Ji et al. Journal of Inequalities and Applications (2017) 2017:86 DOI 10.1186/s13660-017-1357-4
RESEARCH Open Access New applications of the existence of solutions for equilibrium equations with Neumann type boundary condition
Zhaoqi Ji1,TaoLiu2,HongTian3 and Tanriver Ülker4*
*Correspondence: [email protected] Abstract 4Departamento de Matemática, Universidad Austral, Paraguay 1950, Using the existence of solutions for equilibrium equations with a Neumann type Rosario, S2000FZF, Argentina boundary condition as developed by Shi and Liao (J. Inequal. Appl. 2015:363, 2015), Full list of author information is we obtain the Riesz integral representation for continuous linear maps associated available at the end of the article with additive set-valued maps with values in the set of all closed bounded convex non-empty subsets of any Banach space, which are generalizations of integral representations for harmonic functions proved by Leng, Xu and Zhao (Comput. Math. Appl. 66:1-18, 2013). We also deduce the Riesz integral representation for set-valued maps, for the vector-valued maps of Diestel-Uhl and for the scalar-valued maps of Dunford-Schwartz. Keywords: Neumann type boundary condition;ARTICLE set-valued measures; integral representation; topology
1 Introduction The Riesz-Markov-Kakutani representation theorem states that, for every positive func-
tional L on the space Cc(T) of continuous compact supported functional on a locally com- pact Hausdorff space T, there exists a unique Borel regular measure μ on T such that L(f )= fdμ for all f ∈ Cc(T). Riesz’s original form [] was proved in for the unit in- terval (T = [; ]). Successive extensions of this result were given, first by Markov in to some non-compact space (see []), by Radon for compact subset of Rn (see []), by Ba- nach in note II of Saks’ book (see []) and by Kakutani in to a compact Hausdorff space []. Other extensions for locally compact spaces are due to Halmos [], Hewith [], Edward []andBourbaki[]. Singer [, ], Dinculeanu [, ] and Diestel-Uhl [] gave an integral representation for functional on the space C(T, E) of vector-valued con- tinuous functions. Recently Leng, Xu and Zhao (see []) gave the integral representation for continuous functionals defined on the space C(T) of all continuous real-valued func- tions on T; as an application, Shi and Liao (see []) also gave short solutions for the full and truncated K-moment problem. The set-valued measures, which are natural extensions of RETRACTEDthe classical vector measures, have been the subject of many theses. In the school of Pallu DeLaBarrierewehavetheonesofThiam[], Cost [], Siggini [], in the school of Cas- taing the one of Godet-Thobie [], and in the school of Thiam the ones of Dia []and Thiam []. Investigations are undertaken for the generalization of results for set-valued
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 2 of 8
measures in particular the Radon-Nikodym theorem for weak set-valued measures [, ] and the integral representation for additive strictly continuous set-values maps with reg- ular set-valued measures. The work of Rupp in the two cases, T arbitrary non-empty set and T compact, allowed one to generalize the Riesz integral representation of additive and σ -additive scalar measures to the case of additive and σ -additive set-valued measures (see [, ]). He has proved among others that if T is a non-empty set and A the algebra of subsets of T, for all continuous linear maps l defined on the space B(T; R)ofalluniform limits of finite linear combinations of characteristic functions of sets in A associated with an additive set-valued map with values in the space ck(Rn) of convex compact non-empty R subsets of n, there exists a unique bounded additive set-valued measure M from A to the space ck(Rn)suchthatδ∗(·|l(f )) = δ∗(·| fM) and conversely. In this paper we extend this resulttothecaseofanyBanachspaceE.WededucetheRieszintegralrepresentationfor additive set-valued maps with values in the space of all closed bounded convex non-empty subsets of E; for vector-valued maps (see [], Theorem , p.) and for scalar-valued maps (see []).
2 Notations and definitions Let E be a Banach space and E itsdualspace.Wedenoteby·the norm on E and E.If X and Y are subsets of E we shall denote by X + Y the family of all elements of the form x + y with x ∈ X and y ∈ Y ,andbyX +˙ Y or adh(X + Y)theclosureofX + Y .Theclosed convex hull of X is denoted by co(X). The support function of X is the function δ∗(·|X) ARTICLE from E to ] – ∞;+∞]definedby ∗ δ (y|X)=sup y(x); x ∈ X .
We denote by cfb(E) the set of all closed bounded convex non-empty subsets of E.We endowed cfb(E) with the Hausdorff distance denoted by δ and the structures +andthe˙ multiplication by positive real numbers. For all K ∈ cfb(E)andforallK ∈ cfb(E), we have ∗ ∗ δ K; K = sup δ (y|K)–δ y|K ; y ∈ E, y≤ .
Recall that (cfb(E); δ)isacompletemetricspace(see[], Theorem , p.). We denote by Ch(E) the space of all continuous real-valued map defined on E and positively homo- geneous. If u ∈ Ch(E), then we have
u(λy)=λu(y)
for all y ∈ E and for all λ ∈ R,whereλ ≥ . We endowed Ch(E)withthenorm u = sup u(y); y ∈ E ; y≤ .
{ ∗ | ∈ } ˜ ˜ RETRACTEDPut C = δ (y B); B cfb(E) and put C = C – C;then C is a subspace of the vector h space C (E ) generated by C.LetT be a non-empty set, let A be an algebra consisting of subsets of T and let B(T; R) be the space of all bounded real-valued functions defined on T, endowed with the topology of uniform convergence. We denote by S(T; R)thesub- B R ∈ R space of (T; ) consisting of simple functions (i.e. of the form αiAi where αi ; ∈ { · } Ai A; A, A, , An a partition of A and Ai the characteristic function of Ai.) We de- note by B(T, R)theclosureinB(T; R)ofS(T; R); S+(T; R)(resp.B+(T; R)) the subspace Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 3 of 8
of S(T; R)(resp.B(T; R)) consisting of positive functions. We endowed B(T; R)withthe induced topology. Notes that if A is the Borel σ -algebra, then B(T; R)isthespaceofall bounded measurable real-valued functions. Let M be a set-valued map from A to cfb(E). We say that M is additive if M(∅)={} and
M(A ∪ B)=M(A) +˙ M(B)
for all disjoint sets A, B in A.Theset-valuedmeasureM is said to be bounded if {M(A), A ∈ A} is a bounded subset of E.ThesemivariationofM is the map M(·)from A to [; +∞]definedby M(A)=sup f δ y|M(·) (A); y ∈ E , y≤ ,
where |δ(y|M(·))|(A) denotes the total variation of the scalar measure δ∗(y|M(·)) on A de- fined by ∗ δ y|M(·) |(A)=sup δ y|M(Ai) ; i
the supremum is taken over all finite partition (Ai)ofA; Ai ∈ A.IfM(T)<+∞,then M will be called a set-valued measure of finite semivariation.ARTICLE We denote by M(A; cfb(E)) the space of all bounded set-valued measures defined on A with values in cfb(E). Let m be avectormeasurefromA to E.Wesaythatm is a bounded additive vector measure if its verifies similar conditions of bounded additive set-valued measures. We denote by m the semivariation of m defined by m(A)=sup{|y◦m|(A); y ∈ E; y≤} where |y◦m|(A) denotes the total variation of the scalar measure y ◦ m on A defined by |y ◦ m|(A)=sup y m(Ai) i
for all A ∈ A; the supremum is taken over all finite partition (Ai)ofA; Ai ∈ A.Let
L : B+(T; R) → cfb(E)beaset-valuedmap.WesaythatL is an additive (resp. positively
homogeneous) if for all f , g ∈ B+(T; R) (resp. for all λ ≥ ), L(f + g)=L(f ) +˙ L(g)(resp. L(λf )=λL(f )).WedenotebyL(B(T, R); Ch(E)) the space of all linear continuous maps defined on B(T, R)withvaluesinCh(E). If l ∈ L(B(T, R); Ch(E)); we put
l = sup l(f ) ; f ∈ B+(T, R), f ≤ ,
where f = sup{|f (t); t ∈ T|}. For a numerical function f defined on T,wesetf + = sup(f ,) and f – = sup(–f ,).
∈ L B R h B R → RETRACTEDDefinition . Let l ( (T, , C (E ))) and let L : +(T, ) cfb(E) be an additive, positively homogeneous and continuous set-valued map. We say that l is associated with ∗ L if l(f )=δ (·|L(f )) for all f ∈ B+(T; R). Then we have ∗ + ∗ – l(f )=δ ·|L f – δ ·|L f ∈ C
for all f ∈ B(T; R). Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 4 of 8
3 Lemmas In order to prove our main results, we need the following lemmas.
Lemma . Let M : A → cfb(E) be an additive set-valued measure. Then M is bounded if and only if it is finite semivariation.
Proof The set-valued measure M is bounded if there exists a nonnegative real number c such that ∗ sup sup δ y|M(A) ≤ c. A∈A y≤
| ∗ | |≤ | ∗ | · | We have supA∈A supy≤ δ (y M(A)) supy≤ δ (y M( )) (T)= M (T). On the other hand, by Lemma (of [], p.) one has ∗ ∗ δ y|M(·) (T) ≤ supδ y|M(A) A∈A
for all y ∈ E.Then ∗ ∗ sup δ y|M(·) (T) ≤ sup sup δ y|M(A) . y≤ A∈A y≤ ARTICLE Therefore ∗ ∗ sup sup δ y|M(A) ≤ sup sup δ y|M(A) . A∈A y≤ A∈A y≤
∗ h Lemma . Let C be the set {δ (·|B); B ∈ cfb(E)} and let l : B(T; R) → C (E ) be a con- tinuous linear map. Then l is associated with an additive, positively homogeneous and
continuous set-valued map if and only if l(f ) ∈ C for all f ∈ B+(T, R).
Proof The necessary condition is obvious. Now assume that l(f ) ∈ C for all f ∈ B+(T, R). ∗ Let consider the map j : cfb(E) ← C(B → δ (·|B)); then j is an isomorphism, more a home- omorphism (see [], Theorem , p.). Let l be the restriction of l to B+(T, R). If we put L = j– ◦ l,thenitiseasytoseethatL is additive, positively homogeneous and continuous.
Therefore for all f ∈ B+(T, R), we have ∗ l(f )=δ ·|L(f ) ∈ C.
→ ∈ S R Let M : A cfb(E) be a bounded additive set-valued measure. For all h +(T, )such ∈ A that h = aiBi and for all A ,theintegral A hM of h with respect to M is defined by adh ∩ ∩ ··· ∩ A hM = (aM(A B)+aM(A B)+ + anM(A Bn)). This integral is uniquely RETRACTEDdefined. Moreover, for all y ∈ E, δ∗(y| hM)= hδ∗(y|M (·)). The map: h → hM from A A A S+(T, R)tocfb(E) is uniformly continuous. Indeed, for all f , g ∈ S+(T; R), one has ∗ δ fM, gM = sup (f – g)δ y|M(·) A A y≤ A ∗ ≤ sup f – gδ y|M(A) ≤f – gM(T)<+∞. y≤ Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 5 of 8
Since S+(T, R)isdenseonB+(T, R)andcfb(E) is a complete metric space, it has a unique B R ∈ B R S R extension to +(T, ): let f +(T, )andlet( hn)beasequencein +(T, )converging uniformly to f on T; Therefore the integral A fM of f is uniquely defined by fM = lim h M. → ∞ n A n + A
Moreover, ∗ ∗ δ y fM = f δ y|M(·) A A
for all y ∈ E , A ∈ A and for all f ∈ B+(T, R). The map
B+(T, R) → cfb(E) f → fM
is additive, positively homogeneous, and uniformly continuous. If m is a vector measure
defined on A, then the integral will be defined in the same manner. Denote L(B(T, R)), Ch(E)thesubspaceofL(B(T, R), Ch(E)) consisting of functions that verify the condition
l(f ) ∈ C for all f ∈ B+(T, R).
M ARTICLE Lemma . Let (A, cfb(E)) be the space of all bounded additive set-valued from A ∈ L B R h to cfb(E). Let l ( (T, ), C (E )). Then there exists a unique set-valued measure ∈ M ∗ ·| ∈ B R ∈ M (A, cfb(E)) such that l(f )=δ ( fM) for all f +(T, ). Conversely for all M M(A, cfb(E)), the mapping: f → δ∗(·| f +M)–δ∗(·| f –M) from B(T, R) to Ch(E) is an h element of L(B(T, R), C (E )). Moreover, l = M(M).
h Proof Let l ∈ L(B(T, R), C (E )). Let us prove the uniqueness of the set-valued mea- sure M. Assume that there exist two set-valued measures M, M ∈ M(A, cfb(E)) such that ∗ ∗ δ · fM = l(f )=δ · fM
∗ ∗ ∗ for all f ∈ B+(T, R). Then, for all A ∈ A, δ (·| AM)=l(A)=δ (·| AM )(ieδ (·|M(A)) = ∗ h δ (·|M (A))). Hence M(A)=M (A) for all A ∈ A.Sincel ∈ L(B(T, R), C (E )) then l is as- sociated with an additive, positively homogeneous and continuous set-valued map L from
B+(T, R)tocfb(E). Let M : A → cfb(E)betheset-valuedmapdefinedbyM(A)=L(A) for all A ∈ A.ThenM is additive. It follows from the continuity of L that M is bounded. Moreover, RETRACTEDhM = L(h) ∈ S R ∈ B R S R for all h +(T, ). Let f +(T, )andlet(hn)beasequencein +(T, )converging uniformly to f on T. It follows from the definition of the integral fM of f associated with M and the continuity of L that
L(f )= lim L(hn)= lim ∈ hn(M)= fM. n→+∞ n→+∞ Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 6 of 8
Hencewehave(Pan[]) ∗ l(f )=δ · fM
h for all f ∈ B+(T, R). Conversely let M ∈ M(A, cfb(E)). Then the map θ : B+(T, R) → C (E ) defined by ∗ ∗ θ(f )=δ · f +M – δ · f –M
verifies the condition θ(f ) ∈ C for all f ∈ B+(T, R). Let j be the isomorphism from cfb(E) ∗ · B R to C defined byj(B)=δ ( B)andletL be the set-valued map from +(T, )tocfb(E) defined by L(f )= fM for all f ∈ B+(T, R). Then j and L are continuous; therefore θ = j ◦ L
is continuous on B+(T, R)andthenonB(T, R). Let us prove now that l = M(T). On the one hand, for all y ∈ E
l = sup l(f ) f ≤ ∗ ∗ ≤ sup sup δ y f +M – δ y f –M |y|≤ f ≤ ARTICLE ∗ ∗ ≤ sup sup f +δ y|M(·) – f –δ y|M(·) |y|≤ f ≤ ∗ ≤ sup sup f δ y|M(·) . |y|≤ f ≤
On the other hand we have ∗ M(T)=supδ y|M(·) (T). |y|≤ | ∗ | · | | ∗ | · | Then it suffices to prove the equality supf ≤ f δ (y M( )) = δ (y M( )) (T), which is a classic result.
4 Main results and their proofs Theorem . Let L be an additive, positively homogeneous and continuous set-valued map
from B+(T, R) to cfb(E). Then there is a unique bounded additive set-valued measure M from A to cfb(E) such that RETRACTEDL(f )= fM ∈ B R → for all f +(T, ). Conversely for all bounded additive set-valued measure M : A cfb(E), the map: f → fM from B+(T, R) to cfb(E) is an additive, positively homogeneous and continuous set-valued map.
Proof The second part follows from the definition of the integral with respect to M.Let
L : B+(T, R) → cfb(E) be an additive, positively homogeneous and continuous set-valued Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 7 of 8
map and let ∗ j : cfb(E) → C B → j(B)=δ (·|B) .
We denote by l the unique extension of j ◦ L to B(T, R) for all f ∈ B(T, R), where ∗ ∗ l(f )=j ◦ L f + – j ◦ L f – = δ ·|L f + – δ ·|L f – .
∗ ·| ∈ ∈ B R We have l(f )=δ ( L(f )) C for all f +(T, ); then there exists a unique bounded ∗ ·| ∈ B R additive set-valued M from A to cfb(E)suchthatl(f )=δ ( fM) for all f (T, ). Hence L(f )= fM for all f ∈ B+(T, R).
The following corollary is partly known (see [], Theorem , p.).
Theorem . Let L(B(T, R), E) be the space of all continuous linear maps from B(T, R) to E and let M(A, E) be the space of all bounded additive vector measures from A to E. Let ∈ L B R ∈ M l ( (T, ), E). Then there exists a unique vector measure m (A, E) such that l(f )= ∈ B R ∈ M fm for all f (T, ). Conversely, given a vector measure m (A, E), the mapping f → fm from B(T, R) to E is an element of L(B(T, R), E). Moreover, l = m(T).
Proof Put E = {{x}; x ∈ E}.ThenE is a closed subspace of cfb(E). Let j be the map from ARTICLE E to E defined by j(x)={x}.Thenj is an isomorphism more a homeomorphism. Let l be the restriction of j ◦ l to B+(T, R). Then l is additive, positively homogeneous and contin- ∈ M uous. Therefore by Lemma . there exists a unique set-valued measure m (A, cfb(E)) such that l (f )= fm for all f ∈ B+(T, R). It follows from this equality that m (A) ∈ E for ∈ – ◦ ∈ M ∈ all A A.Putm = j m .Then m (A; E)andverifiesm (A)=j(m(A)) for all A A. ∈ B R – ◦ We deduce that fm = j( fm) for all f +(T, ); then fm = j l (f )=l(f ) for all f ∈ B+(T, R) and consequently l(f )= fm for all f ∈ B(T, R). The second part of corollary is proved as in Lemma ..Theequalityl = m(T) is a particular case of Theorem ..
By putting E = R, we have the following result.
Theorem . ([], Theorem , p.) Let M(A, R) be the space of all bounded addi- tive real-valued measures defined on A. Let l be a continuous linear functional defined B R ∈ M R on (T, ). Then there exists a unique measure μ (A, ) such that l(f )= fdμ for all f ∈ B(T, R). Conversely, for all measure μ ∈ M(A, R), the mapping: f → fdμ is a continuous linear functional defined on B(T, R). Moreover, l = |μ|(T).
5Conclusions In this paper, we discussed the Riesz integral representation for continuous linear maps RETRACTEDassociated with additive set-valued maps only using the existence of solutions for equi- librium equations with a Neumann type boundary condition. They inherited the advan- tages of the Shi-Liao type conjugate gradient methods for solving solutions for equilibrium equations with values in the set of all closed bounded convex non-empty subsets of any Banach space, but they had a broader application scope. Moreover, we also deduced the Riesz integral representation for set-valued maps, for the vector-valued maps of Diestel- Uhl and for the scalar-valued maps of Dunford-Schwartz (see []). Ji et al. Journal of Inequalities and Applications (2017) 2017:86 Page 8 of 8
Competing interests The authors declare that they have no competing interests.
Authors’ contributions The work presented here was carried out in collaboration between all authors. TU found the motivation of this paper. ZJ suggested the outline of the proofs. TL provided many good ideas for completing this paper. HT helped TU finish the proof of the main theorem. ZJ, TL and HT helped TU correct small typos and revise the manuscript based on the referee reports. All authors have contributed to, read, and approved the manuscript.
Author details 1College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao, 266590, China. 2College of Environmental Science and Engineering, Ocean University of China, Qingdao, 266100, China. 3College of Earth Science and Engineering, Shandong University of Science and Technology, Qingdao, 266590, China. 4Departamento de Matemática, Universidad Austral, Paraguay 1950, Rosario, S2000FZF, Argentina.
Acknowledgements The authors would like to thank the Editor, the Associate Editor and the anonymous referees for their careful reading and constructive comments, which have helped us to significantly improve the presentation of the paper.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 29 December 2016 Accepted: 31 March 2017
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