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

()

provided with the following Dirichlet exterior boundary condition:

()

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

()

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 c_σ is defined as

()

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

, which appears to be true for σ ∈ (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 λℐ and the Hammerstein operator, is expressed as follows:

()

The operator

is the composition of two operators: the linear integral nonlocal operator

—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

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 λ ∈ ℝ, t ∈ [−1,1], G : P → ℝ, P = [−1,1] × [−1,1], h : [−1,1] × ℝⁿ → ℝⁿ, n ≥ 1, and

. By

we will denote

, 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

()

where L² = L²([−1,1], ℝⁿ) 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 to be a diffeomorphism; that is, , and that there exists an inverse operator 𝒯⁻¹ while both 𝒯, 𝒯⁻¹ are Fréchet differentiable at every point from . In other words, 𝒯 is Fréchet differentiable at every point and for every there exists a unique solution to the equation 𝒯(x) = z depending continuously on z and such that the operator 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 . In the proof of Lemma 12 we have used the compactness of the embedding of the space into the space of continuous functions C. This compact embedding implies that every weakly convergent sequence in 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 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:

()

for any e ∈ X such that e ≠ 0 and

()

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 ,
(ii)
there is a point such that ψ(e) ≤ 0,

then

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
()

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

()

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 defined by (9) to be a diffeomorphism, we precede with a few lemmas.

Lemma 4. For any one has

()

Proof. By the Schwarz inequality, for t ∈ [−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 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 , that is, for every ρ > 0 there exists l_ρ > 0 such that for any t ∈ [−1,1] and |x(t)| ≤ ρ:
  ()
- (b)
  the function G_t(t, ·)h(·, x(·)) is integrable and for every ρ > 0 there exist l_ρ > 0 such that
  ()
  for 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) = 0 for a.e. τ ∈ [−1,1];
(A4) for any and t ∈ [−1,1];
(A5) One has the following:
- (a)
  |h(τ, x)| ≤ a(τ)|x| + b(τ) where τ ∈ [−1,1], x ∈ ℝⁿ, a, b ∈ L²([−1,1], ℝ⁺),
- (b)
  , <∞.

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 with values in .

Proof. Let us choose any . By (A3), 𝒯x₀(−1) = 𝒯x₀(1) = 0. It suffices to show that the function

()

is absolutely continuous and y^′ ∈ L². Observe, by (A2)(b) and

that one has

()

where −1 ≤ t₁ < t₂ < ⋯<t_i < t_i+1 < ⋯<t_N < t_N+1 ≤ 1. As a result, for any

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

()

so that y^′ ∈ L².

Now we present some sufficient conditions for 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 while for and t ∈ [−1,1]

()

Proof. It is sufficient to show that the operator

()

is Fréchet differentiable. The Mean Value Theorem (cf. [24]) yields, for t ∈ [−1,1] and some θ ∈ [0,1],

()

From (15) in Lemma 4 one has

()

Since the strong convergence in

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

, to

()

and thus,

()

where

, which completes the proof.

4. Local Solvability: Analysis of Linearized System

Let

be a fixed but an arbitrary function and

be a linear operator defined, for any

and t ∈ [−1,1], by

()

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

, consider the following sequence of iterations:

()

We will prove the following lemma.

Lemma 8. Under assumptions (A1)(c), (A2)(c), and (A3) one has the following estimates:

()

where l_ρ > 0 is defined by (A2)(c),

()

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

()

To finish the proof we proceed by induction to get estimate (32).

Now, let us consider the linear integral equation

()

where

and

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

()

and the complementary set—the spectrum of T defined as

()

For any bounded and continuous operator T, we can define the spectral radius of T by the formula

()

which must be finite, for example, due to the following estimate:

()

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, .

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

Lemma 10. For any , ρ > 0 such that and any , (35) possesses a unique solution in provided that the functions G and h satisfy (A1), (A2), (A3), and weaker, local version of (A4), with constant l_ρ such that

()

Proof. Our proof starts with observation that (35) can be written in the form

()

where

()

From (29), (30), and (31) it follows that for any k = 1,2, …, t ∈ [−1,1] and

we have

()

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

for any

and k = 1,2, …. By (15) from Lemma 4 and (32) from Lemma 8, we get, with M = ∥x∥_∞, the following estimate:

()

and hence by an arbitrary choice of

we get

()

Consequently,

()

which means that the spectral radius r(T) is less or equal to 2l_ρ. Since, by Theorem 9,

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

. Thus, for any |λ| > 2l_ρ,

, and

, there exists a unique solution

()

which ends the proof. Indeed, by definition of T, there exists a unique solution to

()

5. Palais-Smale Condition Guaranteeing Global Diffeomorphism

Let us consider, for an arbitrary function

, the functional

of the form

()

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

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

Proof. Since the functional Ψ_y is coercive for any 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

()

From (A5)(b) and the above estimate it follows that Ψ₀(x) → ∞ if

. Consequently, for any

we have Ψ_y(x) → ∞ as

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

Proof. Fix . Recall that the functional Ψ_y has the form

()

Straightforward calculation leads to

()

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

is given, for

, by

()

Let 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 . Moreover, the weak convergence of the sequence {x_k} in the space 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, in L² and as being a weakly convergent sequence it has to be bounded. It remains to prove that the sequence {x_k} converges to x₀ in . By (53), a direct calculation leads to

()

where

()

Since

and x_k⇀x₀ weakly in

. We will prove that, for i = 1,2, …, 6, lim _k→∞Gⁱ(x_k) = 0. By the Schwarz inequality, (A1)(a), and (A2)(b) we get

()

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 if ∥x_k − x₀∥_∞ ≤ ε. Similar estimates can be applied to other terms; thus, one can prove that Gⁱ(x_k) → 0 as k → ∞ for i = 3,4, 5,6. Hence, from (54), it follows that x_k → x₀ in .

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 defined by (9) is a diffeomorphism of on .

Proof. Set . From Lemma 10 we infer that the operator 𝒯 satisfies assumption (a1) of Theorem 2, while Lemma 12 ascertains that for any the functional satisfies Palais-Smale condition so that assumption (b1) of Theorem 2 is fulfilled. Therefore, 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 the nonlinear integral equation

()

possesses a unique solution

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 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 and σ ∈ (1,2] there exists a unique solution of

()

By [2, Corollary 3.2] we have for the Green function of (−Δ) ^σ/2 the following estimates:

()

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:

()

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

()

the following estimate holds:

()

Similarly, since 1 + z² ≤ 2z we have the estimate for

()

reading

()

Let us define and b(τ) ≡ 0. Then a, b ∈ L²([−1,1], ℝ⁺) and condition (A5)(a) is fulfilled. Assuming

()

we can guarantee that assumption (A5)(b) is satisfied.

Consequently, if we assume that

()

then condition (A4) holds. Thus, the functions G and h satisfy assumptions (A1)–(A5) and Theorem 14 implies that the equation, for any

()

possesses a unique solution

and

is continuously Fréchet differentiable.

7. Summary

We have considered the nonlinear integral operator of Hammerstein type 𝒯 defined on the Sobolev space 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 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 into the space C and the reflexivity of 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.

References

1 Bogdan K. and Byczkowski T., Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Mathematica. (1999) 133, no. 1, 53–92, MR1671973.
10.4064/sm-133-1-53-92
Web of Science® Google Scholar
2 Bogdan K. and Byczkowski T., Potential theory of Schrödinger operator based on fractional Laplacian, Probability and Mathematical Statistics. (2000) 20, no. 2, 293–335, MR1825645, ZBL0996.31003.
Google Scholar
3 Bogdan K., Byczkowski T., Kulczycki T., Ryznar M., Song R., and Vondraček Z., Potential Analysis of Stable Processes and Its Extensions, 2009, 1980, Springer, Berlin, Germany, Lecture Notes in Mathematics, https://doi.org/10.1007/978-3-642-02141-1, MR2569321.
10.1007/978-3-642-02141-1
Google Scholar
4 Caffarelli L. A., Further regularity for the Signorini problem, Communications in Partial Differential Equations. (1979) 4, no. 9, 1067–1075, https://doi.org/10.1080/03605307908820119, MR542512, ZBL0427.35019.
10.1080/03605307908820119
Google Scholar
5 Vázquez J. J., Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, 2012, 7, 271–298, Abel Symposia.
10.1007/978-3-642-25361-4_15
Google Scholar
6 Baleanu D., Diethelm K., Scalas E., and Trujillo J. J., Fractional Calculus: Models and Numerical Methods, 2012, 3, World Scientific, Singapore, Series on Complexity, Nonlinearity and Chaos, https://doi.org/10.1142/9789814355216, MR2894576.
10.1142/8180
Google Scholar
7 Ma X.-J., Srivastava H. M., Baleanu D., and Yang X.-J., A new Neumann series method for solving a family of local fractional Fredholm and Volterra integral equations, Mathematical Problems in Engineering. (2013) 2013, 6, 325121, https://doi.org/10.1155/2013/325121.
10.1155/2013/325121
Web of Science® Google Scholar
8 Kilbas A. A., Srivastava H. M., and Trujillo J. J., Theory and Applications of Fractional Differential Equations, 2006, 204, Elsevier Science B.V., Amsterdam, The Netherlands, North-Holland Mathematics Studies, MR2218073.
10.1016/S0304-0208(06)80001-0
Google Scholar
9 Ros-Oton X. and Serra J., The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, Journal de Mathématiques Pures et Appliquées. In press.
Google Scholar
10 Ros-Oton X. and Serra J., Fractional Laplacian: Pohozaev identity and nonexistence results, Comptes Rendus Mathématique. (2012) 350, no. 9-10, 505–508, https://doi.org/10.1016/j.crma.2012.05.011, MR2929057.
10.1016/j.crma.2012.05.011
Web of Science® Google Scholar
11 Corduneanu C., Integral Equations and Applications, 1991, Cambridge University Press, Cambridge, UK, https://doi.org/10.1017/CBO9780511569395, MR1109491.
10.1017/CBO9780511569395
Google Scholar
12 Gripenberg G., Londen S.-O., and Staffans O., Volterra Integral and Functional Equations, 1990, 34, Cambridge University Press, Cambridge, UK, Encyclopedia of Mathematics and Its Applications, https://doi.org/10.1017/CBO9780511662805, MR1050319.
10.1017/CBO9780511662805
Google Scholar
13 Brezis H. and Browder F. E., Existence theorems for nonlinear integral equations of Hammerstein type, Bulletin of the American Mathematical Society. (1975) 81, 73–78, MR0407669, https://doi.org/10.1090/S0002-9904-1975-13641-X, ZBL0298.47031.
10.1090/S0002-9904-1975-13641-X
Web of Science® Google Scholar
14 Brézis H. and Browder F. E., Nonlinear integral equations and systems of Hammerstein type, Advances in Mathematics. (1975) 18, no. 2, 115–147, MR0394341, https://doi.org/10.1016/0001-8708(75)90155-3, ZBL0318.45011.
10.1016/0001-8708(75)90155-3
Web of Science® Google Scholar
15 Chidume C. E. and Zegeye H., Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space, Proceedings of the American Mathematical Society. (2005) 133, no. 3, 851–858, https://doi.org/10.1090/S0002-9939-04-07568-9, MR2113936, ZBL1054.47046.
10.1090/S0002-9939-04-07568-9
Web of Science® Google Scholar
16 Chidume C. E. and Shehu Y., Strong convergence theorem for approximation of solutions of equations of Hammerstein type, Nonlinear Analysis: Theory, Methods & Applications. (2012) 75, no. 14, 5664–5671, https://doi.org/10.1016/j.na.2012.05.014, MR2942944, ZBL06082083.
10.1016/j.na.2012.05.014
Web of Science® Google Scholar
17 El-Ameen M. A. and El-Kady M., A new direct method for solving nonlinear Volterra-Fredholm-Hammerstein integral equations via optimal control problem, Journal of Applied Mathematics. (2012) 2012, 10, 714973, https://doi.org/10.1155/2012/714973, MR2915712, ZBL1244.65239.
10.1155/2012/714973
Google Scholar
18 de Figueiredo D. G. and Gupta C. P., On the variational method for the existence of solutions of nonlinear equations of Hammerstein type, Proceedings of the American Mathematical Society. (1973) 40, 470–476, MR0318988.
10.1090/S0002-9939-1973-0318988-X
Web of Science® Google Scholar
19 Stańczy R., Hammerstein equations with an integral over a noncompact domain, Annales Polonici Mathematici. (1998) 69, no. 1, 49–60, MR1630228, ZBL0919.45004.
10.4064/ap-69-1-49-60
Google Scholar
20 Pascali D. and Sburlan S., Nonlinear Mappings of Monotone Type, 1978, Editura Academiae, Bucharest, Romania, MR531036.
10.1007/978-94-009-9544-4_3
Web of Science® Google Scholar
21 Ambrosetti A. and Rabinowitz P. H., Dual variational methods in critical point theory and applications, Journal of Functional Analysis. (1973) 14, 349–381, MR0370183, ZBL0273.49063.
10.1016/0022-1236(73)90051-7
Google Scholar
22 Rabinowitz P. H., Minimax Methods in Critical Point Theory with Applications to Differential Equations, 1986, 65, American Mathematical Society, Providence, RI, USA, CBMS Regional Conference Series in Mathematics, MR845785.
10.1090/cbms/065
Google Scholar
23 Idczak D., Skowron A., and Walczak S., On the diffeomorphisms between Banach and Hilbert spaces, Advanced Nonlinear Studies. (2012) 12, no. 1, 89–100, MR2895945, ZBL1244.57056.
10.1515/ans-2012-0105
Web of Science® Google Scholar
24 Ioffe A. D. and Tihomirov V. M., Theory of Extremal Problems, 1979, 6, North-Holland, Amsterdam, The Netherlands, Studies in Mathematics and its Applications, MR528295.
Google Scholar
25 Reed M. and Simon B., Methods of Modern Mathematical Physics. I, 1980, 2nd edition, Academic Press, New York, NY, USA, MR751959.
Google Scholar
26 Yosida K., Functional Analysis, 1965, Springer, Berlin, Germany.
Google Scholar
27 Sánchez-Pena R. S. and Sznaier M., Robust Systems Theory and Applications, 1998, Wiley-Interscience, New York, NY, USA.
Google Scholar

Citing Literature

All articles

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

Abstract

1. Introduction

2. Global Diffeomorphism by Use of Mountain Pass Theorem

3. Auxiliary Facts and Used Assumptions

4. Local Solvability: Analysis of Linearized System

5. Palais-Smale Condition Guaranteeing Global Diffeomorphism

6. Main Results and Applications

7. Summary

Conflict of Interests

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

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

Abstract

1. Introduction

2. Global Diffeomorphism by Use of Mountain Pass Theorem

3. Auxiliary Facts and Used Assumptions

4. Local Solvability: Analysis of Linearized System

5. Palais-Smale Condition Guaranteeing Global Diffeomorphism

6. Main Results and Applications

7. Summary

Conflict of Interests

References

Citing Literature

References

Related

Information