Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian

Abstract

Some sufficient conditions for the nonlinear integral operator of the Hammerstein type to be a diffeomorphism defined on a certain Sobolev space are formulated. The main result assures the invertibility of the Hammerstein operator and in consequence the global solvability of the nonlinear Hammerstein equations. The applications of the result to nonlinear Dirichlet BVP involving the fractional Laplacian and to some specific Hammerstein equation are presented.

1. Introduction

Consider, for any 𝜎∈(1,2], an arbitrary real number 𝜆, a given function 𝑧, and a nonlinear term ℎ, the Dirichlet boundary value problem involving one-dimensional fractional Laplacian which reads as 𝜆(−Δ)^𝜎/2𝑥(𝑡)+ℎ(𝑡,𝑥(𝑡))=(−Δ)^𝜎/2𝑧(𝑡),𝑡∈(−1,1), (1) provided with the following Dirichlet exterior boundary condition: 𝑥(𝑡)=0,𝑡∈(−∞,−1]∪[1,∞). (2) The problems with the fractional Laplacian attracted lot of attention in recent years as they naturally arise in various areas of applications to mention only, see [1–5] and references therein:(i)Probability—Mathematical Finance—as infinitesimal generators of stable Lévy processes,(ii)Mechanics—Elastostatics—in Signorini obstacle problem originating from linear elasticity,(iii)Fluid Mechanics—appearing in quasi-geostrophic fractional Navier-Stokes equation,(iv)Hydrodynamics—describing some porous media flows in the hydrodynamical model.

For fractional derivatives in various senses one can also see the books and articles like [6–8].

The problem (1) can be transformed into the operator equation 𝜆𝑥+(−Δ)_𝐷^𝜎/2−1ℎ(⋅,𝑥)=𝑧, (3) where the inverse of the fractional Laplacian with Dirichlet boundary condition (2) is defined by (−Δ)_𝐷^𝜎/2−1𝑔(𝑡)=¹₋₁𝐺(𝑡,𝜏)𝑔(𝜏)𝑑𝜏, (4) where the Green function for the Dirichlet fractional Laplace operator is defined, for example, in [2], as𝐺(𝑡,𝜏)=𝑐_𝜎|𝑡−𝜏|^𝜎−1₀^{𝑤(𝑡,𝜏)}𝑟^𝜎/2−1(𝑟+1)^−1/2𝑤()=𝑑𝑟,𝑡,𝜏1−𝑡²1−𝜏²|𝑡−𝜏|⁻², (5) and the constant 𝑐_𝜎 is defined as 𝑐_𝜎=Γ(1/2)2^𝜎𝜋^1/2Γ²().𝜎/2 (6) It should be underlined that only in the case 𝜎=2 the derivative of the Green function is nonsingular, but as soon as 𝜎<2 the singularity for the derivative of the Green function 𝐺_𝑡 appears (cf. [9, 10]) so we should allow in our theory to treat also singular integrals if we want to guarantee the operator on the right hand side of (3) to be a diffeomorphism in 𝐻¹₀, which appears to be true for 𝜎∈(1,2].

Consider, to address the solvability of (3), the general equation of the form 𝒯𝑥=𝑧, (7) where in the leading example (3) the operator 𝒯, being a sum of the rescaled identity operator 𝜆ℐ and the Hammerstein operator, is expressed as follows: 𝒯𝑥=𝜆𝑥+(−Δ)_𝐷^𝜎/2−1ℎ(⋅,𝑥). (8) The operator ((−Δ)_𝐷^𝜎/2)⁻¹ℎ(⋅,𝑥) is the composition of two operators: the linear integral nonlocal operator ((−Δ)_𝐷^𝜎/2)⁻¹—the inverse of the fractional Laplacian equipped with the Green function kernel 𝐺 given by (5) and the nonlinear Nemitskii operator 𝑥↦ℎ(⋅,𝑥) defined by the nonlinear function ℎ. We will show that (7) is globally solvable. In fact it can be proved that under suitable assumptions the operator 𝒯 is the global diffeomorphism on the Sobolev space 𝐻¹₀([−1,1]) of absolutely continuous functions; hence, apart from the solvability (7) also the differentiable continuous dependence on data follows.

In the sequel we will therefore consider the nonlinear integral operators of Hammerstein type of the following form: 𝒯𝑥(𝑡)=𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏, (9)

where 𝜆∈ℝ, 𝑡∈[−1,1], 𝐺∶𝑃→ℝ, 𝑃=[−1,1]×[−1,1], ℎ∶[−1,1]×ℝ^𝑛→ℝ^𝑛, 𝑛≥1, and 𝑥∈𝐻¹₀. By 𝐻¹₀ we will denote 𝐻¹₀([−1,1],ℝ^𝑛), the space of absolutely continuous functions defined on [−1,1] such that 𝑥(−1)=𝑥(1)=0, with the square-integrable derivative; that is, 𝑥^′∈𝐿², endowed with the norm ‖𝑥‖²_𝐻10=¹₋₁𝑥^′(𝑡)²𝑑𝑡, (10) where 𝐿²=𝐿²([−1,1],ℝ^𝑛) is the space of square-integrable functions.

Under some appropriate assumptions imposed on the functions 𝐺 and ℎ to be specified later, it is feasible to formulate some sufficient conditions for the operator 𝒯∶𝐻¹₀→𝐻¹₀ to be a diffeomorphism; that is, 𝒯(𝐻¹₀)=𝐻¹₀, and that there exists an inverse operator 𝒯⁻¹ while both 𝒯, 𝒯⁻¹ are Fréchet differentiable at every point from 𝐻¹₀. In other words, 𝒯 is Fréchet differentiable at every point 𝑥∈𝐻¹₀ and for every 𝑧∈𝐻¹₀ there exists a unique solution 𝑥_𝑧∈𝐻¹₀ to the equation 𝒯(𝑥)=𝑧 depending continuously on 𝑧 and such that the operator 𝐻¹₀∋𝑧→𝑥_𝑧∈𝐻¹₀ is Fréchet differentiable.

It should be underlined that integral operators and integral equations are most commonly considered in the space of square-integrable functions. Under suitable conditions one usually proves some existence and uniqueness theorems for integral equations. In this paper the integral operator 𝒯 is defined on the space 𝐻¹₀. In the proof of Lemma 12 we have used the compactness of the embedding of the space 𝐻¹₀ into the space of continuous functions 𝐶. This compact embedding implies that every weakly convergent sequence in 𝐻¹₀ is uniformly convergent in 𝐶 in the supremum norm. Apparently in the case of 𝐿² space such an implication does not hold. Therefore, one cannot prove, at least with the method applied herein, that the operator 𝒯∶𝐿²→𝐿² is a diffeomorphism.

Integral equations originate from models appearing in various fields of science including elasticity, plasticity, heat and mass transfer, epidemics, fluid dynamics, and oscillation theory; see, for example, books by Corduneanu [11] and by Gripenberg et al. [12]. Various kinds of integral operators considered therein include those of Fedholm, Hammerstein, Volterra and Wiener-Hopf type. Recall that we will establish global solvability of integral equations of Hammerstein type by stating sufficient conditions for Hammerstein operator to be a diffeomorphism. For references on Hammerstein equations see, for example, among others, [13–19] and references therein. Interest in Hammerstein equation, being the special case of Fredholm equation, stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems, whose linear parts possess the inverse defined via the Green’s function, can, as a rule, be transformed into equation involving Hammerstein integral operator. Among these, we mention the problem of the forced oscillations of finite amplitude of a pendulum; see, for example, [20] or for the BVP’s on real line of Hammerstein and Wiener-Hopf type, see, for example, [19], or for optimal problems for Hammerstein and Volterra equations, see, for example, [17].

2. Global Diffeomorphism by Use of Mountain Pass Theorem

Let 𝑋 be a real Banach space and let 𝜓∶𝑋→ℝ be a 𝐶¹-mapping. A sequence {𝑥_𝑘}_𝑘∈ℕ is referred to as a Palais-Smale sequence for functional 𝜓 if for some 𝑀>0, any 𝑘∈ℕ, |𝜓(𝑥_𝑘)|≤𝑀 and 𝜓^′(𝑥_𝑘)→0 as 𝑘→∞. We say that 𝜓 satisfies Palais-Smale condition if any Palais-Smale sequence possesses a convergent subsequence. Moreover, a point 𝑥^∗∈𝑋 is called a critical point of 𝜓 if 𝜓^′(𝑥^∗)=0. In such a case 𝜓(𝑥^∗) is referred to as a critical value of 𝜓.

Let us introduce the following sets used in the Mountain Pass Theorem: 𝑊_𝑒=𝑈⊂𝑋∶𝑈isopen,0∈𝑈,𝑒∉𝑈, (11) for any 𝑒∈𝑋 such that 𝑒≠0 and 𝐵_𝜌=𝑥∈𝑋∶‖𝑥‖_𝑋<𝜌, (12) for any 𝜌>0.

In the proof of the forthcoming diffeomorphism theorem the well-known variational Mountain Pass Theorem is used as the main tool. For more details we refer the reader to vast literature on the subject, for example, among others [21, 22].

Theorem 1 (Mountain Pass Theorem). Let 𝜓∶𝑋→ℝ be a 𝐶¹-mapping satisfying Palais-Smale condition and let 𝜓(0)=0. If (i)there are some constants 𝜌, 𝛼>0 such that 𝜓|_𝜕𝐵𝜌≥𝛼,(ii)there is a point 𝑒∈𝑋⧵𝐵_𝜌 such that 𝜓(𝑒)≤0,then 𝑐=sup_{𝑈∈𝑊𝑒}inf_{𝑥∈𝜕𝑈}𝜓(𝑥) is the critical value of 𝜓 and 𝑐≥𝛼.

Applying the above theorem it is possible, as was done in [23], to prove the following theorem on a global diffeomorphism.

Theorem 2. Let 𝑋 be a real Banach space and let 𝐻 be a real Hilbert space. If 𝒯∶𝑋→𝐻 is a 𝐶¹-mapping such that(a1) for any 𝑥∈𝑋 the equation 𝒯^′(𝑥)ℎ=𝑔 possesses a unique solution for any 𝑔∈𝐻,(b1) for any 𝑦∈𝐻 the functional Ψ_𝑦1∶𝑋∋𝑥⟶2𝒯𝑥−𝑦²_𝐻∈ℝ⁺=[0,∞) (13)satisfies Palais-Smale condition, then 𝒯 is a diffeomorphism.

Remark 3. By (a1) and the bounded inverse theorem, for any 𝑥∈𝑋, there exists 𝛾_𝑥>0 such that 𝒯^′(𝑥)ℎ_𝐻≥𝛾_𝑥‖ℎ‖_𝑋, (14) for any ℎ∈𝑋. Therefore, the above theorem is equivalent in other notations to Theorem 3.1 in [23].

3. Auxiliary Facts and Used Assumptions

The presentation of the proof of the main result of this paper, which formulates sufficient conditions for 𝒯∶𝐻¹₀→𝐻¹₀ defined by (9) to be a diffeomorphism, we precede with a few lemmas.

Lemma 4. For any 𝑥∈𝐻¹₀ one has |𝑥(𝑡)|≤(𝑡+1)^1/2‖𝑥‖_𝐻¹₀𝑓𝑜𝑟𝑡∈[−1,1₁₋₁],|𝑥(𝑡)|²𝑑𝑡≤2‖𝑥‖²_𝐻10. (15)

Proof. By the Schwarz inequality, for 𝑡∈[−1,1], one obtains |𝑥(𝑡)|≤^𝑡₋₁𝑥^′(𝜏)𝑑𝜏≤(𝑡+1)^1/2‖𝑥‖_𝐻¹₀. (16) Consequently, ¹₋₁|𝑥(𝑡)|²𝑑𝑡≤‖𝑥‖²_𝐻10¹₋₁(𝑡+1)𝑑𝑡=2‖𝑥‖²_𝐻10, (17) and this is precisely the second assertion of the lemma.

In what follows, we will use the following assumptions imposed on the functions 𝐺 and ℎ. (A1) One has the following: (a)the functions 𝐺(⋅,𝜏) and ℎ(𝜏,⋅) are continuous for a.e. 𝜏∈[−1,1], (b)there exists continuous derivative 𝐺_𝑡(⋅,𝜏) on (−1,1)⧵{𝜏} for a.e. 𝜏∈[−1,1], (c)there exists derivative ℎ_𝑥(𝜏,⋅) and it is continuous for a.e. 𝜏∈[−1,1]; (A2) One has the following: (a)the function 𝐺(𝑡,⋅)ℎ(⋅,𝑥(⋅)) is integrable and this integral is locally bounded with respect to 𝑥∈𝐻¹₀, that is, for every 𝜌>0 there exists 𝑙_𝜌>0 such that for any 𝑡∈[−1,1] and |𝑥(𝑡)|≤𝜌: ¹₋₁|𝐺(𝑡,𝜏)||ℎ(𝜏,𝑥(𝜏))|𝑑𝜏<2𝑙_𝜌, (18) (b)the function 𝐺_𝑡(𝑡,⋅)ℎ(⋅,𝑥(⋅)) is integrable and for every 𝜌>0 there exist 𝑙_𝜌>0 such that ¹₋₁𝐺_𝑡(𝑡,𝜏)|ℎ(𝜏,𝑥(𝜏))|𝑑𝜏<2𝑙_𝜌, (19) for 𝑥∈𝐻¹₀ such that |𝑥(𝑡)|≤𝜌 for 𝑡∈[−1,1], (c)the function 𝐺(𝑡,⋅)ℎ_𝑥(⋅,𝑥(⋅)) satisfies (A2)(a) with ℎ_𝑥 instead of ℎ whereas the function 𝐺_𝑡(𝑡,⋅)ℎ_𝑥(⋅,𝑥(⋅)) satisfies (A2)(b) with ℎ_𝑥 instead of ℎ; (A3) 𝐺 satisfies the Dirichlet boundary conditions 𝐺(−1,𝜏)=𝐺(1,𝜏)=0for a.e. 𝜏∈[−1,1]; (A4) ∫¹₋₁|𝐺_𝑡(𝑡,𝜏)||ℎ_𝑥(𝜏,𝑥(𝜏))|𝑑𝜏<|𝜆| for any 𝑥∈𝐻¹₀ and 𝑡∈[−1,1]; (A5) One has the following: (a)|ℎ(𝜏,𝑥)|≤𝑎(𝜏)|𝑥|+𝑏(𝜏) where 𝜏∈[−1,1], 𝑥∈ℝ^𝑛, 𝑎,𝑏∈𝐿²([−1,1],ℝ⁺), (b)‖𝐺_𝑡(⋅,⋅)𝑎(⋅)‖_{𝐿2(𝑃,ℝ)}<√2|𝜆|/4, ‖𝐺_𝑡(⋅,⋅)𝑏(⋅)‖_{𝐿2(𝑃,ℝ)}<∞.

Remark 5. Besides regularity (A1),  (A2), and technical (A3) assumptions, we must finally impose on the functions 𝐺 and ℎ some growth and quantitative global assumptions: (A4) and (A5).

Lemma 6. If the functions 𝐺 and ℎ satisfy (A1)(a), (A1)(b), (A2)(a), (A2)(b), and (A3), then the operator 𝒯 is well defined by (9) on the space 𝐻¹₀ with values in 𝐻¹₀.

Proof. Let us choose any 𝑥₀∈𝐻¹₀. By (A3), 𝒯𝑥₀(−1)=𝒯𝑥₀(1)=0. It suffices to show that the function 𝑦(𝑡)=𝒯𝑥₀(𝑡)−𝜆𝑥₀(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝑑𝜏 (20) is absolutely continuous and 𝑦^′∈𝐿². Observe, by (A2)(b) and 𝑦(𝑡_𝑖)=∫^𝑡𝑖₋₁𝑦^′(𝑡)𝑑𝑡 that one has ^𝑁_𝑖=1𝑦𝑡_𝑖+1−𝑦𝑡_𝑖=^𝑁_𝑖=1^𝑡𝑖+1_𝑡𝑖¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝑑𝜏𝑑𝑡≤2𝑙_𝜌^𝑁_𝑖=1𝑡_𝑖+1−𝑡_𝑖, (21) where −1≤𝑡₁<𝑡₂<⋯<𝑡_𝑖<𝑡_𝑖+1<⋯<𝑡_𝑁<𝑡_𝑁+1≤1. As a result, for any 𝑥₀∈𝐻¹₀ the function 𝑦 is absolutely continuous and therefore for almost any 𝑡∈(−1,1) there exists 𝑦^′(𝑡) and its square integral can be estimated by (A2)(b) as ¹₋₁𝑦^′(𝑡)²𝑑𝑡≤¹₋₁¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝑑𝜏²𝑑𝑡≤8𝑙²_𝜌 (22) so that 𝑦^′∈𝐿².

Now we present some sufficient conditions for 𝒯∶𝐻¹₀→𝐻¹₀ to be Fréchet differentiable.

Lemma 7. Suppose that functions 𝐺 and ℎ satisfy (A1)(a), (A1)(c), (A2)(a), (A2)(c), and (A3). Then the operator 𝒯 defined by (9) is Fréchet differentiable at any 𝑥₀∈𝐻¹₀ while for 𝑥∈𝐻¹₀ and 𝑡∈[−1,1]𝒯^′𝑥₀𝑥(𝑡)=𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏. (23)

Proof. It is sufficient to show that the operator 𝒯⁰(𝑥)(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏 (24) is Fréchet differentiable. The Mean Value Theorem (cf. [24]) yields, for 𝑡∈[−1,1] and some 𝜃∈[0,1], 𝒯⁰𝑥₀+𝑥(𝑡)−𝒯⁰𝑥₀(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)(𝜏)()ℎ+𝑥−𝐺𝑡,𝜏𝜏,𝑥₀(𝜏)=𝑑𝜏₁₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏𝑥(𝜏)+)𝑑𝜏₁₋₁₁₀𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)(𝜏)()ℎ+𝜃𝑥𝑑𝜃−𝐺𝑡,𝜏_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏. (25)
From (15) in Lemma 4 one has ¹₋₁¹₀𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)(𝜏)()ℎ+𝜃𝑥𝑑𝜃−𝐺𝑡,𝜏_𝑥𝜏,𝑥₀(𝜏𝑥(𝜏)≤√)𝑑𝜏2‖𝑥‖_𝐻¹₀¹₋₁¹₀𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)(𝜏)()ℎ+𝜃𝑥−𝐺𝑡,𝜏_𝑥𝜏,𝑥₀(𝜏)𝑑𝜃𝑑𝜏. (26) Since the strong convergence in 𝐻¹₀ implies the uniform convergence in 𝐶 and since the assumptions (A1)(c) and (A2)(c) of this lemma are satisfied, the Lebesgue Theorem leads, if we take ‖𝑥‖_𝐻¹₀→0, to ¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)(𝜏)()ℎ+𝜃𝑥−𝐺𝑡,𝜏_𝑥𝜏,𝑥₀(𝜏)𝑑𝜏⟶0, (27) and thus, 𝒯⁰𝑥₀+𝑥(𝑡)−𝒯⁰𝑥₀(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏+𝑜(𝑥), (28) where 𝑜(𝑥)/‖𝑥‖_𝐻¹₀→0 as ‖𝑥‖_𝐻¹₀→0, which completes the proof.

4. Local Solvability: Analysis of Linearized System

Let 𝑥₀∈𝐻¹₀ be a fixed but an arbitrary function and 𝑇∶𝐻¹₀→𝐻¹₀ be a linear operator defined, for any 𝑥∈𝐻¹₀ and 𝑡∈[−1,1], by (𝑇𝑥)(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏, (29) where the functions 𝐺 and ℎ define, respectively, the kernel and the nonlinearity of operator 𝒯 defined in (9).

Next, for any 𝑘∈ℕ, 𝑡∈[−1,1], and 𝑥∈𝐻¹₀, consider the following sequence of iterations: 𝑇⁰𝑥(𝑡)(𝑡),𝑇=𝑥₁𝑥(𝑡)=𝑇𝑇⁰𝑥(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑇𝑑𝜏,₂𝑥(𝑡)=𝑇𝑇¹𝑥(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)×𝑇¹𝑥(𝜏)𝑇𝑑𝜏,⋯,_𝑘+1𝑥(𝑡)=𝑇𝑇^𝑘𝑥(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)×𝑇^𝑘𝑥(𝜏)𝑑𝜏. (30)(31) We will prove the following lemma.

Lemma 8. Under assumptions (A1)(c), (A2)(c), and (A3) one has the following estimates: 𝑇^𝑘𝑥(𝑡)≤2^𝑘𝑙^𝑘_𝜌𝑀,𝑓𝑜𝑟𝑘=0,1,2,…,𝑡∈[−1,1], (32) where 𝑙_𝜌>0 is defined by (A2)(c), 𝑀=‖𝑥‖_∞∶=sup_{[]𝑡∈−1,1}|𝑥(𝑡)|, (33) and {𝑇^𝑘𝑥} is a sequence defined iteratively by (30) and (31).

Proof. First, from (29)–(31) and the assumptions of the lemma, we obtain subsequently 𝑇¹𝑥(𝑡)≤2𝑙_𝜌𝑇𝑀,₂𝑥(𝑡)≤¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑇¹𝑥(𝜏)𝑑𝜏≤2𝑙_𝜌2𝑙_𝜌𝑀=4𝑙²_𝜌𝑇𝑀,₃𝑥(𝑡)≤¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑇²𝑥(𝜏)𝑑𝜏≤8𝑙³_𝜌𝑀. (34) To finish the proof we proceed by induction to get estimate (32).
Now, let us consider the linear integral equation 𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)(𝑡)[],𝑑𝜏=𝑦for𝑡∈−1,1 (35) where 𝑥₀∈𝐻¹₀ and 𝑦∈𝐻¹₀ are fixed. For (35), we will prove the existence and uniqueness result, see Lemma 10. Since, in the proof of this lemma, we will perform spectral analysis we now present some introductory notions and recall some functional analytic theorems and tools on spectral radius.
Let 𝑇 be a bounded, continuous operator in a Banach space 𝑋. Then we can decompose ℂ into the resolvent of the operator 𝑇 defined by 𝜌(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼isbijectionon𝑋}, (36) and the complementary set—the spectrum of 𝑇 defined as 𝜎(𝑇)=ℂ⧵𝜌(𝑇)={𝜆∈ℂ∶𝑇−𝜆𝐼isbijectionon𝑋}. (37) For any bounded and continuous operator 𝑇, we can define the spectral radius of 𝑇 by the formula 𝑟(𝑇)=lim_𝑘→∞sup𝑇^𝑘1/𝑘, (38) which must be finite, for example, due to the following estimate: 𝑟(𝑇)≤‖𝑇‖. (39) Moreover, we have, following, for example, [25, Theorem VI.6] and [26, Theorem VIII.2.3], the theorem.

Theorem 9. For any |𝜆|>𝑟(𝑇), one has 𝜆∈𝜌(𝑇), which means complementarily that the spectrum of 𝑇 is contained in the closed ball of radius 𝑟(𝑇); that is, 𝜎(𝑇)⊂𝐵_𝑟(𝑇)={𝜆∈ℂ∶|𝜆|≤𝑟(𝑇)}.

Now, we are ready to formulate the lemma on solvability of the linear integral equation (35).

Lemma 10. For any 𝑥₀∈𝐻¹₀, 𝜌>0 such that ‖𝑥₀‖_∞≤𝜌 and any 𝑦∈𝐻¹₀, (35) possesses a unique solution in 𝐻¹₀ provided that the functions 𝐺 and ℎ satisfy (A1), (A2), (A3), and weaker, local version of (A4), with constant 𝑙_𝜌 such that |𝜆|>2𝑙_𝜌≥sup_{[],|𝑥(𝑡𝑡∈−1,1)|<𝜌}¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥(𝜏,𝑥(𝜏))𝑑𝜏. (40)

Proof. Our proof starts with observation that (35) can be written in the form 𝜆𝑥+𝑇𝑥=𝑦, (41) where (𝑇𝑥)(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏. (42) From (29), (30), and (31) it follows that for any 𝑘=1,2,…, 𝑡∈[−1,1] and 𝑥∈𝐻¹₀ we have 𝑇^𝑘+1𝑥(𝑡)=¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑇^𝑘𝑥(𝜏)𝑇𝑑𝜏,₀𝑥(𝑡)=𝑥(𝑡). (43) By (A1), (A2), (A3), and inequality (32) from Lemma 8, the analysis similar to that in the proof of Lemma 6 leads, if we apply induction, to the fact that 𝑇^𝑘𝑥∈𝐻¹₀ for any 𝑥∈𝐻¹₀ and 𝑘=1,2,…. By (15) from Lemma 4 and (32) from Lemma 8, we get, with 𝑀=‖𝑥‖_∞, the following estimate: 𝑇^𝑘𝑥²_𝐻10=¹₋₁𝑑𝑇𝑑𝑡^𝑘𝑥(𝑡)²=𝑑𝑡₁₋₁𝑑𝑑𝑡¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)×𝑇^𝑘−1𝑥(𝜏)𝑑𝜏²=𝑑𝑡₁₋₁₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏×𝑇_𝑘−1)𝑥(𝜏)𝑑𝜏²≤𝑑𝑡₁₋₁2𝑙_𝜌2^𝑘−1𝑙_𝜌^𝑘−1𝑀²𝑑𝑡=2^2𝑘+1𝑙_𝜌^2𝑘𝑀²≤2^2𝑘+2𝑙_𝜌^2𝑘‖𝑥‖²_𝐻10, (44) and hence by an arbitrary choice of 𝑥∈𝐻¹₀ we get 𝑇^𝑘1/𝑘≤2^1+(1/𝑘)𝑙_𝜌. (45) Consequently, 𝑟(𝑇)=limsup_𝑘→∞𝑇^𝑘1/𝑘≤2𝑙_𝜌 (46) which means that the spectral radius 𝑟(𝑇) is less or equal to 2𝑙_𝜌. Since, by Theorem 9, 𝜎(𝑇)⊂𝐵_𝑟(𝑇) with 𝑟(𝑇) is defined by (46). Then, in particular, for all 𝜆∈ℝ such that |𝜆|>𝑟(𝑇) we have 𝜆∈𝜌(𝑇). Therefore, we can conclude that, for all 𝜆∈ℝ and |𝜆|>2𝑙_𝜌, the operator 𝑇−𝜆𝐼 is bijective on 𝐻¹₀. Thus, for any |𝜆|>2𝑙_𝜌, ‖𝑥₀‖_∞≤𝜌, and 𝑦∈𝐻¹₀, there exists a unique solution 𝑥∈𝐻¹₀ to (𝑇+𝜆𝐼)𝑥=𝑦, (47) which ends the proof. Indeed, by definition of 𝑇, there exists a unique solution to 𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)𝑥(𝜏)𝑑𝜏=𝑦(𝑡). (48)

5. Palais-Smale Condition Guaranteeing Global Diffeomorphism

Let us consider, for an arbitrary function 𝑦∈𝐻¹₀, the functional Ψ_𝑦∶𝐻¹₀→ℝ⁺ of the form Ψ_𝑦(𝑥)=12𝒯𝑥−𝑦²_𝐻10=12¹₋₁𝑑((𝑡𝑑𝑡𝒯𝑥))−𝑦^′(𝑡)²=1𝑑𝑡2¹₋₁𝜆𝑥^′(𝑡)+¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏−𝑦^′(𝑡)²𝑑𝑡. (49)

To prove the main results of the paper we will need some sufficient conditions under which for any 𝑦∈𝐻¹₀ the functional Ψ_𝑦 is coercive; that is, for any 𝑦∈𝐻¹₀, Ψ_𝑦(𝑥)→∞ provided that ‖𝑥‖_𝐻¹₀→∞.

Lemma 11. If the functions 𝐺 and ℎ satisfy (A1)(a), (A1)(b), (A2)(a), (A2)(b), (A3), and (A5), then for any 𝑦∈𝐻¹₀ the functional Ψ_𝑦 is coercive.

Proof. Since the functional Ψ_𝑦 is coercive for any 𝑦∈𝐻¹₀ if and only if the functional Ψ_𝑦 is coercive for 𝑦=0, we first observe that the functional Ψ₀ is bounded from below. By the Schwarz inequality and the assumptions of this lemma together with the last estimate from Lemma 4, we obtain Ψ₀(𝑥)=12¹₋₁𝜆𝑥^′(𝑡)+¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏²≥‖𝑥‖𝑑𝑡_2𝐻1012|𝜆|²−√2|𝜆|𝐺_𝑡𝑎_{𝐿2(𝑃,ℝ)}−‖𝑥‖_𝐻¹₀√2|𝜆|𝐺_𝑡𝑏_{𝐿2(𝑃,ℝ)}. (50) From (A5)(b) and the above estimate it follows that Ψ₀(𝑥)→∞ if ‖𝑥‖_𝐻¹₀→∞. Consequently, for any 𝑦∈𝐻¹₀ we have Ψ_𝑦(𝑥)→∞ as ‖𝑥‖_𝐻¹₀→∞.

Lemma 12. For any 𝑦∈𝐻¹₀ the functional Ψ_𝑦 satisfies Palais-Smale condition provided that assumptions (A1), (A2), (A3), (A4), and (A5) are satisfied.

Proof. Fix 𝑦∈𝐻¹₀. Recall that the functional Ψ_𝑦 has the form Ψ_𝑦(𝑥)=12¹₋₁𝜆𝑥^′(𝑡)+¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏−𝑦^′(𝑡)²𝑑𝑡. (51)
Straightforward calculation leads to Ψ_𝑦(𝑥)=12¹₋₁|𝜆|²𝑥^′(𝑡)²+2𝜆𝑥^′(𝑡),¹₋₁𝐺_𝑡()ℎ((𝜏+𝑡,𝜏𝜏,𝑥))𝑑𝜏₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏²−2𝜆𝑥^′(𝑡),𝑦^′(𝑡)−2¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏,𝑦^′(𝑡+𝑦_′)(𝑡)²𝑑𝑡. (52) The functional Ψ_𝑦 defined by (49), being a superposition of two 𝐶¹-mappings, is also of the same regularity 𝐶¹ type and its differential Ψ^′_𝑦(𝑥) at 𝑥∈𝐻¹₀ is given, for 𝑣∈𝐻¹₀, byΨ^′_𝑦(𝑥)𝑣=¹₋₁|𝜆|²𝑥^′(𝑡),𝑣^′(𝑡+)𝜆𝑣^′(𝑡),¹₋₁𝐺_𝑡()ℎ((𝜏+𝑡,𝜏𝜏,𝑥))𝑑𝜏𝜆𝑥^′(𝑡),¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥((𝜏𝑣(𝜏)+𝜏,𝑥))𝑑𝜏₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏,₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥((𝜏𝑣(𝜏)−𝜏,𝑥))𝑑𝜏𝜆𝑣^′(𝑡),𝑦^′(𝑡)−¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥((𝜏(𝜏)𝜏,𝑥))×𝑣𝑑𝜏,𝑦^′(𝑡)𝑑𝑡. (53)
Let {𝑥_𝑘}⊂𝐻¹₀ be a Palais-Smale sequence for some fixed but an arbitrary 𝑀≥0; that is, |Ψ_𝑦(𝑥_𝑘)|≤𝑀 and Ψ_𝑦(𝑥_𝑘)→0. Applying Lemma 11 we obtain that Ψ_𝑦 is coercive, and hence the sequence {𝑥_𝑘} is weakly compact as a bounded sequence in a reflexive space. Passing, if necessary, to a subsequence, one can assume that 𝑥_𝑘⇀𝑥₀ weakly in 𝐻¹₀. Moreover, the weak convergence of the sequence {𝑥_𝑘} in the space 𝐻¹₀ implies the uniform convergence in 𝐶; that is, 𝑥_𝑘(𝑡)⇉𝑥₀(𝑡) uniformly with respect to 𝑡∈[−1,1] as well as the weak convergence of its derivatives in 𝐿²; that is, 𝑥^′_𝑘→𝑥^′₀ in 𝐿²and as being a weakly convergent sequence it has to be bounded. It remains to prove that the sequence {𝑥_𝑘} converges to 𝑥₀ in 𝐻¹₀. By (53), a direct calculation leads to Ψ^′_𝑦𝑥_𝑘−Ψ^′_𝑦𝑥₀,𝑥_𝑘−𝑥₀=|𝜆|²𝑥_𝑘−𝑥₀²_𝐻10+⁶_𝑖=1𝐺^𝑖𝑥_𝑘, (54) where 𝐺¹𝑥_𝑘=¹₋₁𝜆𝑥^′_𝑘(𝑡)−𝑥^′₀(𝑡),₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥_𝑘(𝜏)−𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝐺𝑑𝜏𝑑𝑡,₂𝑥_𝑘=¹₋₁𝜆𝑥^′_𝑘(𝑡),¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥_𝑘(𝜏)×𝑥_𝑘(𝜏)−𝑥₀(𝜏)×𝑑𝜏₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥_𝑘(𝜏𝐺)𝑑𝑡,₃𝑥_𝑘=¹₋₁¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥_𝑘(𝜏)𝑑𝜏,₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥_𝑘(𝜏)×𝑥_𝑘(𝜏)−𝑥₀(𝜏)𝐺𝑑𝜏𝑑𝑡,₄𝑥_𝑘=−¹₋₁¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥_𝑘(𝜏)−𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏×𝑥_𝑘)(𝜏)−𝑥₀(𝜏)𝑑𝜏,𝑦^′(𝑡𝐺)𝑑𝑡,₅𝑥_𝑘=−¹₋₁𝜆𝑥^′₀(𝑡),¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)×𝑥_𝑘(𝜏)−𝑥₀(𝜏)𝐺𝑑𝜏𝑑𝑡,₆𝑥_𝑘=−¹₋₁¹₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝑑𝜏,₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ_𝑥𝜏,𝑥₀(𝜏)×𝑥_𝑘(𝜏)−𝑥₀(𝜏)𝑑𝜏𝑑𝑡. (55) Since Ψ^′_𝑦(𝑧_𝑘)→0 and 𝑥_𝑘⇀𝑥₀ weakly in 𝐻¹₀, lim_𝑘→∞⟨Ψ^′_𝑦(𝑥_𝑘)−Ψ^′_𝑦(𝑥₀),𝑥_𝑘−𝑥₀⟩=0. We will prove that, for 𝑖=1,2,…,6, lim_𝑘→∞𝐺^𝑖(𝑥_𝑘)=0. By the Schwarz inequality, (A1)(a), and (A2)(b) we get 𝐺¹𝑥_𝑘²≤|𝜆|²¹₋₁𝑥^′_𝑘(𝑡)−𝑥^′₀(𝑡)²×𝑑𝑡₁₋₁₁₋₁𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥_𝑘(𝜏)−𝐺_𝑡(𝑡,𝜏)ℎ𝜏,𝑥₀(𝜏)𝑑𝜏²𝑑𝑡. (56)
The first factor above is bounded, whereas the second one, by (A4), is convergent to zero, and therefore, 𝐺¹(𝑥_𝑘)→0 as 𝑘→∞. Next, 𝐺²(𝑥_𝑘) can be estimated by 𝜀𝜆²√2‖𝑥^′_𝑘‖_𝐿² if ‖𝑥_𝑘−𝑥₀‖_∞≤𝜀. Similar estimates can be applied to other terms; thus, one can prove that 𝐺^𝑖(𝑥_𝑘)→0 as 𝑘→∞ for 𝑖=3,4,5,6. Hence, from (54), it follows that 𝑥_𝑘→𝑥₀ in 𝐻¹₀.

6. Main Results and Applications

Applying formerly presented lemmas and Theorem 2 we prove the main result of this paper.

Theorem 13. If the functions 𝐺 and ℎ satisfy assumptions (A1), (A2), (A3), (A4), and (A5), then the nonlinear Hammerstein operator 𝒯∶𝐻¹₀→𝐻¹₀ defined by (9) is a diffeomorphism of 𝐻¹₀ on 𝐻¹₀.

Proof. Set 𝑋=𝐻=𝐻¹₀. From Lemma 10 we infer that the operator 𝒯 satisfies assumption (a1) of Theorem 2, while Lemma 12 ascertains that for any 𝑦∈𝐻¹₀ the functional Ψ_𝑦(𝑥)=1/2‖𝒯𝑥−𝑦‖²_𝐻10 satisfies Palais-Smale condition so that assumption (b1) of Theorem 2 is fulfilled. Therefore, 𝒯∶𝐻¹₀→𝐻¹₀ defined by (9) is a diffeomorphism.

Theorem 13 can be formulated in the following equivalent version focusing on the solvability, uniqueness, and continuous dependence issues, following from the diffeomorphism property.

Theorem 14. If the functions 𝐺 and ℎ satisfy assumptions of Theorem 13, then for any 𝑧∈𝐻¹₀ the nonlinear integral equation 𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ℎ(𝜏,𝑥(𝜏))𝑑𝜏=𝑧(𝑡),𝑡∈[−1,1], (57) possesses a unique solution 𝑥=𝑥_𝑧∈𝐻¹₀ and moreover the solution operator 𝐻¹₀∋𝑧⟶𝑥_𝑧∈𝐻¹₀ (58) is continuously Fréchet differentiable.

Next, we will present the application of our general theorem to the equation involving the fractional Laplacian operator for 𝑛=1.

Example 15. Assume that the nonlinear term ℎ satisfy the Green function 𝐺 estimates (A1)–(A5). This is the case if, for example, the function ℎ is smooth, that is, 𝐶¹, and it satisfies the linear growth conditions (A4)-(A5). Then for any 𝑧∈𝐻¹₀ and 𝜎∈(1,2] there exists a unique solution 𝑥∈𝐻¹₀ of 𝜆(−Δ)^𝜎/2𝑥(𝑡)((𝑡=()+ℎ𝑡,𝑥))−Δ^𝜎/2𝑧(𝑡),𝑡∈(−1,1). (59) By [2, Corollary 3.2] we have for the Green function of (−Δ)^𝜎/2 the following estimates: 𝑐_𝛼𝛿^𝜎/2(𝑡)𝛿^𝜎/2(𝜏)|𝑡−𝜏|∧𝛿^(𝜎−1)/2(𝑡)𝛿^(𝜎−1)/2(𝜏)≤𝐺(𝑡,𝜏)≤𝐶_𝛼𝛿^𝜎/2(𝑡)𝛿^𝜎/2(𝜏)|𝑡−𝜏|∧𝛿^(𝜎−1)/2×(𝑡)𝛿^(𝜎−1)/2(𝜏), (60) where 𝛿(𝜏)=dist(𝜏,{−1,1}), 𝑎∧𝑏=min(𝑎,𝑏). It should be noted (cf. [2]) that the Green function for 𝜎∈(1,2] is bounded and continuous. For estimates on 𝐺_𝑡 and regularity see [2, 9, 10]. One can recall or show directly that continuous 𝐺 behaves like (⋅)^𝜎/2 and 𝐺_𝑡 behaves like (⋅)^𝜎/2−1 at the boundary, that is, at −1 or 1, while 𝐺 is like (⋅)^𝜎−1 and 𝐺_𝑡 is like (⋅)^𝜎−1 at 𝑡=𝜏. Therefore, the integrability assumptions are satisfied for mild singularity; that is, only if 𝜎∈(1,2] but not in the range of stronger singularity when 𝜎∈(0,1].

Finally, we will present the application of the main theorem to some specific nonlinear integral Hammerstein operator this time with smooth kernel.

Example 16. Let us consider the following operator: (𝑡)(𝑡)+𝒯𝑥=𝜆𝑥₁₋₁𝐺(𝑡,𝜏)ln1+𝐵(𝜏)𝑥²(𝜏)[],𝑑𝜏,𝑡∈−1,1 (61) with functions 𝐵∈𝐶¹([−1,1],ℝ) satisfying 𝐵(𝜏)>0 on [−1,1] and 𝐺∈𝐶¹(𝑃,ℝ) with 𝑃=[−1,1]² such that 𝐺(−1,𝜏)=𝐺(1,𝜏)=0 for 𝜏∈[−1,1].
Since ln(1+𝑧²)≤|𝑧|, for the function ℎ(𝜏,𝑥)=ln1+𝐵(𝜏)𝑥² (62) the following estimate holds: |ℎ(𝜏,𝑥)|≤𝐵(𝜏)|𝑥|. (63) Similarly, since 1+𝑧²≤2𝑧 we have the estimate for ℎ_𝑥()=(𝜏)𝑥𝜏,𝑥2𝐵1+𝐵(𝜏)𝑥² (64) reading ℎ_𝑥(𝜏,𝑥)≤𝐵(𝜏). (65)
Let us define 𝑎(𝜏)=𝐵(𝜏) and 𝑏(𝜏)≡0. Then 𝑎,𝑏∈𝐿²([−1,1],ℝ⁺) and condition (A5)(a) is fulfilled. Assuming 𝐺_𝑡(𝑡,𝜏)𝑎(𝜏)_{𝐿2(𝑃,ℝ)}=𝐺_𝑡(𝑡,𝜏)𝐵(𝜏)_{𝐿2(𝑃,ℝ)}<√24|𝜆|, (66) we can guarantee that assumption (A5)(b) is satisfied.
Consequently, if we assume that ¹₋₁𝐺_𝑡(𝑡,𝜏)𝐵(𝜏)𝑑𝜏<|𝜆|, (67) then condition (A4) holds. Thus, the functions 𝐺 and ℎ satisfy assumptions (A1)–(A5) and Theorem 14 implies that the equation, for any 𝑧∈𝐻¹₀, 𝜆𝑥(𝑡)+¹₋₁𝐺(𝑡,𝜏)ln1+𝐵(𝜏)𝑥²(𝜏)(𝑡)[]𝑑𝜏=𝑧for𝑡∈−1,1 (68) possesses a unique solution 𝑥=𝑥_𝑧∈𝐻¹₀ and 𝐻¹₀∋𝑧→𝑥_𝑧∈𝐻¹₀ is continuously Fréchet differentiable.

7. Summary

We have considered the nonlinear integral operator of Hammerstein type 𝒯 defined on the Sobolev space 𝐻¹₀ with some application to the nonlocal Dirichlet BVP involving the fractional Laplacian. The key point in the proof of the main result of this paper is the application of the theorem on global diffeomorphism. In particular, we have shown that that the assumptions (A1), (A2), (A3), (A4), and (A5) imply some sufficient conditions for the operator 𝒯∶𝐻¹₀→𝐻¹₀ defined by (9) to be a diffeomorphism, compare Theorem 13. Equivalently, we have obtained the existence and uniqueness result for the nonlinear Hammerstein equation (57) and the differentiable dependence of the solution on parameters as well, see Theorem 14. Thus, in other words, our problem is well-posed and robust, compare [27]. It should be emphasized that in the proof of Lemma 12 we have used the compactness of the embedding of the space 𝐻¹₀ into the space 𝐶 and the reflexivity of 𝐻¹₀ and these properties are crucial in the method of the proof applied therein. Finally, in Section 6 we have proposed some examples of the nonlinear Hammerstein operators for which Theorems 13 and 14 are applicable, including the one originating from the BVP involving the fractional Laplacian.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.