Abstract

We try to generalize the concept of a spectrum in the nonlinear case starting from its splitting into several subspectra, not necessarily disjoint, following the classical decomposition of the spectrum. To obtain an extension of spectrum with rich properties, we replace the identity map by a nonlinear operator 𝐽 acting between two Banach spaces 𝑋 and π‘Œ, which takes into account the analytical and topological properties of a given operator 𝐹, although the original definitions have been given only in the case 𝑋=π‘Œ and 𝐽=𝐼. The FMV spectrum reflects only asymptotic properties of 𝐹, while the Feng's spectrum takes into account the global behaviour of 𝐹 and gives applications to boundary value problems for ordinary differential equations or for the second-order differential equations, which are referred to as three-point boundary value problems with the classical or the periodic boundary conditions.

1. Introduction

Let us first recall the concept of a spectrum for linear operators acting in a complex Banach space 𝑋. We denote by 𝐿(𝑋) the algebra of all bounded linear operators on 𝑋 and the resolvent set of 𝐿 is defined byξ€½πœŒ(𝐿)=πœ†βˆˆβŠ„/(πœ†πΌβˆ’πΏ)βˆ’1ξ€ΎβˆˆπΏ(𝑋)(1.1) and the spectrum of 𝐿 by𝜎(𝐿)=βŠ„/𝜌(𝐿).(1.2)

This spectrum consists of all complex scalars such that πœ†πΌβˆ’πΏ is not invertible because the inverse does not exists as a bounded operator. This can happen when πœ†πΌβˆ’πΏ is not one to one, which means that there exists π‘₯β‰ 0 such that 𝐿π‘₯=πœ†π‘₯. Such a value of πœ† is called an eigenvalue of 𝐿 and the set of all eigenvalues is called the point spectrum.

For a linear compact operator 𝐿, the spectrum 𝜎(𝐿) has some remarkable properties: it is compact, nonempty and it is countable, so has an empty interior. Moreover, it is bounded by the spectral radius given by Gelfandβ€²s formula:π‘Ÿ(𝐿)=limπ‘›β†’βˆžπ‘›βˆšβ€–πΏπ‘›β€–(1.3) and it commutes with any polynomial 𝑝, that is,𝜎(𝑝(𝐿))=𝑝(𝜎(𝐿))(1.4) by the spectral mapping theorem.

Finally, the map 𝜌(𝐿)βˆ‹πœ†β†’(πœ†πΌβˆ’πΏ)βˆ’1∈𝐿(𝑋) is analytic and the multivalued map 𝐿(𝑋)βˆ‹πΏβ†’πœŽ(𝐿)∈2βŠ„ is upper semicontinuous.

It is imposible to have a theory for nonlinear operators which collect all the useful properties of a spectrum that are satisfied by linear maps; see [1]. The spectrum of a nonlinear map contains rather little information about the map itself and may be empty. There are various notions of a spectrum for different classes of nonlinear maps that are useful in the study of nonlinear equations; see [2].

So, given a continuous nonlinear operator πΉβˆΆπ‘‹β†’π‘‹, one should try to define a spectrum 𝜎(𝐹) such that 𝜎(𝐹) has the usual properties like nonemptiness, compactness, and so forth, as in the linear case; 𝜎(𝐹) contains the point spectrum πœŽπ‘(𝐹) of 𝐹 as in the linear case (where πœŽπ‘(𝐹)={πœ†βˆˆβŠ„/𝐹(𝑒)=πœ†π‘’}, for some 𝑒≠0); 𝜎(𝐹) has reasonable applications, for instance, in existence and uniqueness problems, to boundary value problems, bifurcation problems; see [3].

We try to generalize the concept of a spectrum in the nonlinear case starting from its splitting into several subspectra, not necessarily disjoint. We follow the classical decomposition of the spectrum:𝜎(𝐿)=πœŽπ‘(𝐿)βˆͺπœŽπ‘‘(𝐿)βˆͺ𝜎co(𝐿),(1.5) where πœŽπ‘(𝐿) is the point spectrum of 𝐿(πœ†πΌβˆ’πΏ is not 1βˆ’1); πœŽπ‘‘(𝐿) is the defect spectrum of 𝐿(πœ†πΌβˆ’πΏ is not onto); 𝜎co(𝐿) is the compression spectrum of 𝐿(πœ†πΌβˆ’πΏ is not proper).

One way of defining an apropiate spectrum is to restrict attention to specific classes of maps and to replace the algebra 𝐿(𝑋) in (1.1) by other classes of continuous nonlinear operators. More general, let 𝑋 be a Banach space over a field 𝐾(𝑅orβŠ„) and 𝑀(𝑋) denote a class of continuous maps which contains the identity operator 𝐼. This leads to the Rhodius resolvent spectrum [4]. One of the first such definition was given by Neuberger (1969), who took 𝑀(𝑋)=𝐢1(𝑋), the FrΓ©chet differentiable maps on 𝑋. The corresponding spectrum is allways nonempty but may not be closed. Another possible choice is that of Lipschitz continuous maps πΉβˆΆπ‘‹β†’π‘‹, which leads to the Kachurovskij spectrum [5], which is closed but may be empty.

All these spectra have β€œbad” properties, they do not satisfy the above minimal requirements (see [3, 6–9], for a comparison between these spectra).

Later, generalizations of spectrum are based on the Kuratowski measure of noncompactness 𝛼(𝑀) of a bounded set π‘€βŠ‚π‘‹. It is defined as infimum of all πœ€>0 such that 𝑀 may be covered by finitely sets which have at most diameter πœ€. The name is motivated by the fact that 𝛼(𝑀)=0 if and only if 𝑀 has a compact closure.

A nonlinear operator πΉβˆΆπ‘‹β†’π‘Œ satisfy two conditions:𝛼𝛼(𝐹(𝑀))β‰€π‘˜π›Ό(𝑀),(𝐹(𝑀))β‰₯𝐾𝛼(𝑀),(βˆ€)π‘‹βŠƒπ‘€bounded.(1.6)

The smallest constant π‘˜ denoted by [𝐹]𝐴 and the largest constant 𝐾 denoted by [𝐹]π‘Ž are the first (metric) characteristics. So [𝐹]𝐴=0 if and only if 𝐹 is compact and [𝐹]π‘Ž>0 implies that 𝐹 is proper on closed bounded sets.

A major contribution was made in 1978 by Furi et al. see [10]. They employed the concept of stably solvable maps together with the asymptotic characteristics:[𝐹]𝑄=limsupβ€–π‘₯β€–β†’βˆž(‖𝐹π‘₯)β€–,[𝐹]β€–π‘₯β€–π‘ž=liminfβ€–π‘₯β€–β†’βˆž(‖𝐹π‘₯)β€–β€–π‘₯β€–,(1.7) the upper and lower quasinorm, respectively.

A continuous function πΉβˆΆπ‘‹β†’π‘Œ, is stably solvable if, for any compact map πΊβˆΆπ‘‹β†’π‘Œ with [𝐺]𝑄=0, the coincidence equation 𝐹(𝑒)=𝐺(𝑒) has a solution π‘’βˆˆπ‘‹. Taking 𝐺(𝑒)=𝑣 for a fixed π‘£βˆˆπ‘Œ it is clear that the stable solvability is equivalent to surjectivity (only if 𝐹 is linear); see [11].

To obtain an extension of spectrum with rich properties, we replace the identity map by a nonlinear operator π½βˆΆπ‘‹β†’π‘Œ which takes into account the analytical and topological properties of the given operator 𝐹. We can define spectra for pair of operators (𝐹,𝐽) between two Banach spaces 𝑋 and π‘Œ, although the original definitions have been given only the case 𝑋=π‘Œ and 𝐽=𝐼, see [1].

The Furi-Martelli-Vignoli spectrum of the pair (𝐹,𝐽)βˆΆπ‘‹β†’π‘Œ is defined by the union:𝜎FMV(𝐹,𝐽)=πœŽπ‘†π‘†(𝐹,𝐽)βˆͺπœŽπ‘ž(𝐹,𝐽)βˆͺπœŽπ‘Ž(𝐹,𝐽),(1.8) where πœ†βˆˆπœŽπ‘†π‘†(𝐹,𝐽)ifπœ†π½βˆ’πΉ is not stably solvable; πœ†βˆˆπœŽπ‘ž(𝐹,𝐽)if[πœ†π½βˆ’πΉ]π‘ž=0; πœ†βˆˆπœŽπ‘Ž(𝐹,𝐽)if[πœ†π½βˆ’πΉ]π‘Ž=0.

Relating to the previous decomposition for 𝐿∈𝐿(π‘₯), we get the relations:πœŽπ‘†π‘†(𝐿,𝐼)=πœŽπ‘‘(𝐿),πœŽπ‘ž(𝐿,𝐼)βŠ‡πœŽπ‘(𝐿),πœŽπ‘Ž(𝐿,𝐼)βŠ†πœŽco(𝐿).(1.9)

The FMV spectrum is closed and upper semicontinuous, as we can see in [10], but has one defect: did not contain the point spectrum, in the sense that it did not contain the eigenvalue πœ† such that πœ†π‘₯βˆ’πΉ(π‘₯)=0 from some nonzero π‘₯.

The FMV theory was so successful for many developments until 1997, when Feng, see [12], introduced a spectrum defined in a similar way, but with other concepts of solvability and characteristics which contains the classical point spectrum.

The Feng spectrum takes into account the global behaviour of 𝐹, while the FMV spectrum reflects only asymptotic properties of 𝐹.

Feng developed an attractive theory and was able to use the theory to give applications to boundary value problems.

2. Application for Ordinary Differential Equations Involving Spectral Methods

We consider the problem:Μ‡π‘₯(𝑑)βˆ’π΄(𝑑)π‘₯(𝑑)=πœ€g(𝑑,π‘₯(𝑑)),0≀𝑑≀𝑇𝐿π‘₯=πœƒ,(2.1) where 𝐴∢[0,𝑇]→𝑅𝑛×𝑛 is continuous matrix valued function; π‘”βˆΆ[0,𝑇]×𝑅𝑛→𝑅𝑛 is a CarathΓ©odory function; 𝐿∢𝐢([0,𝑇],𝑅𝑛)→𝑅𝑛 is a bounded linear operator which associates to each continuous function π‘₯∢[0,𝑇]→𝑅𝑛 a vector 𝐿π‘₯βˆˆπ‘…π‘› and πœ€β‰ 0 is a scalar parameter.

Putting𝐷π‘₯(𝑑)=𝑑π‘₯π‘‘π‘‘βˆ’π΄(𝑑)π‘₯,(2.2)𝐺(π‘₯)(𝑑)=g(𝑑,π‘₯(𝑑)),(2.3) we may write (2.1) as an operator equation:𝐷π‘₯=πœ€πΊ(π‘₯)(2.4) in the Banach space 𝑋={π‘₯∈(𝐢([0,1],𝑅𝑛)/𝐿π‘₯=πœƒ}.

By π‘ˆ(𝑑,𝑠) we denote the Cauchy function of the operator family 𝐴(𝑑) which means the unique solution of the linear Volterra integral equation:ξ€œπ‘ˆ(𝑑,𝑠)=𝐼+𝑑𝑠𝐴(πœ‹)π‘ˆ(πœ‹,𝑠)π‘‘πœ‹,0≀𝑑,𝑠≀𝑇(2.5) and byξ€œπΈπ‘§(𝑑)=𝑑0π‘ˆ(𝑑,𝑠)𝑧(𝑠)𝑑𝑠,0≀𝑑≀𝑇(2.6) the associated evolution operator.

It is clear the fact that 𝐷𝐸=𝐼, that is, the operator (2.6) is the right inverse to the differential operator (2.2).

Assume that the composition πΏπ‘ˆ=πΏπ‘ˆ0 of the boundary operator 𝐿 in (2.1) and the operator π‘ˆ0βˆΆπ‘…π‘›β†’πΆ([0,𝑇],𝑅𝑛) defined by (π‘ˆ0π‘₯)(𝑑)=π‘ˆ(𝑑,0)π‘₯,π‘₯βˆˆπ‘…π‘› is an isomorphism in 𝑅𝑛.

The nonlinear operator 𝐹 defined by𝐹(π‘₯)=πΌβˆ’π‘ˆ0πΏπ‘ˆβˆ’1𝐿𝐸𝐺(π‘₯)(2.7) maps the Banach space 𝑋 into itself.

For any π‘₯βˆˆπ‘‹, we have𝐿𝐹(π‘₯)=𝐿𝐸𝐺(π‘₯)βˆ’πΏπ‘ˆ0πΏπ‘ˆβˆ’1𝐿𝐸𝐺(π‘₯)=𝐿𝐸𝐺(π‘₯)βˆ’πΏπΈπΊ(π‘₯)=πœƒβŸΉπΉ(π‘₯)βˆˆπ‘‹.(2.8)

We put𝑀=sup0≀𝑑,π‘ β‰€π‘‡β€–π‘ˆ(𝑑,𝑠)β€–(2.9) and we denote byπœ‡πΊ(π‘Ÿ)=supβ€–π‘₯β€–β‰€π‘Ÿβ€–πΊ(π‘₯)β€–(2.10) the growth function of the Nemytskij operator (2.3).

Proposition 2.1. Suppose that the nonlinearity g∢[0,𝑇]×𝑅𝑛→𝑅𝑛 satisfies a growth condition: ||||g(𝑑,𝑒)β‰€π‘Ž(𝑑)+𝑏(𝑑)|𝑒|,0≀𝑑≀𝑇,π‘’βˆˆπ‘…π‘›(2.11) for some π‘Ž,π‘βˆˆπΏ1([0,𝑇]). Define a scalar function πœ‘βˆΆ(0,∞)β†’(0,∞) by πœ‘(π‘Ÿ)=𝑀2β€–β€–πΏπ‘ˆβˆ’1β€–β€–β€–πΏβ€–πœ‡πΊ(π‘Ÿ),π‘Ÿ>0(2.12) with 𝑀 given by (2.9) and πœ‡πΊ(π‘Ÿ) given by (2.10). Then the following linear problem: πœ†π‘₯βˆ’πΏπ‘₯=𝑦,π‘¦βˆˆπ‘‹(2.13) admits a solution π‘₯βˆˆπ‘‹ if and only if 1/πœ€ belongs to the point spectrum of the operator (2.7). Moreover, the asymptotic point spectrum of this operator satisfies the inclusion πœŽπ‘žξ‚»π‘…(𝐹)βŠ†πœ†βˆˆπœ†ξ‚΅βˆ’π‘€exp‖𝑏‖1πœ†ξ‚Άβ‰€limπ‘Ÿβ†’βˆžπœ‘(π‘Ÿ)π‘Ÿξ‚Ό.(2.14)

Proof. We put πœ†=1/πœ€. It is well known the fact that every solution of the boundary value problem (2.1) solves the eigenvalue equation 𝐹(π‘₯)=πœ†π‘₯ and vice versa. We only have to prove (2.14).
Let π‘₯βˆˆπ‘‹ be a solution of nonlinear equation, see [2]: πœ†π‘₯βˆ’πΉ(π‘₯)=𝑦,π‘¦βˆˆπ‘‹(2.15) for some πœ†>0. Then ||πœ†||||||≀||||+||||+||π‘ˆπ‘₯(𝑑)𝑦(𝑑)𝐸𝐺(π‘₯)(𝑑)0πΏπ‘ˆβˆ’1||≀||||𝐿𝐸𝐺(π‘₯)(𝑑)𝑦(𝑑)+π‘€β€–π‘Žβ€–1ξ€œ+𝑀𝑑0||||𝑏(𝑠)π‘₯(𝑠)𝑑𝑠+𝑀2β€–β€–πΏπ‘ˆβˆ’1β€–β€–||||‖𝐿‖g(𝑑,π‘₯(𝑑))(2.16) so |π‘₯(𝑑)|β‰€π‘π‘Ÿβˆ«+(𝑀/πœ†)𝑑0𝑏(𝑠)|π‘₯(𝑠)|𝑑𝑠,β€–π‘₯β€–βˆžβ‰€π‘Ÿ, where π‘π‘Ÿ=1πœ†ξ€Ίβ€–π‘¦β€–βˆž+π‘€β€–π‘Žβ€–1+𝑀2β€–β€–πΏπ‘ˆβˆ’1β€–β€–β€–πΏβ€–πœ‡πΊξ€»(π‘Ÿ).(2.17)
Applying Gronwall’s lemma to (2.16) we have ||||π‘₯(𝑑)β‰€π‘π‘Ÿξ‚΅π‘€exp‖𝑏‖1πœ†ξ‚Ά(2.18) hence ξ€·πœ†expβˆ’π‘€β€–π‘β€–1ξ€Έπœ†β‰€π‘π‘Ÿπœ†β€–π‘₯β€–βˆž+β€–πœ†π‘₯βˆ’πΉ(π‘₯)β€–β€–π‘₯β€–βˆž+π‘€β€–π‘Žβ€–1β€–π‘₯β€–βˆž+πœ‘(π‘Ÿ)β€–π‘₯β€–βˆž,for0<β€–π‘₯β€–βˆžβ‰€π‘Ÿ.(2.19)
Passing to the limit π‘Ÿβ†’βˆž we conclude that [πœ†πΌβˆ’πΉ]π‘ž>0 for any πœ† such that limπ‘Ÿβ†’βˆžπœ‘(π‘Ÿ)π‘Ÿξ‚΅βˆ’π‘€<πœ†exp‖𝑏‖1πœ†ξ‚Ά(2.20) (this means that the nonlinear operator 𝐹 maps the Banach space 𝑋 into itself); see [13].
The last hypotesis is easily checked by asuming that g(𝑑,𝑒)=π‘Ž(𝑑)+𝑏(𝑑)𝑒 with π‘Ž,π‘βˆˆπΏ1[0,𝑇]. The growth function in this case satisfies the trivial estimate: πœ‡πΊ(π‘Ÿ)β‰€β€–π‘Žβ€–1+‖𝑏‖1π‘Ÿ(2.21) so, the condition (2.20) becomes 𝑀2β€–β€–πΏπ‘ˆβˆ’1‖‖‖𝐿‖‖𝑏‖1<1πœ€ξ€·expβˆ’π‘€πœ€β€–π‘β€–1ξ€Έ.(2.22)
Putting 𝑀‖𝑏‖1πœ€=πœ‚ and πœ”(πœ‚)=1/πœ‚π‘’πœ‚ we can rewrite (2.22) as πœ”(πœ‚)>π‘€β€–πΏπ‘ˆβˆ’1‖‖𝐿‖.
But the function πœ”βˆΆ(0,∞)β†’(0,∞) is strictly decreasing, hence invertible, with limπœ‚β†’0+πœ”(πœ‚)=∞,limπœ‚β†’βˆžπœ”(πœ‚)=0 and (2.22) is true for 0<πœ€<πœ”βˆ’1(π‘€β€–πΏπ‘ˆβˆ’1‖‖𝐿‖)/𝑀‖𝑏‖1.
Passing to πœ†=1/πœ€, this gives an explicit bound of the type (2.14) for an asymptotic point spectrum πœŽπ‘ž(𝐹).

3. Another Type of Boundary Value Problems for the Second-Order Differential Equation

We consider the second-order differential equation..π‘₯(𝑑)+g(𝑑)𝑓(π‘₯(𝑑))=0(3.1) with the conditionπ‘₯(0)=0,π‘₯(1)=𝛼π‘₯(πœ‚)(3.2) orΜ‡π‘₯(0)=0,π‘₯(1)=𝛼π‘₯(πœ‚),(3.3) where πœ‚βˆˆ(0,1) is fixed.

This kind of problems are referred to as three-point boundary value problems. Many existence results have been obtained and is known that, when π›Όπœ‚β‰ 1 in (3.2) or 𝛼≠1 in (3.3), these boundary value problems may be transformed equivalently in a Hammerstein integral equation:ξ€œπ‘₯(𝑠)=10π‘˜(𝑠,𝑑)g(𝑑)𝑓(π‘₯(𝑑))𝑑𝑑,(3.4) where the Kernel function 𝐾 depends on the boundary condition (3.2) or (3.3).

In case of the boundary condition (3.2), the Kernel 𝐾 (Greenβ€²s function) from (3.4) is given by𝐾(𝑠,𝑑)=𝑠(1βˆ’π‘‘)1βˆ’π›Όπœ‚βˆ’π‘™(𝑠,𝑑,𝛼,πœ‚),(3.5) where⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘™(𝑠,𝑑,𝛼,πœ‚)=𝛼𝑠(πœ‚βˆ’π‘‘)1βˆ’π›Όπœ‚+π‘ βˆ’π‘‘,𝑑≀min{πœ‚,𝑠},𝛼𝑠(πœ‚βˆ’π‘‘)1βˆ’π›Όπœ‚,𝑠<𝑑<πœ‚,sβˆ’π‘‘,πœ‚<𝑑≀𝑠,0,𝑑>max{πœ‚,s}.(3.6)

In case of the boundary condition (3.3), the Kernel is given byπ‘˜(𝑠,𝑑)=1βˆ’π‘‘1βˆ’π›Όβˆ’π‘š(𝑠,𝑑,𝛼,πœ‚),(3.7) where⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘š(𝑠,𝑑,𝛼,πœ‚)=𝛼(πœ‚βˆ’π‘‘)1βˆ’π›Ό+π‘ βˆ’π‘‘,𝑑≀min{πœ‚,𝑠},𝛼(πœ‚βˆ’π‘‘)1βˆ’π›Ό,𝑠<π‘‘β‰€πœ‚,π‘ βˆ’π‘‘,πœ‚<𝑑≀𝑠,0,𝑑>max{πœ‚,𝑠}.(3.8)

We define the scalar function π‘˜ byπ‘˜(𝑑)=max0≀𝑠≀1||||π‘˜(𝑠,𝑑),0≀𝑑≀1.(3.9)

The function 𝑓 in (3.1) is continuous and positive and satisfy the growth condition:||||𝑓(𝑑,𝑒)β‰€π‘Ž(𝑑)+𝑏(𝑑)|𝑒|,0≀𝑑≀1,π‘’βˆˆπ‘…,(3.10) where we may suppose that the functions π‘Ž and 𝑏 are constant and we can rewrite (3.10) as||||𝑓(𝑒)β‰€π‘Ž+𝑏|𝑒|.(3.11)

Solving the three-point boundary value problems (3.1) with condition (3.2), or (3.1) with condition (3.3) can be reduced to solving a Hammerstein integral equation of the from:ξ€œπœ†π‘₯(𝑠)βˆ’10π‘˜(𝑠,𝑑)𝑓(𝑑,π‘₯(𝑑))𝑑𝑑=𝑦(𝑠),0≀𝑠≀1(3.12) with πœ†=1 and 𝑦(𝑠)=0 that is (3.4).

We have the following four propositions (for proofs, see [Nonlinear Spectral Theory [13, pages 355–358]]).

Proposition 3.1. Suppose that π›Όπœ‚β‰ 1 and π‘“βˆΆπ‘…β†’π‘… is a CarathΓ©odory function which satisfies the growth condition (3.11). Then the boundary value problem (3.1) with condition (3.2) has at least one solution provided that ‖𝑏𝑔‖1<⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩4(1βˆ’π›Όπœ‚)1βˆ’π›Ό,π›Όπœ‚β‰€0,4(1βˆ’π›Όπœ‚)max{𝛼,1},0<π›Όπœ‚<1,4(π›Όπœ‚βˆ’1)𝛼,π›Όπœ‚>1,(3.13) or ‖𝑏𝑔‖2<⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩√30(1βˆ’π›Όπœ‚)√1βˆ’π›Ό,π›Όπœ‚β‰€0,30(1βˆ’π›Όπœ‚)√max{𝛼,1},0<π›Όπœ‚<1,30(π›Όπœ‚βˆ’1)𝛼,π›Όπœ‚>1.(3.14)

Proposition 3.2. Suppose that 𝛼≠1 and π‘“βˆΆπ‘…β†’π‘… is a CarathΓ©odory function which satisfies the growth condition (3.11). Then the boundary value problem (3.1) with condition (3.3) has at least one solution provided that ‖𝑏𝑔‖1<⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩1,𝛼≀0,1βˆ’π›Ό,0<𝛼<1,π›Όβˆ’1𝛼,𝛼>1,(3.15) or ‖𝑏𝑔‖2<⎧βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺ⎩√√3,𝛼≀0,√3(1βˆ’π›Ό),0<𝛼<1,3(π›Όβˆ’1)𝛼,𝛼β‰₯1.(3.16)

Proposition 3.3. Let π›Όπœ‚β‰ 1. Then the following alternative holds. (i)The quasilinear there-point boundary value problem: ..π‘₯(𝑑)=πœ‡π‘“(𝑑,π‘₯(𝑑),Μ‡π‘₯(𝑑))+𝑦(𝑑),π‘₯(0)=0,π‘₯(1)=𝛼π‘₯(πœ‚),(3.17) where π‘¦βˆˆπΆ[0,1] is given and πœ‡β‰ 0 has a solution for πœ‡=1 and any function π‘¦βˆˆπΏ1[0,1].(ii)There exist some πœ‡β‰€1 such that the boundary value problem (3.17) has a nontrivial solution for 𝑦(𝑑)=0.(We are interested in solutions π‘₯ of (3.17) in the Sobolev space π‘Š21[0,1] of all absolutely continuous functions π‘₯ such that Μ‡π‘₯ is also absolutely continuous and ..π‘₯∈𝐿1[0,1]).

Proposition 3.4. Let π›Όπœ‚β‰ 1. Then the bounduary value problem (3.17) has a solution for any function π‘¦βˆˆπΏ1[0,1] provided that ||πœ‡||‖𝑝‖1+β€–π‘žβ€–1ξ€Έ<1𝑐(π›Όπœ‚),(3.18) where 𝑐(𝛼,πœ‚)=1+π›Όπœ‚+1||||1βˆ’π›Όπœ‚(3.19) or ⎧βŽͺ⎨βŽͺ⎩2𝑐(𝛼,πœ‚)=1βˆ’π›Όπœ‚,π›Όπœ‚<1,2π›Όπœ‚π›Όπœ‚βˆ’1,π›Όπœ‚>1.(3.20)
We can foccus on the equation ..π‘₯(𝑑)=πœ‡π‘“(𝑑,π‘₯(𝑑),Μ‡π‘₯(𝑑))+𝑦(𝑑),(3.21) where π‘“βˆΆ[0,1]×𝑅𝑛×𝑅𝑛→𝑅𝑛 and π‘¦βˆΆ[0,1]→𝑅𝑛 are supposed to be continuous vector functions and πœ‡β‰ 0.
We also can consider (3.21) together with the classical boundary condition π‘₯(0)=π‘₯(1)=0(3.22) or with the periodic boundary conditions π‘₯(0)=π‘₯(1),Μ‡π‘₯(0)=Μ‡π‘₯(1).(3.23)

Now we have two different problems. The boundary value problem (3.21) with condition (3.22) may be studied by means of the Feng or FMV spectrum, while the second boundary value problem given by (3.21) with condition (3.23) requires the semilinear spectra. For the first problem we put 𝑋=π‘₯∈𝐢2[]ξ€Ύ,[]0,1/π‘₯(0)=π‘₯(1)=0π‘Œ=𝐢0,1(3.24) and we define the operators 𝐿,πΉβˆΆπ‘‹β†’π‘Œ by𝐿π‘₯(𝑑)=..π‘₯(𝑑),𝐹(π‘₯)(𝑑)=𝑓(𝑑,π‘₯(𝑑),Μ‡π‘₯(𝑑)).(3.25)

Then 𝐿 is invertible on π‘Œ with inverseπΏβˆ’1ξ€œπ‘¦(𝑠)=10π‘˜(𝑠,𝑑)𝑦(𝑑)𝑑𝑑,(3.26) whereξƒ―π‘˜(𝑠,𝑑)=𝑠(π‘‘βˆ’1),0≀𝑠≀𝑑≀1,𝑑(π‘ βˆ’1),0≀𝑑≀𝑠≀1(3.27) is the classical Greenβ€²s function of 𝐿.

The solvability of semilinear equation, see [14],πœ†πΏπ‘₯βˆ’πΉ(π‘₯)=𝑦,π‘¦βˆˆπ‘Œ(3.28) reduces to the solvability of the classical eigenvalue equationπœ†π‘₯βˆ’πΏβˆ’1𝐹(π‘₯)=𝑧,π‘§βˆˆπ‘‹.(3.29)

For the second problem,see [2, 14], we put𝑋=π‘₯∈𝐢2[]ξ€Ύ,[].0,1/π‘₯(0)=π‘₯(1),Μ‡π‘₯(0)=Μ‡π‘₯(1)π‘Œ=𝐢0,1(3.30)

In this case, the operator 𝐿 is not invertible and again we define the operators 𝐿,πΉβˆΆπ‘‹β†’π‘Œ by (3.25).

We have𝑁(𝐿)={π‘₯βˆˆπ‘‹/π‘₯(𝑑)=const}≅𝑅𝑛,π‘Œπ‘…(𝐿)={π‘¦βˆˆπ‘Œ/𝑄𝑦=πœƒ}≅𝑅𝑛,(3.31) whereξ€œπ‘„π‘¦=10𝑦(𝑑)𝑑𝑑.(3.32)

For the projection π‘ƒβˆΆπ‘‹β†’π‘… we may choose 𝑃π‘₯=π‘₯(0), so we have the decompositions𝑋=π‘…π‘›βŠ•π‘‹0andπ‘Œ=π‘…π‘›βŠ•π‘Œ0.(3.33)

So dim𝑁(𝐿)=π‘π‘œdim𝑅(𝐿)=𝑛 which shows that 𝐿 is a Fredholm operator of index zero.

The restriction of (3.26) to the range of 𝐿 is the operator given byπΏπ‘βˆ’1=𝐿/𝑋0ξ€Έβˆ’1βˆΆπ‘…(𝐿)βŸΆπ‘‹0.(3.34)

The linear operator (the natural quotient map) Ξ βˆΆπ‘Œβ†’π‘Œ/𝑅(𝐿) and Ξ›βˆΆπ‘Œ/𝑅(𝐿)→𝑁(𝐿) (the natural isomorphism induced by 𝐿) are given by[𝑦]Ξ›[𝑦]Π𝑦=={Μƒπ‘¦βˆˆπ‘Œ/𝑄̃𝑦=𝑄𝑦},=𝑄𝑦.(3.35)

As a canonical homeomorphism β„ŽβˆΆπ‘Œ/𝑅(𝐿)β†’π‘Œ0 we may choose β„Ž[𝑦]=𝑄𝑦. So the linear isomorphism 𝐿+β„ŽΞ›βˆ’1π‘ƒβˆΆπ‘‹β†’π‘Œ is given by𝐿+β„ŽΞ›βˆ’1𝑃π‘₯(𝑑)=..π‘₯(𝑑)+π‘₯(0)(3.36) and its inverse ΛΠ+𝐾𝑃𝑄=ΛΠ+πΏπ‘ƒβˆ’1(πΌβˆ’π‘„)βˆΆπ‘Œβ†’π‘‹, isΛΠ+πΎπ‘ƒπ‘„ξ€Έξ€œπ‘¦(𝑠)=10ξ‚΅ξ€œπ‘˜(𝑠,𝑑)𝑦(𝑑)𝑑𝑑+1βˆ’10ξ€œ(π‘˜(𝑠,𝑑)𝑑𝑑)10𝑦(𝑑)𝑑𝑑.(3.37)

If 𝐿 is a bijection between 𝑋 and π‘Œ we have 𝑋0=𝑋,π‘Œ0=πœƒ,𝑃π‘₯=𝑄𝑦=πœƒ and 𝐾𝑃𝑄=πΏβˆ’1.

Acknowledgment

The author expresses her gratitude to the supervisor Professor D. Dan Pascali for his hard-working support in the accomplishment of this paper.