Abstract
An abstract Hammerstein equation is an equation of the form . A new method is introduced to prove the existence of a solution of this equation where and are nonlinear accretive (monotone) operators. The method does not involve the complicated technique of factorizing a linear map via a Hilbert space and does not involve the use of deep variational techniques.
1. General Introduction
Let be a real normed space and let . The space is said to have Gâteaux differentiable norm if the limit exists for all ; in this case is said to be smooth. is said to have uniformly Gâteaux differentiable norm if, for each , the limit is attained uniformly for . Further, is said to be uniformly smooth if the limit is attained uniformly for . The modulus of smoothness of , , is defined by is equivalently said to be smooth if . Let ; is said to be -uniformly smooth (or to have a modulus of smoothness of power type ) if there exists such that .
, , and the Sobolev space , , are all -uniformly smooth. In factFurthermore (see, e.g., [1]), Let denote the generalized duality mapping from to defined bywhere denotes the dual space of and denotes the generalized duality pairing. It is well known (see, e.g., Xu [2]) that where denotes (called the normalized duality mapping). It is well known that if is strictly convex, is single-valued. For more information and examples concerning (generalized) duality mappings, one may see the book of Cioranescu [3] and its review by Reich [4]. In the sequel, we will denote the single-valued duality map by .
A map is called accretive if, for all , there exists such that the following inequality holds:If is a real Hilbert space, the map is called monotone. In this case, satisfies the following condition:The map is called strongly accretive if there exists such that, for all , there exists , such that
A nonlinear integral equation of Hammerstein type (see, e.g., Hammerstein [5]) has the formwhere is a -finite measure on ; the kernel is defined on , is a real-valued function defined on and is, in general, nonlinear, and is a function on . Settingand on , then integral equation (9) can be put in abstract operator form as follows:where, without loss of generality, we have taken .
Interest in (9) stems mainly from the fact that several problems that arise in differential equations, for instance, elliptic boundary value problems whose linear parts possess Green’s function, can, as a rule, be transformed into the form of (9).
Furthermore, equations of Hammerstein type play crucial role in the theory of optimal control systems, in automation, and in network theory (see, e.g., Dolezale [6]).
Several existence theorems for the solution of (9) have been proved by a host of distinguished mathematicians using various techniques (see, e.g., Browder and Gupta [7, 8], Chepanovich [9], and Petryshyn and Fitzpatrick [8]). In the remaining part of this section, we highlight the techniques used by Browder and Gupta [7] and Petryshyn and Fitzpatrick [8]. To do this, we first give definitions of some terms which are required in the theorems.
In the sequel, the symbol “” denotes strong convergence while “” denotes weak convergence.
Definition 1 (see, e.g., [7]). A mapping is said to be hemicontinuous if it is continuous from each line segment of to the weak topology of . That is, , , and such that and for sufficiently large and we have .
Definition 2 (see, e.g., [7]). Let be a bounded monotone linear mapping. is said to be angle-bounded with constant if, for all , in , . (This is well defined since and by the linearity and monotonicity of .)
In [7] Browder and Gupta proved the following theorem.
Theorem 3 (Browder-Gupta [7]). Let be a real Banach space and its conjugate dual space. Let be a monotone angle-bounded continuous linear mapping of into with constant of angle-boundedness . Let be a hemicontinuous (possibly nonlinear) mapping of into such that, for a given constant ,for all and in . Suppose finally that there exists a constant with such that for in Then, there exists exactly one solution in of the nonlinear equation
The main tool used by the authors in proving Theorem 3 is that of splitting the linear operator via a Hilbert space and then applying a deep result of Minty [10]. Precisely, they proved that if is a real Banach space, is its dual space, and is a bounded linear mapping of into which is monotone and angle-bounded, then there exist a Hilbert space , a continuous linear mapping of into with adjoint injective, and a bounded skew-symmetric linear mapping of into such that(see Figure 1).
This factorization enabled the authors to transform the problem into another problem in a Hilbert space such that Hammerstein equation (11) has a solution if and only if the new problem has a solution in a real Hilbert space. They set , , the closed unit ball in , and showed that is hemicontinuous and monotone and satisfies . With these facts, they used the following result of Minty [10] to prove Theorem 3 (see [10] for definitions of terms).
Theorem 4 (Minty [10]). Let be bounded and surround ; let contain and surround every point of densely. Letbe monotone and hemicontinuous at every point of and supposeThen, there exists such that .
Petryshyn and Fitzpatrick employed deep variational techniques to prove the existence of a solution to (11). They proved the following theorems.
Theorem 5 (Petryshyn-Fitzpatrick [8]). Let be a reflexive Banach space and let be a linear, monotone, and symmetric mapping of into . Suppose is a weakly (sequential) lower semicontinuous functional on such thatwhere , , , and . Suppose also that is such that . Then, has a solution in .
Theorem 6 (Petryshyn-Fitzpatrick [8]). Let be a reflexive Banach space with linear, monotone, and symmetric. Let be potential and have a Gâteaux derivative which satisfies the inequalityand is continuous in for u and v fixed, where . Then, (19) has a solution in .
In this paper, we introduce a new method, perhaps simpler than methods used so far in the literature, of proving existence of solutions of Hammerstein equation in certain cases. To achieve this, we recast (11) into a fixed point problem and use a technique recently introduced by Chidume and Zegeye [11], some existence results of Deimling [12] for zeros of accretive maps, and some surjectivity results of Browder [13] for Lipschitz strongly accretive maps. No linearity assumption is imposed on any of our maps.
2. Preliminaries
Let be a normed linear space and let be a convex subset of . For , the inward set, , of relative to , is defined as follows: A mapping is said to be inward if for each and weakly inward if belongs to the closure of for each .
A relationship between the weak inward condition and the conditionfor a map is given in Lemma 11. Further relationship between condition (22), the weak inward condition, and Lemma 11 can be found in [14].
In the sequel, is a -uniformly smooth real Banach space, , and with If is a real Hilbert space, we will denote by .
If and are maps from to such that range of is contained in domain of , that is, , Chidume and Zegeye [11] defined a map as follows:for all and observed that if and only ifso that solves (11). System (25) can be recast as a fixed point problem as follows:We will use the ideas of map on .
In Lemmas 9 and 10, we use the following variant definition of accretive maps as given by Deimling [12].
Definition 7 (accretive map in the sense of Deimling [12]). Let be real Banach space. A map is said to be accretive (in the sense of Deimling) ifwhere
It is evident that, in any real Banach space, an accretive map is also accretive in the sense of Deimling. The converse is true in any real Banach whose dual is strictly convex or whose normalized duality map is single-valued. This is certainly the case when is -uniformly smooth, .
Definition 8 (see, e.g., [15]). A bounded convex subset of a Banach space is said to have normal structure if every convex subset of having more than one element contains at least one nondiametral point; that is, there exists such thatThe Banach space is said to have normal structure if every bounded convex subset of has normal structure.
Lemma 9 (Deimling [12]). Let be a reflexive real Banach space with normal structure and let be a closed convex bounded subset of . Let be a Lipschitz and accretive map satisfying condition (22). Then, .
Lemma 10 (Deimling [12]). Let be real Banach space and let be a closed convex subset of . Let be an accretive continuous map such that for all with for some or as . Suppose satisfies condition (22) and suppose that is closed. Then, .
Lemma 11 (Caristi [16]). Let be a convex subset of a normed linear space and let be a map. Then condition (22) holds if and only if is weakly inward and is the identity map on .
Remark 12. In view of Lemma 11, if in Lemma 10, then condition (22) can be dropped.
Lemma 13 (Xu [2]). Let and a smooth real Banach space. Then the following are equivalent. (i) is -uniformly smooth.(ii)There exists a constant such that, for all , (iii)There exists a constant such that for all and where .
From now on, and denote the constants appearing in Lemma 13.
Lemma 14 (Chidume [15], p. 173). Let be a -uniformly smooth real Banach space. Let be maps with surjective such that the following conditions hold:(i)there exists such that, for each ,(ii)there exists such that, for each ,(iii).Let a map be defined by (24). Then, for each ,
Lemma 15. Let be a real Hilbert space. Let , be two monotone maps such that . Then the map defined by (24) is monotone.
Proof. The proof follows from the lines of argument of the proof of Lemma 14 (see Chidume and Zegeye [11]).
Lemma 16 (Chidume [15], p. 173). Let be a -uniformly smooth real Banach space and let , be two Lipschitz maps such that . Let be a map such that and defined by (24). Then, is Lipschitz.
We need the following definition which was given by Browder [17].
Definition 17 (Browder [17]). Let and be real Banach spaces with the conjugate space of . Let be a mapping of into such that is dense in withfor all , . The mapping is said to be strongly -accretive if there exists such that, for all and in ,
It follows from this definition that if is a real Banach space such that the normalized duality map is single-valued and is dense in (e.g., when is a reflexive and smooth real Banach space), then a strongly accretive map is -strongly accretive.
Theorem 18 (Browder [13]). Let and be Banach spaces with uniformly convex and suppose is a strongly -accretive mapping satisfying a Lipschitz condition on each bounded subset of . Then, .
The following corollary follows from Theorem 18.
Corollary 19. Let be a real Banach space with uniformly convex dual and suppose is a strongly accretive Lipschitz mapping. Then, .
3. Main Results
Let , , and let with for arbitrary . For spaces, , the following estimate has been established (see, e.g., Chidume [15], p. 183):We begin with a proof of the following theorem for spaces, , which is new.
Theorem 20. Let ; let be mappings such that and the following conditions hold:(a)there exists such that, for each , (b)there exists such that, for each , (c) with .Let and define by (24) for all . Then, for arbitrary , the following inequality holds:
Proof. We compute as follows: completing proof of the theorem.
Remark 21. Observe that the condition implies .
We now prove the following existence theorems.
3.1. The Case of Hilbert Spaces
Theorem 22. Let be a real Hilbert space and let , be two Lipschitz monotone maps such that and are closed, convex, and bounded and . Let be a map such that and is defined by (24). Suppose that satisfies condition (22). Then, Hammerstein equation (11) has a solution.
Proof. The fact that and are Lipschitz and monotone implies that is Lipschitz and monotone (Lemmas 15 and 16). Since the normalized duality map is the identity map in real Hilbert spaces, monotonicity of is equivalent to accretivity in the sense of Deimling. Also is closed and convex since and are. Therefore, by Lemma 9, ; that is, there exists such that and . So solves (11). This completes the proof.
Theorem 23. Let be a real Hilbert space and let , be two continuous monotone maps such that and are closed and convex and . Let be a map such that and is defined by (24). Suppose that for all with for some or as and suppose that satisfies condition (22). Suppose that is closed. Then, Hammerstein equation (11) has a solution.
Proof. The fact that and are monotone implies that is monotone (Lemma 15). The fact that and are closed and convex implies that is closed and convex. Also since is a real Hilbert space and the normalized duality map of any real Hilbert space is the identity map, we have for all . Therefore, the assumptions on and together with Lemma 10 give that ; that is, there exists such that and . So solves (11). This completes the proof.
Corollary 24. Let be a real Hilbert space and let be two continuous monotone maps defined on . Let be a map defined by (24). Suppose that for all with for some or as . Suppose that is closed. Then, Hammerstein equation (11) has a solution.
Proof. Since is defined on , it satisfies condition (22). Therefore, the result follows from Theorem 23.
3.2. The Case of Spaces,
Theorem 25. Let and be two Lipschitz mappings satisfying the following conditions:(a)there exists such that, for each , (b)there exists such that, for each , Let and be closed, convex, and bounded such that . Let and let be a map such that and is defined by (24). Suppose that satisfies condition (22). Let . If or , then Hammerstein equation (11) has a solution.
Proof. The fact that and are Lipschitz implies that is Lipschitz by Lemma 16. Also is closed and convex since and are.
Case 1 (). In this case is -uniformly smooth space and (see, e.g., [2]). Therefore, andfor . This implies by Lemma 14 that is accretive. Therefore, is accretive in the sense of Deimling. Hence, using Lemma 9, we have that ; that is, there exists such that and . So solves (11).
Case 2 (). The condition implies that . Hence, by Theorem 20, is accretive. We conclude as in Case 1. This completes the proof.
Theorem 26. Let , be two continuous mappings satisfying the following conditions:(a)there exists such that, for each , (b)there exists such that, for each , Let and be closed and convex, such that . Let and let be a mapping such that and is defined by (24) for . Suppose that for all with for some or as and suppose satisfies condition (22). Let . If or , then Hammerstein equation (11) has a solution.
Proof. Evidently, continuity of and gives the continuity of . Also is closed and convex since and are. The rest follows as in the proof of Theorem 25. This completes the proof.
Corollary 27. Let , be two continuous accretive mappings satisfying the following conditions:(a)there exists such that, for each , (b)there exists such that, for each , Let . Let and let be a mapping defined by (24) . Suppose that for all with for some or as . Let . If or , then Hammerstein equation (11) has a solution.
Proof. Since is defined on , it satisfies condition (22) of Theorem 26. Also is closed and convex. Therefore, the result follows from Theorem 26.
3.3. The Case of Hilbert Spaces with Lipschitz Strongly Monotone Mappings
Theorem 28. Let be a real Hilbert space and let , be two Lipschitz strongly monotone mappings with constants , respectively. Let be a mapping defined by (24) for . Then, Hammerstein equation (11) has a solution.
Proof. Using Lemma 16 we have that is Lipschitz. Also since every real Hilbert space is -uniformly smooth with , , we have that . Also . Therefore, is strongly monotone by Lemma 14. Since is a real Hilbert space and every real Hilbert space is uniformly convex, we invoke Corollary 19 to obtain that . So there exists such that ; that is, . Hence solves (11). This completes the proof.
3.4. The Case of Spaces, , with Lipschitz Strongly Accretive Mappings
Theorem 29. Let , be two Lipschitz mappings satisfying the following conditions:(a)there exists such that, for each , (b)there exists such that, for each , Let and let be a mapping defined by (24). Let . If or , then, Hammerstein equation (11) has a solution.
Proof. Using Lemma 16 we have that is Lipschitz.
Case 1 (). In this case is -uniformly smooth space and (see, e.g., [2]). Therefore, and for . This implies by Lemma 14 that is strongly accretive. Since every space, , is uniformly convex, by Corollary 19, . Therefore there exists such that ; that is, and . So solves (11).
Case 2 (). The inequality implies that . Hence by Theorem 20 is strongly accretive. The result now follows as in Case 1 since every space, , is uniformly convex. This completes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.