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New Applications of the Existence of Solutions for Equilibrium Equations with Neumann Type Boundary Condition

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New Applications of the Existence of Solutions for Equilibrium Equations with Neumann Type Boundary Condition 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.
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