# Compactness Conditions in the Study of Functional, Differential, and Integral Equations

Authors: Józef Banaś; Kishin Sadarangani
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss some existence results for various types of functional, differential, and integral equations which can be obtained with the help of argumentations based on compactness
conditions. We restrict ourselves to some classical compactness conditions appearing in fixed point theorems due to Schauder, Krasnosel’skii-Burton, and Schaefer. We present also the technique associated with measures of noncompactness and we illustrate its applicability in proving the solvability of some functional integral equations. Apart from this, we discuss the application of the mentioned technique to the theory of ordinary differential equations in Banach spaces.


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## Body

## 1. Introduction

The concept of the compactness plays a fundamental role in several branches of mathematics such as topology, mathematical analysis, functional analysis, optimization theory, and nonlinear analysis [1–5]. Numerous mathematical reasoning processes depend on the application of the concept of compactness or relative compactness. Let us indicate only such fundamental and classical theorems as the Weierstrass theorem on attaining supremum by a continuous function on a compact set, the Fredholm theory of linear integral equations, and its generalization involving compact operators as well as a lot of fixed point theorems depending on compactness argumentations [6, 7]. It is also worthwhile mentioning such an important property saying that a continuous mapping transforms a compact set onto compact one.Let us pay a special attention to the fact that several reasoning processes and constructions applied in nonlinear analysis depend on the use of the concept of the compactness [6]. Since theorems and argumentations of nonlinear analysis are used very frequently in the theories of functional, differential, and integral equations, we focus in this paper on the presentation of some results located in these theories which can be obtained with the help of various compactness conditions.We restrict ourselves to present and describe some results obtained in the last four decades which are related to some problems considered in the theories of differential, integral, and functional integral equations. Several results using compactness conditions were obtained with the help of the theory of measures of noncompactness. Therefore, we devote one section of the paper to present briefly some basic background of that theory.Nevertheless, there are also successfully used argumentations not depending of the concept of a measure on noncompactness such as Schauder fixed point principle, Krasnosel’skii-Burton fixed point theorem, and Schaefer fixed point theorem.Let us notice that our presentation is far to be complete. The reader is advised to follow the most expository monographs in which numerous topics connected with compactness conditions are broadly discussed [6, 8–10].Finally, let us mention that the presented paper has a review form. It discusses some results described in details in the papers which will be cited in due course.


## 2. Selected Results of Nonlinear Analysis Involving Compactness Conditions

In order to solve an equation having the form(1)x=Fx,
where F is an operator being a self-mapping of a Banach space E, we apply frequently an approach through fixed point theorems. Such an approach is rather natural and, in general, very efficient. Obviously, there exists a huge number of miscellaneous fixed point theorems [6, 7, 11] depending both on order, metric, and topological argumentations.The most efficient and useful theorems seem to be fixed point theorems involving topological argumentations, especially those based on the concept of compactness. The reader can make an acquaintance with the large theory of fixed point-theorems involving compactness conditions in the above, mentioned monographs, but it seems that the most important and expository fixed point theorem in this fashion is the famous Schauder fixed point principle [12]. Obviously, that theorem was generalized in several directions but till now it is very frequently used in application to the theories of differential, integral, and functional equations.At the beginning of our considerations we recall two well-known versions of the mentioned Schauder fixed point principle (cf. [13]). To this end assume that (E,||·||) is a given Banach space.Theorem 1.
LetΩ be a nonempty, bounded, closed, and convex subset of E and let F:Ω→Ω be a completely continuous operator (i.e., F is continuous and the image FΩ is relatively compact). Then F has at least one fixed point in the set Ω (this means that the equation x=Fx has at least one solution in the set Ω).Theorem 2.
IfΩ is a nonempty, convex, and compact subset of E and F:Ω→Ω is continuous on the set Ω, then the operator F has at least one fixed point in the set Ω.Observe that Theorem1 can be treated as a particular case of Theorem 2 if we apply the well-known Mazur theorem asserting that the closed convex hull of a compact subset of a Banach space E is compact [14]. The basic problem arising in applying the Schauder theorem in the version presented in Theorem 2 depends on finding a convex and compact subset of E which is transformed into itself by operator F corresponding to an investigated operator equation.In numerous situations, we are able to overcome the above-indicated difficulty and to obtain an interesting result on the existence of solutions of the investigated equations (cf. [7, 15–17]). Below we provide an example justifying our opinion [18].To do this, let us denote byℝ the real line and put ℝ+=[0,+∞). Further, let Δ={(t,s)∈ℝ2:0≤s≤t}.Next, fix a functionp=p(t) defined and continuous on ℝ+ with positive real values. Denote by Cp=C(ℝ+,p(t)) the space consisting of all real functions defined and continuous on ℝ+ and such that
(2)sup{|x(t)|p(t):t≥0}<∞.
It can be shown that Cp forms the Banach space with respect to the norm
(3)∥x∥=sup{|x(t)|p(t):t≥0}.
For our further purposes, we recall the following criterion for relative compactness in the space Cp [10, 15].Theorem 3.
LetX be a bounded set in the space Cp. If all functions belonging to X are locally equicontinuous on the interval ℝ+ and if limT→∞{sup{|x(t)|p(t):t≥T}}=0 uniformly with respect to X, then X is relatively compact in Cp.In what follows, ifx is an arbitrarily fixed function from the space Cp and if T>0 is a fixed number, we will denote by νT(x,ε) the modulus of continuity of x on the interval [0,T]; that is,
(4)νT(x,ε)=sup{|x(t)-x(s)|:t,s∈[0,T],|t-s|≤ε}.Further on, we will investigate the solvability of the nonlinear Volterra integral equation with deviated argument having the form(5)x(t)=d(t)+∫0tv(t,s,x(φ(s)))ds,
where t∈ℝ+. Equation (5) will be investigated under the following formulated assumptions.(i)
v:Δ×ℝ→ℝis a continuous function and there exist continuous functions n:Δ→ℝ+,a:ℝ+→(0,∞),b:ℝ+→ℝ+ such that
(6)|v(t,s,x)|≤n(t,s)+a(t)b(s)|x|
for all (t,s)∈Δ and x∈ℝ.In order to formulate other assumptions, let us put(7)L(t)=∫0ta(s)b(s)ds,t≥0.
Next, take an arbitrary number M>0 and consider the space Cp, where p(t)=[a(t)exp(ML(t)+t)]-1. Then, we can present other assumptions.(ii)
The functiond:ℝ+→ℝ is continuous and there exists a nonnegative constant D such that |d(t)|≤Da(t)exp(ML(t)) for t≥0.(iii)
There exists a constantN≥0 such that
(8)∫0tn(t,s)ds≤Na(t)exp(ML(t))
for t≥0.(iv)
φ:ℝ+→ℝ+ is a continuous function satisfying the condition
(9)L(φ(t))-L(t)≤K
for t∈ℝ+, where K≥0 is a constant.(v)
D+N<1 and a(φ(t))/a(t)≤M(1-D-N)exp(-MK) for all t≥0.Now, we can formulate the announced result.Theorem 4.
Under assumptions (i)–(v), (5) has at least one solution x in the space Cp such that |x(t)|≤a(t)exp(ML(t)) for t∈ℝ+.We give the sketch of the proof (cf. [18]). First, let us define the transformation F on the space Cp by putting
(10)(Fx)(t)=d(t)+∫0tv(t,s,x(φ(s)))ds,t≥0.
In view of our assumptions, the function (Fx)(t) is continuous on ℝ+.Further, consider the subsetG of the space Cp consisting of all functions x such that |x(t)|≤a(t)exp(ML(t)) for t∈ℝ+. Obviously, G is nonempty, bounded, closed, and convex in the space Cp. Taking into account our assumptions, for an arbitrary fixed x∈G and t∈ℝ+, we get
(11)|(Fx)(t)|≤|d(t)|+∫0t|v(t,s,x(φ(s)))|ds≤Da(t)exp(ML(t))+∫ot[n(t,s)+a(t)b(s)|x(φ(s))|]ds≤Da(t)exp(ML(t))+Na(t)exp(ML(t))+a(t)∫0tb(s)a(φ(s))exp(ML(φ(s)))ds≤(D+N)a(t)exp(ML(t))+(1-D-N)a(t)∫0tMa(s)b(s)×exp(ML(s))exp(-MK)exp(MK)ds≤(D+N)a(t)exp(ML(t))+(1-D-N)a(t)exp(ML(t))=a(t)exp(ML(t)).
This shows that F transforms the set G into itself.Next we show thatF is continuous on the set G.To this end, fixε>0 and take x,y∈G such that ||x-y||≤ε. Next, choose arbitrary T>0. Using the fact that the function v(t,s,x) is uniformly continuous on the set [0,T]2×[-α(T),α(T)], where α(T)=max{a(φ(t))exp(ML(φ(t))):t∈[0,T]}, for t∈[0,T], we obtain
(12)|(Fx)(t)-(Fy)(t)|≤∫0t|v(t,s,x(φ(s)))-v(t,s,y(φ(s)))|ds≤β(ε),
where β(ε) is a continuous function with the property limε→0β(ε)=0.Now, taket≥T. Then we get
(13)|(Fx)(t)-(Fy)(t)|[a(t)exp(ML(t)+t)]-1≤{|(Fx)(t)|+|(Fy)(t)|}×[a(t)exp(ML(t))]-1·e-t≤2e-t.
Hence, for T sufficiently large, we have
(14)|(Fx)(t)-(Fy)(t)|p(t)≤ε
for t≥T. Linking (12) and (14), we deduce that F is continuous on the set G.The next essential step in our proof which enables us to apply the Schauder fixed point theorem (Theorem1) is to show that the set FG is relatively compact in the space Cp. To this end let us first observe that the inclusion FG⊂G and the description of G imply the following estimate: (15)|(Fx)(t)|p(t)≤e-t.
This yields that
(16)limT→∞{sup{|(Fx)(t)|p(t):t≥T}}=0
uniformly with respect to the set G.On the other hand, for fixedε>0,T>0 and for t,s∈[0,T] such that |t-s|≤ε, in view of our assumptions, for x∈G, we derive the following estimate:
(17)|(Fx)(t)-(Fx)(s)|≤|d(t)-d(s)|+|∫0tv(t,τ,x(φ(τ)))dτ-∫0sv(t,τ,x(φ(τ)))dτ|+|∫0sv(t,τ,x(φ(τ)))dτ-∫0sv(s,τ,x(φ(τ)))dτ|≤νT(d,ε)+εmax{×p(φ(τ)):0≤τ≤t≤Tn(t,τ)+a(τ)b(τ)νT(d,ε)+εmax×p(φ(τ)):0≤τ≤t≤T}+TνT(v(ε,T,α(T))),
where we denoted
(18)νT(v(ε,T,α(T)))=sup{|v(t,u,v)-v(s,u,v)|:t,s∈[0,T],|t-s|≤ε,u∈[0,T],|v|≤α(T)}.
Taking into account the fact that limε→0νT(d,ε)=limε→0νT(v(ε,T,α(T)))=0, we infer that functions belonging to the set FG are equicontinuous on each interval [0,T]. Combining this fact with (16), in view of Theorem 3, we conclude that the set FG is relatively compact. Applying Theorem 1, we complete the proof.Another very useful fixed point theorem using the compactness conditions is the well-known Krasnosel’skii fixed point theorem [19]. That theorem was frequently modified by researchers working in the fixed point theory (cf. [6, 7, 20]), but it seems that the version due to Burton [21] is the most appropriate to be used in applications.Below we formulate that version.Theorem 5.
LetS be a nonempty, closed, convex, and bounded subset of the Banach space X and let A:X→X and B:S→X be two operators such that(a)
A is a contraction, that is, there exists a constant k∈[0,1) such that ||Ax-Ay||≤k||x-y|| for x,y∈X,(b)
B is completely continuous,(c)
x=Ax+By⇒x∈S for all y∈S.
Then the equationAx+Bx=x has a solution in S.In order to show the applicability of Theorem5, we will consider the following nonlinear functional integral equation [22]:
(19)x(t)=f(t,x(α1(t)),…,x(αn(t)))+∫0β(t)g(t,s,x(γ1(s)),…,x(γm(s)))ds,
where t∈ℝ+. Here we assume that f and g are given functions.The above equation will be studied in the spaceBC(ℝ+) consisting of all real functions defined, continuous, and bounded on the interval ℝ+ and equipped with the usual supremum norm
(20)∥x∥=sup{|x(t)|:t∈ℝ+}.
Observe that the space BC(ℝ+) is a special case of the previously considered space Cp with p(t)=1 for t∈ℝ+. This fact enables us to adapt the relative compactness criterion contained in Theorem 3 for our further purposes.In what follows we will impose the following requirements concerning the components involved in (19):(i)
The functionf:ℝ+×ℝn→ℝ is continuous and there exist constants ki∈[0,1)(i=1,2,…,n) such that
(21)|f(t,x1,x2,…,xn)-f(t,y1,y2,…,yn)|≤∑i=1n‍ki|xi-yi|
for all t∈ℝ+ and for all (x1,x2, …,xn), (y1,y2,…,yn)∈ℝn.(ii)
The functiont→f(t,0,…,0) is bounded on ℝ+ with F0=sup{|f(t,0,…,0)|:t∈ℝ+}.(iii)
The functionsαi,γj:ℝ+→ℝ+ are continuous and αi(t)→∞ as t→∞(i=1,2,…,n;j=1,2,…,m).(iv)
The functionβ:ℝ+→ℝ+ is continuous.(v)
The functiong:ℝ+2×ℝm→ℝis continuous and there exist continuous functions q:ℝ+2→ℝ+ and a,b:ℝ+→ℝ+ such that
(22)|g(t,s,x1,…,xm)|≤q(t,s)+a(t)b(s)∑i=1m‍|xi|
for all t,s∈ℝ+ and (x1,x2,…,xm)∈ℝm. Moreover, we assume that
(23)limt→∞∫0β(t)q(t,s)ds=0,limt→∞a(t)∫0β(t)b(s)ds=0.Now, let us observe that based on assumption (v), we conclude that the functionsv1,v2:ℝ+→ℝ+ defined by the formulas
(24)v1(t)=∫0β(t)q(t,s)ds,v2(t)=a(t)∫0β(t)b(s)ds
are continuous and bounded on ℝ+. Obviously this implies that the constants M1,M2 defined as:
(25)Mi=sup{vi(t):t∈ℝ+}(i=1,2)
are finite.In order to formulate our last assumption, let us denotek=∑i=1n‍ki, where the constants ki(i=1,2,…,n) appear in assumption (i).(vi)
k+mM2<1. Then we have the following result [22] which was announced above.Theorem 6.
Under assumptions (i)–(vi), (19) has at least one solution in the space BC(ℝ+). Moreover, solutions of (19) are globally attractive.Remark 7.
In order to recall the concept of the global attractivity mentioned in the above theorem (cf. [22]), suppose that Ω is a nonempty subset of the space BC(ℝ+) and Q:Ω→BC(ℝ+) is an operator. Consider the operator equation
(26)x(t)=(Qx)(t),t∈ℝ+.
We say that solutions of (26) are globally attractive if for arbitrary solutions x,y of this equation, we have that
(27)limt→∞(x(t)-y(t))=0.
Let us mention that the above-defined concept was introduced in [23, 24].Proof of Theorem6.
We provide only the sketch of the proof. Consider the ballBr in the space X=BC(ℝ+) centered at the zero function θ and with radius r=(F0+M1)/[1-(k+mM2)]. Next, define two mappings A:X→X and B:Br→X by putting
(28)(Ax)(t)=f(t,x(α1(t)),…,x(αn(t))),(Bx)(t)=∫0β(t)g(t,s,x(γ1(s)),…,x(γm(s)))ds,
for t∈ℝ+. Then (19) can be written in the form
(29)x(t)=(Ax)(t)+(Bx)(t),t∈ℝ+.
Notice that in view of assumptions (i)–(iii), the mappingA is well defined, and for arbitrarily fixed function x∈X, the function Ax is continuous and bounded on ℝ+. Thus A transforms X into itself. Similarly, applying assumptions (iii)–(v), we deduce that the function Bx is continuous and bounded on ℝ+. This means that B transforms the ball Br into X.
Now, we check that operatorsA and B satisfy assumptions imposed in Theorem 5. To this end take x,y∈X. Then, in view of assumption (i), for a fixed t∈ℝ+, we get
(30)|(Ax)(t)-(Ay)(t)|≤∑i=1n‍ki|x(αi(t))-y(αi(t))|≤k∥x-y∥.
This implies that ||Ax-Ay||≤k||x-y||, and in view of assumption (vi), we infer that A is a contraction on X.
Next, we prove thatB is completely continuous on the ball Br. In order to show the indicated property of B, we fix ε>0 and we take x,y∈Br with ||x-y||≤ε. Then, taking into account our assumptions, we obtain
(31)|(Bx)(t)-(By)(t)|≤∫0β(t)[|g(t,s,x(γ1(s)),…,x(γm(s)))|+|g(t,s,y(γ1(s)),…,y(γm(s)))|]ds≤2(v1(t)+rmv2(t)).
Hence, keeping in mind assumption (v), we deduce that there exists T>0 such that v1(t)+rmv2(t)≤ε/2 for t≥T. Combining this fact with (31), we get
(32)|(Bx)(t)-(By)(t)|≤ε
for t≥T.
Further, for an arbitraryt∈[0,T], in the similar way, we obtain
(33)|(Bx)(t)-(By)(t)|≤∫0β(t)ωrT(g,ε)ds≤βTωrT(g,ε),
where we denoted βT=sup{β(t):t∈[0,T]} and
(34)ωrT(g,ε)=sup{|g(t,s,x1,x2,…,xm)-g(t,s,y1,y2,…,ym)|:t∈[0,T],s∈[0,βT],xi,yi∈[-r,r],|xi-yi|≤ε(i=1,2,…,m)}.
Keeping in mind the uniform continuity of the function g on the set [0,T]×[0,βT]×[-r,r]m, we deduce that ωrT(g,ε)→0 as ε→0. Hence, in view of (32) and (33), we conclude that the operator B is continuous on the ball Br.
The boundedness of the operatorB is a consequence of the inequality
(35)|(Bx)(t)|≤v1(t)+rmv2(t)
for t∈ℝ+. To verify that the operator B satisfies assumptions of Theorem 3 adapted to the case of the space BC(ℝ+), fix arbitrarily ε>0. In view of assumption (v), we can choose T>0 such that v1(t)+rmv2(t)≤ε for t≥T. Further, take an arbitrary function x∈Br. Then, keeping in mind (35), for t≥T, we infer that
(36)|(Bx)(t)|≤ε.
Next, take arbitrary numbers t,τ∈[0,T] with |t-τ|≤ε. Then we obtain the following estimate:
(37)|(Bx)(t)-(Bx)(τ)|≤|∫β(τ)β(t)|g(t,s,x(γ1(s)),…,x(γm(s)))|ds|+∫0β(τ)|g(t,s,x(γ1(s)),…,x(γm(s)))-g(τ,s,x(γ1(s)),…,x(γm(s)))|ds≤|∫β(τ)β(t)[q(t,s)+a(t)b(s)∑i=1m‍|x(γi(s))|]ds|+∫0β(τ)ω1T(g,ε;r)ds≤|∫β(τ)β(t)q(t,s)ds|+a(t)mr|∫β(τ)β(t)b(s)ds|+βTω1T(g,ε;r),
where we denoted
(38)ω1T(g,ε;r)=sup{|g(t,s,x1,…,xm)-g(τ,s,x1,…,xm)|:t,τ∈[0,T],|t-s|≤ε,s∈[0,βT],|x1|≤r,…,|xm|≤r}.
From estimate (37), we get
(39)|(Bx)(t)-(Bx)(τ)|≤qTνT(β,ε)+rmaTbTνT(β,ε)+βTω1T(g,ε;r),
where qT=max{q(t,s):t∈[0,T],s∈[0,βT]}, aT=max{a(t):t∈[0,T]}, bT=max{b(t):t∈[0,βT]}, and νT(β,ε) denotes the usual modulus of continuity of the function β on the interval [0,T].
Now, let us observe that in view of the standard properties of the functionsβ=β(t), g=g(t,s,x1,…,xm), we infer that νT(β,ε)→0 and ω1T(g,ε;r)→0 as ε→0. Hence, taking into account the boundedness of the image BBr and estimates (36) and (39), in view of Theorem 3, we conclude that the set BBr is relatively compact in the space BC(ℝ+);  thatis,B is completely continuous on the ball Br.
In what follows fix arbitraryx∈BC(ℝ+) and assume that the equality x=Ax+By holds for some y∈Br. Then, utilizing our assumptions, for a fixed t∈ℝ+, we get
(40)|x(t)|≤|(Ax)(t)|+|(By)(t)|≤|f(t,x(α1(t)),…,x(αn(t)))-f(t,0,…,0)|+|f(t,0,…,0)|+∫0β(t)q(t,s)ds+m∥y∥a(t)∫0β(t)b(s)ds≤k∥x∥+F0+M1+mrM2.
Hence we obtain
(41)∥x∥≤F0+M1+mrM21-k.
On the other hand, we have that r=(F0+M1+mrM2)/(1-k). Thus ||x||≤r or, equivalently, x∈Br. This shows that assumption (c) of Theorem 5 is satisfied.
Finally, combining all of the above-established facts and applying Theorem5, we infer that there exists at least one solution x=x(t) of (19).
The proof of the global attractivity of solutions of (19) is a consequence of the estimate
(42)|x(t)-y(t)|≤∑i=1n‍kimax{|x(αi(t))-y(αi(t))|:i=1,2,…,n}+2∫0β(t)q(t,s)ds+a(t)∫0β(t)(b(s)∑i=1m‍|x(γi(s))|)ds+a(t)∫0β(t)(b(s)∑i=1m‍|y(γi(s))|)ds≤kmax{|x(αi(t))-y(αi(t))|:i=1,2,…,n}+2v1(t)+m(∥x∥+∥y∥)v2(t),
which is satisfied for arbitrary solutions x=x(t), y=y(t) of  (19).
Hence we get(43)limsupt→∞|x(t)-y(t)|≤kmax1≤i≤n{limsupt→∞|x(αi(t)-y(αi(t)))|}+2limsupt→∞v1(t)+m(∥x∥+∥y∥)limsupt→∞v2(t)=klimsupt→∞|x(t)-y(t)|
which implies that limsupt→∞|x(t)-y(t)|=limt→∞|x(t)-y(t)|=0. This means that the solutions of (19) are globally attractive (cf. Remark 7).It is worthwhile mentioning that in the literature one can encounter other formulations of the Krasnosel’skii-Burton fixed point theorem (cf. [6, 7, 20, 21]). In some of those formulations and generalizations, there is used the concept of a measure of noncompactness (both in strong and in weak sense) and, simultaneously, the requirement of continuity is replaced by the assumption of weak continuity or weak sequential continuity of operators involved (cf. [6, 20], for instance).In what follows we pay our attention to another fixed point theorem which uses the compactness argumentation. Namely, that theorem was obtained by Schaefer in [25].Subsequently that theorem was formulated in other ways and we are going here to present two versions of that theorem (cf. [11, 26]).Theorem 8.
Let(E,||·||) be a normed space and let T:E→E be a continuous mapping which transforms bounded subsets of E onto relatively compact ones. Then either(i)
the equationx=Tx has a solution
or(ii)
the set⋃0≤λ≤1{x∈E:x=λTx} is unbounded.The below presented version of Schaefer fixed point theorem seems to be more convenient in applications.Theorem 9.
LetE,||·|| be a Banach space and let T:E→E be a continuous compact mapping (i.e., T is continuous and T maps bounded subsets of E onto relatively compact ones). Moreover, one assumes that the set
(44)⋃0≤λ≤1‍{x∈E:x=λTx}
is bounded. Then T has a fixed point.It is easily seen that Theorem9 is a particular case of Theorem 8.Observe additionally, that Schaefer fixed point theorem seems to be less convenient in applications than Schauder fixed point theorem (cf. Theorems1 and 2). Indeed, Schaefer theorem requires a priori bound on utterly unknown solutions of the operator equation x=λTx for λ∈[0,1]. On the other hand, the proof of Schaefer theorem requires the use of the Schauder fixed point principle (cf. [27], for details).It is worthwhile mentioning that an interesting result on the existence of periodic solutions of an integral equation, based on a generalization of Schaefer fixed point theorem, may be found in [26].


## 3. Measures of Noncompactness and Their Applications

Let us observe that in order to apply the fundamental fixed point theorem based on compactness conditions, that is, the Schauder fixed point theorem, say, the version of Theorem2, we are forced to find a convex and compact subset of a Banach space E which is transformed into itself by an operator F. In general, it is a hard task to encounter a set of such a type [16, 28]. On the other hand, if we apply Schauder fixed point theorem formulated as Theorem 1, we have to prove that an operator F is completely continuous. This causes, in general, that we have to impose rather strong assumptions on terms involved in a considered operator equation.In view of the above-mentioned difficulties starting from seventies of the past century mathematicians working on fixed point theory created the concept of the so-called measure of noncompactness which allowed to overcome the above-indicated troubles. Moreover, it turned out that the use of the technique of measures of noncompactness allows us also to obtain a certain characterization of solutions of the investigated operator equations (functional, differential, integral, etc.). Such a characterization is possible provided we use the concept of a measure of noncompactness defined in an appropriate axiomatic way.It is worthwhile noticing that up to now there have appeared a lot of various axiomatic definitions of the concept of a measure of noncompactness. Some of those definitions are very general and not convenient in practice. More precisely, in order to apply such a definition, we are often forced to impose some additional conditions on a measure of noncompactness involved (cf. [8, 29]).By these reasons it seems that the axiomatic definition of the concept of a measure of noncompactness should be not very general and should require satisfying such conditions which enable the convenience of their use in concrete applications.Below we present axiomatics which seems to satisfy the above-indicated requirements. That axiomatics was introduced by Banaś and Goebel in 1980 [10].In order to recall that axiomatics, let us denote by𝔐E the family of all nonempty and bounded subsets of a Banach space E and by 𝔑E its subfamily consisting of relatively compact sets. Moreover, let B(x,r) stand for the ball with the center at x and with radius r. We write Br to denote the ball B(θ,r), where θ is the zero element in E. If X is a subset of E, we write X¯,ConvX to denote the closure and the convex closure of X, respectively. The standard algebraic operations on sets will be denoted by X+Y and λX, for λ∈ℝ.As we announced above, we accept the following definition of the concept of a measure of noncompactness [10].Definition 10.
A functionμ:𝔐E→ℝ+=[0,∞) is said to be a measure of noncompactness in the space E if the following conditions are satisfied. (1o)
The familykerμ={X∈𝔐E:μ(X)=0} is nonempty and kerμ⊂𝔑.(2o)
X⊂Y⇒μ(X)≤μ(Y).(3o)
μ(X¯)=μ(ConvX)=μ(X).(4o)
μ(λX+(1-λ)Y)≤λμ(X)+(1-λ)μ(X) for λ∈[0,1].(5o)
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 the set X∞=⋂n=1∞‍Xn is nonempty.Let us pay attention to the fact that from axiom(5o) we infer that μ(X∞)≤μ(Xn) for any n=1,2,…. This implies that μ(X∞)=0. Thus X∞ belongs to the family kerμ described in axiom (1o). The family kerμ is called the kernel of the measure of noncompactnessμ.The property of the measure of noncompactnessμ mentioned above plays a very important role in applications.With the help of the concept of a measure of noncompactness, we can formulate the following useful fixed point theorem [10] which is called the fixed point theorem of Darbo type.Theorem 11.
LetΩ be a nonempty, bounded, closed, and convex subset of a Banach space E and let F:Ω→Ω be a continuous operator which is a contraction with respect to a measure of noncompactness μ; that is, there exists a constant k, k∈[0,1), such that μ(FX)≤kμ(X) for any nonempty subset X of the set Ω. Then the operator F has at least one fixed point in the set Ω.In the sequel we show an example of the applicability of the technique of measures of noncompactness expressed by Theorem11 in proving the existence of solutions of the operator equations.Namely, we will work on the Banach spaceC[a,b] consisting of real functions defined and continuous on the interval [a,b] and equipped with the standard maximum norm. For sake of simplicity, we will assume that [a,b]=[0,1]=I, so the space on question can be denoted by C(I).One of the most important and handy measures of noncompactness in the spaceC(I) can be defined in the way presented below [10].In order to present this definition, take an arbitrary setX∈𝔐C(I). For x∈X and for a given ε>0, let us put
(45)ω(x,ε)=sup{|x(t)-x(s)|:t,s∈I,|t-s|≤ε}.
Next, let us define
(46)ω(X,ε)=sup{ω(x,ε):x∈X},ω0(X)=limε→0ω(X,ε).
It may be shown that the function ω0(X) is the measure of noncompactness in the space C(I) (cf. [10]). This measure has also some additional properties. For example, ω0(λX)=|λ|ω0(X) and ω0(X+Y)≤ω0(X)+ω0(Y) provided X,Y∈𝔐C(I) and λ∈ℝ[10].In what follows we will consider the nonlinear Volterra singular integral equation having the form(47)x(t)=f1(t,x(t),x(a(t)))+(Gx)(t)∫0tf2(t,s)(Qx)(s)ds,
where t∈I=[0,1], a:I→I is a continuous function, and G, Q are operators acting continuously from the space C(I) into itself. Apart from this, we assume that the function f2 has the form
(48)f2(t,s)=k(t,s)g(t,s),
where k:Δ→ℝ is continuous and g is monotonic with respect to the first variable and may be discontinuous on the triangle Δ={(t,s):0≤s≤t≤1}.Equation (47) will be considered in the space C(I) under the following assumptions (cf. [30]).(i)
a:I→I is a continuous function.(ii)
The functionf1:I×ℝ×ℝ→ℝ is continuous and there exists a nonnegative constant p such that
(49)|f1(t,x1,y1)-f1(t,x2,y2)|≤pmax{|x1-x2|,|y1-y2|}
for any t∈I and for all x1,x2,y1,y2∈ℝ.(iii)
The operatorG transforms continuously the space C(I) into itself and there exists a nonnegative constant q such that ω0(GX)≤qω0(X) for any set X∈𝔐C(I), where ω0 is the measure of noncompactness defined by (46).(iv)
There exists a nondecreasing functionφ:ℝ+→ℝ+ such that ||Gx||≤φ(||x||) for any x∈C(I).(v)
The operatorQ acts continuously from the space C(I) into itself and there exists a nondecreasing function Ψ:ℝ+→ℝ+ such that ||Qx||≤Ψ(||x||) for any x∈C(I).(vi)
f2:Δ→ℝ has the form (48), where the function k:Δ→ℝ is continuous.(vii)
The functiong(t,s)=g:Δ→ℝ+ occurring in the decomposition (48) is monotonic with respect to t (on the interval [s,1]), and for any fixed t∈I, the function s→g(t,s) is Lebesgue integrable over the interval [0,t]. Moreover, for every ε>0, there exists δ>0 such that for all t1,t2∈I with t1<t2 and t2-t1≤δ the following inequalities are satisfied:
(50)|∫0t1[g(t2,s)-g(t1,s)]ds|≤ε,∫t1t2g(t2,s)ds≤ε.The main result concerning (47), which we are going to present now, will be preceded by a few remarks and lemmas (cf. [30]). In order to present these remarks and lemmas, let us consider the function h:I→ℝ+ defined by the formula
(51)h(t)=∫0tg(t,s)ds.
In view of assumption (vii), this function is well defined.Lemma 12.
Under assumption (vii), the functionh is continuous on the interval I.For proof, we refer to [30].In order to present the last assumptions needed further on, let us define the constantsk¯, f1¯, h¯ by putting
(52)k¯=sup{|k(t,s)|:(t,s)∈Δ},f1¯=sup{|f1(t,0,0)|:t∈I},h¯=sup{h(t):t∈I}.
The constants k¯ and f1¯ are finite in view of assumptions (vi) and (ii), while the fact that h¯<∞ is a consequence of assumption (vii) and Lemma 12.Now, we formulate the announced assumption.(viii)
There exists a positive solutionr0 of the inequality
(53)pr+f1¯+k¯h¯φ(r)Ψ(r)≤r
such that p+k¯h¯qΨ(r0)<1.For further purposes, we definite operators corresponding to (47) and defined on the space C(I) in the following way:
(54)(F1x)(t)=f1(t,x(t),x(α(t))),(F2x)(t)=∫0tf2(t,s)(Qx)(s)ds,(Fx)(t)=(F1x)(t)+(Gx)(t)(F2x)(t),
for t∈I. Apart from this, we introduce two functions M and N defined on ℝ+ by the formulas
(55)M(ε)=sup{|∫0t1[g(t2,s)-g(t1,g)]ds|:t1,t2∈I,t1<t2,t2-t1≤ε{|∫0t1[g(t2,s)-g(t1,g)]ds|}},(56)N(ε)=sup{∫t1t2g(t2,s)ds:t1,t2∈I,t1<t2,t2-t1≤ε}.
Notice that in view of assumption (vii), we have that M(ε)→0 and N(ε)→0 as ε→0.Now, we can state the following result.Lemma 13.
Under assumptions(i)–(vii), the operator F transforms continuously the space C(I) into itself.Proof.
Fix a functionx∈C(I). Then F1x∈C(I), which is a consequence of the properties of the so-called superposition operator [9]. Further, for arbitrary functions x,y∈C(I), in virtue of assumption (ii), for a fixed t∈I, we obtain
(57)|(F1x)(t)-(F1y)(t)|≤pmax{|x(t)-y(t)|,|x(a(t))-y(a(t))|}.
This estimate in combination with assumption (i) yields
(58)∥F1x-F1y∥≤p∥x-y∥.
Hence we conclude that F1 acts continuously from the space C(I) into itself.
Next, fixx∈C(I) and ε>0. Take t1,t2∈I such that |t2-t1|≤ε. Without loss of generality, we may assume that t1≤t2. Then, based on the imposed assumptions, we derive the following estimate:
(59)|(F2x)(t2)-(F2x)(t1)|≤|∫0t2k(t2,s)g(t2,s)(Qx)(s)ds-∫0t1k(t2,s)g(t2,s)(Qx)(s)ds|+|∫0t1k(t2,s)g(t2,s)(Qx)(s)ds-∫0t1k(t1,s)g(t2,s)(Qx)(s)ds|+|∫0t1k(t1,s)g(t2,s)(Qx)(s)ds-∫0t1k(t1,s)g(t1,s)(Qx)(s)ds|≤∫t1t2|k(t2,s)|g(t2,s)|(Qx)(s)|ds+∫0t1|k(t2,s)-k(t1,s)|g(t2,s)|(Qx)(s)|ds+∫0t1|k(t1,s)||g(t2,s)-g(t1,s)||(Qx)(s)|ds≤k¯Ψ(∥x∥)N(ε)+ω1(k,ε)Ψ(∥x∥)∫0t2g(t2,s)ds+k¯Ψ(∥x∥)∫0t1|g(t2,s)-g(t1,s)|ds,
where we denoted
(60)ω1(k,ε)=sup{|k(t2,s)-k(t1,s)|:(t1,s),(t2,s)∈Δ,|t2-t1|≤ε}.
Since the function t→g(t,s) is assumed to be monotonic, by assumption (vii) we get
(61)∫0t1|g(t2,s)-g(t1,s)|ds=|∫0t1[g(t2,s)-g(t1,s)]ds|.
This fact in conjunction with the above-obtained estimate yields
(62)ω(F2x,ε)≤k¯Ψ(∥x∥)N(ε)+h¯Ψ(∥x∥)ω1(k,ε)+k¯Ψ(∥x∥)M(ε),
where the symbol ω(y,ε) denotes the modulus of continuity of a function y∈C(I).
Further observe thatω1(k,ε)→0 as ε→0, which is an immediate consequence of the uniform continuity of the function k on the triangle Δ. Combining this fact with the properties of the functions M(ε) and N(ε) and taking into account (62), we infer that F2x∈C(I). Consequently, keeping in mind that F1:C(I)→C(I) and assumption (iii), we conclude that the operator F is a self-mapping of the space C(I).
In order to show thatF is continuous on C(I), fix arbitrarily x0∈C(I) and ε>0. Next, take x∈C(I) such that ||x-x0||≤ε. Then, for a fixed t∈I, we get
(63)|(Fx)(t)-(Fx0)(t)|≤|(F1x)(t)-(F1x0)(t)|+|(Gx)(t)(F2x)(t)-(Gx0)(t)(F2x0)(t)|≤∥F1x-F1x0∥+∥Gx∥|(F2x)(t)-(F2x0)(t)|+∥F2x0∥∥Gx-Gx0∥.
On the other hand, we have the following estimate:
(64)|(F2x)(t)-(F2x0)(t)|≤∫0t|k(t,s)|g(t,s)|(Qx)(s)-(Qx0)(s)|ds≤k¯(∫0tg(t,s)ds)∥Qx-Qx0∥≤kh¯∥Qx-Qx0∥.
Similarly, we obtain
(65)|(F2x0)(t)|≤k¯(∫0tg(t,s)ds)∥Qx0∥≤kh¯∥Qx0∥,
and consequently
(66)∥F2x0∥≤kh¯∥Qx0∥.
Next, linking (63)–(66) and (58), we obtain the following estimate:
(67)∥Fx-Fx0∥≤p∥x-x0∥+∥Gx∥kh¯∥Qx-Qx0∥+kh¯∥Gx-Gx0∥∥Qx0∥.
Further, taking into account assumptions (iv) and (v), we derive the following inequality
(68)∥Fx-Fx0∥≤pε+φ(∥x0∥+ε)kh¯∥Qx-Qx0∥+kh¯Ψ(∥x0∥)∥Gx-Gx0∥.
Finally, in view of the continuity of the operators G and Q (cf. assumptions (iii) and (v)), we deduce that operator F is continuous on the space C(I). The proof is complete.Now, we can formulate the last result concerning (47) (cf. [30]).Theorem 14.
Under assumptions (i)–(viii), (47) has at least one solution in the space C(I).Proof.
Fixx∈C(I) and t∈I. Then, evaluating similarly as in the proof of Lemma 13, we get
(69)|(Fx)(t)|≤|f1(t,x(t),x(a(t)))-f1(t,0,0)|+|f1(t,0,0)|+|Gx(t)||∫0tk(t,s)g(t,s)(Qx)(s)ds|≤f1¯+p∥x∥+kh¯φ(∥x∥)Ψ(∥x∥).
Hence, we obtain(70)∥Fx∥≤f1¯+p∥x∥+kh¯φ(∥x∥)Ψ(∥x∥).
From the above inequality and assumption (viii), we infer that there exists a number r0>0 such that the operator F maps the ball Br0 into itself and p+kh¯qΨ(r0)<1. Moreover, by Lemma 13, we have that F is continuous on the ball Br0.
Further on, take a nonempty subsetX of the ball Br0 and a number ε>0. Then, for an arbitrary x∈X and t1,t2∈I with |t2-t1|≤ε, in view of (66) and the imposed assumptions, we obtain
(71)|(Fx)(t2)-(Fx)(t1)|≤|f1(t2,x(t2),x(a(t2)))-f1(t1,x(t2),x(a(t2)))|+|f1(t1,x(t2),x(a(t2)))-f1(t1,x(t1),x(a(t1)))|+|(F2x)(t2)||(Gx)(t2)-(Gx)(t1)|+|(Gx)(t1)||(F2x)(t2)-(F2x)(t1)|≤ωr0(f1,ε)+pmax{|x(t2)-x(t1)|,ωr0(f1,ε)+pmax|x(a(t2))-x(a(t1))|}+kh¯Ψ(r0)ω(Gx,ε)+φ(r0)ω(F2x,ε),
where we denoted
(72)ωr0(f1,ε)=sup{|f1(t2,x,y)-f1(t1,x,y)|:t1,t2∈I,|t2-t1|≤ε,x,y∈[-r0,r0]}.
Hence, in virtue of (62), we deduce the estimate
(73)ω(Fx,ε)≤ωr0(f1,ε)+pmax{ω(x,ε),ω(x,ω(a,ε))}+kh¯Ψ(r0)ω(Gx,ε)+φ(r0)Ψ(r0)[k¯N(ε)+h¯ω1(k,ε)+k¯M(ε)].
Finally, taking into account the uniform continuity of the function f1 on the set I×[-r0,r0]2 and the properties of the functions k(t,s), a(t), M(ε), and N(ε) and keeping in mind (46), we obtain
(74)ω0(FX)≤pω0(X)+kh¯Ψ(r0)ω0(GX).
Linking this estimate with assumption (iii), we get
(75)ω0(FX)≤(p+kh¯qΨ(r0))ω0(X).
The use of Theorem 11 completes the proof.


## 4. Existence Results Concerning the Theory of Differential Equations in Banach Spaces

In this section, we are going to present some classical results concerning the theory of ordinary differential equations in Banach space. We focus on this part of that theory in a which the technique associated with measures of noncompactness is used as the main tool in proving results on the existence of solutions of the initial value problems for ordinary differential equations. Our presentation is based mainly on the papers [31, 32] and the monograph [33].The theory of ordinary differential equations in Banach spaces was initiated by the famous example of Dieudonné [34], who showed that in an infinite-dimensional Banach space, the classical Peano existence theorem is no longer true. More precisely, Dieudonné showed that if we consider the ordinary differential equation
(76)x′=f(t,x)
with the initial condition
(77)x(0)=x0,
where f:[0,T]×B(x0,r)→E and E is an infinite-dimensional Banach space, then the continuity of f (and even uniform continuity) does not guarantee the existence of solutions of problem (76)-(77).In light of the example of Dieudonné, it is clear that in order to ensure the existence of solutions of (76)-(77), it is necessary to add some extra conditions. The first results in this direction were obtained by Kisyński [35], Olech [36], and Ważewski [37] in the years 1959-1960. In order to formulate those results, we need to introduce the concept of the so-called Kamke comparison function (cf. [38, 39]).To this end, assume thatT is a fixed number and denote J=[0,T], J0=(0,T]. Further, assume that Ω is a nonempty open subset of ℝn and x0 is a fixed element of Ω. Let f:J×Ω→ℝn be a given function.Definition 15.
A functionw:J×ℝ+→ℝ+ (or w:J0×ℝ+→ℝ+) is called a Kamke comparison function provided the inequality
(78)∥f(t,x)-f(t,y)∥≤w(t,∥x-y∥)
for x,y∈Ω and t∈J (or t∈J0), together with some additional assumptions concerning the function w, guarantees that problem (76)-(77) has at most one local solution.
In the literature, one can encounter miscellaneous classes of Kamke comparison functions (cf. [33, 39]). We will not describe those classes, but let us only mention that they are mostly associated with the differential equation u′=w(t,u) with initial condition u(0)=0 or the integral inequality u(t)≤∫0tw(s,u(s))ds for t∈J0 with initial condition limt→0u(t)/t=limt→0u(t)=0. It is also worthwhile recalling that the classical Lipschitz or Nagumo conditions may serve as Kamke comparison functions [39].
The above-mentioned results due to Kisyński et al. [35–37] assert that if f:J×B(x0,r)→E is a continuous function satisfying condition (78) with an appropriate Kamke comparison function, then problem (76)-(77) has exactly one local solution.
Observe that the natural translation of inequality (78) in terms of measures of noncompactness has the form
(79)μ(f(t,X))≤w(t,μ(X)),
where X denotes an arbitrary nonempty subset of the ball B(x0,r). The first result with the use of condition (79) for w(t,u)=Cu (C is a constant) was obtained by Ambrosetti [40]. After the result of Ambrosetti, there have appeared a lot of papers containing existence results concerning problem (76)-(77) (cf. [41–44]) with the use of condition (79) and involving various types of Kamke comparison functions. It turned out that generalizations of existence results concerning problem (76)-(77) with the use of more and more general Kamke comparison functions are, in fact, only apparent generalizations [45], since the so-called Bompiani comparison function is sufficient to give the most general result in the mentioned direction.
On the other hand, we can generalize existence results involving a condition like (79) taking general measures of noncompactness [31, 32]. Below we present a result coming from [32] which seems to be the most general with respect to taking the most general measure of noncompactness.
In the beginning, let us assume thatμ is a measure of noncompactness defined on a Banach space E. Denote by Eμ the set defined by the equality
(80)Eμ={x∈E:{x}∈kerμ}.
The set Eμ will be called the kernel set of a measure μ. Taking into account Definition 10 and some properties of a measure of noncompactness (cf. [10]), it is easily seen that Eμ is a closed and convex subset of the space E.
In the case when we consider the so-calledsublinear measure of noncompactness [10], that is, a measure of noncompactness μ which additionally satisfies the following two conditions:(6o)
μ(X+Y)≤μ(X)+μ(Y),(7o)
μ(λX)=|λ|μ(X) for λ∈ℝ, then the kernel set Eμ forms a closed linear subspace of the space E.
Further on, assume thatμ is an arbitrary measure of noncompactness in the Banach space E. Let r>0 be a fixed number and let us fix x0∈Eμ. Next, assume that f:J×B(x0,r)→E (where J=[0,T]) is a given uniformly continuous and bounded function; say, ||f(t,x)||≤A.
Moreover, assume thatf satisfies the following comparison condition of Kamke type:
(81)μ(x0+f(t,X))≤w(t,μ(X))
for any nonempty subset X of the ball B(x0,r) and for almost all t∈J.
Here we will assume that the functionw(t,u)=w:J0×ℝ+→ℝ+(J0=(0,T]) is continuous with respect to u for any t and measurable with respect to t for each u. Apart from this, w(t,0)=0 and the unique solution of the integral inequality
(82)u(t)≤∫0tw(s,u(s))ds(t∈J0)
such that limt→0u(t)/t=limt→0u(t)=0, is u≡0.
The following formulated result comes from [32].Theorem 16.
Under the above assumptions, if additionallysup{t+a(t):t∈J}≤1, where a(t)=sup{||f(0,x0)-f(s,x)||:s≤t,||x-x0||≤As} and A is a positive constant such that AT≤r, the initial value problem (76)-(77) has at least one local solution x=x(t) such that x(t)∈Eμ for t∈J.The proof of the above theorem is very involved and is therefore omitted (cf. [32, 33]). We restrict ourselves to give a few remarks.At first, let us notice that in the case whenμ is a sublinear measure of noncompactness, condition (81) is reduced to the classical one expressed by (79) provided we assume that x0∈Eμ. In such a case, Theorem 16 was proved in [10].The most frequently used type of a comparison functionw is this having the form w(t,u)=p(t)u, where p(t) is assumed to be Lebesgue integrable over the interval J. In such a case comparison, condition (81) has the form
(83)μ(x0+f(t,X))≤p(t)μ(X).
An example illustrating Theorem 16 under condition (83) will be given later.Further, observe that in the case whenμ is a sublinear measure of noncompactness such that x0∈Eμ, condition (83) can be written in the form
(84)μ(f(t,X))≤p(t)μ(X).
Condition (84) under additional assumption ||f(t,x)||≤P+Q||x||, with some nonnegative constants P and Q, is used frequently in considerations associated with infinite systems of ordinary differential equations [46–48].Now, we present the above-announced example coming from [33].Example 17.
Consider the infinite system of differential equations having the form(85)xn′=an(t)xn+fn(xn,xn+1,…),
where n=1,2,… and t∈J=[0,T]. System (85) will be considered together with the system of initial conditions
(86)xn(0)=x0n
for n=1,2,…. We will assume that there exists the limit limn→∞x0n=a(a∈ℝ).
Problem (85)-(86) will be considered under the following conditions.(i)
an:J→ℝ(n=1,2,…) are continuous functions such that the sequence (an(t)) converges uniformly on the interval J to the function which vanishes identically on J.(ii)
There exists a sequence of real nonnegative numbers(αn) such that limn→∞αn=0 and |fn(xn,xn+1,…)|≤αn for n=1,2,… and for all x=(x1,x2,x3,…)∈l∞.(iii)
The functionf=(f1,f2,f3,…) transforms the space l∞ into itself and is uniformly continuous.Let us mention that the symboll∞ used above denotes the classical Banach sequence space consisting of all real bounded sequences (xn) with the supremum norm; that is, ||(xn)||=sup{|xn|:n=1,2,…}.Under the above hypotheses, the initial value problem (85)-(86) has at least one solution x(t)=(x1(t),x2(t),…) such that x(t)∈l∞ for t∈J and limn→∞xn(t)=a uniformly with respect to t∈J, provided T≤1(J=[0,T]).As a proof, let us take into account the measure of noncompactness in the spacel∞ defined in the following way:
(87)μ(X)=limsupn→∞{supx∈X|xn-a|}
for X∈𝔐l∞ (cf. [10]). The kernel kerμ of this measure is the family of all bounded subsets of the space l∞ consisting of sequences converging to the limit equal to a with the same rate.Further, take an arbitrary setX∈𝔐l∞. Then we have
(88)μ(x0+f(t,X))=limsupn→∞{supx∈X|x0n+an(t)xnlimsup+fn(xn,xn+1,…)-a|{supx∈X|x0n+an(t)xn}}≤limsupn→∞{supx∈X[|an(t)||xn|limsup+|fn(xn,xn+1,…)+x0n-a|]{supx∈X[|an(t)||xn|}}≤limsupn→∞{supx∈Xp(t)|xn-a|}+limsupn→∞{supx∈X|an(t)||a|}+limsupn→∞{supx∈X|fn(xn,xn+1,…)|}+limsupn→∞{supx∈X|x0n-a|},
where we denoted p(t)=sup{|an(t)|:n=1,2,…} for t∈J. Hence we get
(89)μ(x0+f(t,X))≤p(t)μ(X),
which means that the condition (84) is satisfied.Combining this fact with assumption (iii) and taking into account Theorem16, we complete the proof.



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