AMS Subj. Classiﬁcation: 47J10, 47H30, 47H10We study some possibilities of nonlinear spectral theories for solving nonlinear
operator equations. The main aim is to research a spectrum and establish some
kind of nonlinear Fredholm alternative for Hammerstein operator KF

Halilović, Sanela

Bulgarian Digital Mathematics Library at IMI-BAS

M a t h e m a t i c aB a l k a n i c a—————————New Series Vol. 24, 2010, Fasc.3-4Nonlinear Spectral Theories and Solvability ofNonlinear Hammerstein EquationsSanela Halilovic´We study some possibilities of nonlinear spectral theories for solving nonlinearoperator equations. The main aim is to research a spectrum and establish somekind of nonlinear Fredholm alternative for Hammerstein operator KF.AMS Subj. Classification: 47J10, 47H30, 47H10Key words: spectrum, Hammerstein equation, solvability, superposition operator, nonlinearFredholm alternativeIt is well-known that the major methods for studying solvability of linearor nonlinear equations in literature are: the variational method, the method ofa vector field rotation and the fixed point methods.”The brachistochrone problem” is usually considered to be the begin-ning of variational calculus and nonlinear analysis. It was first introduced byJ. Bernoulli in the 17th century and was first solved by Isaac Newton. An-other method for obtaining existence and uniqueness results has been built ona topological method known as the degree theory, the index theory or rota-tion of the vector fields. The founders of the index theory were: M. Atiyah,R. Bott, F. Hirzebruch and Is. Singer. The first claim, which is an equivalentto the Brouwer fixed-point theorem, was given and proved by H. Poincare´ in1883 and the next by P. Bole in 1904 (the first one-dimensional equivalent isthe well-known Bolcano’s theorem on the zeros of a continuous function, provedin 1817). In 1909 Brouwer proved the theorem on fixed points in the case ofthe three-dimensional space. In 1910 Adamar proved a similar statement foran arbitrary finite-dimensional space, by using a Cronecker index. The samestatement was proved by Brouwer in 1912 by using simplex approximation andthe notion of mapping degree.Although Poincare´ and Bole gave direct applications of their results in thetheory of differential equations, as well as in spacial and analytical mechanics,322 Sanela Halilovic´since then there have not been serious applications of the Brouwer theorem in themathematical analysis, except one Shauder’s result in 1927 about the existenceof solution of elliptic partial equations. The situation reversed when John vonNeumann applied this theorem to proving the existence of solution for matrixgames. These results, which present a base of the classic game theories, increasedthe mathematicians´s interest in studying the applications of this theorem invarious areas of analysis.Modern research in contractive type conditions started with the Banachfixed point theorem which is one of the classic statements in functional analysis.The following two facts have enabled broad applications of this theorem:1. solving many kinds of numerical and functional equations can be doneby finding fixed points of some mappings;2. the Banach theorem provides effective calculation (construction) offixed point and also gives the possibility for estimating error i.e. finding maximaldistance from approximative to the accurate solution.Given a Banach space X over the field C and a bounded linear operatorA : X → X. If some x0 6= 0 is a fix point of the operator A, i.e. Ax = x has anontrivial solution x = x0, we can also say(∃x 6= 0)(I −A)x = 0.It means that operator I−A is not a bijection. More generally, we may considerwhether the operator λI − A is a bijection and it leads us to the notion ofspectrum which is the setσ(A) = {λ ∈ C : λI −A is not a bijection}.The importance of the spectral theory for linear operators is well-known([3],[5]). Various attempts have been made to define and study spectra alsofor nonlinear operators. In the very beginning, the term spectrum was usedfor nonlinear operators just in the sense of point spectrum (i.e. the set ofeigenvalues) e.g. by Nemytskij [7] , [8] or Krasnosel’skij [6]. Later it became clearthat a more complete description requires, as in the linear case, other (”non-discrete”) spectral sets. Starting from the late sixties, this led to a number ofdefinitions of nonlinear spectra which are all different. It was assumed that areasonable definition of a spectrum of a continuous nonlinear operator shouldsatisfy some minimal requirements, namely:• It should be reduced to the familiar spectrum in case of linear operators• It should share some of the usual properties with the linear spectrum(e.g. compactness)• It should contain the eigenvalues of the operator involvedNonlinear Spectral Theories... 323• It should have nontrivial applications, i.e. those which may not be ob-tained by other known meansFrom the viewpoint of these four requirements, any definition of a spec-trum should focus on its analytical and topological properties and, of courseon applications. Important special classes of continuous (nonlinear) operatorsare: the Fre´chet differentiable operators, the Lipschitz continuous operators,the quasibounded operators and the linearly bounded operators. These are theoperators for which many important results have been proved in the nonlinearspectral theory. Spectra have something to do with the ”lack of invertibility”of operators. For some class of continuous nonlinear operators M (X) we candefine a resolvent setρ (F ) ={λ ∈ K : λI − F is bijective and (λI − F )−1 ∈ M (X)}and a spectrumσ (F ) = K\ρ (F ) , F ∈ M (X) .If we take M (X) = C (X) we get Rhodius spectrum ([14]), and if we takeM (X) = C1 (X) we get Neuberger spectrum([15]). The Rhodius spectrum(for continuous operators) may be noncompact or empty, while the Neubergerspectrum( for C1 operators) is always nonempty (in the complex case), but itneeds be neither closed nor bounded. The Do¨rfner and Kachurovski spectrumis defined for Lipschitz continuous operators ([16],[17]). In contrast to the Neu-berger spectrum, the Kachurovski spectrum is compact, but it may be empty.the Do¨rfner spectrum in turn is always closed, but it may be unbounded orempty. All four spectra considered so far are reduced to the familiar spectrumin the linear case, and they all contain the eigenvalues of the operator involved.A spectrum for certain special continuous operators (stably solvable operators)was introduced by Furi, Martelli and Vignoli in 1978 ([18]). That spectrumis always closed, sometimes even compact and it has many interesting applica-tions. It need not contain the point spectrum in the nonlinear case. A certainmodification of this spectrum has been recently given by Appell, Giorgieri andVa¨th. In 1997 Feng introduced a new spectrum of nonlinear operator (for epiand k-epi operators)([19],[2]). Roughly speaking, one may say that the Furi-Martelli-Vignoli spectrum takes into account the ”asymptotic” properties of anoperator, while the Feng spectrum reflects its ”global” properties. This is alsoone reason why the latter contains the eigenvalues, but the former does not.The range of applications of spectral theory is vast. It is a useful toolfor solving nonlinear operator equations. When we want to apply some spectraltheory to specific nonlinear problem, we have to choose carefully a spectrumwhich has at least some of the needed features for solving the problem.324 Sanela Halilovic´For each spectral theory there is some associated eigenvalue theory deal-ing, to be more precise, with nontrivial solutions of the equationF (x) = λx.The spectral theory for homogeneous nonlinear operators may be usedto derive a certain nonlinear Fredholm alternative which provides existence andperturbation results for the p-Laplace equation ([1]).Below we consider the nonlinear Hammerstein integral equations(1) x(t) =∫Ωk(s, t)f(s, x(s))dµ(s) + g(t), (t ∈ Ω)where Ω ⊂ RN is a bounded domain, k (s, t) : Ω×Ω→ R is a measurable kerneland f (s, u) : Ω × R → R is a Caratheodory function (measurable in s, for allu ∈ R and continuous in u, for almost all s ∈ Ω). A discrete analogon of (1) isnonlinear Hammerstein system of equations:(2) x(t) =∞∑s=1k(s, t)f(s, x(s)) + g(t), (t ∈ N)with kernel k(s, t)(s, t ∈ N). In system (2) k : N × N → R defines a linearbounded operator(3) Kx(t) =∞∑s=1k(s, t)x(s)and f : N × R → R is a real function which generates a nonlinear operatorsuperposition F(4) Fx (s) = f (s, x (s)) (s ∈ N)The methods of studying solvability of nonlinear Hammerstein equationsin the literature are: the variational method (for example, the Golomb func-tional Φ; gradΦ = F ), the rotation of the vector field and the fixed point meth-ods ([4],[6]). The necessary and sufficient conditions have been found for thenonlinear superposition operator F , for its action from one Banach space toanother (Lp → Lq ; lp → lq), as well as the conditions for its boundedness,continuity, absolute boundedness ([9]). Some results of the solvability of theHammerstein integral equations (1) in Banach spaces Lp are given in [2]. Theconditions of solvability and uniqueness of the Hammerstein system of equa-tions (2) in Banach spaces lp are given in [10]. For the nonlinear operator F thefunctionµF (r) = sup‖x‖≤r‖Fx‖Nonlinear Spectral Theories... 325is called the growth function of the operator F in normed space X. The functionνF (r) = inf{‖a‖q + brpq : |f(s, u)| ≤ a(s) + b|u| pq , |u| ≤ r}also gives us some useful information about the operator F .If F is a linear operator, thenµF (r) = ‖F‖rSince the growth function µF (r) and νF (r) are logarithmically convex functionson the convex set L(F, act.)([11]), we could define a norm of the superpositionoperator F as‖F‖ = µF (r)ror ‖F‖ = νF (r)r.In that case, these norms would be topologically equivalent norms (see Theorem3, in [9]). We note here that the reviewer of the paper [11] has given onecounterexample that L(F, act.) is not the convex set. But in that example, thesetL(F, act.) = {(1/p, 1/q) : 4 ≤ p <∞, q ≥ p/2} ∪ {(1/p, 1/q) : 1 ≤ p ≤ 4, q ≥ 2}is convex (see [13]), and therefore it is not in contradiction with the Theorem 2given in [11].The conception of the spectral radius for the linear operator L and re-lation rs(L) ≤ ‖L‖, could be carried over onto the spectrum of the nonlinearsuperposition operator F with rs(F ) ≤ ‖F‖.As for the equations (2), recently we have extended the results from [10]on the weighted Banach spaces lp,σ(1 ≤ σ ≤ ∞). Applying the fixed pointtheorem for monotone operators, we have gained (see [10],[12]) the conditionsfor the solvability of the system (2) in the spaces lp,σ and they are:Theorem 1. Let the operator K, defined by (3), be P-positive. Suppose thatthe generator f : N × R → R of superposition operator F given by (4), satisfiescondition (u− v)(f(s, u)− f(s, v)) ≤ c(u− v)2 for some cf > 0 and f(s, 0) = 0for all s ∈ N. If cfµK < 1 then, for arbitrary g ∈ N(l2,τ ) the equation:x = KFx+ ghas a solution xˆ ∈ N(l2,τ ). If g = Nl for some l ∈ l2,τ , then there exists hˆ ∈ l2,τ ,such that xˆ = Nhˆ, and‖hˆ‖ ≤ ‖l‖1− cfµKMoreover,the solution xˆ is unique in the space lp,σ.326 Sanela Halilovic´Theorem 2. Let the operator K, defined by (3), be P-quasi-positive in l2,τ .Suppose that the generator f : N× R→ R of superposition operator F given by(4), satisfies (u−v)(f(s, u)−f(s, v)) ≤ c(u−v)2 for some cf > 0 and f(s, 0) = 0for all s ∈ N. If cfνK < −1, whereνK = sup{ν > 0, ‖Nh‖ ≥√νPh (h ∈ l2,τ )},then, for arbitrary g ∈ N(l2,τ ) the equation:x = KFx+ ghas a solution xˆ ∈ N(l2,τ ). If g = Nl for some l ∈ l2,τ , then there exists hˆ ∈ l2,τ ,such that xˆ = Nhˆ, and‖hˆ‖ ≤ − ‖l‖1 + cfνKMoreover,the solution xˆ is unique in the space lp,σ.The problem of solvability of the equations (1) or (2) is equivalent to theproblem of solvability of the operator equation(5) x = KFx+ g, or (I −KF )x = g.Since the solvability of the equation (5) is closely related to the propertiesof the spectrum of the nonlinear Hammerstein operator KF , from the theoremsthat are given in [9] and [10], we can conclude that a spectrum of the operatorKF is nonempty. On the other side, we could considered the spectral radius tobe rs(F ) ≤ µF (r)r = ‖F‖.We are looking forward to finding some kind of nonlinear Fredholm alter-native for this Hammerstein operator. The aims of our research are: extendingnonlinear spectral theories, finding new statements and new applications.References[1] J .A pp e l l , E . P a s c a l e a n d A .V i g n o l i , Nonlinear Spectral Theory,Walter de Gruyter Berlin, New York, 2004.[2] J .A pp e l l (Editor), Recent Trends in Nonlinear Analysis, Birkha¨user,Basel, 2000.[3] V . S . S u n d e r , Functional Analysis: Spectral Theory, Birkha¨user Ad-vanced Texts, Hardcover,1998.[4] H . B r e z i s , K .C .Ch an g , S . J . L i , P .R a b i n ow i t z , TopologicalMethods, Variational Methods, and Their Applications: ICM 2002 SateliteConference on Nonlinear Functional Analysis Taiyuan, Shan Xi, 2002.[5] W . A r v e s o n , A Short Course on Spectral Theory, Hardcover, 2001.Nonlinear Spectral Theories... 327[6] M .A .K r a s n o s e l´ s k i j , On a topological method in the problem ofeigenfunctions for nonlinear operators, Doklady Akad. Nauk SSSR, 74,1950, 5-7.[7] V .V .N emy t s k i j, Some problems concerning the structure of the spec-trum of completely continuous nonlinear operators, Doklady Akad. NaukSSSR, 80, 1951, 161-164.[8] V .V .N emy t s k i j, Structure of the spectrum of completely continuousnonlinear operators, Mat.Sb., 33 , 1953, 545-558.[9] F . D e d a g i c a n d P .P . Z a b r e j k o, Ob operatorah superpozicii vprostranstvah lp, Sibirskij matematieskij zˇurnal, XXVIII, No1, 1987.[10] F . D e d a g i c, On the discrete nonlinear Hammerstein systems with non-symetric kernels, Sarajevo Journal of Mathematics, 5, No2, 2009.[11] F . D e d a g i c , N . Ok i c i c, On the L- -characteristic of nonlinear super-position operators in lp,σ-spaces, Mathematica Balkanica, 19, 2005.[12] F .D e d a g i c , S .H a l i l o v i c , E .B a r a k o v i c, On the solvability of dis-crete nonlinear Hammerstein systems, MICOM 2009, Ohrid , 2009.[13] J .A pp e l l , P . P . Z a b r e j k o , U¨ber die L-charakteristik nichtlinearer op-eratoren in ra¨umen integrierbarer funktionen, Manuscripta Math., 1988,355-367.[14] A . Rho d i u s , U¨ber numerische Werterbereiche und Spektralwertab-scha¨tzungen, Acta Sci. Math. 47, 1984, 465-470.[15] J .W . N e ub e r g e r , Existence of a spectrum for nonlinear transforma-tions, Pacific.J. Math. 31, 1969, 157-159.[16] R . I . K a c hu r o v s k i j , Regular points, spectrum and eigenfunctions ofnonlinear operators, Dokl. Akad. Nauk SSSR, 188, 1969, 274-277.[17] M . D o¨ r f n e r , Spektraltheorie fu¨r nichtlineare Operatoren, Ph.D.thesis,Universita¨t Wu¨rzburg, 1997.[18] M . F u r i , M . Ma r t e l l i , A . V i g n o l i , Contributions to the spectraltheory for nonlinear operators in Banach spaces, Ann. Mat. Pura Appl.,118, 1978, 229-294.[19] W . F e n g , A new spectral theory for nonlinear operators and its applica-tions, Abstr.Appl.Anal., 2, 1997, 163-183.Department of MathematicsFaculty of Science and MathematicsUniversity of TuzlaBosnia and Herzegovinaemail: sanela.halilovic@untz.ba

Nonlinear Spectral Theories and Solvability of  Nonlinear Hammerstein Equations

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Nonlinear Spectral Theories and Solvability of Nonlinear Hammerstein Equations

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