џ 1. Introduction - Issues of Analysis

Trudy Petrozavodskogo Gosudarstvennogo UniversitetaSeria “Matematika” Vipusk 12, 2005ÓÄÊ 517THE THEOREM ON EXISTENCE OF SINGULARSOLUTIONS TO NONLINEAR EQUATIONSA. Prusinska, A. Tret'yakovThe aim of this paper is to present some applications of p-regularity theory to investigations of nonlinear multivalued mappings.The main result addresses to the problem of existence of solutionsto nonlinear equations in the degenerate case when the linear partis singular at the considered initial point. We formulate conditionsfor existence of solutions of equation F (x) = 0 when rst p − 1derivatives of F are singular.Ÿ 1. IntroductionIn this paper we consider the problem of the solutions existence to thenonlinear equation:F (x) = 0, (1)where F : X→Y and X, Y be Banach spaces. The problem (1) is calledregular at the point x ∗ if ImF ′ (x ∗ ) = Y. Otherwise, the problem (1) iscalled irregular (nonregular, degenerate) at the point x ∗ .The construction of p-regularity introduced in [9] (see also [4, 10])gives new possibilities for description and investigation of solutions in thedegenerate case.Let p be a natural number and let B : X ×. . .×X → Y be a continuousp-multilinear mapping. A p-form associated to B is the map B[·] p : X → Ydened by B[x] p = B(x, . . . , x) for x ∈ X. Alternatively, we may simply} {{ }pview B[·] p as homogeneous polynomial map B : X → Y of degree p, i.e.B(αx) = α p · B(x).c○ A. Prusinska, A. Tret'yakov, 2005

The theorem on existence of singular solutions 23Throughout this paper we assume that the mapping F : X → Yis continuous and p-times Frechet dierentiable on X and its p-th orderderivative at point x ∈ X will be denoted as F (p) (x) (a symmetricmultilinear map of p copies of X to Y ) and the associated p-form isF (p) (x)[h] p = F (p) (x)[h, . . . , h].} {{ }pFurthermore, we use the following notationKer p F (p) (x) ={}h ∈ X : F (p) (x)[h] p = 0for the p-kernel of the mapping F (p) (x) (the zero locus of F (p) (x)). Denotealso L(X, Y ) as a space of continuous linear operators from X to Y .The setM(x ∗ ) = {x ∈ U : F (x) = F (x ∗ )}is called the solution set for the mapping F in neighborhood U.We call h a tangent vector to a set M ⊆ X at x ∗ ∈ M if there existε > 0 and a function r : [0, ε] → X with the property that for t ∈ [0, ε],we have x ∗ + th + r(t) ∈ M and‖r(t)‖lim = 0.t→0 tThe collection of all tangent vectors at x ∗ is called the tangent cone to Mat x ∗ and it is denoted by T 1 M(x ∗ ) (see e. g. [1]).Let X and Y be sets. We denote by 2 Y the set of all subsets of the setY . Any mapping Φ : X→2 Y is said to be multimapping (or a multivaluedmapping) from X into Y . For a linear operator Λ : X → Y, we denote byΛ −1 its right inverse, that is Λ −1 : Y → 2 X , which any element y ∈ Ymaps on its complete inverse image at the mapping Λ :Λ −1 y = {x ∈ X : Λx = y} .Of course,ΛΛ −1 = I Y .Furthermore, we shall use the "norm"∥ Λ−1 ∥ ∥ = sup‖y‖=1inf {‖x‖ : Λx = y, x ∈ X} . (2)

24 A. Prusinska, A. Tret'yakovNote, that when Λ is one-to-one, ‖Λ −1 ‖ can be considered as the usualnorm of the element Λ −1 in the space L(Y, X).In our further considerations, under notion Λ −1 we shall mean justright inverse operator (multivalued) with the norm dened by (2). Moreover,unless otherwise stated we will assume that X and Y are Banachspaces.Ÿ 2. Elements of p-regularity theoryLet U be a neighborhood of a point x ∗ ∈ X. If F : U → Y is p-timesdierentiable mapping in U andF (i) (x ∗ ) = 0, for all i = 1, . . . , p − 1,then we say that F is completely degenerate at x ∗ up to order p.A mapping F is called regular at some point x ∗ ifImF ′ (x ∗ ) = Y. (3)The mapping F is called nonregular (irregular, degenerate) if regularitycondition is not satised.In this paper we consider the case when the regularity condition (3)does not hold, but the mapping F is p-regular. First of all, let us remindthe denition of p-regularity and construction of p-factor-operator.Definition 1. Let F : U → Y, U ⊆ X, and F ∈ C p (U). Then forany x ∈ U and for any element h ∈ X, h ≠ 0 we can associate a linearoperator F (p) (x)[h] p−1 ∈ L(X, Y ), dened by the formula:The linear operatorF (p) (x)[h] p−1 ξ = F (p) (x)[h] p−1 [ξ], for any ξ ∈ X.Ψ p (x, h) = F (p) (x)[h] p−1 ∈ L(X, Y )is called the p-factor operator.A generalization of the regularity condition is given by the followingcondition of p-regularity:

The theorem on existence of singular solutions 25Definition 2. Let F : U → Y , U ⊆ X where F ∈ C p (U) and iscompletely degenerate at x ∗ . Then F is called p-regular at the point x ∗ ifeitherImΨ p (x ∗ , h) = Y, for any h ∈ Ker p F (p) (x ∗ ) \ {0},or Ker p F (p) (x ∗ ) = {0}.Definition 3. We say that the mapping F is p-regular at x ∗ along (on)the element h, if ImΨ p (x 0 , h) = Y.The description of a solution set in degenerate case is given by thefollowing theorem:Theorem (Generalized Lyusternik Theorem) [9] Let U be aneighborhood of a point x ∗ ∈ X. Assume that F : X→Y, where F ∈ C p (U)is p-regular at x ∗ . ThenT 1 M(x ∗ ) = Ker p F (p) (x ∗ ).Let Z be a metric space with distance ρ. If A 1 ⊂ Z and A 2 ⊂ Z, thenthe numberσ(A 1 , A 2 ) = supz∈A 1ρ(z, A 2 ) = supz∈A 1inf ρ(z, ω)ω∈A 2is called the deviation of the set A 1 from the set A 2 . The maximum of thedeviations σ(A 1 , A 2 ) and σ(A 2 , A 1 ),h(A 1 , A 2 ) = max {σ(A 1 , A 2 ), σ(A 2 , A 1 )} ,is called the Hausdor distance between the sets A 1 and A 2 . It follows atonce from the denition that, if h(A 1 , A 2 ) < α, then, for every z 1 ∈ A 1 ,there exists a z 2 ∈ A 2 such that ρ(z 1 , z 2 ) < α.Let Φ be a multimapping from the space Z into itself. We shall saythat it is a contraction on a set A ⊂ Z if there exists a number θ with0 < θ < 1 such that the inequalityholds for any z 1 and z 2 from A.h(Φ(z 1 ), Φ(z 2 )) ≤ θρ(z 1 , z 2 )

26 A. Prusinska, A. Tret'yakovWe shall give three auxiliary lemmas. The rst of these lemmas is a"multivalued"generalization of the contraction mapping principle, and isindependent interest.Lemma 1. (contraction multimapping principle) [3] Let Z be acomplete metric space with distance ρ. Assume that we are given a multimappingΦ : U ε (z 0 ) → 2 Z ,on a ball U ε (z 0 ) = {z : ρ(z, z 0 ) < ε} (ε > 0) where the sets Φ(z) are nonemptyand closed for any z ∈ U ε (z 0 ). Further, assume that there exists anumber θ, 0 < θ < 1, such that1) h(Φ(z 1 ), Φ(z 2 )) ≤ θρ(z 1 , z 2 ) for any z 1 , z 2 ∈ U ε (z 0 ),2) ρ(z 0 , Φ(z 0 )) < (1 − θ)ε.Then, for every number ε 1 which satises the inequalityρ(z 0 , Φ(z 0 )) < ε 1 < (1 − θ)ε,there exists an element z ∈ B ε1 /(1−θ)(z 0 ) = {ω : ρ(ω, z 0 ) ≤ ε 1 /(1 − θ)}such thatz ∈ Φ(z) (4)Lemma 2. [3] Let M 1 and M 2 be linear manifolds in X which are translationsof a single subspace L. Thenh(M 1 , M 2 ) = σ(M 1 , M 2 ) = σ(M 2 , M 1 ) == inf {‖x 1 − x 2 ‖ : x 1 ∈ M 1 , x 2 ∈ M 2 } .Lemma 3. [3] Let Λ ∈ L(X, Y ). We setIf ImΛ = Y, then C(Λ) < ∞.C(Λ) = sup inf {‖x‖ : x ∈ X, Λx = y} .‖y‖=1Ÿ 3. Regular caseWe quote one modication of the theorem on existence of solutions tothe equations with non-degenerate mappings (see in [2, 5, 6]) with proof

The theorem on existence of singular solutions 27which the main idea is based on the similar construction to this in singularcase.Consider a mapping F : X → Y and the problem of existence of suchpoint x ∗ that F (x ∗ ) = 0, i.e. x ∗ is a solution of (1). This equation is solvedby using non-degenerated condition of operator F ′ (x). Throughout thissection we assume that [F ′ (x 0 )] −1 is the multivalued mapping, and thatF ′ (x 0 ) is nonsingular, i.e. F ′ (x 0 )X = Y.Theorem 1. Let ‖F (x 0 )‖ = η and ∥ ∥[F ′ (x 0 )] −1∥ ∥ = δ, and assumethatsup ‖F ′′ (x)‖ = C,x∈U ε (x 0 )where F ∈ C 2 (U ε (x 0 )). Moreover, assume the following inequalities:1) δ · C · ε ≤ 6,12) δ · η ≤ 2,ε3) ε ≤ 1.Then the equation F (x) = 0 has a solution x ∗ ∈ U ε (x 0 ).Proof. Dene a multivalued mappingΨ : U ε (x 0 ) → 2 X ,U ε (x 0 ) ⊂ XΨ(x) = x − [F ′ (x 0 )] −1 F (x), x ∈ U ε (x 0 ).We shall show now that there exists a number q, 0 < q < 1, such thatfor any s 1 , s 2 ∈ U ε (x 0 ).By Lemma 2 and Lemma 3, we haveh(Ψ(s 1 ), Ψ(s 2 )) ≤ q · ‖s 1 − s 2 ‖, (5)h(Ψ(s 1 ), Ψ(s 2 )) = inf {‖z 1 − z 2 ‖ : z i ∈ Ψ(s i ), i = 1, 2} == inf {‖z 1 − z 2 ‖ : F ′ (x 0 )z i = F ′ (x 0 )s i − F (s i ), i = 1, 2} == inf {‖z‖ : F ′ (x 0 )z = F ′ (x 0 )[s 1 − s 2 ] − F (s 1 ) + F (s 2 )} == δ · ‖F (s 1 ) − F (s 2 ) − F ′ (x 0 ) · [s 1 − s 2 ]‖ .From the Taylor expansion of F, we haveF (s 1 ) − F (s 2 ) = F ′ (s 2 )[s 1 − s 2 ] + ω(s 1 , s 2 ),

28 A. Prusinska, A. Tret'yakovwhereand moreover‖ω(s 1 , s 2 )‖ ≤∥sup ∥F ′′ (x)[s 1 − s 2 ] 2∥ ∥ ≤ C · ‖s 1 − s 2 ‖ 2 ,x∈U ε (x 0 )whereF ′ (s 2 ) = F ′ (x 0 ) + ξ(s 2 , x 0 ),‖ξ(s 2 , x 0 )‖ ≤sup ‖F ′′ (x)[s 2 − x 0 ]‖ ≤ C · ‖s 2 − x 0 ‖.x∈U ε (x 0 )Taking into account the above and the assumption 1 we obtainh(Ψ(s 1 ), Ψ(s 2 )) = δ · ‖ξ(s 2 , x 0 )[s 1 − s 2 ] + ω(s 1 , s 2 )‖ ≤Now, we claim thatIndeed,≤ δ · (‖ξ(s 2 , x 0 )‖ · ‖s 1 − s 2 ‖ + ‖ω(s 1 , s 2 )‖) ≤≤ δ · (C· ‖s 2 − x 0 ‖ · ‖s 1 − s 2 ‖ + C · ‖s 1 − s 2 ‖ 2)

The theorem on existence of singular solutions 29We conclude the discussion of the regular case by the following example,which is very simple, but serves to illustrate the basic idea of Theorem 1.Example. Consider the problem (1) with (x 0 , y 0 ) = (0, 0) andF : R 2 → R,where F (x, y) = x 3 + x − y + 11100. Moreover, ‖F (0, 0)‖ = η =100 ,√ ∥ [F ′ (0, 0)] −1∥ ∥ 2 = δ =2 , sup ‖F ′′ (x, y)‖ ≤ C = 6 · ε,(x,y)∈U ε (0,0)where ε = 0, 1. The assumptions of Theorem 1 are fullled, so there existssuch point (x ∗ , y ∗ ), that F (x ∗ , y ∗ ) = 0, and for instance (x ∗ , y ∗ ) = (0,1100 ).Ÿ 4. p-regular (singular) caseLet us mention one of the consequences of the Mean Value Theorem,which is important for our further investigations. For the proof we referthe reader to [7, 8].Lemma 4. Let F : U → Y, where U ⊆ X, such that [a, b] ⊂ U, then‖F (b) − F (a) − Λ(b − a)‖ ≤ sup ‖F ′ (ξ) − Λ‖ · ‖a − b‖,ξ∈[a,b]for any Λ ∈ L(X, Y ).Next lemma follows from homogeneity properties of p-form (see [4]).Lemma 5. [4] Let U(x 0 ) neighborhood of x 0 in X, F : U(x 0 ) → Y,F ∈ C p+1 (U(x 0 )), and F (i) (x 0 ) = 0, for i = 1, . . . , p − 1. Then for every(R = max{1, 2 ɛ · 1(p−1)!∥∥F (p) (x 0 ) ∥ })ɛ > 0 there exist δ > 0 and R > 0such that for any x ∈ X, ‖x‖ ≤ δ and for any x 1 , x 2 ∈ X, ‖x j ‖ ≤‖x‖/R, j = 1, 2, the following condition∥ F (x 10 + x + x 1 ) − F (x 0 + x + x 2 ) −(p − 1)! F (p) (x 0 )[x] p−1 (x 1 − x 2 )∥ ≤≤ ɛ · ‖x‖ p−1 · ‖x 1 − x 2 ‖ holds.Definition 4. (Banach condition) Let F : X → Y, and let F ∈C p (X). Then for any y ∈ Y, ‖y‖ = 1 there exists x ∈ X, such thatF (p) (x 0 )[x] p = y, ‖x‖ ≤ c,

30 A. Prusinska, A. Tret'yakovwhere c > 0 is a constant independent of y.Let us introduce following additional notationsĥ =δ = ‖F (x 0 )‖ ̸= 0,{ [ h‖h‖ , where ] −1 1 h ∈ p! F (p) (x 0 ) (−F (x 0 ))}, h ≠ 0,as a generalization of (2) we have[η =∥F (p) (x 0 )] −1∥ ∥∥∥= sup‖y‖=1C ={}inf ‖x‖ : F (p) (x 0 )[x] p = y, x ∈ X ,supx∈U ε (x 0 )∥∥F (p+1) (x) ∥ ,[∥C 1 =∥ F (p) (x 0 )[ĥ]p−1] −1 ∥∥and∥C 2 = ∥F (p) (x 0 ) ∥ .We can now formulate our main result which is a generalization of Theorem 1in the degenerate case.Theorem 2. Let F : X → Y and assume that for F ∈ C p+1 (U ε (x 0 ))the Banach condition holds and F is p-regular mapping at the point x 0along ĥ and F (i) (x 0 ) = 0, for i = 1, . . . , p − 1. Moreover assume thefollowing inequalities1) p! · η · δ 1 p ≤ε2 ,2) 4 p · (p − 1)! · C · C 1 · ε ≤ 1 2 ,3) ε < 1.Then the equation F (x) = 0 has a solution x ∗ ∈ U ε (x 0 ).Äîêàçàòåëüñòâî.As in the proof of Theorem 1 consider a multivalued mappingΨ : U ε (x 0 ) → 2 X , U ε (x 0 ) ⊂ X,

The theorem on existence of singular solutions 31[] −11Ψ(x) = x −(p − 1)! F (p) (x 0 )[h] p−1 [F (x)], x ∈ U ε (x 0 ).Let us rst prove that there exists a number q, 0 < q < 1, such thath(Ψ(s 1 ), Ψ(s 2 )) ≤ q · ‖s 1 − s 2 ‖, (7)for any s 1 , s 2 ∈ U ε (x 0 ), such that s 1 = x 0 + h + u 1 , s 2 = x 0 + h + u 2 ,where ‖u i ‖ ≤ ‖h‖R , i = 1, 2 and {R = max 21,ε·C · ∥1(p−1)!∥F (p) (x 0 ) ∥ } .By Lemmas 2 and 3, we haveh(Ψ(s 1 ), Ψ(s 2 )) = inf {‖z 1 − z 2 ‖ : z i ∈ Ψ(s i ), i = 1, 2} === inf{‖z 1 − z 2 ‖ :1(p − 1)! F (p) (x 0 )[h] p−1 [z i ] =1(p − 1)! F (p) (x 0 )[h] p−1 [s i ] − F (s i ), i = 1, 2} ≤≤ inf{‖z‖ :1(p − 1)! F (p) (x 0 )[h] p−1 [z]1=(p − 1)! F (p) (x 0 )[h] p−1 [s 1 − s 2 ] − F (s 1 ) + F (s 2 )} ≤(p − 1)!≤‖h‖ p−1 C 1 ·∥ F (s 11) − F (s 2 ) −(p − 1)! F (p) (x 0 )[h] p−1 [s 1 − s 2 ]∥ .Further, taking into account Lemma 4, we have≤ supθ∈[0,1]∥ F (s 11) − F (s 2 ) −(p − 1)! F (p) (x 0 )[h] p−1 [s 1 − s 2 ]∥ ≤∥ F ′ (s 2 + θ(s 1 − s 2 )) −From the Taylor expansion, we obtain1(p − 1)! F (p) (x 0 )[h] p−1 ∥ ∥∥∥· ‖s 1 − s 2 ‖ . (8)F ′ (s 2 + θ(s 1 − s 2 )) = F ′ (x 0 ) + . . . ++ 1(p−1)! F (p) (x 0 )[s 2 − x 0 + θ(s 1 − s 2 )] p−1 + ω(h, u 1 , u 2 , θ) == 1(p−1)! F (p) (x 0 )[s 2 − x 0 + θ(s 1 − s 2 )] p−1 + ω(h, u 1 , u 2 , θ), (9)

32 A. Prusinska, A. Tret'yakovwhere‖ω(h, u 1 , u 2 , θ)‖ ≤supx∈U ε (x 0 )On account of R and ‖u i ‖, i = 1, 2, we have∥∥∥F (p+1) (x)[h + u 2 + θ(s 1 − s 2 )] p ∥∥ .‖h + u 2 + θ(s 1 − s 2 )‖ ≤ 4 · ‖h‖,and from assumption, ‖h‖ ≤ ε 2. By the above,Moreover,‖ω(h, u 1 , u 2 , θ)‖ ≤ C · ‖h‖ p ≤ 4 p · C · ε2 · ‖h‖p−1 . (10)F (p) (x 0 )[h + u 2 + θ(s 1 − s 2 )] p−1 =∑p−1Cp−1F i (p) (x 0 )[h] p−1−i [u 2 + θ(s 1 − s 2 )] i =i=0∑p−1F (p) (x 0 )[h] p−1 +i=1C i p−1F (p) (x 0 )[h] p−1−i [u 2 + θ(s 1 − s 2 )] i , (11)where‖u 2 + θ(s 1 − s 2 )‖ ≤ 3 · ‖h‖/R. (12)Taking into account the choice of R,∥ ∑p−1∥∥∥∥ C i∥p−1F (p) (x 0 )[h] p−1−i [u 2 + θ(s 1 − s 2 )] i ≤i=1∥≤ ∥F (p) ∑p−1(x 0 ) ∥ · Cp−1‖h‖ i p−1−i (3 · ‖h‖) i /R i ≤i=1∥≤ ∥F (p) (x 0 ) ∥ · ‖h‖ p−1 · 4 p−1 /R ≤ 4 p · (p − 1)! · ε2 · C · ‖h‖p−1 . (13)And nally, applying (9)(13) in (8) we obtain∥ F (s 11) − F (s 2 ) −(p − 1)! F (p) (x 0 )[h] p−1 · [s 1 − s 2 ]∥ ≤≤ 4 p · ε · C · ‖h‖ p−1 · ‖s 1 − s 2 ‖.

The theorem on existence of singular solutions 33Henceh(Ψ(s 1 ), Ψ(s 2 )) ≤(p − 1)!‖h‖ p−1 · 4p · C · C 1 · ε · ‖h‖ p−1 · ‖s 1 − s 2 ‖ == 4 p · (p − 1)! · C · C 1 · ε · ‖s 1 − s 2 ‖ ≤ q · ‖s 1 − s 2 ‖, where q = 1 2 .Letx 1 ∈ x 0 −[ 1p! F (p) (x 0 )] −1(F (x 0 )) (14)Thus this element x 1 is such that ‖x 1 − x 0 ‖ ≤ ε 2 . Indeed,[ ] ∥ −1 1 ∥∥∥∥‖x 1 − x 0 ‖ ≤∥ p! F (p) (x 0 ) · ‖F (x 0 )‖ 1 1 εp ≤ p! · δp · η ≤2 ,so x 1 ∈ U ε (x 0 ). We have to verify thatρ(Ψ(x 1 ), x 1 ) < (1 − q) · ε,where q = 1 2 . So ρ(Ψ(x 1 ), x 1 ) = inf {‖z‖ : z ∈ Ψ(x 1 ) − x 1 } ={[] }−11= inf ‖z‖ : z ∈ x 1 −(p − 1)! F (p) (x 0 )[h] p−1 F (x 1 ) − x 1 ={ [] }−11= inf ‖z‖ :(p − 1)! F (p) (x 0 )[h] p−1 z = −F (x 1 ) ≤(p − 1)!‖h‖ p−1 · C 1 · ‖F (x 1 )‖.From the Taylor expansion of F (x 1 ) and under an assumption of completedegeneration of F:whereF (x 1 ) = F (x 0 ) + F (p) (x 0 )[x 1 − x 0 ] p + ω 1 (x 1 , x 0 ),p!‖ω 1 (x 1 , x 0 )‖ ≤supx∈U ε(x 0)∥∥∥F (p+1) (x)[h] p+1 ∥∥ ≤ C · ‖h‖ p+1 ,

34 A. Prusinska, A. Tret'yakov[−11and x 1 − x 0 ∈p! F (p) (x 0 )](−F (x0 )), hence 1 p! F (p) (x 0 )[x 1 − x 0 ] p =−F (x 0 ), we nally obtain:(p − 1)!ρ(Ψ(x 1 ), x 1 ) ≤‖h‖ p−1 ·C 1·∥ F (x 0) + F (p) (x 0 )[x 1 − x 0 ] p + ω 1 (x 1 , x 0 )p!∥ =≤=(p − 1)!‖h‖ p−1 · C 1 · ‖F (x 0 ) − F (x 0 ) + ω 1 (x 1 , x 0 )‖ ≤≤(p − 1)!‖h‖ p−1 · C 1 · C · ‖ω 1 (x 1 , x 0 )‖ ≤(p − 1)!‖h‖ p−1 · C 1 · C · ‖h‖ p+1 ≤ (p − 1)! · C 1 · C · ‖x 1 − x 0 ‖ 2 ≤≤ (p − 1)! · C · C 1 · ε24 = 1 8 · ε.It follows that all conditions of Lemma 1 and inclusion (4), so there existsan element x ∗ such that x ∗ ∈ Ψ(x ∗ ). It means thathence[] −1x ∗ ∈ x ∗ 1+(p − 1)! F (p) (x 0 )[h] p−1 (−F (x ∗ )),[] −110 ∈(p − 1)! F (p) (x 0 )[h] p−1 (−F (x ∗ ))and1(p−1)! F (p) (x 0 )[h] p−1 0 = −F (x ∗ ), further 0 = −F (x ∗ ) and nallyF (x ∗ ) = 0. ✷As in regular case we give simple examples illustrating the applicationof Theorem 2. More detailed and profound applications we omit withrespect to the limit of this paper.Example 1. Consider situation, where the p-regularity condition isfullled, but the Banach conditionF ′′ (x 0 )[X] 2 = Y (15)does not hold.Namely, let F : R → R, and F (x) = x 2 + ε, where ε > 0. Moreover,if x 0 = 0, and p = 2, we have that F is p-regular at x 0 , but the equation

The theorem on existence of singular solutions 35F (x) = 0 has no solution. This follows from the fact that the condition(15) fails.Example 2. Let F : R 2 → R, and F (x, y) = x 2 − y 2 + x 3 + x 7 + 110.Consider the problem (1) with 6 (x 0 , y 0 ) = (0, 0), and p = 2. It is easy toprove that the mapping F is 2-regular at the point (x 0 , y 0 ). Moreover, wehave δ = 1110, h = (0, 6 10), ĥ = (0, 1), η = 1, C = 6 + 3 210ε4 , C 1 = 1 2 ,C 2 = 2ε 2 . All of the assumptions of the Theorem 2 are fullled. Hencethere exists a solution (x ∗ , y ∗ ) of the problem (1), and one of such pointsfrom a ε-neighborhood U ε (0), where ε = 1100 , is (0, 110). 3

36 A. Prusinska, A. Tret'yakovÑïèñîê ëèòåðàòóðû[1] Alexeev V. M. Optimal Control / V. M. Alexeev, V. M. Tihomirov, S. V.Fomin. N. Y.: Consultants Bureau, 1987.[2] Demidovitch B. P. Basis of Computational Mathematics / B. P.Demidovitch, I. A. Maron. M.: Nauka, 1973. (Russian).[3] Ioe A. D. Theory of extremal problems / A. D. Ioe, V. M. Tihomirov.Amsterdam: Studies in Mathematics and its Applications, 1979.[4] Izmailov A. F. Factor-Analysis of Non-Linear Mapping / A. F. Izmailov,A. A. Tret'yakov. M.: Nauka, 1994. (Russian).[5] Kantorovich L. V. Functional Analysis / L. V. Kantorovitch, G. P. Akilov.Oxford: Pergamon Press, 1982.[6] Krasnosel'skii M. A. Approximated Solutions of Operator Equations /M. A. Krasnosel'skii, G. M. Wainikko, P. P. Zabreiko, Yu. B. Rutitcki,V. Yu. Stetsenko. Groningen: Walters-Noordho Publishing, 1972.[7] Maurin K. Analysis. Part I. Elements / K. Maurin. Warsow: PWN, 1971.(Polish).[8] Ortega J. M. Iterative Solution of Non-Linear Equations in SeveralVariables / J. M. Ortega, W. C. Reinboldt. N. Y.: Academic Press, 1970.[9] Tret'yakov A. A. Necessary and Sucient Conditions for Optimality ofp-th Order / A. A. Tret'yakov // USSR Comput. Math. and Math Phys.V. 24. 1984. P. 123127.[10] Tret'yakov A. A. Factor-analysis of nonlinear mappings: p-regularity theory/ A. A. Tret'yakov, J. E. Marsden // Communications on Pure and AppliedAnalysis. 2003. V. 2. N 4. P. 425445.A. Prusinska, University of Podlasie,Department of Natulal Sciences,ul. 3 Maja 54, 08110 Siedlce,PolandE-mail: aprus@ap.siedlce.plA. Tret'yakov, Academy of Sciences,System Research Institute,ul. Newelska 6, 01447 WarszawaPoland E-mail: tret@ap.siedlce.plRuusian Academy of Sciences, Computing Center,Vavilova 40, Moskow, GSP1, Russia