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

and Applied Analysis 3 Theorem 1 (Mountain Pass Theorem). Let ψ : X → R be a C -mapping satisfying Palais-Smale condition and let ψ(0) = 0. If (i) there are some constants ρ, α > 0 such that ψ| ∂B ρ ≥ α, (ii) there is a point e ∈ X \ B ρ such that ψ(e) ≤ 0, then c = sup U∈W e 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. IfT : X → H is a C-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

For fractional derivatives in various senses one can also see the books and articles like [6][7][8].
The problem (1) can be transformed into the operator equation where the inverse of the fractional Laplacian with Dirichlet boundary condition (2) is defined by where the Green function for the Dirichlet fractional Laplace operator is defined, for example, in [2], as and the constant is defined as = Γ (1/2) 2 1/2 Γ 2 ( /2) .
It should be underlined that only in the case = 2 the derivative of the Green function is nonsingular, but as soon as < 2 2 Abstract and Applied Analysis the singularity for the derivative of the Green function 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 1 0 , which appears to be true for ∈ (1,2]. Consider, to address the solvability of (3), the general equation of the form where in the leading example (3) the operator T, being a sum of the rescaled identity operator I and the Hammerstein operator, is expressed as follows: The operator ((−Δ) /2 ) −1 ℎ(⋅, ) is the composition of two operators: the linear integral nonlocal operator ((−Δ) /2 ) −1 the inverse of the fractional Laplacian equipped with the Green function kernel given by (5) and the nonlinear Nemitskii operator → ℎ(⋅, ) 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 1 0 ([−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: where ∈ R, ∈ [−1, 1], : → R, = [−1, 1]×[−1, 1], ℎ : [−1, 1] × R → R , ≥ 1, and ∈ 1 0 . By 1 0 we will denote 1 0 ([−1, 1], R ), the space of absolutely continuous functions defined on [−1, 1] such that (−1) = (1) = 0, with the square-integrable derivative; that is, ∈ 2 , endowed with the norm Under some appropriate assumptions imposed on the functions and ℎ to be specified later, it is feasible to formulate some sufficient conditions for the operator T : 1 0 → 1 0 to be a diffeomorphism; that is, T( 1 0 ) = 1 0 , and that there exists an inverse operator T −1 while both T, T −1 are Fréchet differentiable at every point from 1 0 . In other words, T is Fréchet differentiable at every point ∈ 1 0 and for every ∈ 1 0 there exists a unique solution ∈ 1 0 to the equation T( ) = depending continuously on and such that the operator 1 0 ∋ → ∈ 1 0 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 1 0 . In the proof of Lemma 12 we have used the compactness of the embedding of the space 1 0 into the space of continuous functions . This compact embedding implies that every weakly convergent sequence in 1 0 is uniformly convergent in in the supremum norm. Apparently in the case of 2 space such an implication does not hold. Therefore, one cannot prove, at least with the method applied herein, that the operator T : 2 → 2 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][14][15][16][17][18][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].

Global Diffeomorphism by Use of Mountain Pass Theorem
Let be a real Banach space and let : → R be a 1mapping. A sequence { } ∈N is referred to as a Palais-Smale sequence for functional if for some > 0, any ∈ N, | ( )| ≤ and ( ) → 0 as → ∞. We say that satisfies Palais-Smale condition if any Palais-Smale sequence possesses a convergent subsequence. Moreover, a point * ∈ is called a critical point of if ( * ) = 0. In such a case ( * ) is referred to as a critical value of .
Let us introduce the following sets used in the Mountain Pass Theorem: for any ∈ such that ̸ = 0 and for any > 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 :
→ R be a 1 -mapping satisfying Palais-Smale condition and let (0) = 0. If Applying the above theorem it is possible, as was done in [23], to prove the following theorem on a global diffeomorphism.

Theorem 2. Let be a real Banach space and let be a real Hilbert space. If T : →
is a 1 -mapping such that (a1) for any ∈ the equation T ( )ℎ = possesses a unique solution for any ∈ , (b1) for any ∈ the functional satisfies Palais-Smale condition, then T is a diffeomorphism.
Remark 3. By (a1) and the bounded inverse theorem, for any ∈ , there exists > 0 such that for any ℎ ∈ . Therefore, the above theorem is equivalent in other notations to Theorem 3.1 in [23].

Auxiliary Facts and Used Assumptions
The presentation of the proof of the main result of this paper, which formulates sufficient conditions for T : 1 0 → 1 0 defined by (9) to be a diffeomorphism, we precede with a few lemmas.

Lemma 4.
For any ∈ 1 0 one has Proof. By the Schwarz inequality, for ∈ [−1, 1], one obtains Consequently, and this is precisely the second assertion of the lemma.
In what follows, we will use the following assumptions imposed on the functions and ℎ.
Now we present some sufficient conditions for T : 1 0 → 1 0 to be Fréchet differentiable.

Local Solvability: Analysis of Linearized System
Let 0 ∈ 1 0 be a fixed but an arbitrary function and : 1 0 → 1 0 be a linear operator defined, for any ∈ 1 0 and ∈ [−1, 1], by where the functions and ℎ define, respectively, the kernel and the nonlinearity of operator T defined in (9). Next, for any ∈ N, ∈ [−1, 1], and ∈ 1 0 , consider the following sequence of iterations: Abstract and Applied Analysis 5 We will prove the following lemma.
Proof. First, from (29)-(31) and the assumptions of the lemma, we obtain subsequently To finish the proof we proceed by induction to get estimate (32). Now, let us consider the linear integral equation where 0 ∈ 1 0 and ∈ 1 0 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 be a bounded, continuous operator in a Banach space . Then we can decompose C into the resolvent of the operator defined by and the complementary set-the spectrum of defined as For any bounded and continuous operator , we can define the spectral radius of by the formula which must be finite, for example, due to the following estimate: Proof. Our proof starts with observation that (35) can be written in the form where ( ) ( ) = ∫  Abstract and Applied Analysis apply induction, to the fact that ∈ 1 0 for any ∈ 1 0 and = 1, 2, . . .. By (15) from Lemma 4 and (32) from Lemma 8, we get, with = ‖ ‖ ∞ , the following estimate: and hence by an arbitrary choice of ∈ 1 0 we get Consequently, which means that the spectral radius ( ) is less or equal to 2 . Since, by Theorem 9, ( ) ⊂ ( ) with ( ) is defined by (46). Then, in particular, for all ∈ R such that | | > ( ) we have ∈ ( ). Therefore, we can conclude that, for all ∈ R and | | > 2 , the operator − is bijective on 1 0 . Thus, for any | | > 2 , ‖ 0 ‖ ∞ ≤ , and ∈ 1 0 , there exists a unique solution ∈ 1 0 to ( + ) = ,

Palais-Smale Condition Guaranteeing Global Diffeomorphism
Let us consider, for an arbitrary function ∈ 1 0 , the functional Ψ : 1 0 → R + of the form To prove the main results of the paper we will need some sufficient conditions under which for any ∈ 1 0 the functional Ψ is coercive; that is, for any ∈ 1 0 , Ψ ( ) → ∞ provided that ‖ ‖ 1 0 → ∞. Proof. Since the functional Ψ is coercive for any ∈ 1 0 if and only if the functional Ψ is coercive for = 0, we first observe that the functional Ψ 0 is bounded from below. By the Schwarz inequality and the assumptions of this lemma together with the last estimate from Lemma 4, we obtain From (A5)(b) and the above estimate it follows that Ψ 0 ( ) → ∞ if ‖ ‖ 1 0 → ∞. Consequently, for any ∈ 1 0 we have Ψ ( ) → ∞ as ‖ ‖ 1 0 → ∞.

Main Results and Applications
Applying formerly presented lemmas and Theorem 2 we prove the main result of this paper. satisfies Palais-Smale condition so that assumption (b1) of Theorem 2 is fulfilled. Therefore, T : (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.
possesses a unique solution = ∈ 1 0 and moreover the solution operator is continuously Fréchet differentiable.
Next, we will present the application of our general theorem to the equation involving the fractional Laplacian operator for = 1.
Finally, we will present the application of the main theorem to some specific nonlinear integral Hammerstein operator this time with smooth kernel.
we can guarantee that assumption (A5)(b) is satisfied.

Summary
We have considered the nonlinear integral operator of Hammerstein type T defined on the Sobolev space 1 0 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 : 1 0 → 1 0 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 1 0 into the space and the reflexivity of 1 0 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.