Existence of Some New Classes of Semilinear Unbounded Perturbed Operator Equations

In this paper, the adapted approach of existence based on Leray-Schauder degree theorem. The necessary theorems of nonexpansive perturbed operators, lemmas and propositions for existence and uniqueness of proposed classes of semilinear perturbed unbounded operator equations have been adapted and developed with proofs and supported by some illustrative examples


INTRODUCTION
et H be a real Hilbert space endowed with the inner product , 〈⋅ ⋅〉 and the norm . .
In [13], general class of nonlinear operators ( ) F x x µ = , 0 µ > in a Banach space, which we call projectionally-compact (P-compact) and which, among others, contains completely continuous, quasi-compact, and monotone operators and general fixed point theorem for deduce in a simple way the fixed point theorems of Schauder, Rothe, Kaniel and others,have been established.
To study the collection of nonlinear operator (.) F and linear operator A to get the semilinear operator equation, the following literatures are very useful. Locker in [7], the semilinear equation Equations of this type can be considered as abstract formulations for a variety of problems in differential equations as examples: 1.
The solutions x of the elliptic boundary value problem ( , ) Where Ω is a bounded smooth domain in .

 
In each case we assume that the nonlinearities are given smooth functions.
Okazawa in [11] has studied some properties on A B using only the Brouwer degree theory and the finite dimensional Morse theory has been presented and studied in [8].
The solvability of the equation Au Tu Cu f − + = is studied under various assumptions of monotonicity and compactness on the operators A, T and C, which maps subsets of reflexive Banach space X in to dual space, [5]. Mortici in [9], [10] is a linear maximal monotone, strongly positive operator, : J H H → is a duality map and F is a Lipschitz operator.
In this paper, we focus on study the types of densely define unbounded linear operator is self-adjoint or generator of nonexpansive perturbed semigroup and suggested the basic condition for solving semilinear perturbed unbounded operator equations properties. The solvability of some such classes with perturbation dependent on the some estimator establish on the values of the resolvent set and using different type of fixed point theorems, such as Leray-Schauder mapping degree have been presented.

Preliminaries:
In this section, we review some basic definitions and theorems that we needed later on.
A semigroup of bounded linear operator, T(t), is:

Definition (2.2), [1]:
The linear operator A defined on the domain: Let ( ) T t be C  -semigroup and let A be its generator, then: i. ii.

Theorem (2.1), [12]:
A linear operator A is the generator of a uniformly continuous semigroup if and only if A is a bounded linear operator.

Definition (2.3), [4]:
Let X and Y be two normed spaces and Mbe a subspace of X, then the linear operator T defined on M into Y is called closed linear operator if for every convergent sequence { } n x of points of M with the limit x∈X, such that the sequence { } n Tx is convergent sequence with the limit y,x∈M and y Tx = .
We define the duality set

Definition (2.4), [12]:
A linear operator A is dissipative if for every

Lemma(2.1), [12]:
Let X be a Banach space and A generates a strongly continuous semigroup of The resolvent set ( ) A ρ of A contains the ray { λ : Im is bounded linear operator in X, where Imλ is standing for imaginary part of a complex eigenvalue.

Theorem (2.2), [12]:
Let X be a Banach space and A be the unbounded linear generator of a C semigroup ) ( T t on X, satisfying:

Definition (2.7) "Compact Operator", [19]:
Let X and Y be normed spaces, the operator A is continuous, and 2.
A transforms bounded subset M of X into relatively compact subset in Y ( ( ) A M is compact).

Lemma (2.5) "Compactness of Product", [4]: Let
: A X X → be a compact linear operator and : B X X → a bounded linear operator on a normed space X. Then AB and BA are compact.

Lemma (2.6) "Compactness Perturbation", [18]:
Let f,g, h : X X → be mappings on a Banach space X, then f is called compact perturbation of the mapping g if and only if f g h = + and h is compact.
If dim X = ∞ , then the identity operator : iii.
If dim X < ∞ , then the identity operator :

2.
Let X and Y be Banach spaces and : A X Y → be an operator: i. If A is a compact linear operator, then A is completely continuous. ii.
If X is reflexive and A (need not be linear) is completely continuous then A is compact, [3]. 3.
If A andB are completely continuous operators, then 1 is also a completely continuous operator, where 1 2 , α α are scalar values.

Definition (2.9), [18]:
Let X and Y be real Hilbert space, and let : A X X * → be an operator. Then 1.
A is called monotone if and only if A is called strictly monotone if and only if A is called strongly monotone if and only if there is a constant A is called maximal monotone if and only if A is monotone and , A has no proper monotone extension.

2.
A is called accretive if and only if ( A A is monotone and R(I + A) = H.

3.
A is maximal monotone.

Lemma (2.9),[11]:
Let A be a linear operator with domainD(A) and rangeR(A) in a Banach space X. Let B be a linear operator in X, such that Then the following conditions are mutually equivalent:

1.
A is the generator of a linear nonexpansivesemigroup.

Definition (2.12) "Leray-Schauder type (LS)", [2]:
The operator of the type I T − in infinite dimensional space, with I the identity mapping and T compactness called compact displacements of the identity.  3. The degree for the identity map :

Solvability of Semilinear Perturbed Unbounded Class by Using Leray-Schauder Degree Theory
The proposed results of the following theorem depends on the results of Leray-Schauder degree, homotopy theorem of compact perturbation theorem and nonexpansive perturbed semigroup theory with some necessary conditions on perturbation operator, so the solvability is guaranteed , but the uniqueness is not necessary, so the existence of at least one solution is discussed. In the following lemmas, we suggest important assumptions which make sense in some useful properties to develop and generalize the results later on.
Where , a b and c are nonnegative constants, with (3.8)

X.
The following lemma is important in the following result and its proof is helping us to go further, so it has been imposed for benefit point of view.  Our aim to examine the existence solution of the following system (3.14) From lemma (2.8), we have that ( ) A B − + is a unique maximal monotone on H.

µ =
Now, the Equation (3.14) can be written as ) . By using Leray-Schauder, we have that (

Concluding remark (3.2):
The selection of λ in theorem (3.4), will not effect the solvability result. The solution is independent of chose of λ .  ∈ Ω 0 p a > > 0 is studied in [10], under the assumption that the nonlinear part is quasi-positive, we have that

Fx t g t x =
The problem (3.22) can be written in the abstract form The new results of the following theorem depend on the results of Leray-Schauder degree, homotopy theorem of compact perturbation theorem of nonlinear operators and some necessary conditions of a self-adjoint densely defined linear operator, so the solvability is guaranteed, but the uniqueness is not necessary, so the existenece of at least one solution is discussed.