function f is independent on x (cf. [3,Theorem 3.17]). This remark allows us to reduce the number of the considered cases.
Let S = S(I) denotes the set of measurable (in Lebesgue sense) functions on I and let meas stand for the Lebesgue measure in R. Identifying the functions equal almost everywhere the set S furnished with the metric we obtain a complete metric space. Moreover, the convergence in measure on I is equivalent to the convergence with respect the metric to d (Proposition 2.14 in [46]). The compactness in such a space is called a "compactness in measure" and such sets have very nice properties when considered as subsets of L p -spaces of integrable functions (p ≥ 1).
The following theorems give different sufficient conditions for compactness in measure that will be more convenient for our discussion ( [8,37]). Theorem 2.3. Let X be a bounded subset of L 1 (I) and suppose that there is a family of measurable subsets (Ω c ) 0≤c≤1 of the interval I such that measΩ c = c for every c ∈ Iand for x ∈ X Then this family is equimeasurable and the set X is compact in measure in L 1 (I).

It is clear that by putting Ω
where E is a set with measure zero, this family contains nonincreasing functions (possibly except for a set E). We will call the functions from this family "a.e. nonincreasing" functions. This is the case, when we choose an integrable and nonincreasing function y and the all the functions equal a.e. to y satisfies the above condition. Thus we can write that elements from L 1 (I) belong to this class of functions. Due to the compactness criterion in the space of measurable functions (with the topology convergence in measure) (see Lemma 4.1 in [8]) we have a desired theorem concerning the compactness in measure of a subset X of L 1 (I) (cf. [8,Corollary 4.1] or [29, Section III.2]). Let us recall, in metric spaces the set U 0 is compact if and only if each sequence from U 0 has a subsequence that converges in U 0 (i.e. sequentially compact). Proof. Let R > 0 be such that X ⊂ B R ⊂ L p (I). It is known that X is compact in measure as a subset of S. Since the compactness in measure is equivalent to sequential compactness, we are interested in studying the properties of the latter on. By taking an arbitrary sequence (x n ) in X we obtain that there exists a subsequence (x n k ) convergent in measure to some x in the space S. Since the balls in L p (I) spaces (p ≥ 1) are closed in the topology of convergence in measure, we obtain x ∈ B R ⊂ L p (I) and finally x ∈ X.
In the paper we will need to distinguish between two different cases: when an operator take their values in Lebesgue spaces L p (I) or in a space of essentially bounded functions L ∞ (I) (for Nemytskii operators see Theorem 2.2). For Urysohn operators the continuity is not "automatic" as in the case of superposition operators. Let us recall an important sufficient condition: The first two conditions are satisfied when I R h (t, s)ds ∈ L q (I), for instance.
We will use also the majorant principle for Urysohn operators (cf.
then U is a continuous operator.
We mention also that some particular conditions guaranteeing the continuity of the operator U may be found in [47,48].
Let us recall some properties of operators preserving monotonicity properties of functions.
is a.e. nonincreasing on R for any t ∈ I. Then the superposition operator F generated by f transforms functions being a.e. nonincreasing on I into functions having the same property.
We will use the fact that the superposition operator takes the bounded sets compact in measure into the sets with the same property. Proof. Let V be a bounded and compact in measure subset of L p (I). By our assumption F (V ) ⊂ L q (I). It is known that as a subset of S the set F (V ) is compact in measure (cf. [8]). It was noted that the topology of convergence in measure is metrizable, so the compactness of the set is equivalent with the sequential compactness. By taking an arbitrary sequence (y n ) ⊂ F (V ) we get a sequence (x n ) in V such that y n = F (x n ). Since (x n ) ⊂ V , as follows from Lemma 2.8 F transforms this sequence into the sequence convergent in measure. Thus (y n ) is compact in measure, so is F (V ). Next, we give some definitions and results which will be needed further on. Assume that (E, · ) is an arbitrary Banach space with zero element θ. Denote by B(x, R) the closed ball centered at x and with radius R. The symbol B R stands for the ball B(θ, R).
If X is a subset of E, thenX and convX denote the closure and convex closure of X, respectively. We denote the standard algebraic operations on sets by the symbols λ X and X + Y . Moreover, we denote by M E the family of all nonempty and bounded subsets of E and N E its subfamily consisting of all relatively compact subsets. Now we present the concept of a regular measure of noncompactness: is said to be a measure of noncompactness in E if it satisfies the following conditions: If X n is a sequence of nonempty, bounded, closed subsets of E such that X n+1 ⊂ X n , n = 1, 2, 3, . . ., and lim n→∞ μ(X n ) = 0, then the set X ∞ = ∞ n=1 X n is nonempty. An example of such a mapping is the following: Definition 2.12. [13] Let X be a nonempty and bounded subset of E. The Hausdorff measure of noncompactness χ(X) is defined as Another regular measure was defined in the space L 1 (I) (cf. [17]). For any ε > 0, let c be a measure of equiintegrability of the set X (the so-called Sadovskii functional [3, p. 39]) i.e.
Restricted to the family compact in measure subsets of this space it forms a regular measure of noncompactness (cf. [32]).
An importance of such a kind of functions can be clarified by using the contraction property with respect to this measure instead of compactness in the Schauder fixed point theorem. Namely, we have the theorem due to Darbo ([13,26]

Main Result
Denote by H the operator associated with the right hand side of equation (1.1) which takes the form This operator will be written as ( a 1 1 + K 0 L∞(I) a 2 1 a 3 1 ), where and let R denotes a positive solution of the quadratic equation Then we can prove the following theorem.
If L < 1, then the equation (1.1) has at least one integrable solution a.e. nonincreasing on I.
Proof. First of all observe that by assumption (i) and Theorem (2.2) we have that F φ1,f1 and F f2 are continuous mappings from L 1 (I) into itself. By assumption (iii) and Theorem 2.6 we can deduce that U maps L 1 (I) into L ∞ (I). From the Hölder inequality the operator A maps L 1 (I) into itself continuously. Finally, for a given x ∈ L 1 (I) the function H(x) belongs to L 1 (I) and is continuous. Thus By our assumption (vi) , it follows that there exists a positive constant R being the positive solution of the equation from assumption (vi) and such that H maps the ball B R into itself. Further, let Q R stand for the subset of B R consisting of all functions which are a.e. nonincreasing on I. Similarly as claimed in [10] we are able to show that this set is nonempty, bounded (by R), convex and closed in L 1 (I). Only the last property needs some comments. Let (y n ) be a sequence of elements in Q R convergent in L 1 (I) to y. Then the sequence is convergent in measure and as a consequence of the Vitali convergence theorem and of the characterization of convergence in measure (the Riesz theorem) we obtain the existence of a subsequence (y n k ) of (y n ) which converges to y almost uniformly on I. Moreover, y is nonincreasing a.e. on I which means that y ∈ Q R and so the set Q R is closed. Now, in view of Theorem 2.4 the set Q R is compact in measure. To see this it suffices to put Ω c = [0, c] \ P for any c ≥ 0, where P denotes a suitable set with measP = 0.

Monotonic Integrable Solutions 919
Now, we will show that H preserve the monotonicity of functions. Take x ∈ Q R , then x(t) and x(φ i (t)) are a.e. nonincreasing on I and consequently each f i is also of the same type by virtue of the assumption (i) and Theorem 2.7. Further, Ux(t) is a.e. nonincreasing on I due to assumption (ii). Moreover, F φ1,f1 , A(x)(t) are also of the same type. Thus we can deduce that H(x) = F φ1,f1 + A(x) is also a.e. nonincreasing on I. This fact, together with the assertion H : B R → B R gives that H is also a self-mapping of the set Q R . From the above considerations it follows that H maps continuously Q R into Q R .
From now we will assume that X is a nonempty subset of Q R and the constant ε > 0 is arbitrary, but fixed. Then for an arbitrary x ∈ X and for a set D ⊂ I, measD ≤ ε we obtain Hence, taking into account the equalities and by the definition of c(X) (cf. Section 2) we get Recall that L = b1 M1 +b 2 K 0 L∞(I) ( a 3 1 + b3 M2 R) < 1 and then the inequality obtained above together with the properties of the operator H and the fact that the set Q R is compact in measure allows us to apply Theorem 2.13 which completes the proof.
Remark 3.2. Let us recall that in the proof we utilize the following fact: U maps L 1 (I) into L ∞ (I) and F 2 maps L 1 (I) into itself. This allows us to use the Hölder inequality. In this situation, we prove the existence of a.e. monotonic Mediterr. J. Math.
solutions which are integrable. Sometimes we need more information about the solution, namely if a solution is in some subspace of L 1 (I) (the space L p , for instance). In such a case we are able to use also the same type of inequality. Namely we need only to modify the growth conditions and consequently the spaces in which our operators act. As claimed in the introductory part of our paper we can repeat our proof with appropriate changes for considered operators: F 2 maps L p (I) into L q (I) and U maps L p (I) into L r (I), where 1 r + 1 q = 1 p . Whence we obtain an existence result for L p -solutions. It should be noted that in some papers, their authors consider the existence of solutions in L p spaces simultaneously for p ≥ 1. As claimed above it cannot be done for quadratic equations. Here is a version for p > 1. An interesting (and motivating) remark about the solutions in L p spaces for integral equations (by using similar method of the proof) can be found in [31, page 93]. However, by considering the measure of noncompactness c(X) = lim sup meas D→0 {sup x∈X xχ D Lp(I) } introduced by Erzakova ( [32]) (restricted to the family of sets compact in measure) instead of usually considered ones based on Kolomogorov or Riesz criteria of compactness (cf. [13]) we are able to examine by the same manner the case of L p (I) spaces.
Assume that p > 1 and 1 p1 + 1 p2 = 1 p . Denote by q the value min(p 1 , p 2 ) and by r the value max(p 1 , p 2 ). This implies, in particular, that q ≤ 2p. We shall treat the equation (1.1) under the following set of assumptions presented below.
(i) Assume that functions f i : R + ×R → R satisfy Carathéodory conditions and there are positive constants b i (i = 1, 2) and positive functions a 1 ∈ L p (I), a 2 ∈ L q (I) such that , for all t ∈ I and x ∈ R. Moreover, f i (i = 1, 2) are assumed to be nonincreasing with respect to both variable t and x separately. (ii) u : R + × R + × R → R satisfies Carathéodory conditions. The function u is nonincreasing with respect to each variable, separately. Suppose that for arbitrary non-negative z(t) ∈ L q (I) Let us note, that in the assumption (v) we consider the equation of the type A + Bt + Ct p q + Dt 2p q = t. The case p = q leads to the quadratic equation (considered in our first theorem). Altough the case p < q seems to be more complicated, it should be noted that since p q < 1 and 2p q < 2 this equation has a solution in (0, 1]. In some papers the assumption of this type is described by using auxiliary functions. In such a formulation the problem of existence of functions is unclear. Let us note, that for arbitrary pair of spaces L p (I) and L q (I) we are able to solve our problem.
Indeed, if 2p q ≥ 1, then for t ∈ I we have A + Bt + Ct p q + Dt 2p q ≤ A + Bt + C + Dt and our inequality has a solution in (0, 1] whenever A+C 1−B−D < 1. In the case 2p q < 1, we have the following estimation: A + Bt + Ct p q + Dt 2p q ≤ A + Bt + C + D and then A+C+D 1−B < 1 form a sufficient condition for the existence of solutions of our inequality in (0, 1]. Thus the set of functions satisfying our assumptions is nonempty (cf. also some interesting Examples in [11]). Let us recall that the first case is considered in the paper.
We would like to pay attention, that the condition (ii) implies that the kernels k(t, s) should be of Hille-Tamarkin classes i.e. k(t, ·) q r and k(·, s) q r it is sufficient to assume that they are finite being at the same time the upper bounds for K 0 , where q and r are conjugated with q and r, respectively.
Moreover, it is worthwhile to note that by the same manner we can extend our main result for other subspaces of L 1 (I) for which we are able to check the required properties of considered operators (some Orlicz spaces, for instance) cf. [25].