# A Family of Measures of Noncompactness in the Locally Sobolev Spaces and Its Applications to Some Nonlinear Volterra Integrodifferential Equations

Authors: H. Mehravaran; M. Khanehgir; R. Allahyari
Journal: Journal of Mathematics
 (2018)

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## Abstract

We study a new family of measures of noncompactness in the locally Sobolev spaceWlocn,1(R+N), equipped with a suitable topology. As an application of the technique associated with this family of measures of noncompactness, we study the existence of solutions for a class of nonlinear Volterra integrodifferential equations. Further, we give an illustrative example to verify the effectiveness and applicability of our results.


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## Body

## 1. Introduction

A locally integrable function is a function which is integrable on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar toLp spaces, but its members are not required to satisfy any growth restriction on their behavior at infinity: in other words, locally integrable functions can grow arbitrarily fast at infinity but are still manageable in a way similar to ordinary integrable functions.Sobolev spaces [1], that is, the class of functions with derivatives in Lp, play an important role in the analysis. In the recent times, there have been considerable efforts to study these spaces. The significant impact of them arises from the fact that solutions of partial differential equations are naturally found in Sobolev spaces. They also can be applied to approximation theory, calculus of variations, differential geometry, spectral theory, and so on.The degree of noncompactness of a set is measured by means of functions called measures of noncompactness. Measures of noncompactness are very useful tools in Banach spaces. They are widely used in fixed point theory, differential equations, functional equations, integral and integrodifferential equations, optimization, and so on [2–8].In this paper, we define two topological structures on the locally Sobolev spaceWlocn,1(R+N): the Fréchet metric topology τlocF given by a sequence of seminorms and the topology τlocω generated by the family of projections,(1)πT:Wlocn,1R+N⟶Wn,10,TN,T≥0,where the spaces Wn,1([0,T]N) are furnished with weak topologies. Next, we introduce a weak measure of noncompactness in Wn,1([0,T]N). Thereafter, we define a family of measures of noncompactness {μT}T≥0 in Wlocn,1(R+N) with the topology τlocω and we investigate the basic properties of {μT}T≥0.As an application, we study the existence of solutions for a class of nonlinear Volterra integrodifferential equations(2)xt1,…,tN=ft1,…,tN,∫0t1⋯∫0tNϑt1,…,tN,s1,…,sN,xξs1,…,sN,∂x∂s1ξs1,…,sN,…,∂x∂sNξs1,…,sN,T0xs1,…,sNds1⋯dsN.Finally, we give an illustrative example to verify the effectiveness and applicability of our results.


## 2. Preliminaries

Letm(D) denote the Lebesgue measure of a Lebesgue measurable subset D⊂R+N. For subset D⊂[0,T]N of a fixed N-cell [0,T]N, we write D′=[0,T]N∖D.Suppose thatn∈N. We recall that the Sobolev space Wn,1([0,T]N) (T≥0) is defined to consist of those real measurable functions f which, together with all their distributional derivatives Dαf of order |α|≤n, belong to L1([0,T]N). Here α=(α1,…,αn) is a multi-index; that is, each αj is a nonnegative integer, |α|=α1+⋯+αn, and Dα=∂|α|/∂x1α1⋯∂xnαn.If0≤S≤T, then the symbol πST stands for the operator of restriction (3)πST:Wn,10,TN⟶Wn,10,SN,πSTf≔f0,SN,where f|[0,S]N is the restriction of the function f∈Wn,1([0,T]N) to the N-cell [0,S]N.The locally Sobolev spaceWlocn,1(R+N) (shortly Wlocn,1) is defined to consist of those real measurable functions f which, together with all their distributional derivatives Dαf of order |α|≤n, are locally integrable on R+N; that is, fn,1T<∞ for each T≥0, where(4)fn,1T=maxα≤n⁡∫0,TNDαfdμ.Furthermore, we denote by πT the operator of restriction(5)πT:Wlocn,1R+N⟶Wn,10,TN,πTf≔f0,TN.We consider the space Wlocn,1 with the Fréchet metric topology τlocF.Definition 1 (see [9]).
The metrizable topology induced inWlocn,1 by the family of seminorms (4), that is, the topology defined by the distance (6)df,g≔∑T=1∞12Tf-gn,1T1+f-gn,1T, or equivalently (7)d1f,g≔sup⁡2-Tf-gn,1T,T≥0, will be called Fréchet metric topology in Wlocn,1 and it will be denoted by τlocF.The convergence and compactness in the topologyτlocF are characterized in the following proposition.Proposition 2.
A sequence{fk}⊂Wlocn,1 is convergent to f∈Wlocn,1 in the topology τlocF if and only if limk→∞⁡‖fk-f‖n,1T=0 for each T≥0. A set X⊂Wlocn,1 is relatively compact in the topology τlocF if and only if πT(X) is relatively compact in the Banach space Wn,1([0,T]N) for T≥0.Proof.
The proof is easy.We conclude this section by recalling Dunford-Pettis Theorem (see [10]) which will be used in proving our results in next section.Theorem 3.
LetF be a bounded set in L1([0,T]N). Then F has compact closure in the weak topology τω if and only if F is equi-integrable; that is, (8)∀ε>0,∃δ>0suchthat∫Dfdμ<ε for all measurable sets D⊂[0,T]N with m(D)<δ and all f∈F.


## 3. Main Results

In this section, we introduce a weak measure of noncompactness inWn,1([0,T]N). For this purpose, we need the following result.Proposition 4.
Assume thatF is a bounded set in Wn,1([0,T]N). Then F has compact closure with the weak topology τω if and only if (9)∀ε>0,∃δ>0suchthatmaxα≤n⁡∫DDαfdμ<εfor all measurable sets D⊂[0,T]N with m(D)<δ and all f∈F.Proof.
Takek=∑i=0nNi and let (L1([0,T]N))k be a Banach space with norm (10)f1,…,fkmax=max⁡fi1T,i=1,…,k. Define Φ:Wn,1([0,T]N)→(L1([0,T]N))k({α:|α|≤n}={β1,β2,…,βk-1}) by (11)Φf=f,Dβ1f,Dβ2f,…,Dβk-1f. Obviously Φ is one-to-one, continuous linear operator and Φ(Wn,1([0,T]N)) is closed in (L1([0,T]N))k. Thus, Wn,1([0,T]N) and Φ(Wn,1([0,T]N)) are isometrically isomorphic. The proof is completed by Dunford-Pettis Theorem.ConsiderNWn,1([0,T]N)ω the subfamily of Wn,1([0,T]N) consisting of relatively compact sets in the topology τω and MWn,1([0,T]N) the family of all nonempty and bounded subsets (in norm) of Wn,1([0,T]N). Now, we introduce a weak measure of noncompactness in Wn,1([0,T]N).LetX be a bounded subset of the space Wn,1([0,T]N). For f∈X and ε>0 let us denote (12)CTf,ε=sup⁡maxα≤n⁡∫DDαfdμ:D⊂0,TN:mD<ε,CTX,ε=sup⁡CTf,ε:f∈X,μX≔CTX=limε→0⁡CTX,ε.The essential properties of this measure are contained in the following lemma.Lemma 5.
The weak measure of noncompactnessμ:MWn,1([0,T]N)→R+ satisfies the following conditions:(1)
ker⁡μ=NWn,1([0,T]N)ω.(2)
IfX⊂Y, then μ(X)≤μ(Y).(3)
μ(λX)=|λ|μ(X).(4)
μ(X+Y)≤μ(X)+μ(Y).(5)
μ(X)=μ(X¯ω)=μ(conv⁡X¯) where X¯ω is the weak closure of a set X and conv⁡X¯ is the closed convex hull (with respect to the norm topology) of X.(6)
If{Xn} is a sequence of nonempty, bounded (in norm), weakly closed subsets of Wn,1([0,T]N) and X1⊃X2⊃⋯ with limn→∞⁡μ(Xn)=0, then the intersection set X∞≔⋂n=1∞Xn is nonempty and weakly compact in Wn,1([0,T]N).Proof.
TakeX∈MWn,1([0,T]N) such that μ(X)=0. Then limε→0⁡CT(X,ε)=0. Therefore, for any ϱ>0 there exists δ>0 such that (13)maxα≤n⁡∫DDαfdμ<ϱ for all D⊂[0,T]N with m(D)<δ and all f∈X. So, using Proposition 4  X is relatively compact in the weak topology and ker⁡μ⊆NWn,1([0,T]N)ω. In order to verify ker⁡μ⊃NWn,1([0,T]N), suppose that X∈NWn,1([0,T]N)ω. Applying Proposition 4 for any ε>0 there exists δ>0 such that (14)maxα≤n⁡∫DDαfdμ<ε for all f∈X and D⊂[0,T]N such that m(D)<δ.
Now, for allf∈X we have (15)CTf,δ=sup⁡maxα≤n⁡∫DDαfdμ:D⊂0,TN,mD<δ<ε. It enforces that CT(X,δ)<ε and thus we get limδ→0⁡CT(X,δ)=0. It implies that μ(X)=0, and so NWn,1([0,T]N)ω⊂ker⁡μ. We conclude that (1) holds.
The fact thatμ satisfies conditions (2)–(4) is obvious.
Now, we check that condition (5) holds. For this purpose, suppose thatX∈MWn,1([0,T]N) and f∈X¯ω. Thus, there is a sequence such that fm→f weakly in Wn,1([0,T]N). On the other hand, by the definition of CT(X,ε) we have (16)maxα≤n⁡∫DDαfmdμ⩽CTX,ε for any m∈N and D⊂[0,T]N with m(D)<ε.
Lettingm→∞ and using Fatou’s Lemma we yield(17)maxα≤n⁡∫DDαfdμ≤CTX,ε for any D⊂[0,T]N with m(D)<ε.
Hence(18)CTX¯ω,ε⩽CTX,ε,limε→0⁡CTX¯ω,ε⩽limε→0⁡CTX,ε. This implies that (19)μX¯ω⩽μX. Moreover, we obtain by condition (2) that μ(X¯ω)=μ(X). Particularly, μ(conv⁡X¯ω)=μ(conv⁡X). By [11, Theorem 3.12] we have that (20)μX=μconv⁡X=μconv⁡X¯ω=μconv⁡X¯. It remains to prove (6); suppose that {Xm} is a sequence of the weakly closed and nonempty sets of MWn,1([0,T]N) such that Xm+1⊂Xm for m=1,2,…, and limm→∞⁡μ(Xm)=0. Now, for any m∈N, take fm∈Xm. We claim that B={fm}¯ω is a weakly compact set in Wn,1([0,T]N). To prove this assertion, we need to check that μ(B)=0. Let ε>0 be fixed. Since limm→∞⁡μ(Xm)=0, then there exists m1∈N such that μ(Xm1)<ε.
Hence, we can findδ1>0 small enough that (21)CTXm1,δ1<ε. Thus, for all m≥m1, we have(22)sup⁡maxα≤n⁡∫DDαfmdμ:D⊂0,TN,mD<δ1<ε.As we know that the set {f1,…,fm1-1} is weakly compact. By Proposition 4, there exists δ2>0 such that (23)maxα≤n⁡∫DDαfidμ<ε for all i=1,…,m1-1 and D⊂[0,T]N such that m(D)<δ2. Therefore(24)sup⁡maxα≤n⁡∫DDαfidμ:i=1,…,m1-1,D⊂0,TN,mD<δ2<ε.Using (22) and (24) we obtain CT(B,δ)<ε for δ=min⁡{δ1,δ2}. It means that μ(B)=CT(B)=0.
Thus, there is a subsequence such thatfmj→f0 weakly in B. Since fm∈Xm,  Xm+1⊂Xm, and Xm is weakly closed for all m∈N, we get (25)f0∈⋂j=1∞Xmj=⋂m=1∞Xm=X∞, which completes the proof of (6).


## 4. The Topologyτloc⁡ω and the Family of Measures of Noncompactness

In this section, we will define the topologyτlocω in Wlocn,1. Furthermore, we will introduce the family of measures of noncompactness {μT}T≥0 in Wlocn,1 with the topology τlocω and we will describe its properties.Definition 6.
The topologyτloc⁡ω is the weakest topology on Wloc⁡n,1 such that all operators πT:Wloc⁡n,1(R+N)→Wn,1([0,T]N)(T≥0) defined in (5) are continuous.A base forτloc⁡ω is the collection of all sets of the form πT-1(G), where T≥0 and G is an arbitrary open subset of Wn,1([0,T]N) with the weak topology. Also, there is the natural topological homeomorphism h from Wloc⁡n,1 onto h(Wloc⁡n,1)⊆∏T≥0Wn,1([0,T]N), given by formula(26)hx≔πTxT≥0forx∈Wlocn,1.In this note, we say that a subsetX⊂Wloc⁡n,1 is bounded, if X is bounded for every seminorm ‖·‖n,1T,T≥0. Furthermore, we denote by MWloc⁡n,1 the family of all nonempty and bounded subsets of Wloc⁡n,1 and by NWloc⁡n,1 its subfamily consisting of all relatively compact sets in the topology τloc⁡ω.Now, we are in a position to introduce a family of measures of noncompactness inWlocn,1 with the topology τlocω.Definition 7.
The family of mappings{μT}T≥0,μT:MWloc⁡n,1→R+, defined by formula(27)μTX≔limε→0+⁡supf∈X⁡sup⁡maxα≤n⁡∫DDαfdμ:D⊂0,TN,mD≤ε,is said to be the family of measures of noncompactness in Wloc⁡n,1 with topology τloc⁡ω. Indeed, (28)μTX=CTπTX, where CT is defined in (12).Theorem 8.
The family of measures of noncompactness{μT}T≥0,  μT:MWlocn,1→R+ defined by (27), fulfills the following conditions: (1∘)
The familyker⁡{μT}≔{X∈MWlocn,1:μT(X)=0 for T≥0} is nonempty and ker⁡{μT}=NWlocn,1.(2∘)
IfX⊂Y, then μT(X)≤μT(Y) for T≥0.(3∘)
μT(λX)=|λ|μT(X) for T≥0.(4∘)
μT(X+Y)≤μT(X)+μT(Y) for T≥0.(5∘)
μT(X)=μT(X¯ω)=μT(conv⁡X¯F) for T≥0 where X¯ω and conv⁡X¯F are the closure of the set X in the topology τlocω and the closed convex hull of the set X with respect to the topology τlocF, respectively.(6∘)
If{Xn} is a sequence of nonempty, bounded, weakly closed subsets of Wlocn,1 and X1⊃X2⊃⋯ with limn→∞⁡μT(Xn)=0 for each T≥0 then the intersection set X∞≔⋂n=1∞Xn is nonempty and compact in τlocω.Proof.
First we show that(1∘) holds. To do this, take X∈ker⁡{μT}. Therefore for each T≥0,μT(X)=CT(πT(X))=0. Now, by using Lemma 5, πT(X)¯ω is weakly compact in Wn,1([0,T]N) for T≥0. Applying Tychonoff’s Theorem we conclude that ∏T≥0πT(X)¯ω is weakly compact in ∏T≥0Wn,1([0,T]N).
On the other hand, it is easy to see that∏T≥0(πT(X))=h(X). It in turns implies that (29)∏T≥0πTX¯ω=∏T≥0πTX¯ω=hX¯ω=hX¯ω. Now, since h is a homeomorphism, it follows that X¯ω is compact in τlocω and X∈NWlocn,1.
Conversely, suppose thatX∈NWlocn,1. Thus, X¯ω is compact in τloc⁡ω, which implies that πT(X¯ω) is weakly compact in Wn,1([0,T]N).
By Lemma5 we have (30)μTX=CTπTX≤CTπTX¯ω=μπTX¯ω=0,T≥0;that is, X∈ker⁡{μT}.
Parts(2∘)–(4∘) are easy to verify. To prove (5∘), suppose that X⊂Wlocn,1. Using again Lemma 5 and inclusion πT(X¯ω)⊂πT(X)¯ω, we get (31)CTπTX=CTπTX¯ω≥CTπTX¯ω≥CTπTX. Hence, CT(πT(X))=CT(πT(X¯ω)) and μT(X)=μT(X¯ω).
We know thatπT(Y¯F)⊂πT(Y)¯ for Y∈Wlocn,1; therefore we have(32)CTπTX≤CTπTconv⁡X≤CTπTconv⁡X¯F≤CTπTconv⁡X¯=CTconv⁡πTX¯=CTπTX. This means that μT(X)=μT(X¯ω)=μT(conv⁡X¯F).
It remains to prove(6∘). Suppose that {Xn} is a sequence of nonempty and weakly closed sets of MWlocn,1 such that Xn+1⊂Xn for n=1,2,… and limn→∞⁡μT(Xn)=0 for T≥0. By applying Lemma 5, we have (33)limn→∞⁡CTπTXn¯ω=limn→∞⁡CTπTXn=limn→∞⁡μTXn=0. Therefore, by Lemma 5, we have that ⋂n=1∞πT(Xn)¯ω is nonempty and weakly compact in Wn,1([0,T]N) for T≥0. It implies that ∏T≥0(⋂n=1∞πT(Xn)¯ω) is nonempty and weakly compact.
It is routine to see that⋂n=1∞h(Xn)=∏T≥0(⋂n=1∞πT(Xn)). Now, suppose that π^T:h(Wn,1loc)→Wn,1([0,T]N) is a mapping defined by (34)π^Thx=πTx. By [12, Proposition 2.5.5], the family of all sets π^T-1(UT), where UT is a neighborhood of πT(x) in the space Wn,1([0,T]N), is a base for the subspace h(Wn,1loc) at the point h(x). Also, we know that if f is a continuous mapping between two topological spaces, then(35)f⋂n=1∞Bn¯⊆⋂n=1∞fBn¯for all subsets Bn,n=1,2,… contained in the domain of f.
Takef=π^T and Bn=h(Xn),n=1,2,…. Then from (35) we have (36)π^T⋂n=1∞hXn¯ω⊆⋂n=1∞π^ThXn¯ω=⋂n=1∞πTXn¯ω, which implies that(37)⋂n=1∞hXn¯ω⊆∏T≥0⋂n=1∞πTXn¯ω.
Conversely, let{fT}∈∏T≥0⋂n=1∞πT(Xn)¯ω. Then for each T≥0,fT∈⋂n=1∞πTXn¯ω=⋂n=1∞π^T(h(Xn))¯ω. Hence, for each n∈N,fT∈π^T(h(Xn))¯ω(T≥0).
Suppose thatUT is a neighborhood of fT in the space Wn,1([0,T]N). Then (38)UT⋂π^ThXn≠∅. This implies that π^T(UT)⋂h(Xn)≠∅ and {fT}∈h(Xn)¯ω. Since this is true for each n∈N, then (39)fT∈⋂n=1∞hXn¯ω. Hence(40)∏T≥0⋂n=1∞πTXn¯ω⊆⋂n=1∞hXn¯ω.Therefore using (37) and (40) we get ⋂n=1∞h(Xn)¯ω=∏T≥0⋂n=1∞πT(Xn)¯ω and so ⋂n=1∞h(Xn)¯ω is nonempty and weakly compact. Since h is a homeomorphism, we get (41)⋂n=1∞hXn¯=⋂n=1∞hXn¯ω=h⋂n=1∞Xn¯ω; hence, ⋂n=1∞Xn¯ω=⋂n=1∞Xn=X∞ is nonempty and compact in Wloc⁡n,1.As an immediate consequence of Theorem8(1∘), we have the following corollary.Corollary 9.
A subsetX⊂Wloc1,1 is relatively compact in the topology τlocω if and only if πT(X) is relatively weakly compact in W1,1([0,T]N) for each T>0.We recall that a subsetX⊂W1,1([0,T]N) is a.e. equicontinuous on [0,T]N, if, for each ε>0, there exists a closed subset Dε⊂[0,T]N such that m(Dε′)≤ε and the restriction of all functions from X to the set Dε form an equicontinuous family of functions. Furthermore, X is said to be countable a.e. equicontinuous on [0,T]N if every sequence {xn}⊂X has a subsequence {x¯n}, which is a.e. equicontinuous on [0,T]N.Using some ideas from [8] we have the following results.Lemma 10.
Suppose that a subsetX⊂W1,1([0,T]N) is bounded in norm. Then X is relatively compact if and only if X is relatively weakly compact and countable a.e. equicontinuous on [0,T]N.Proof.
We first suppose thatX is a relatively compact subset of the Banach space W1,1([0,T]N). Since the set X is also relatively weakly compact, then it suffices to show that X is countable a.e. equicontinuous. Regarding the proof of Corollary 9 of [13], X is totally bounded in W1,1([0,T]N) if and only if X and D1X={∂f/∂xi:i=1,…,N,f∈X} are totally bounded in L1([0,T]N). Then X and D1X are relatively compact in L1([0,T]N). From [8, Lemma 3.7], it follows that X and D1X are countable a.e. equicontinuous on [0,T]N with respect to ·1T. It can be easily shown that X is countable a.e. equicontinuous on [0,T]N with respect to ·1,1T. Conversely, suppose that X is relatively weakly compact in W1,1([0,T]N) and countable a.e. equicontinuous on [0,T]N. Obviously, X and D1X are countable a.e. equicontinuous on [0,T]N with respect to ·1T. Using Proposition 4 and Theorem 3, it yields that X and D1X are relatively weakly compact in L1([0,T]N). Applying again [8, Lemma 3.7], it obtains that X and D1X are relatively compact in L1([0,T]N) and so X is relatively compact in W1,1([0,T]N).Lemma 11.
Suppose that a subsetX⊂Wloc1,1 is bounded. Then X is relatively compact in the topology τlocF if and only if X is relatively compact in the topology τlocω and X is countable a.e. equicontinuous on N-cells of R+N.Proof.
Suppose thatX is relatively compact in the topology τlocF. According to Proposition 2, πT(X) is relatively compact in W1,1([0,T]N) for each T>0. Applying Lemma 10, we obtain πT(X) is relatively weakly compact for all T>0 and X is countable a.e. equicontinuous on N-cells of R+N. From Corollary 9, we conclude that X is relatively compact in the topology τlocω.
Conversely, suppose thatX is relatively compact in the topology τlocω and X is countable a.e. equicontinuous on N-cells of R+N. From Corollary 9, we deduce that πT(X) is relatively weakly compact for each T>0. Hence from Lemma 10 and Proposition 2, we find that X is relatively compact in the topology τlocF.Lemma 12 (see [5]).
Assume that(S,d) is a complete metric space. Let X,Xn,  n=1,2,… be nonempty subsets of S such that Xn are relatively compact for n=1,2,… and limn→∞⁡D(X,Xn)=0, where D(·,·) denotes the Hausdorff distance between sets. Then the set X is also relatively compact.


## 5. Example of Applications of the Family of Measures of Noncompactness{μT}T≥0

In this section, we study the applicability of the family of measures of noncompactness{μT}T≥0 in the locally Sobolev space Wloc1,1 with the topology τlocω. For this purpose, we investigate the existence solutions to a special nonlinear Volterra integrodifferential equation of the form (42)xt1,…,tN=ft1,…,tN,∫0t1⋯∫0tNϑt1,…,tN,s1,…,sN,xξs1,…,sN,∂x∂s1ξs1,…,sN,…,∂x∂sNξs1,…,sN,T0xs1,…,sNds1⋯dsN.We start with some preliminary facts which will be needed further on.Suppose thatJ⊂RN is a given measurable subset and S is a metric space. We say that a function f((t1,…,tN),x(t1,…,tN))=f:J×S→R satisfies Carathéodory conditions if it is measurable in (t1,…,tN) for any x∈S and is continuous in x for almost all (t1,…,tN)∈J.Definition 13.
Assume thatJ⊂R+N is a given set and f:J×R+→R+ is a function satisfying the Carathéodory conditions. Then, to every positive real-valued function x=x(t1,…,tN) which is measurable on J, we may assign the function (Fx)(t1,…,tN)=f((t1,…,tN),x(t1,…,tN)), (t1,…,tN)∈J. The operator F defined in such a way is said to be the superposition (or Nemytskii) operator generated by the function f. It is well known that the function Fx is also measurable on J.Definition 14.
We say that a functionf:R→R is locally absolutely continuous if for each nonempty compact interval [a,b], the restriction of f to [a,b] is absolutely continuous.Now, letΩ⊂RN. Recall that the space AC=AC(Ω) consists of all real-valued measurable functions f on Ω such that, for every line τ parallel to one of the si-coordinate axes {(s1,…,sN):s1=⋯=si-1=si+1=⋯=sN=0}, f is locally absolutely continuous on Ω∩τ.Given a spaceX of measurable functions over Ω, we denote by X¯ the space of all measurable functions f such that there exists f¯∈X with f=f¯ almost everywhere in Ω. In particular, for f∈AC¯ we denote by ∂/∂ti¯f the derivative ∂/∂tif¯ where f¯∈AC and f=f¯ almost everywhere in Ω.Following Marcus and Mizel [14] we call a function f on Ω×R a locally absolutely continuous Carathéodory function if f(·,u) belongs to AC¯ for every u∈R, and f(s,·) is continuous and locally absolutely continuous for almost all s∈Ω.By using some facts from [15] we can prove the following theorem.Theorem 15.
LetN>1 be a natural number and let f:R+N×R→R be a locally absolutely continuous Carathéodory function. Then the superposition operator F generated by f((t1,…,tN),x) maps the space Wloc1,1 into itself and it is bounded and continuous with the Fréchet topology τlocF if and only if the partial derivative ∂f/∂ti(i=1,…,N) generates superposition operator from LN/N-1 into L1, and the partial derivative ∂f/∂x generates a superposition operator from LN/N-1 into L∞; that is, for each x∈Wloc1,1 and i=1,…,N, we have (43)∂∂ti¯ft1,…,tN,xt1,…,tN≤at1,…,tN+bxt1,…,tNN/N-1,∂f∂xt1,…,tN,xt1,…,tN≤ct1,…,tN+d,where (t1,…,tN)∈R+N,a∈Lloc1(R+N),  c∈L∞(R+N), and b,d≥0.Proof.
See [15, Theorem 9.5] and comment after it (page 185) and apply it for the special case p=q=1.Finally, we proceed by establishing some facts concerning the linear Volterra integral operator inWloc1,1.We denote byΔ the area Δ={((t1,…,tN),(s1,…,sN)):0≤si≤ti} and assume κ:Δ→R+ is a given function which is measurable with respect to both variables. For an arbitrary function x∈Wlocn,1 put (44)Kxt1,…,tN≔∫0t1⋯∫0tNκt1,…,tN,s1,…,sNxs1,…,sNds1⋯dsN,where (t1,…,tN)∈R+N. The operator K is defined in such a way which is called the linear Volterra integral operator. The following lemma gives us a fundamental property of the Volterra integral operator (see [16]).Lemma 16.
If the linear Volterra integral operatorK maps Wloc1,1 into itself, then it is continuous in the Fréchet metric topology τlocF.Now, let us assume thatKT≔K|Wn,1([0,T]N); that is, KT is the restriction of the operator K to the Banach space Wn,1([0,T]N). The operator KT:Wn,1([0,T]N)→Wn,1([0,T]N) is continuous (cf. [16, 17]) and the norm of the linear operator KT we denote by ‖KT‖ for T≥0.Equation (42) will be discussed later under the following assumptions:(1)
N>1 is a natural number and ξ:R+N→R+N is a continuous function such that, for each A⊂R+N,ξ(A)⊂A.(2)
f:R+N×R→R is a locally absolutely continuous Carathéodory function and moreover, for each x∈Wloc1,1,  (t1,…,tN)∈R+N, and i=0,1,…,N, we have(45)∂∂ti¯ft1,…,tN,xt1,…,tN≤at1,…,tN+bxt1,…,tNN/N-1,(46)∂f∂xt1,…,tN,xt1,…,tN≤ct1,…,tN+d,wherea∈Lloc1(R+N),  c∈L∞(R+N) with c¯=esssup|c| and b,d≥0.(3)
T0:Wloc1,1→Wloc1,1 is a continuous operator and for each x∈Wloc1,1, and i=0,1,…,N(47)∂∂siT0xs1,…,sN≤∂∂sixs1,…,sN.(4)
ϑ:R+N×R+N×RN+2→R+ satisfies Carathéodory conditions; that is, the function ϑ(·,(t1,…,tN),(s1,…,sN+2)) is measurable on R+N and the function ϑ((t1,…,tN),·,·) is continuous a.e. on R+N×RN+2.(5)
V:Wloc1,1→Wloc1,1 is an operator given by the formula (48)Vxt1,…,tN≔∫0t1⋯∫0tNϑt1,…,tN,s1,…,sN,xξs1,…,sN,∂x∂s1ξs1,…,sN,…,∂x∂sNξs1,…,sN,T0xs1,…,sNds1⋯dsN,wherex∈Wloc1,1 and it satisfies in the following conditions:(49)iVxt1,…,tNN/N-1≤∫0t1⋯∫0tNκt1,…,tN,s1,…,sNa1ξs1,…,sN+Mx,s1,…,sNds1⋯dsN,(50)ii∂Vxt1,…,tN∂ti≤∫0t1⋯∫0tNκt1,…,tN,s1,…,sNa1ξs1,…,sN+Mx,s1,…,sNds1⋯dsN,wherei=1,…,N,  a1:R+N→[1,∞) is in Wloc1,1,κ:R+N→R+ satisfies Carathéodory conditions, and (51)Mx,s1,…,sN=max⁡xξs1,…,sN,∂x∂s1ξs1,…,sN,…,∂x∂sNξs1,…,sN,T0xs1,…,sN.(6)
For allT≥0,‖KT‖(b+c¯+d)<1.We study the existence result in the following theorem.Theorem 17.
Under assumptions (1)–(6), (42) has at least one solution x=x(t1,…,tN) which belongs to the space Wloc1,1.Proof.
Let us consider the operatorH:Wloc1,1→Wloc1,1 defined by the right side of (42). Then we can assume that H=FV, where F is the superposition operator generated by f. Fix arbitrary x∈Wloc1,1. For i=1,…,N, we have (52)∂∂siHxs1,…,sN=∂∂sifs1,…,sN,Vxs1,…,sN=∂f∂sis1,…,sN,Vxs1,…,sN+∂f∂Vxs1,…,sN,Vxs1,…,sN∂Vx∂sis1,…,sN. Hence, (53)max⁡∫0,TN∂∂siHxs1,…,sNds1⋯dsN:i=1,…,N≤max⁡∫0,TN∂f∂sis1,…,sN,Vxs1,…,sNds1⋯dsN:i=1,…,N+max⁡∫0,TN∂f∂Vxs1,…,sN,Vxs1,…,sN∂Vx∂sis1,…,sNds1⋯dsN:i=1,…,N.On the other hand, by (45), (49), and condition (3) we have (54)max⁡∫0,TN∂f∂sis1,…,sN,Vxs1,…,sNds1⋯dsN:i=1,…,N≤∫0,TNas1,…,sN+bVxs1,…,sNN/N-1ds1⋯dsN≤a1T+b∫0,TN∫0s1⋯∫0sNκs1,…,sN,τ1,…,τNa1ξτ1,…,τN+Mx,τ1,…,τNdτ1⋯dτNds1⋯dsN≤a1T+b∫0,TNKa1s1,…,sN+b∫0,TNKMx,s1…,sNds1⋯dsN≤a1T+bKTa11,1T+x1,1T,where a1T=∫[0,T]N|a(s1…,sN)|ds1⋯dsN.
Applying (46), (50), and condition (3) we have(55)max⁡∫0,TN∂f∂Vxs1,…,sN,Vxs1,…,sN∂Vx∂sis1,…,sNds1⋯dsN:i=1,…,N≤max⁡∫0,TN∂Vx∂sis1,…,sNcs1,…,sN+dds1⋯dsN:i=1,…,N≤c¯+d∫0,TN∫0s1⋯∫0sNκs1,…,sN,τ1,…,τNa1ξτ1,…,τN+Mx,τ1,…,τNdτ1⋯dτNds1⋯dsN≤c¯+d∫0,TNKa1s1,…,sN+Mx,s1,…,sNds1⋯dsN≤c¯+dKTa11,1T+x1,1T.Thus, combining (53), (54), and (55) it can be deduced that (56)max⁡∫0,TN∂∂siHxs1,…,sNds1⋯dsN:i=1,…,N≤a1T+b+c¯+dKTa11,1T+x1,1T.In addition, from (45) and using the same techniques as above, one can show that (57)∫0,TNHxs1,…,sNds1⋯dsN≤a1T+bKTa11,1T+x1,1T.Now (56) and (57) imply that(58)Hx1,1T≤AT+BTx1,1T,where A(T)=‖a‖1T+(b+c¯+d)‖KT‖‖a1‖1,1T and B(T)=(b+c¯+d)‖KT‖.
Next, we define the nonnegative, nondecreasing functionr:R+→R+ and the subset B of Wloc1,1 as follows:(59)rt≔At1-Bt,B≔x∈Wloc1,1:x1,1t≤rtfort≥0. The set B is bounded and convex in Wloc1,1. Inequality (58) ensures that H maps B into itself; that is,(60)H:B⟶B.Furthermore, the set (61)πTB=x∈W1,10,TN:x1,1t≤rtfort∈0,T is convex and closed in the Banach space W1,1([0,T]N), so it is weakly closed in W1,1([0,T]N) for T≥0. Since(62)B=⋂T≥0πT-1πTB,then B is closed in the topology τlocω; that is,(63)B¯ω=B.Regarding condition (5) and applying majorant principle (cf. [16, 17]) we infer that the operator V|W1,1([0,T]N):W1,1([0,T]N)→W1,1([0,T]N) is continuous. In aid to this fact, together with Proposition 2, we conclude that(64)V:Wloc1,1⟶Wloc1,1is  continuous  in  the  topology  τlocF.
Further, take a nonempty subsetX of the set B and arbitrary number T≥0. Fix a number ε>0 and a nonempty subset D⊂[0,T]N such that D is measurable and m(D)≤ε. Then, for arbitrarily fixed x∈X we have(65)max⁡∫DDiHxs1,…,sNds1⋯dsN:i=0,1,…,N≤∫Das1,…,sNds1⋯dsN+b+c¯+dKT∫Da1s1,…,sNds1⋯dsN+∫DMx,s1,…,sNds1⋯dsN.
Taking into account absolute continuity of the integrals of the functionsa and a1 and letting ε→0, we get(66)μTHX≤KTb+c¯+dμTX.Put B0=B,  Bn=conv⁡H(Bn-1)¯ω,  n=1,2,…. In view of (60) and (63), we have Bn⊂Bn-1 for all n∈N, by which together with (66) we get (67)μTBn≤KTb+c¯+bnμTB,T≥0. It follows then from condition (6) that limn→∞⁡μT(Bn)=0. Due to condition (6∘) in Theorem 8 we deduce that the set B∞=⋂n=1∞Bn is nonempty, convex, and compact in τlocω. This means (68)limε→0⁡supx∈B∞⁡sup⁡maxi=0,…,N∫DDixt1,…,tNdt1⋯dtN:D⊂0,TN,mD≤ε=0for each T≥0.
In the sequel, let us consider the sequence of the operatorsPn:B→B,n=1,2,…, defined by formula (69)Pnxt1,…,tN≔xt1,…,tN,xt1,…,tN≤n,nsgn⁡xt1,…,tN,xt1,…,tN>n. It is readily verified that(70)DiPnxt1,…,tN≤Dixt1,…,tNfori=0,…,N.We now apply Lemma 12 to prove VB∞ is relatively compact in Wloc1,1 with the topology τlocF. For the sake of convenience, we fulfill it in 2 steps.
Step 1. We are going to show that VPnB∞ is relatively compact in the topology τlocF. To do this, we apply Lemma 11. First we show that μT(VPnB∞)=0; that is, (71)limε→0⁡supx∈B∞⁡sup⁡maxα≤1⁡∫DDαVPnxs1,…,sNds1⋯dsN:D⊂0,TN,μD<ε=0. Let us fix a number T>0 and let us put (72)Ax,n≔t1,…,tN∈0,TN:xt1,…,tN≠Pnxt1,…,tNforx∈B∞,n∈N.
In view of (70) and condition (5) we derive that (73)∫DVPnxs1,…,sNds1⋯dsN=∫Ax,n∩DVPnxs1,…,sNds1⋯dsN+∫D-Ax,nVxs1,…,sNds1⋯dsN≤∫Ax,n∩DKa1ξs1,…,sN+MPnx,s1,…,sNN-1/Nds1⋯dsN+∫D-Ax,nKa1ξs1,…,sN+Mx,s1,…,sNN-1/Nds1⋯dsN≤∫DKa1s1,…,sN+Mx,s1,…,sNN-1/Nds1⋯dsN≤KTN-1/N∫Da1s1,…,sNds1⋯dsN+∫DDαxs1,…,sNds1⋯dsN. Taking into account absolute continuity of the integral of the function a1, it yields that(74)∫DVPnxs1,…,sNds1⋯dsN≤KTN-1/N∫DDαxs1…,sNds1⋯dsN.On the other hand, using again (70) and condition (5) we have that (75)∫D∂∂siVPnxs1,…,sNds1⋯dsN=∫Ax,n∩D∂∂siVPnxs1,…,sNds1⋯dsN+∫D-Ax,n∂∂siVxs1,…,sNds1⋯dsN≤KTmaxα≤1⁡∫DDαxs1,…,sNds1⋯dsNfori=1,…,n.Regarding (74) and (75) we obtain (76)μTVPnB∞≤max⁡KT,KTN-1/NμTB∞=0,forT≥0. Hence μT(VPnB∞)=0, for T≥0. It enforces that the bounded set VPnB∞ is relatively compact in the topology τlocω. It remains to prove that VPnB∞ is countably a.e. equicontinuous on N-cells. Let us fix T>0 and ε>0. According to Luzin’s Theorem, there is a closed subset Dε⊂[0,T]N such that m(Dε′)≤ε, and the functions ϑ|Dε×R+N×RN+2 and κ|Dε×R+N are continuous.
Let us take arbitraryn-tuples (t1,…,tN),(s1,…,sN)∈R+N such that, for each i=1,…,N,si>ti. Then, for x∈B∞, by condition (5) we have (77)VPnxs1,…,sN-VPnxt1,…,tN≤TNωTϑ,s1,…,sN-t1,…,tN×a11,1T+x1,1T+∫t1s1⋯∫tNsNϑs1,…,sN,τ1,…,τN,Pnxξτ1,…,τN,∂Pnx∂τ1ξτ1,…,τN,…,∂Pnx∂τNξτ1,…,τN,T0Pnxτ1,…,τNdτ1⋯dτN≤TNωTϑ,s1,…,sN-t1,…,tN×a11,1T+x1,1T+∫0s1⋯∫0sNκs1,…,sN,τ1,…,τNa1ξτ1,…,τN+MPnx,τ1,…,τNdτ1⋯τNN-1/N≤TNωTϑ,s1,…,sN-t1,…,tN×a11,1T+x1,1T+κ¯∫0s1⋯∫0sNa1τ1,…,τNdτ1⋯dτN+κ¯max⁡∫0s1⋯∫0sNDixτ1,…,τNdτ1⋯dτN:i=0,…,NN-1/N, where ωT(ϑ,·) denotes the modulus of continuity of the function ϑ on the set Dε×[0,T]N×[-n,n]N+2 and (78)κ¯=max⁡κt1,…,tN,s1,….sN:t1,…,tN,s1,…,sN∈Dε×0,TN.From uniform continuity of the function ϑ on the compact set Dε×[0,T]N×[-n,n]N+2, absolute continuity of the integral of the function a1, and μT(B∞)=0, we yield(79)VPnxs1,…,sN-VPnxt1,…,tN,is arbitrarily small provided the number ‖(s1,…,sN)-(t1,…,tN)‖ is small enough. This proves that VPnB∞ is countably a.e. equicontinuous on N-cells.
Note that if(t1,…,tN),(s1,…,sN)∈R+N are arbitrary, then it is enough to break down the integration intervals into suitable subintervals and applying the same reasoning as above, to see that VPnB∞ is countably a.e. equicontinuous on N-cells. We then conclude by applying Lemma 11 that VPnB∞ is relatively compact in τlocF.
Step 2. We are going to prove that limn→∞D(VB∞,VPnB∞)=0, where D(·,·) denotes the Hausdorff distance in the Fréchet metric space (Wloc1,1,d). Since all functions of the set B∞ are uniformly bounded in ‖·‖1,1T, (80)limn→∞⁡sup⁡mAx,n:x∈B∞=0.Using our assumptions, for arbitrarily fixed x∈B∞, we have(81)Vx-VPnx1,1T=maxi≤1⁡∫0,TNDiVxt1,…,tN-VPnxt1,…,tNdt1⋯dtN=maxi≤1⁡∫Ax,nDiVxt1,…,tN-VPnxt1,…,tNdt1⋯dtN≤maxi≤1⁡∫Ax,nDiVxt1,…,tNdt1⋯dtN+maxi≤1⁡∫Ax,nDiVPnxt1,…,tNdt1⋯dtN≤max⁡∫Ax,nKa1ξt1,…,tN+Mx,t1,…,tNN-1/Ndt1⋯dtN,∫Ax,nKa1ξt1,…,tN+Mx,t1,…,tNdt1⋯dtN+max⁡∫Ax,nKa1ξt1,…,tN+MPnx,t1,…,tNN-1/Ndt1⋯dtN,∫Ax,nKa1ξt1,…,tN+MPnx,t1,…,tNdt1⋯dtN≤2KT∫Ax,na1t1,…,tN+Mx,t1,…,tNdt1⋯dtN.By absolute continuity of integral of function a1 and applying (68) and (80) we derive that (82)limn→∞⁡supx∈B∞⁡Vx-VPnx1,1T=0,forT≥0. This equality together with inclusion VPnB∞⊂VB∞ implies that(83)limn→∞⁡DVB∞,VPnB∞=0.
Regarding (83), Step 1, and Lemma 12 we derive that VB∞ is also relatively compact in τlocF. Taking into account the continuity of F:Wloc1,1→Wloc1,1 (see Theorem 15), we get that HB∞=FVB∞ is also relatively compact in τlocF. Let us take Y=conv⁡HB∞¯F. We observe that Y is compact and convex subset of the space Wloc1,1 with the topology τlocF. From continuity of V and in view of Theorem 15 the operator H=FV transforms continuously the set Y into itself. So, applying the classical Tychonoff fixed point principle we conclude that there is a solution x∈B∞⊂Wloc1,1 of (42). This completes the proof of our theorem.Now, we present an example which verifies the effectiveness and applicability of Theorem17.Example 18.
Consider the following Volterra integrodifferential equation:(84)xt1,t2=1t12+1t22+1+arctan⁡1100t13t23+1100t1t2∫0t1∫0t2sin2⁡xs1,s210+e∂xs1,s2/∂s1+cosh⁡∂xs1,s2∂s2+14xs1,s22ds1ds2.Equation (84) is a special case of (42) with(85)ξt1,t2=t1,t2,ft1,t2,x=1t12+1t22+1+arctan⁡x,T0x=12x,ϑt1,t2,s1,s2,x1,x2,x3,x4=1100t12t22+t1t2sin2⁡x110+ex2+cosh⁡x3+x42,κt1,t2,s1,s2=15. It is easy to see that ξ satisfies in condition (1) of Theorem 17. Also, f satisfies Carathéodory conditions and if we define a(t1,t2)=2(t1+t2)/(t12+1)(t22+1), b=d=1/1000, and c(t1,t2)=1, then condition (2) of Theorem 17 holds. Moreover, let us take (86)Vxt1,t2=1100t13t23+1100t1t2∫0t1∫0t2sin2⁡xs1,s210+e∂xs1,s2/∂s1+cosh⁡∂xs1,s2∂s2+14xs1,s22ds1ds2,a1s1,s2=36s1s25+50s1s24+16s1s23+18s1s22+s12s2+4s1+s2+1. Then we have (87)Vx2≤1104t1t26+2t1t25+t1t24,∂Vx∂t1≤11003t12t23+2t1t22∂Vx∂t2≤11003t13t22+2t12t2. It can be easily shown that condition (5) of Theorem 17 holds, too. On the other hand, ‖KT‖(b+c¯+d)=1002/5000<1.
Consequently, all the conditions of Theorem17 are satisfied. It implies that the integrodifferential equation (84) has at least one solution in the space Wloc1.1.



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