# Noncoercive Perturbed Densely Defined Operators and Application to Parabolic Problems

Authors: Teffera M. Asfaw
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

LetX be a real locally uniformly convex reflexive separable Banach space with locally uniformly convex dual space X∗. Let T:X⊇D(T)→2X∗ be maximal monotone and S:X⊇D(S)→X∗ quasibounded generalized pseudomonotone such that there exists a real reflexive separable Banach space W⊂D(S), dense and continuously embedded in X. Assume, further, that there exists d≥0 such that 〈v∗+Sx,x〉≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx. New surjectivity results are given for noncoercive, not everywhere defined, and possibly unbounded operators of the type T+S. A partial positive answer for Nirenberg's problem on surjectivity of expansive mapping is provided. Leray-Schauder degree is applied employing the method of elliptic superregularization. A new characterization of linear maximal monotone operator L:X⊇D(L)→X∗ is given as a result of surjectivity of L+S, where S is of type (M) with respect to L. These results improve the corresponding theory for noncoercive and not everywhere defined operators of pseudomonotone type. In the last section, an example is provided addressing existence of weak solution in X=Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem of the type ut-∑i=1n(∂/∂xi)ai(x,t,u,∇u)=f(x,t),  (x,t)∈Q; u(x,t)=0,  (x,t)∈∂Ω×(0,T); u(x,0)=0,  x∈Ω, where p>1, Ω is a nonempty, bounded, and open subset of RN, ai:Ω×(0,T)×R×RN→R  (i=1,2,…,n) satisfies certain growth conditions, and f∈Lp′(Q), Q=Ω×(0,T), and p′ is the conjugate exponent of p.


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## Body

## 1. Introduction—Preliminaries

In what follows,X is a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X∗. The norm of the space X, and any other normed spaces herein, will be denoted by ·. For x∈X and x∗∈X∗, the pairing x∗,x denotes the value x∗(x). Let X and Y be real Banach spaces. For a multivalued mapping T:X→2Y, we define the domain D(T) of T by D(T)={x∈X:Tx≠∅} and the range R(T) of T by R(T)=∪x∈D(T)Tx. We also denote the graph of T by G(T)={(x,Tx):x∈D(T)}. A mapping T:X⊃D(T)→Y is “demicontinuous” if it is continuous from the strong topology of D(T) to the weak topology of Y. A multivalued mapping T:X⊃D(T)→2Y is “bounded” if it maps bounded subsets of D(T) to bounded subsets of Y. It is “compact” if it is strongly continuous and maps bounded subsets of D(T) to relatively compact subset of Y. It is “finitely continuous” if it is upper semicontinuous from each finite dimensional subspace F of X to the weak topology of Y. It is “quasibounded” if for every M>0 there exists K(M)>0 such that [x,w∗]∈G(T) with x≤M and w∗,x≤Mx imply w∗≤K(M). It is “strongly quasibounded” if for every M>0 there exists K(M)>0 such that [x,w∗]∈G(T) with x≤M and w∗,x≤M imply w∗≤K(M). In what follows, a mapping will be called “continuous” if it is strongly continuous.Letψ:[0,∞)→[0,∞) be continuous strictly increasing function such that ψ(0)=0 and ψ(t)→∞ as t→∞. The duality mapping corresponding to ψ denoted by Jψ:X→2X∗ is defined by (1)Jψx=x∗∈X∗:x∗,x=x∗x,x∗=ψx.It is well-known that, for each x∈X, the Hahn-Banach Theorem implies Jψ(x)≠∅. Since X and X∗ are locally uniformly convex, Jψ is single-valued, bounded monotone of type (S+) and bicontinuous. If ψ(t)=t for t≥0, then Jψ is denoted by J and is called the normalized duality mapping.An operatorT:X⊃D(T)→2X∗ is said to be “monotone” if, for every x∈D(T), y∈D(T), and every u∗∈Tx, v∗∈Ty, we have u∗-v∗,x-y≥0. A monotone mapping T:X⊃D(T)→2X∗ is “maximal monotone” if R(T+λJ)=X∗ for every λ>0; that is, T is maximal monotone if and only if T is monotone and u∗-u0∗,x-x0≥0 for every (x,u∗)∈G(T) implies x0∈D(T) and u0∗∈Tx0. If T is maximal monotone, the operator Tt:X→X∗, t∈(0,∞), defined by Ttx=(T-1+tJ-1)-1x, is bounded, continuous, maximal monotone and such that Ttx⇀T(0)x as t→0+, for every x∈D(T), where T(0)x=inf{y∗:y∗∈Tx}. The “resolvent” Jt:X→D(T), defined by Jtx=x-tJ-1(Ttx), is continuous and Ttx∈T(Jtx) for every x∈X. Moreover, limt→0Jtx=x for all x∈coD(T)¯, where coD(T) is the convex hull of the set D(T). An operator A:X⊇D(A)→2X∗ is called “coercive” if either D(A) is bounded or there exists a function ψ:[0,∞)→(-∞,∞) such that ψ(t)→∞ as t→∞ and y∗,x≥ψ(x)x for all x∈D(A) and y∗∈Ax. For an operator A:X⊇D(A)→2X∗ and x∈D(A), we denote |Ax|=inf{v∗:v∗∈Ax}. It is called weakly coercive if either D(A) is bounded or Ax→∞ as x→∞.The following definitions are used throughout the paper. In arbitrary Banach spaceX, 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 (M) with domain all of X, is due to Brézis [2].Definition 1.
An operatorS:X⊇D(S)→X∗ is called (i)
“generalized pseudomonotone” if, for each sequence{xn} in D(S) with xn⇀x0 and Sxn⇀v0∗ as n→∞ such that limsupn→∞Sxn,xn-x0≤0, then x0∈D(S), Sx0=v0∗, and Sxn,xn→Sx0,x0 as n→∞.(ii)
“type(M)” if, for each sequence {xn} in D(S) with xn⇀x0 in X and Sxn⇀v0∗ as n→∞ such that limsupn→∞Sxn,xn-x0≤0, then x0∈D(S) and Sx0=v0∗.(iii)
“α-expansive” if there exists α>0 such that v∗-u∗≥αx-y for all x∈D(S), y∈D(S), v∗∈Sx, and u∗∈Sy. It is called expansive if α=1.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 X=H, if T:H⊇D(T)→2H is monotone, we see that, for each λ>0, T+λI is multivalued expansive with domain D(T).The following definition gives a larger class of operators of monotone type, which can be found in Kartsatos and Skrypnik [4].Definition 2.
LetT:X⊇D(T)→2X∗ be maximal monotone and A:X⊇D(A)→X∗. Let L⊆D(M)∩D(A) be a linear subspace of X. Then A is said to be (i)
“quasibounded with respect toT” if, for each M>0, there exists K(M)>0 such that (2)Au+u∗,u≤M,u≤M,where u∈L and u∗∈Tu, then Au≤K(M),(ii)
“generalized(S+) with respect to T” if, for each {un} in L with un∗∈Tun, un⇀u0 in X and Aun⇀h0∗ in X∗ as n→∞ such that (3)limsupn→∞Aun,un-u0≤0,un∗+Aun,un≤0for all n, then un→u0∈D(A) and Au0=h0∗,(iii)
“generalized pseudomonotone with respect toT” if, for each {un} in L with un∗∈Tun, un⇀u0 in X and Aun⇀h0∗ in X∗ as n→∞ such that (4)limsupn→∞Aun,un-u0≤0,un∗+Aun,un≤0for all n, then u0∈D(A), Au0=h0∗, and Sun,un→Su0,u0 as n→∞,(iv)
“of type(M) with respect to T” if, for each {un} in L with un∗∈Tun, un⇀u0 in X and Aun⇀h0∗ in X∗ as n→∞ such that (5)limsupn→∞Aun,un-u0≤0,un∗+Aun,un≤0for all n, then u0∈D(A) and Au0=h0∗.By Definition2, it is not difficult to see that 0∈D(T) and A is quasibounded implying that A is quasibounded with respect to T. Furthermore, it follows that the class of generalized (S+) operators with respect to T includes the class of operators of type (S+).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 [12–14], Guan et al. [15], Le [16], Guan and Kartsatos [15, 17], 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→2X∗ 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 fornoncoercive andnot 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 spaceW⊆D(S), dense and continuously embedded in X;(b)
there existsd≥0 such that v∗+Sx,x≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx;(c)
there existα>d and μ≥0 such that v∗+Sx≥αx-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 Section2, 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 while Theorem 6 used S 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 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 of Theorem 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 linear monotone operator L is maximal monotone 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 Lp(0,T;W01,p(Ω)) of a nonlinear parabolic problem, where p>1 and Ω is a nonempty, bounded, and open subset of RN.The following important lemma is due to Brèzis et al. [6].Lemma 3.
LetB be a maximal monotone set in X×X∗. If (un,un∗)∈B such that un⇀u, un∗⇀u∗ as n→∞, and either (6)limsupn,m→∞un∗-um∗,un-um≤0or (7)limsupn→∞un∗-u∗,un-u≤0,then (u,u∗)∈B and un∗,un→u∗,u as n→∞.Browder and Ton [26] gave the following important embedding result.Lemma 4.
LetX be a separable reflexive Banach space. Then there exists a real separable Hilbert space H and a compact injection Q:H→X such that Q(H)¯=X.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 and Dugundji [27, Theorem 5.2, page 123].Lemma 5.
LetC be a convex subset of a normed linear space X and let U be nonempty relatively open in C with 0∈U. Then each compact map F:U¯→C satisfies that either (i)
F has a fixed point in Uor (ii)
there existx∈∂CU and λ∈(0,1) such that x=λF(x), where ∂CU is the boundary of U with respect to the subspace topology on C.


## 2. 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.Theorem 6.
LetT:X⊇D(T)→2X∗ be maximal monotone with 0∈T(0) and S:X⊇D(S)→X∗ quasibounded demicontinuous generalized pseudomonotone. Suppose W⊆D(S) is a real reflexive separable Banach space dense and continuously embedded in X. Assume, further, that there exist μ≥0, d≥0, and α>d satisfying (8)Sx,x≥-dx2for all x∈D(S) and either (i)
(9)v∗+Sx≥αx-μ∀x∈DT∩DS,v∗∈Txor (ii)
there existsϕ:[0,∞)→(-∞,∞) such that ϕ(t)→∞ as t→∞ and (10)v∗+Sx≥ϕxx∀x∈DT∩DS,v∗∈Tx.Then T+S is surjective.Proof.
Letλ>0 be fixed temporarily and Tλ the Yosida approximant of T. For each ɛ>0, by using the inner product condition on S and monotonicity of Tλ (Tλ(0)=0 for all λ>0), we see that(11)Tλx+Sx+δxJx-f∗,x≥δx3-dx2-f∗x=x3δ-dx-f∗x3>0for all x∈D(S)∩∂BRδ(0) for some Rδ>0. Let Gδ=BRδ(0). Let H be a real separable Hilbert space and Q:H→W a compact injection such that Q(H) is dense in W guaranteed by Lemma 4. Let j:W→X be the natural injection and let Q∗:W∗→H∗ and j∗:X∗→W∗ be adjoint of Q and j, respectively. It follows that ψ=jQ:H→X is a compact operator. Let U=Q-1(Gδ∩W). First we show that Gδ∩W is open in W; that is, W∖(Gδ∩W)=W∩(X∖Gδ) is closed in W. To this end, let {xn} be a sequence in W∩(X∖Gδ) such that xn→x0 in W as n→∞. Since W is continuously embedded in X, we get xn→x0 in X as n→∞. Since X∖Gδ is closed in X, it follows that x0∈X∖Gδ; that is, x0∈W∩(X∖Gδ). This shows that W∩(X∖Gδ) is closed in W; that is, Gδ∩W is open in W. The continuity of Q implies that U is open in H. Since W is continuously embedded in X, it follows that (12)Gδ∩W¯W⊆Gδ∩W¯X⊆Gδ¯,where the closures are taken with respect to the spaces W and X, respectively. Since Gδ∩W¯W⊆W, we obtain that(13)Gδ∩W∪∂WGδ∩W=Gδ∩W¯W⊆Gδ¯∩W=Gδ∩W∪∂Gδ∩W.Since the sets Gδ∩W and ∂W(Gδ∩W) are disjoint, we conclude that (14)∂WGδ∩W⊆∂Gδ∩W.For each λ>0, let Tλ be the Yosida approximant of T. Let J1x=xJx,x∈X. It is known that, for each δ>0, Tλ+δJ1 is bounded, continuous, monotone, and of type (S+). Let ψ=jQ and Cɛλ,δ:U¯→H be given by (15)Cɛλ,δv=-ε-1ψ∗Tλ+S+δJ1ψv-f∗,v∈U¯.Since ψ is continuous, it follows that(16)U¯=ψ-1Gδ∩W¯⊆ψ-1Gδ∩W¯W⊆His closed subset of H. We show that Cɛλ,δ is a compact operator. To this end, let xn∈U¯ such that xn→x0 as n→∞. Since Q is continuous from H into X, we have Qxn→Qx0 as n→∞. Since xn∈U¯, the sequence {Qxn} lies in W. Since xn∈H for all n and x0∈U¯, it follows that Qxn∈W and Qx0∈W for all n. Since S and Tλ are demicontinuous, it follows that (Tλ+S)Qxn⇀(Tλ+S)Qx0 as n→∞. By the density of W in X, it is known that j∗ is defined from W∗ into H. As a result, for each w∈W, we see that(17)j∗Tλ+S+δJ1Qxn-j∗Tλ+S+δJ1Qx0,w=Tλ+S+δJ1Qxn,w-Tλ+S+δJ1Qx0,wfor all n. However, the right side expression goes to 0 as n→∞; that is, for each w∈W, it follows that (18)j∗Tλ+S+δJ1Qxn-j∗Tλ+S+δJ1Qx0,w⟶0asn⟶∞.On the other hand, by the density of W in X, for each x∈X, we get (19)j∗Tλ+S+δJ1Qxn-j∗Tλ+S+δJ1Qx0,x⟶0asn⟶∞;that is, j∗(Tλ+S+δJ1)Qxn⇀j∗(Tλ+S+δJ1)Qx0 as n→∞. Since Q∗ is compact linear, which is completely continuous and (jQ)∗=Q∗j∗, we arrive at ψ∗(Tλ+S+δJ1)Qxn→ψ∗(Tλ+S+δJ1)Qx0 as n→∞. This shows that the mapping Cɛδ,λ is continuous. Following similar argument as above, it is not difficult to show that Cɛδ,λ maps any bounded subset of U¯ into relatively compact subset of H. As a result, we conclude that Cɛδ,λ is a compact operator. Fix ɛ>0. In order to use Lemma 5, it is enough to show that (i) of Lemma 5 does not hold; that is, for all μ∈(0,1) and x∈∂HU, we have x≠μCɛδ,λ(x). Suppose this is false; that is, there exist x0∈∂HU and μ0∈(0,1) such that x0=μ0Cɛδ,λ(x0). This yields(20)ɛx0+μ0ψ∗Tλ+S+δJ1ψx0-f∗=0.We notice here that the continuity of Q, property of Q-1, and definition of boundary of an open set imply that(21)∂HU=∂HQ-1Gδ∩W⊆Q-1∂WGδ∩W⊆Q-1∂Gδ∩Wholds. Since x0∈∂HU, it follows that Qx0∈∂G∩W. By (11) and (20), we get (22)ɛμ0x02=-ψ∗Tλ+S+δJ1Qx0-f∗,x0=-Tλ+S+δJ1Qx0-f∗,Qx0≤0,which implies x0=0. But this is impossible because 0∈Gδ∩W. Therefore, by applying Lemma 5, for each ɛ>0, λ>0, and δ>0, we conclude that the compact operator Cɛδ,λ has a fixed point xɛ∈U; that is, (23)ɛxɛ+ψ∗Tλ+S+δJ1ψxɛ-f∗=0.Therefore, for each εn↓0+, there exists xn∈U¯ such that(24)εnxn+ψ∗Tλ+S+δJ1ψxn-f∗=0for all n. Since Gδ is bounded, the sequence {ψxn} is bounded. Since Tλ and J1 are bounded, it follows that the sequence {(Tλ+δJ1)ψxn} is bounded. Since W=Q(H)¯W and W is continuously embedded, we see that W=Q(H)¯W⊆Q(H)¯X, where the closures are with respect to the norms in W and X, respectively. As a result, the density of W in X implies that X=Q(H)¯X. By using (11), the monotonicity of Tλ and J1, and property of ψ∗, we obtain that(25)Sψxn,ψxn=-εnxn2-Tλψxn+δJ1ψxn-Tλ0+δJ10,ψxn+Tλ0+δJ10+f∗,ψxn≤Tλ0+δJ10+f∗ψxn≤T0+f∗ψxnfor all n. Since Qxn∈W, it follows that ψxn=jQxn=Qxn for all n. Consequently, we obtain that (26)SQxn,Qxn≤T0+f∗Qxnfor all n. Since {Qxn} is bounded and S is quasibounded, we conclude that {SQxn} is bounded. Consequently, by using (24), it is not difficult to see that {εnxn2} is bounded. If the sequence {xn} is bounded, then εnxn→0 as n→∞. Otherwise, by using the boundedness of {εnxn2}, we assume without loss of generality that εnxn→0 as n→∞, Qxn⇀x0, and SQxn⇀v0∗ as n→∞. Since Q(H)¯X=X, by choosing a sequence {zm=Qym} such that zm→x0 as m→∞ and using (24) together with the monotonicity of Tλ+δJ1, we get(27)SQxn,Qxn=SQxn,Qym-Qym+Qxn=SQxn,Qym+SQxn,Qxn-Qym=SQxn,Qym+f∗,Qxn-Qym-Tλ+δJ1Qxn-Tλ+δJ1Qym,Qxn-Qym-Tλ+δJ1Qym,Qxn-Qym-εnxn,xn-ym≤SQxn,Qym-Tλ+δJ1Qym,Qxn-Qym+εnxn,ym+f∗,Qxn-Qymfor all n and m. Fixing m and letting n→∞ in (27), we obtain that (28)limsupn→∞SQxn,Qxn≤v0∗,Qym-Tλ+δJ1Qym-f∗,x0-Qym.Since Tλ+δJ1 is demicontinuous, letting m→∞, we arrive at (29)limsupn→∞SQxn,Qxn≤v0∗,x0;that is, (30)limsupn→∞SQxn,Qxn-x0≤0.Since S is generalized pseudomonotone, we conclude that x0∈D(S), Sx0=v0∗, and SQxn,Qxn→v0∗,x0 as n→∞. For any y∈Q(H), applying the monotonicity of Tλ+δJ1, we arrive at(31)Tλy+δJ1y-f∗,y-x0=limn→∞Tλy+δJ1y-f∗,y-Qxn≥liminfn→∞Tλy+δJ1y-TλQxn+δJ1Qxn,y-Qxn+liminfn→∞TλQxn+δJ1Qxn-f∗,y-Qxn≥liminfn→∞TλQxn+δJ1Qxn-f∗,y-Qxn.Moreover, from (24), we obtain that (32)Tλ+δJ1Qxn-f∗,y-Qxn=-εnxn,Q-1y-xn-SQxn,y-Qxn=εnxn2-εnxn,Q-1y-SQxn,y-Qxn≥-εnxn,Q-1y-SQxn,y-Qxnfor all n. As a result, we arrive at(33)liminfn→∞Tλ+δJ1Qxn-f∗,y-Qxn≥-Sx0,y-x0.From (31) and (33), we obtain (34)Tλ+δJ1y+Sx0-f∗,y-x0≥0for all y∈Q(H). By the density of Q(H) in X and the continuity of Tλ+δJ1, we conclude that (35)Tλy+δJ1y+Sx0-f∗,y-x0≥0for all y∈X. Since, for any y∈X, xt=tx0+(1-t)y∈X for all t∈[0,1), using xt in place of y, we obtain that (36)Tλ+δJ1xt+Sx0-f∗,1-ty-x0≥0for all t∈[0,1); that is, (37)Tλ+δJ1xt+Sx0-f∗,y-x0≥0for all t∈[0,1). Since Tλ+δJ1 is continuous and xt→x0 as t→1-, we have Tλxt+δJ1xt→Tλx0+δJ1x0 as t→1-. Letting t→1-, we arrive at (38)Tλ+δJ1x0+Sx0-f∗,y-x0≥0for all y∈X. Since y∈X is arbitrary, setting y+x0 in place of y yields (39)Tλ+δJ1x0+Sx0-f∗,y≥0for all y∈X. Therefore, for each λ>0 (by fixing δ>0 temporarily), we see that there exists xλ∈D(S)∩G¯δ such that Tλxλ+δJ1xλ+Sxλ=f∗. Thus, for each λn↓0+, there exists yn∈D(S)∩G¯δ such that(40)Tλnyn+δJ1yn+Syn-f∗=0for all n. Since G¯δ and J1 are bounded, it follows that {yn} and {J1yn} are bounded. Since S is quasibounded, it is not hard to see that {Syn} is bounded, which implies the boundedness of {Tλnyn}. Assume without loss of generality that yn⇀y0, Syn⇀v0∗ and Tλnyn⇀u0∗ as n→∞. Since S+δJ1 is generalized pseudomonotone with domain D(S), it follows that (41)liminfn→∞Syn+δJ1yn,yn-y0≥0.Consequently, from (40), we arrive at (42)limsupn→∞Tλnyn,yn-y0≤0.Let Jλn be the Yosida resolvent of T. It is well known that Jλnyn∈D(T), Jλnyn=xn-λnJ-1(Tλnyn), and Tλnyn∈T(Jλnyn) for all n. Since yn⇀y0 and {Tλnyn} is bounded, it follows that Jλnyn⇀y0 as n→∞. Thus, we have (43)Tλnyn,Jλnyn-y0=Tλnyn,Jλnyn-yn+Tλnyn,yn-y0=-Tλnyn,λnJ-1Tλnyn+Tλnyn,yn-y0=-tnTλnyn2+Tλnyn,yn-y0≤Tλnyn,yn-y0for all n. Consequently, we have (44)limn→∞Tλnyn,Jλnyn-y0≤limn→∞Tλnyn,yn-y0≤0.By the maximality of T, applying Lemma 3, we obtain x0∈D(T), v0∗∈Tx0, and Tλnyn,Jλnyn→v0∗,x0 as n→∞, which implies (45)limsupn→∞Syn,yn-y0≤0.The generalized pseudomonotonicity of S implies y0∈D(S) and Sy0=h0∗. As a result, letting n→∞ in (40), we conclude that v0∗+Sy0+δJ1y0=f∗. This implies that, for each δn↓0+, there exist zn∈D(T)∩D(S) and vn∗∈Tzn such that(46)vn∗+Szn+δnJ1zn=f∗for all n. Next we will show that {zn} is bounded. Assume without loss of generality that zn→∞ as n→∞. By the inner product condition on S and monotonicity of T with 0∈T(0), we get (47)δnzn3≤dzn2+f∗znfor all n; that is, dividing this inequality by zn for all large n, we get δnzn2≤dzn+f∗ for all large n. By using condition (i) and (46), we get that (48)-μ+αzn≤vn∗+Szn≤δnzn2+f∗≤dzn+2f∗for all n. This gives (α-d)zn≤2f∗+μ for all n. Consequently, the boundedness of {zn} follows. Since S is quasibounded and 0∈D(T), it is not hard to see that {Szn} is bounded. Consequently, the boundedness of {vn∗} follows. Assuming that zn∗⇀z0∗, vn∗⇀v0∗, and Szn⇀h0∗ as n→∞ and using the arguments used in the first half of the proof of Theorem 6, we conclude that z0∈D(T)∩D(S), v0∗∈Tz0, Sz0=h0∗, and v0∗+Sz0=f∗; that is, for each f∗∈X∗, the inclusion problem Tu+Su∋f∗ is solvable in D(T)∩D(S). Since f∗∈X∗ is arbitrary, we obtain the surjectivity of T+S. The proof using condition (ii) can be completed following similar arguments. The details are omitted here. This completes the proof.It is worth mentioning that Theorem6 is a new result because the perturbed operator T+S is noncoercive and S is densely defined such that D(S) contains a dense real separable reflexive Banach space. Under the conditions on T+S, the result was unknown earlier even for coercive operator T+S. The analog of Theorem 6 for single multivalued, finitely continuous, coercive, and quasibounded generalized pseudomonotone operator S such that D(S) contains a dense linear subspace is due to Browder and Hess [1]. If S is quasimonotone with weakly closed graph or graph of T is weakly closed and S is monotone of type (M), the arguments used in the proof of Theorem 6 can be easily carried out to conclude the surjectivity of T+S. The reader is referred to Gupta [28] for a result for T+S, where graph of T is assumed to be weakly closed and S:X⊇D(S)→2X∗ is quasibounded, finitely continuous coercive operator of type (M) such that D(S) 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 [12–14] for maximal monotone perturbations of coercive bounded pseudomonotone operators with domain, all of X. In addition, it can be easily seen that the proof of Theorem 6 can go through if the quasiboundedness of S is omitted and T is assumed to be strongly quasibounded with 0∈T(0). Another observation is that the condition v∗+Sx,x≥-dx2 for all x∈D(T)∩D(S) and v∗∈Tx can be replaced by a stronger condition v∗+Sx,x≥-dx for all x∈D(T)∩D(S) and v∗∈Tx, and the weak coercivity condition (i) can be relaxed to satisfy Tx+Sx→∞ as x→∞. On the other hand, one can easily see that weak coercivity condition on T+S is automatically satisfied if T+S is α-expansive. Consequently, the following corollary is immediate.Corollary 7.
LetT:X⊇D(T)→2X∗ be strongly quasibounded maximal monotone with 0∈T(0) and let S:X⊇D(S)→X∗ be demicontinuous generalized pseudomonotone. Suppose W⊆D(S) is a real reflexive separable Banach space dense and continuously embedded in X. Assume, further, that T+S is α-expansive mapping and there exists d≥0 such that α>d and Sx,x≥-dx2 for all x∈D(S). Then T+S is surjective.Proof.
Since0∈T0 and T is monotone, by the condition on S, it follows that v∗+Sx,x≥-dx2 for all x∈D(T)∩D(S). Furthermore, by the expansiveness of T+S, for some u0∈D(T)∩D(S) and v0∗∈Tu0, we arrive at (49)v∗+Sx≥v∗+Sx-v0∗+Su0-v0∗+Su0≥αx-u0-v0∗+Su0≥αx-αu0-v0∗+Su0=αx-A0,where A0=αu0+v0∗+Su0, for all x∈D(T)∩D(S) and v∗∈Tx. This shows that T+S 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 T instead of quasiboundedness of S, we conclude that T+S is surjective. The details are omitted here.It is worth noticing here that Corollary7 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 X∗, 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 typeL+S, where L:X⊇D(L)→X∗ is linear, densely defined, monotone, and closed, and S:X⊇D(S)→X∗ is quasibounded demicontinuous of type (M) such that D(L) lies in D(S).Theorem 8.
LetL:X⊇D(L)→X∗ be closed, densely defined, and linear monotone, and let S:X⊇D(S)→X∗ be quasibounded demicontinuous of type (M) with respect to L such that D(L) lies in D(S). Assume, further, that there exist μ≥0 and α>d≥0 such that(50)Lx+Sx,x≥-dx2for all x∈D(L) and either (i)
(51)Lx+Sx≥αx-μ∀x∈DLor(ii)
there existsϕ:[0,∞)→(-∞,∞) such that ϕ(t)→∞ as t→∞ and (52)Lx+Sx≥ϕxx∀x∈DL.Then L+S is surjective.Proof.
Fixf∗∈X∗. Let Y=D(L) and let ·Y be the graph norm on Y given by (53)xY=xX+LxX∗,x∈Y.It is well-known that Y equipped with the graph norm becomes a real reflexive separable Banach space. By Lemma 4, let H be a Hilbert space and let Q:H→Y be a compact injection such that Q(H) is dense in Y. Let j:Y→X be the natural embedding of Y into X and let j∗:X∗→Y∗ be its adjoint. It follows that ψ=jQ is a compact injection from H into X. By using the graph norm on Y, it follows that Y is dense and continuously embedded in X. Moreover, by the inner product condition on L+S, for each δ>0, there exists Rδ>0 such that(54)Lx+Sx+δJ1x,x≥δx3-dx2=x3δ-dx>0for all x∈D(L)∩D(S)∩∂BRδ(0). Let Gδ=BRδ(0). By using the arguments used in the first half of the proof of Theorem 6, we see that Gδ∩Y is open in Y and ∂Y(Gδ∩Y)⊆∂Gδ∩Y. Let U=ψ-1(Gδ∩Y). Since j:Y→X and Q:H→Y are continuous, it follows that U is open in H. Since the operator j∗Lj:Y→Y∗ is linear and monotone, it is continuous. By the arguments used in the proof of Theorem 6, using Y in place of X and the closed convex subset U¯ of H, it follows that the mapping Cɛ:U¯→H defined by (55)Cɛv=-ε-1ψ∗L+S+δJ1Jv-f∗,v∈U¯is compact. In addition, we see that (56)∂HU=∂Hψ-1Gδ∩Y⊆ψ-1∂YGδ∩Y⊆ψ-1∂Gδ∩Y.Following the argument as in the proof of Theorem 6, it is not difficult to see that x≠λCɛ(x) for all x∈∂HU and all λ∈(0,1). Consequently, by Lemma 5, we obtain that, for each ɛ>0, Cɛ has a fixed point in U. Therefore, for each εn↓0+, there exists xn∈U such that(57)εnxn+ψ∗L+S+δJ1ψxn=ψ∗f∗for all n; that is,(58)εnxn,x+Q∗j∗L+S+δJ1jQxn,x=Q∗j∗f∗,x∀x∈H.Since j:Y→X and Q:H→Y, by the definition of Q∗ and j∗, we see that (59)εnxn,x+L+S+δJ1Qxn,Qx=f∗,Qx∀x∈H.
Sinceψxn∈Gδ∩Y and Gδ is bounded in X, it follows that the sequence {jxn}={Qxn} is bounded in X. From (57), using the monotonicity L (L(0)=0), boundedness of {Qxn}, and quasiboundedness of S, we get the boundedness of the sequence {SQxn}. This gives(60)εnxn2=-Q∗L+S+δJ1Qxn-Q∗f∗,xn≤-SQxn+δJ1Qxn-f∗,Qxn≤SQxn+δJ1Qxn-f∗Qxn∀n.As a result, we get the boundedness of {εnxn2}. If {xn} is bounded, then εnxn→0 as n→∞. If {xn} is unbounded, by passing into a subsequence, we see that (61)εnxn=εnxn2xn⟶0asn⟶∞.In all cases, we assume without loss of generality that εnxn→0 as n→∞. As a result, we get(62)LQxn,Qx=-εnxn,x-SQxn+δJ1Qxn,Qx+f∗,Qx≤εnxn+SQxn+δJ1Qxn+f∗Qx≤μQx,where μ is an upper bound for the sequence {εnxn+SQxn+δJ1Qxn+f∗}. Since Q(H) is dense in Y, for each y∈Y, there exists a sequence {ym} in H such that Qym→y as n→∞. This gives (63)LQxn,y=limm→∞LQxn,Qym≤limm→∞μQym=μy.By similar argument, the density of Y in X implies that (64)LQxn,x≤μx∀x∈X.By using the uniform boundedness principle, we conclude that {LQxn} is bounded. Assume without loss of generality that Qxn⇀x0 in X, SQxn⇀v0∗, and LQxn⇀h∗ in X∗ as n→∞. Since L is closed linear, it follows that x0∈Y and h∗=Lx0. By following the arguments used in the first half of the proof of Theorem 6 along with (57), we get (65)limsupn→∞SQxn-f∗,Qxn-x0≤0.On the other hand, from (58), by using xn∈H in place of x, we see that(66)LQxn+SQxn-f∗,Qxn≤0∀n.Since S is of type (M) with respect to L, it follows that S-f∗ is also of type (M) with respect to L, which yields x0∈D(S) and v0∗=Sx0. Finally, letting n→∞ in (57), we get ψ∗(L+S+δJ1)x0=ψ∗f∗; that is, Q∗j∗(L+S+δJ1)x0=Q∗j∗f∗. Since Q(H) and Y are dense in Y and X, respectively, it follows that j∗ and Q∗ are one to one. Therefore, we arrive at Lx0+Sx0+δJ1x0=f∗. Consequently, for each δn↓0+, there exists yn∈D(L) such that (67)Lyn+Syn+δnJ1yn=f∗∀n.Since L is closed and S is of type (M), by weak coercivity condition on L+S, 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 Lx+Sx∋f∗ in D(L). Since f∗∈X∗ is arbitrary, we conclude that L+S 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.
LetX be a real separable reflexive Banach space and let L:X⊇D(L)→X∗ be linear operator. Then the following two statements are equivalent: (i)
L is maximal monotone,(ii)
L 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 λ>0. It is sufficient to show that R(L+λJ)=X∗. To this end, we will use Theorem 8. Since L is linear and monotone, that is, Lx,x≥0 for all x∈D(L), and J is monotone, it follows that Lx+λJx,x≥λx2 for all x∈D(L). Therefore, for each λ>0, it follows that (68)Lx+λJx≥λx∀x∈DL.By using J in place of S in Theorem 8, we conclude that R(L+λJ)=X∗ for any λ>0. Thus, L is maximal monotone.It is worth noticing that Brézis proved (i) in arbitrary reflexive Banach space provided thatL∗ is monotone and (ii) holds. As a result, Corollary 9 is an improvement of the result of Brézis when X 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 S of type (M) such that D(S) contains a dense linear subspace of X. However, the result in Theorem 8 is for noncoercive operator S along with weak coercivity of L+S. It is also important to mention here that the results in Theorems 6 and 8 are new even in the case where the operator S 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.


## 3. Example and Discussion

In this section, we demonstrate the existence of weak solution inX=Lp(0,T;W01,p(Ω)) for the parabolic problem of the type(69)∂u∂t-∑i=1N∂∂xiaix,t,u,∇u=fx,tx,t∈Qux,t=0x,t∈∂Ω×0,Tux,0=0x∈Ω,where p>1, p′ is conjugate exponent of p, Q=Ω×(0,T), Ω is a nonempty, bounded, and open subset of RN, and f∈Lp′(Q) such that the following conditions are satisfied: (A1)
ai(x,t,s,ξ)  (i=1,2,…N) satisfies the Carathéodory conditions; that is, for each (s,ξ)∈R×RN, the function (x,t)↦ai(x,t,s,ξ) is measurable and, for almost all (x,t)∈Ω×(0,T), the function (s,ξ)↦ai(x,t,s,ξ) is continuous.(A2)
there exists positive constantsμ1 and μ2 such that (70)∑i=1Naix,t,s,ξ-aix,t,s,ηξi-ηi≥μ1ξ-ηp,aix,t,s,ξ≤μ2sp0+ξp-1+gx,tfor all (x,t,s,ξ)∈Ω×(0,T)×R×RN, where ξ=(ξi)∈RN and η=(ηi)∈RN, g∈Lp′(Q), p0<p-1+2(p-1)/N, and 1<p<N.Let X=Lp(0,T;V), V=W01,p(Ω), and L:X⊇D(L)→X∗ be defined by Lu=u′, where u′ is understood in the sense of distributions; that is,(71)∫0Tu′tψtdt=-∫0Tutψ′tdt,ψ∈C0∞0,T,where D(L)={u∈X:u′∈X∗, u(0)=0}. We notice that(72)Lu,ϕ=∫0Tu′t,ϕtVdt,u∈DL,ϕ∈X.Let A:X⊇D(A)→X∗ be defined by (73)Au,ϕ=∑i=1n∫Qaix,t,u,∇u∂ϕx,t∂xidxdt,ϕ∈X,u∈DA,where (74)DA=u∈X:u∈Lp¯Q,p¯=p0pp-1.It is known from Kartsatos and Skrypnik [4] that A is quasibounded demicontinuous generalized (S+) with respect to L such that D(A) contains D(L). Moreover, it is well-known that L is linear, closed, and densely defined maximal monotone. The operator A is densely defined, that is, not everywhere defined, and coercive. Since p0<p-1 in (A2), operator A may be unbounded. Therefore, by Theorem 8 using A in place of S, for each f∈Lp′(Q), we conclude that the equation Au+Lu=f∗ is solvable in D(L), where f∗:X→R is given by f∗,ϕ=∫Q‍f(x,t)ϕ(x,t)dxdt. Therefore, the parabolic problem (69) admits at least one weak solution in D(L).SinceΩ is bounded and g∈Lp′(Q), it is well known that A is bounded, continuous, everywhere defined, and coercive provided that p0=p-1 in (A2). More precisely, these conditions on A are satisfied if condition (A2) is replaced by(A3):
there exists positive constantsμ1 and μ2 such that (75)∑i=1Naix,t,s,ξ-aix,t,s,ηξi-ηi≥μ1ξ-ηp,aix,t,s,ξ≤μ2sp-1+ξp-1+gx,tfor all (x,t,s,ξ)∈Ω×(0,T)×R×RN, where ξ=(ξi)∈RN and η=(ηi)∈RN, g∈Lp′(Q), and 1<p<N.Abstract existence results concerning nonlinear parabolic problems of the type in (69) under conditions (A1) and (A3) 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 [12–14], 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 Ôtani [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(76)u∈X:u′+Au∋f∗inX∗,u0=u0,where X=Lp(0,T;W01,p(Ω)), p>1, Ω is a nonempty, bounded, and open subset of RN, and A 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 u∈Y:-Δpu+B(u)∋g∗,g∗∈Y∗, where Y=W01,p(Ω) and A and B are possibly noncoercive and densely defined and satisfy conditions of either Theorem 6 or Theorem 8.



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