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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 240863, 10 pages
Global Solvability of Hammerstein Equations with Applications to BVP Involving Fractional Laplacian
Faculty of Mathematics and Computer Science, University of Lodz, Banacha 22, 90-238 Lodz, Poland
Received 19 July 2013; Accepted 6 November 2013
Academic Editor: Juan J. Trujillo
Copyright © 2013 Dorota Bors. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
Consider, for any , an arbitrary real number , a given function , and a nonlinear term , 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.
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 , as and the constant is defined as It should be underlined that only in the case the derivative of the Green function is nonsingular, but as soon as 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 , which appears to be true for .
Consider, to address the solvability of (3), the general equation of the form where in the leading example (3) the operator , being a sum of the rescaled identity operator 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 given by (5) and the nonlinear Nemitskii operator defined by the nonlinear function . We will show that (7) is globally solvable. In fact it can be proved that under suitable assumptions the operator 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 , , , , , , and . By we will denote , the space of absolutely continuous functions defined on such that , with the square-integrable derivative; that is, , endowed with the norm where is the space of square-integrable functions.
Under some appropriate assumptions imposed on the functions and 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 while both , are Fréchet differentiable at every point from . In other words, is Fréchet differentiable at every point and for every there exists a unique solution to the equation depending continuously on 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 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 . This compact embedding implies that every weakly convergent sequence in is uniformly convergent in in the supremum norm. Apparently in the case of space such an implication does not hold. Therefore, one cannot prove, at least with the method applied herein, that the operator 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  and by Gripenberg et al. . 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,  or for the BVP’s on real line of Hammerstein and Wiener-Hopf type, see, for example, , or for optimal problems for Hammerstein and Volterra equations, see, for example, .
2. Global Diffeomorphism by Use of Mountain Pass Theorem
Let be a real Banach space and let be a -mapping. A sequence is referred to as a Palais-Smale sequence for functional if for some , any , and 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 . 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 and for any .
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 be a -mapping satisfying Palais-Smale condition and let . If (i)there are some constants , such that ,(ii)there is a point such that ,then is the critical value of and .
Applying the above theorem it is possible, as was done in , 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 is a -mapping such that for any the equation possesses a unique solution for any , for any the functional satisfies Palais-Smale condition, then is a diffeomorphism.
Remark 3. By and the bounded inverse theorem, for any , there exists such that for any . Therefore, the above theorem is equivalent in other notations to Theorem 3.1 in .
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 , 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 . (A1) One has the following: (a)the functions and are continuous for a.e. , (b)there exists continuous derivative on for a.e. , (c)there exists derivative and it is continuous for a.e. ; (A2) One has the following: (a)the function is integrable and this integral is locally bounded with respect to , that is, for every there exists such that for any and : (b)the function is integrable and for every there exist such that for such that for , (c)the function satisfies (A2)(a) with instead of whereas the function satisfies (A2)(b) with instead of ; (A3) satisfies the Dirichlet boundary conditions for a.e. ; (A4) for any and ; (A5) One has the following: (a) where , , , (b), .
Remark 5. Besides regularity (A1), (A2), and technical (A3) assumptions, we must finally impose on the functions and some growth and quantitative global assumptions: (A4) and (A5).
Lemma 6. If the functions and satisfy (A1)(a), (A1)(b), (A2)(a), (A2)(b), and (A3), then the operator is well defined by (9) on the space with values in .
Proof. Let us choose any . By (A3), . It suffices to show that the function is absolutely continuous and . Observe, by (A2)(b) and that one has where . As a result, for any the function is absolutely continuous and therefore for almost any there exists and its square integral can be estimated by (A2)(b) as so that .
Now we present some sufficient conditions for to be Fréchet differentiable.
Lemma 7. Suppose that functions and satisfy (A1)(a), (A1)(c), (A2)(a), (A2)(c), and (A3). Then the operator defined by (9) is Fréchet differentiable at any while for and
Proof. It is sufficient to show that the operator
is Fréchet differentiable. The Mean Value Theorem (cf. ) yields, for and some ,
From (15) in Lemma 4 one has Since the strong convergence in implies the uniform convergence in 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 as , 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 , by where the functions and define, respectively, the kernel and the nonlinearity of operator defined in (9).
Next, for any , , and , consider the following sequence of iterations: 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 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 be a bounded, continuous operator in a Banach space . Then we can decompose 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: Moreover, we have, following, for example, [25, Theorem VI.6] and [26, Theorem VIII.2.3], the theorem.
Theorem 9. For any , one has , which means complementarily that the spectrum of is contained in the closed ball of radius ; that is, .
Now, we are ready to formulate the lemma on solvability of the linear integral equation (35).
Lemma 10. For any , such that and any , (35) possesses a unique solution in provided that the functions and satisfy (A1), (A2), (A3), and weaker, local version of (A4), with constant 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 , 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 . By (15) from Lemma 4 and (32) from Lemma 8, we get, with , the following estimate: and hence by an arbitrary choice of we get Consequently, which means that the spectral radius is less or equal to . Since, by Theorem 9, with is defined by (46). Then, in particular, for all such that we have . Therefore, we can conclude that, for all and , the operator is bijective on . Thus, for any , , and , there exists a unique solution to which ends the proof. Indeed, by definition of , 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 is coercive; that is, for any , provided that .
Lemma 11. If the functions and satisfy (A1)(a), (A1)(b), (A2)(a), (A2)(b), (A3), and (A5), then for any the functional is coercive.
Proof. Since the functional is coercive for any if and only if the functional is coercive for , we first observe that the functional 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 if . Consequently, for any we have as .
Lemma 12. For any the functional satisfies Palais-Smale condition provided that assumptions (A1), (A2), (A3), (A4), and (A5) are satisfied.
Proof. Fix . Recall that the functional has the form
Straightforward calculation leads to The functional defined by (49), being a superposition of two -mappings, is also of the same regularity type and its differential at is given, for , by
Let be a Palais-Smale sequence for some fixed but an arbitrary ; that is, and . Applying Lemma 11 we obtain that is coercive, and hence the sequence is weakly compact as a bounded sequence in a reflexive space. Passing, if necessary, to a subsequence, one can assume that weakly in . Moreover, the weak convergence of the sequence in the space implies the uniform convergence in ; that is, uniformly with respect to as well as the weak convergence of its derivatives in ; that is, in and as being a weakly convergent sequence it has to be bounded. It remains to prove that the sequence converges to in . By (53), a direct calculation leads to where Since and weakly in , . We will prove that, for , . 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, as . Next, can be estimated by if . Similar estimates can be applied to other terms; thus, one can prove that as for . Hence, from (54), it follows that 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 and 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 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 and 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 .
Example 15. Assume that the nonlinear term satisfy the Green function estimates (A1)–(A5). This is the case if, for example, the function is smooth, that is, , and it satisfies the linear growth conditions (A4)-(A5). Then for any and there exists a unique solution of By [2, Corollary 3.2] we have for the Green function of the following estimates: where , . It should be noted (cf. ) that the Green function for is bounded and continuous. For estimates on and regularity see [2, 9, 10]. One can recall or show directly that continuous behaves like and behaves like at the boundary, that is, at −1 or 1, while is like and is like at . Therefore, the integrability assumptions are satisfied for mild singularity; that is, only if but not in the range of stronger singularity when .
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 satisfying on and with such that for .
Since , for the function the following estimate holds: Similarly, since we have the estimate for reading
Let us define and . Then 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 and satisfy assumptions (A1)–(A5) and Theorem 14 implies that the equation, for any , possesses a unique solution and is continuously Fréchet differentiable.
We have considered the nonlinear integral operator of Hammerstein type 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 . 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 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.
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