Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

and Applied Analysis 3 and Kartsatos and Skrypnik [4] and the references therein. For recent degree theory and applications for solvability of operator inclusions involving bounded pseudomonotone perturbations of maximal monotone operators under general coercivity and Leray-Schauder type boundary conditions, we cite the paper due to Asfaw and Kartsatos [18]. Existence results concerning noncoercive operators of the type T + S, where T : X ⊇ D(T) → 2X ∗ is maximal monotone and S : X → 2 X ∗ is bounded pseudomonotone, can be found in the paper due to Asfaw [19]. For applications of the theory of perturbed monotone type operators to variational and hemivariational inequality problems, the reader is referred to the papers due to Carl and Le [20], Carl et al. [21], Carl [22], and Carl and Motreanu [23] and the references therein. For a separable reflexive Banach space X and a nonempty, closed, and convex subset K of X, Asfaw and Kartsatos [24] gave existence results for locally defined operators of the type T + S, where T : X ⊇ D(T) → 2X ∗ is maximal monotone and S : K → X∗ is demicontinuous and generalized pseudomonotone under coercivity condition on S. The main contribution of the paper is to obtain surjectivity results for noncoercive and not everywhere defined operators of the type (i) T + S, where S : X ⊇ D(S) → X∗ is quasibounded demicontinuous generalized pseudomonotone such that (a) there exists a real reflexive separable Banach space W ⊆ D(S), dense and continuously embedded in X; (b) there exists d ≥ 0 such that ⟨V∗ + Sx, x⟩ ≥ −d‖x‖ 2 for all x ∈ D(T) ∩ D(S) and V∗ ∈ Tx; (c) there exist α > d and μ ≥ 0 such that ‖V∗+Sx‖ ≥ α‖x‖ − μ for all x ∈ D(T) ∩ D(S) and V∗ ∈ Tx, (ii) L + S, where S : X ⊇ D(S) → X∗ is quasibounded demicontinuous of type (M) with D(L) ⊆ D(S) such that (b) and (c) of (i) are satisfied. In Section 2, we proved surjectivity results for T + S and L + S satisfying conditions (i) and (ii), respectively. In Theorem 6, we provide a surjectivity result for operators of the type T + S, where T and S satisfy condition (i).Theorem 6 is new and improves the existing surjectivity results for an operator S, which is single-valued, everywhere defined, bounded, and coercive pseudomonotone. In particular, for a single-valued pseudomonotone operator S, Theorem 6 improves the surjectivity results due to Browder and Hess [1], Kenmochi [12–14], Le [16], Guan and Kartsatos [17], Asfaw and Kartsatos [18], and Asfaw [19, 25] because the results in these references require S to be everywhere defined, bounded, and coercive whileTheorem 6 used S to be densely defined, quasibounded, and noncoercive.Moreover, Browder (cf. Zeidler [9,Theorem 32. A, pages 866–872]) gave themain theorem for perturbations of maximal monotone operator by a single-valued, bounded, demicontinuous, and coercive operator S with D(S) = C, a nonempty, closed, and convex subset of X. In view of this, Theorem 6 gives an analogous result, where D(S) is dense in X, possibly, neither closed nor convex, and S is weakly coercive. It is also known, due to Browder and Hess [1], that every pseudomonotone operator S from X into X∗ with D(S) = X is generalized pseudomonotone. It is also true that S is demicontinuous provided that it is bounded, single-valued, and everywhere defined. Consequently, the arguments used in the proof of Theorem 6 give analogous conclusion if S : X = D(S) → X ∗ is bounded pseudomonotone and T and S satisfy the given hypotheses. As a consequence of Corollary 7, a partial positive answer for Nirenberg’s problem on surjectivity of densely defined demicontinuous generalized pseudomonotone expansive mapping is provided. In addition, Theorem 8 provides surjectivity result for operators of the type T + S, where T and S satisfy condition (ii). As a result ofTheorem 8, a new characterization of linear maximal monotone operator is proved when the space X is separable. It is well known due to Brézis (cf. Zeidler [9,Theorem 32. L, pages 897–899]) that a linearmonotone operator L is maximalmonotone if and only if L is closed and densely defined and the adjoint operator L ∗ is monotone. An interesting result in the present paper is that a linear monotone operator L is maximal monotone if and only if L is closed and densely defined, provided that X is separable. This result weakens the monotonicity condition on L∗ used by Brézis (cf. Zeidler [9, Theorem 32. L, pages 897–899]). To the best of the author’s knowledge, Theorem 8 is a new result and Corollary 9 improves the well-known result of Brézis. In Section 3, we demonstrate the applicability of the results by proving existence of weak solution in


Introduction-Preliminaries
In what follows, is a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space * . The norm of the space , and any other normed spaces herein, will be denoted by ‖ ⋅ ‖. For ∈ and * ∈ * , the pairing ⟨ * , ⟩ denotes the value * ( ). Let and be real Banach spaces. For a multivalued mapping : → 2 , we define the domain ( ) of by ( ) = { ∈ : ̸ = 0} and the range ( ) of by ( ) = ∪ ∈ ( ) . We also denote the graph of by ( ) = {( , ) : ∈ ( )}. A mapping : ⊃ ( ) → is "demicontinuous" if it is continuous from the strong topology of ( ) to the weak topology of . A multivalued mapping : ⊃ ( ) → 2 is "bounded" if it maps bounded subsets of ( ) to bounded subsets of . It is "compact" if it is strongly continuous and maps bounded subsets of ( ) to relatively compact subset of . It is "finitely continuous" if it is upper semicontinuous from each finite dimensional subspace of to the weak topology of . It is "quasibounded" if for every > 0 there exists ( ) > 0 such that [ , * ] ∈ ( ) with ‖ ‖ ≤ and ⟨ * , ⟩ ≤ ‖ ‖ imply ‖ * ‖ ≤ ( ). It is "strongly quasibounded" if for every > 0 there exists ( ) > 0 such that [ , * ] ∈ ( ) with ‖ ‖ ≤ and ⟨ * , ⟩ ≤ imply ‖ * ‖ ≤ ( ). In what follows, a mapping will be called "continuous" if it is strongly continuous. uniformly convex, is single-valued, bounded monotone of type ( + ) and bicontinuous. If ( ) = for ≥ 0, then is denoted by and is called the normalized duality mapping.
The following definitions are used throughout the paper. In arbitrary Banach space , Browder and Hess [1] introduced the definitions of pseudomonotone and generalized pseudomonotone operators. The original definition for single-valued pseudomonotone, generalized pseudomonotone, and operators of type ( ) with domain all of , is due to Brézis [2].
We notice here that the definition of single-valued expansive mapping is due to Nirenberg [3]. In order to enlarge the class of single-valued operators, the multivalued version is introduced in (iii) of Definition 1. It is not hard to notice that every uniformly monotone operator is expansive. Furthermore, in a Hilbert space = , if : ⊇ ( ) → 2 is monotone, we see that, for each > 0, + is multivalued expansive with domain ( ).
The following definition gives a larger class of operators of monotone type, which can be found in Kartsatos and Skrypnik [4].
By Definition 2, it is not difficult to see that 0 ∈ ( ) and is quasibounded implying that is quasibounded with respect to . Furthermore, it follows that the class of generalized ( + ) operators with respect to includes the class of operators of type ( + ).
For basic definitions and further properties of mappings of monotone type, the reader is referred to Barbu [5], Brèzis et al. [6], Brèzis [2], Browder and Hess [1], Pascali and Sburlan [7], Browder [8], and Zeidler [9]. For results concerning perturbations of maximal monotone operators by bounded and everywhere defined pseudomonotone type operators, the reader is referred to Browder and Hess [1], Brèzis [2], Browder [10], Brèzis and Nirenberg [11], Kenmochi [12][13][14], Guan et al. [15], Le [16], Guan and Kartsatos [15,17], Abstract and Applied Analysis 3 and Kartsatos and Skrypnik [4] and the references therein. For recent degree theory and applications for solvability of operator inclusions involving bounded pseudomonotone perturbations of maximal monotone operators under general coercivity and Leray-Schauder type boundary conditions, we cite the paper due to Asfaw and Kartsatos [18]. Existence results concerning noncoercive operators of the type + , where : ⊇ ( ) → 2 * is maximal monotone and : → 2 * is bounded pseudomonotone, can be found in the paper due to Asfaw [19]. For applications of the theory of perturbed monotone type operators to variational and hemivariational inequality problems, the reader is referred to the papers due to Carl and Le [20], Carl et al. [21], Carl [22], and Carl and Motreanu [23] and the references therein. For a separable reflexive Banach space and a nonempty, closed, and convex subset of , Asfaw and Kartsatos [24] gave existence results for locally defined operators of the type + , where : ⊇ ( ) → 2 * is maximal monotone and : → * is demicontinuous and generalized pseudomonotone under coercivity condition on .
In Section 2, we proved surjectivity results for + and + satisfying conditions (i) and (ii), respectively. In Theorem 6, we provide a surjectivity result for operators of the type + , where and satisfy condition (i). Theorem 6 is new and improves the existing surjectivity results for an operator , which is single-valued, everywhere defined, bounded, and coercive pseudomonotone. In particular, for a single-valued pseudomonotone operator , Theorem 6 improves the surjectivity results due to Browder and Hess [1], Kenmochi [12][13][14], Le [16], Guan and Kartsatos [17], Asfaw and Kartsatos [18], and Asfaw [19,25] because the results in these references require to be everywhere defined, bounded, and coercive while Theorem 6 used to be densely defined, quasibounded, and noncoercive. Moreover, Browder (cf. Zeidler [9,Theorem 32. A, pages 866-872]) gave the main theorem for perturbations of maximal monotone operator by a single-valued, bounded, demicontinuous, and coercive operator with ( ) = , a nonempty, closed, and convex subset of . In view of this, Theorem 6 gives an analogous result, where ( ) is dense in , possibly, neither closed nor convex, and is weakly coercive. It is also known, due to Browder and Hess [1], that every pseudomonotone operator from into * with ( ) = is generalized pseudomonotone. It is also true that is demicontinuous provided that it is bounded, single-valued, and everywhere defined. Consequently, the arguments used in the proof of Theorem 6 give analogous conclusion if : = ( ) → * is bounded pseudomonotone and and satisfy the given hypotheses. As a consequence of Corollary 7, a partial positive answer for Nirenberg's problem on surjectivity of densely defined demicontinuous generalized pseudomonotone expansive mapping is provided. In addition, Theorem 8 provides surjectivity result for operators of the type + , where and satisfy condition (ii). As a result of Theorem 8, a new characterization of linear maximal monotone operator is proved when the space is separable. It is well known due to Brézis (cf. Zeidler [9, Theorem 32. L, pages 897-899]) that a linear monotone operator is maximal monotone if and only if is closed and densely defined and the adjoint operator * is monotone. An interesting result in the present paper is that a linear monotone operator is maximal monotone if and only if is closed and densely defined, provided that is separable. This result weakens the monotonicity condition on * used by Brézis (cf. Zeidler [9, Theorem 32. L, pages 897-899]). To the best of the author's knowledge, Theorem 8 is a new result and Corollary 9 improves the well-known result of Brézis. In Section 3, we demonstrate the applicability of the results by proving existence of weak solution in (0, ; 1, 0 (Ω)) of a nonlinear parabolic problem, where > 1 and Ω is a nonempty, bounded, and open subset of R .
The following important lemma is due to Brèzis et al. [6].
Browder and Ton [26] gave the following important embedding result.

Lemma 4. Let be a separable reflexive Banach space. Then there exists a real separable Hilbert space and a compact injection :
→ such that ( ) = .
In this paper, we use the following fixed point result for compact operators, originally due to Leray and Schauder, which may be found in the book of Granas

Main Results
In this section, we prove the following new surjectivity result for maximal monotone perturbation of densely defined noncoercive generalized pseudomonotone operator in separable reflexive Banach spaces.
Proof. Let > 0 be fixed temporarily and the Yosida approximant of . For each > 0, by using the inner product condition on and monotonicity of ( (0) = 0 for all > 0), we see that for all ∈ ( ) ∩ (0) for some > 0. Let = (0). Let be a real separable Hilbert space and : → a compact injection such that ( ) is dense in guaranteed by Lemma 4. Let : → be the natural injection and let * : * → * and * : * → * be adjoint of and , respectively. It follows that = : where the closures are taken with respect to the spaces and , respectively. Since ∩ ⊆ , we obtain that Since the sets ∩ and ( ∩ ) are disjoint, we conclude that For each > 0, let be the Yosida approximant of . Let It is known that, for each > 0, + 1 is bounded, continuous, monotone, and of type ( + ). Let = and , : → be given by Since is continuous, it follows that is closed subset of . We show that , is a compact operator. To this end, let ∈ such that → 0 as → ∞. Since is continuous from into , we have → 0 as → ∞. Since ∈ , the sequence { } lies in . Since ∈ for all and 0 ∈ , it follows that ∈ and 0 ∈ for all . Since and are demicontinuous, it follows that ( + ) ⇀ ( + ) 0 as → ∞. By the density of in , it is known that * is defined from * into . As a result, for each ∈ , we see that for all . However, the right side expression goes to 0 as → ∞; that is, for each ∈ , it follows that Abstract and Applied Analysis 5 On the other hand, by the density of in , for each ∈ , we get that is, * ( + + 1 ) ⇀ * ( + + 1 ) 0 as → ∞. Since * is compact linear, which is completely continuous and ( ) * = * * , we arrive at * ( + + 1 ) → * ( + + 1 ) 0 as → ∞. This shows that the mapping , is continuous. Following similar argument as above, it is not difficult to show that , maps any bounded subset of into relatively compact subset of . As a result, we conclude that , is a compact operator. Fix > 0. In order to use Lemma 5, it is enough to show that (i) of Lemma 5 does not hold; that is, for all ∈ (0, 1) and ∈ , we have ̸ = , ( ). Suppose this is false; that is, there exist 0 ∈ and 0 ∈ (0, 1) such that 0 = 0 , ( 0 ). This yields We notice here that the continuity of , property of −1 , and definition of boundary of an open set imply that holds. Since 0 ∈ , it follows that 0 ∈ ∩ . By (11) and (20), we get which implies 0 = 0. But this is impossible because 0 ∈ ∩ . Therefore, by applying Lemma 5, for each > 0, > 0, and > 0, we conclude that the compact operator , has a fixed point ∈ ; that is, Therefore, for each ↓ 0 + , there exists ∈ such that for all . Since is bounded, the sequence { } is bounded. Since and 1 are bounded, it follows that the sequence {( + 1 ) } is bounded. Since = ( ) and is continuously embedded, we see that = ( ) ⊆ ( ) , where the closures are with respect to the norms in and , respectively. As a result, the density of in implies that = ( ) . By using (11), the monotonicity of and 1 , and property of * , we obtain that for all . Since ∈ , it follows that = = for all . Consequently, we obtain that for all . Since { } is bounded and is quasibounded, we conclude that { } is bounded. Consequently, by using (24), it is not difficult to see that { ‖ ‖ 2 } is bounded. If the sequence { } is bounded, then → 0 as → ∞. Otherwise, by using the boundedness of { ‖ ‖ 2 }, we assume without loss of generality that → 0 as → ∞, ⇀ 0 , and ⇀ V * 0 as → ∞. Since ( ) = , by choosing a sequence { = } such that → 0 as → ∞ and using (24) together with the monotonicity of + 1 , we get for all and . Fixing and letting → ∞ in (27), we obtain that Since + 1 is demicontinuous, letting → ∞, we arrive at that is, Since is generalized pseudomonotone, we conclude that 0 ∈ ( ), 0 = V * 0 , and ⟨ , ⟩ → ⟨V * 0 , 0 ⟩ as → ∞. For any ∈ ( ), applying the monotonicity of + 1 , we arrive at Moreover, from (24), we obtain that for all . As a result, we arrive at From (31) and (33), we obtain for all ∈ ( ). By the density of ( ) in and the continuity of + 1 , we conclude that for all ∈ . Since, for any ∈ , = 0 + (1 − ) ∈ for all ∈ [0, 1), using in place of , we obtain that for all ∈ [0, 1); that is, for all ∈ [0, 1). Since + 1 is continuous and → 0 as → 1 − , we have for all ∈ . Since ∈ is arbitrary, setting + 0 in place of yields for all ∈ . Therefore, for each > 0 (by fixing > 0 temporarily), we see that there exists ∈ ( ) ∩ such that + 1 + = * . Thus, for each ↓ 0 + , there exists ∈ ( ) ∩ such that for all . Since and 1 are bounded, it follows that { } and { 1 } are bounded. Since is quasibounded, it is not hard to see that { } is bounded, which implies the boundedness of { }. Assume without loss of generality that ⇀ 0 , ⇀ V * 0 and ⇀ * 0 as → ∞. Since + 1 is generalized pseudomonotone with domain ( ), it follows that Consequently, from (40), we arrive at Let be the Yosida resolvent of . It is well known that ∈ ( ), = − −1 ( ), and ∈ ( ) for all . Since ⇀ 0 and { } is bounded, it follows that ⇀ 0 as → ∞. Thus, we have for all . Consequently, we have By the maximality of , applying Lemma 3, we obtain 0 ∈ ( ), V * 0 ∈ 0 , and ⟨ , The generalized pseudomonotonicity of implies 0 ∈ ( ) and 0 = ℎ * 0 . As a result, letting → ∞ in (40), we conclude that V * 0 + 0 + 1 0 = * . This implies that, for each ↓ 0 + , there exist ∈ ( ) ∩ ( ) and V * ∈ such that for all . Next we will show that { } is bounded. Assume without loss of generality that ‖ ‖ → ∞ as → ∞. By Abstract and Applied Analysis 7 the inner product condition on and monotonicity of with 0 ∈ (0), we get for all ; that is, dividing this inequality by ‖ ‖ for all large , we get ‖ ‖ 2 ≤ ‖ ‖ + ‖ * ‖ for all large . By using condition ( ) and (46), we get that for all . This gives ( − )‖ ‖ ≤ 2‖ * ‖ + for all . Consequently, the boundedness of { } follows. Since is quasibounded and 0 ∈ ( ), it is not hard to see that { } is bounded. Consequently, the boundedness of {V * } follows.
It is worth mentioning that Theorem 6 is a new result because the perturbed operator + is noncoercive and is densely defined such that ( ) contains a dense real separable reflexive Banach space. Under the conditions on + , the result was unknown earlier even for coercive operator + . The analog of Theorem 6 for single multivalued, finitely continuous, coercive, and quasibounded generalized pseudomonotone operator such that ( ) contains a dense linear subspace is due to Browder and Hess [1]. If is quasimonotone with weakly closed graph or graph of is weakly closed and is monotone of type ( ), the arguments used in the proof of Theorem 6 can be easily carried out to conclude the surjectivity of + . The reader is referred to Gupta [28] for a result for + , where graph of is assumed to be weakly closed and : ⊇ ( ) → 2 * is quasibounded, finitely continuous coercive operator of type ( ) such that ( ) contains a dense linear subspace. Theorem 6 improves and gives unifications of the existing surjectivity results due to Le [16], Asfaw and Kartsatos [18,24], Asfaw [19], and Kenmochi [12][13][14] for maximal monotone perturbations of coercive bounded pseudomonotone operators with domain, all of . In addition, it can be easily seen that the proof of Theorem 6 can go through if the quasiboundedness of is omitted and is assumed to be strongly quasibounded with 0 ∈ (0). Another observation is that the condition ⟨V * + , ⟩ ≥ − ‖ ‖ 2 for all ∈ ( )∩ ( ) and V * ∈ can be replaced by a stronger condition ⟨V * + , ⟩ ≥ − ‖ ‖ for all ∈ ( ) ∩ ( ) and V * ∈ , and the weak coercivity condition (i) can be relaxed to satisfy | + | → ∞ as ‖ ‖ → ∞. On the other hand, one can easily see that weak coercivity condition on + is automatically satisfied if + is -expansive. Consequently, the following corollary is immediate.
Proof. Since 0 ∈ 0 and is monotone, by the condition on , it follows that ⟨V * + , ⟩ ≥ − ‖ ‖ 2 for all ∈ ( ) ∩ ( ). Furthermore, by the expansiveness of + , for some 0 ∈ ( ) ∩ ( ) and V * 0 ∈ 0 , we arrive at where 0 = ‖ 0 ‖ + ‖V * 0 + 0 ‖, for all ∈ ( ) ∩ ( ) and V * ∈ . This shows that + satisfies conditions of Theorem 6. By applying similar arguments as in the last part of the proof of Theorem 6 and using the strong quasiboundedness of instead of quasiboundedness of , we conclude that + is surjective. The details are omitted here.
It is worth noticing here that Corollary 7 gives a partial positive answer for Nirenberg's problem on the surjectivity of expansive mapping in a real separable reflexive Banach space. More precisely, Corollary 7 gives surjectivity of densely defined demicontinuous generalized pseudomonotone expansive mapping. To the best of the authors knowledge, this result was unknown. For related surjectivity results for continuous expansive mappings in a real Hilbert space, we cite the papers by Kartsatos [29] and Xiang [30]. For range result for single continuous quasimonotone expansive mapping defined from arbitrary reflexive Banach space into its dual space * , the reader is referred to the paper due to Asfaw [25].
The content of the following theorem addresses the solvability of operator equations involving operators of the type + , where : ⊇ ( ) → * is linear, densely defined, monotone, and closed, and : ⊇ ( ) → * is quasibounded demicontinuous of type ( ) such that ( ) lies in ( ).
Proof. Fix * ∈ * . Let = ( ) and let ‖ ⋅ ‖ be the graph norm on given by It is well-known that equipped with the graph norm becomes a real reflexive separable Banach space. By Lemma 4, let be a Hilbert space and let : → be a compact injection such that ( ) is dense in . Let : → be the natural embedding of into and let * : * → * be its adjoint. It follows that = is a compact injection from into . By using the graph norm on , it follows that is dense and continuously embedded in . Moreover, by the inner product condition on + , for each > 0, there exists > 0 such that → * is linear and monotone, it is continuous. By the arguments used in the proof of Theorem 6, using in place of and the closed convex subset of , it follows that the mapping : → defined by is compact. In addition, we see that Following the argument as in the proof of Theorem 6, it is not difficult to see that ̸ = ( ) for all ∈ and all ∈ (0, 1). Consequently, by Lemma 5, we obtain that, for each > 0, has a fixed point in . Therefore, for each ↓ 0 + , there exists ∈ such that for all ; that is, Since : → and : → , by the definition of * and * , we see that Since ∈ ∩ and is bounded in , it follows that the sequence { } = { } is bounded in . From (57), using the monotonicity ( (0) = 0), boundedness of { }, and quasiboundedness of , we get the boundedness of the sequence { }. This gives As a result, we get the boundedness of { ‖ ‖ 2 }. If { } is bounded, then → 0 as → ∞. If { } is unbounded, by passing into a subsequence, we see that In all cases, we assume without loss of generality that → 0 as → ∞. As a result, we get where is an upper bound for the sequence {‖ + + 1 + * ‖}. Since ( ) is dense in , for each ∈ , there exists a sequence { } in such that → as → ∞. This gives By similar argument, the density of in implies that By using the uniform boundedness principle, we conclude that { } is bounded. Assume without loss of generality that ⇀ 0 in , ⇀ V * 0 , and ⇀ ℎ * in * as → ∞. Since is closed linear, it follows that 0 ∈ and ℎ * = 0 . By following the arguments used in the first half of the proof of Theorem 6 along with (57), we get lim sup On the other hand, from (58), by using ∈ in place of , we see that Since is of type ( ) with respect to , it follows that − * is also of type ( ) with respect to , which yields 0 ∈ ( ) and V * 0 = 0 . Finally, letting → ∞ in (57), we get * ( + + 1 ) 0 = * * ; that is, * * ( + + 1 ) 0 = * * * . Since ( ) and are dense in and , respectively, it follows that * and * are one to one. Therefore, we arrive at 0 + 0 + 1 0 = * . Consequently, for each ↓ 0 + , there exists ∈ ( ) such that Since is closed and is of type ( ), by weak coercivity condition on + , the same arguments used in the second half of the proof of Theorem 6 can be carried over to conclude the solvability of the equation + ∋ * in ( ). Since * ∈ * is arbitrary, we conclude that + is surjective. The details are omitted here.
The following corollary gives a characterization of linear maximal monotone operator in separable reflexive Banach space.

Corollary 9.
Let be a real separable reflexive Banach space and let : ⊇ ( ) → * be linear operator. Then the following two statements are equivalent: (ii) is monotone, densely defined, and closed.
Proof. The proof of (i) implies (ii) follows by the well-known result due to Brézis (cf. Zeidler [9, Theorem 32. L, page 897]). Next we prove (ii) implies (i). Let > 0. It is sufficient to show that ( + ) = * . To this end, we will use Theorem 8. Since is linear and monotone, that is, ⟨ , ⟩ ≥ 0 for all ∈ ( ), and is monotone, it follows that ⟨ + , ⟩ ≥ ‖ ‖ 2 for all ∈ ( ). Therefore, for each > 0, it follows that By using in place of in Theorem 8, we conclude that ( + ) = * for any > 0. Thus, is maximal monotone.
It is worth noticing that Brézis proved (i) in arbitrary reflexive Banach space provided that * is monotone and (ii) holds. As a result, Corollary 9 is an improvement of the result of Brézis when is separable. It is important to mention that Gupta [28] gave surjectivity result for graph weakly closed maximal monotone perturbations of quasibounded, finitely continuous multivalued coercive operator of type ( ) such that ( ) contains a dense linear subspace of . However, the result in Theorem 8 is for noncoercive operator along with weak coercivity of + . It is also important to mention here that the results in Theorems 6 and 8 are new even in the case where the operator is coercive but not everywhere defined. In conclusion, Theorems 6 and 8 gave improvements over the existing theory for maximal monotone perturbations of coercive and everywhere defined operators of pseudomonotone type.
Abstract existence results concerning nonlinear parabolic problems of the type in (69) under conditions ( 1 ) and ( 3 ) have been intensively studied by many researchers. For some of the basic and relevant references, the reader is referred to the papers by Browder and Hess [1], Brézis [2], Le [16], Kenmochi [12][13][14], Guan and Kartsatos [17], Asfaw and Kartsatos [18], Asfaw [19,25], and the references therein. For further examples and applications of perturbed everywhere defined pseudomonotone type operators to inclusion, variational inequality, and evolution problems, the reader is referred to the papers of Landes and Mustonen [31], Kobayashi and Otani [32], and Mustonen [33] and books of Kinderlehrer and Stampacchia [34], Browder [8], and Naniewicz and Panagiotopoulos [35] and the references therein. The method of sub-supersolution is employed in the papers by Carl and Le [20], Carl et al. [21], Carl [22], Carl and Motreanu [23], and Le [36,37] to study existence and properties of solution(s) for evolution inclusion problems of the type ∈ : + ( ) ∋ * in * , (0) = 0 , where = (0, ; 1, 0 (Ω)), > 1, Ω is a nonempty, bounded, and open subset of R , and is noncoercive but still everywhere defined operator of pseudomonotone type. For further relevant information about sub-supersolution arguments concerning evolution type problems, the reader is referred to the recent book on nonsmooth analysis due to Carl et al. [38] and the references therein. Finally, it is important to indicate the readers that the results in this paper can be conveniently applied to address nonlinear parabolic problems of type (76) as well as elliptic problems of the type ∈ : −Δ + ( ) ∋ * , * ∈ * , where = 1, 0 (Ω) and and are possibly noncoercive and densely defined and satisfy conditions of either Theorem 6 or Theorem 8.

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