Results of a perturbation theory generating a one-parameter semigroup

: This paper consists of the results about ω -order preserving partial contraction mapping using perturbation theory to generate a one-parameter semigroup. We show that adding a bounded linear operator B to an infinitesimal generator A of a semigroup of the linear operator does not destroy A’s property. Furthermore, A is the generator of a one-parameter semigroup, and B is a small perturbation so that A + B is also the generator of a one-parameter semigroup.


Introduction
P erturbation theory comprises methods for finding an approximate solution to a problem; in perturbation theory, the solution is expressed as a power series in a small parameter ε. The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of ε usually become smaller. Assume X is a Banach space, X n ⊆ X is a finite set, T(t) the C 0 -semigroup, ω − OCP n the ω-order preserving partial contraction mapping, M m be a matrix, L(X) be a bounded linear operator on X, P n a partial transformation semigroup, ρ(A) a resolvent set, σ(A) a spectrum of A and A ∈ ω − OCP n is a generator of C 0 -semigroup. This paper consists of results of ω-order preserving partial contraction mapping generating a one-parameter semigroup.
Akinyele et al., [1] introduced perturbation of the infinitesimal generator in the semigroup of the linear operator. Batty [2] established some spectral conditions for stability of one-parameter semigroup and also in [3] Batty et al., revealed some asymptotic behavior of semigroup of the operator. Balakrishnan [4] obtained an operator calculus for infinitesimal generators of the semigroup. Banach [5] established and introduced the concept of Banach spaces. Chill and Tomilov [6] deduced some resolvent approaches to stability operator semigroup. Davies [7] obtained linear operators and their spectra. Engel and Nagel [8] introduced a one-parameter semigroup for linear evolution equations. Räbiger and Wolf [9] deduced some spectral and asymptotic properties of the dominated operator. Rauf and Akinyele [10] introduced ω-order preserving partial contraction mapping and established its properties, also in [11], Rauf et al., deduced some results of stability and spectra properties on semigroup of a linear operator. Vrabie [12] proved some results of C 0 -semigroup and its applications. Yosida [13] established and proved some results on differentiability and representation of one-parameter semigroup of linear operators.
In this paper, we show that adding a bounded linear operator B to an infinitesimal generator A of a semigroup of the linear operator does not destroy A's property. Furthermore, A is the generator of a one-parameter semigroup, and B is a small perturbation so that A + B is also the generator of a one-parameter semigroup.  (ω-OCP n ) [11] A transformation α ∈ P n is called ω-order preserving partial contraction mapping if ∀x, y ∈ Dom α : x ≤ y =⇒ αx ≤ αy and at least one of its transformation must satisfy αy = y such that T(t + s) = T(t)T(s) whenever t, s > 0 and otherwise for T(0) = I.

Definition 3. (Perturbation) [1] Let
A : D(A) ⊆ X → X be the generator of a strongly continuous semigroup (T(t)) t≥0 and consider a second operator B : D(B) ⊆ X → X such that the sum A + B generates a strongly continuous semigroup (S(t)) t≥0 . We say that A is perturbed by operator B or that B is a perturbation of A.

Definition 5. (Perturbation class) [7]
We say that operator B is a class P perturbation of the generator A of the one-parameter semigroup T(t) if: A is a closed operator; Note that BT(t) is bounded for all t > 0 under conditions (1) 1 and (1) 2 by the closed graph theorem.
and let T(t) = e tA , then e tA = e 2t e I e t e 2t .
and let T(t) = e tA , then

Main results
This section present results of one-parameter semigroup generated by ω-OCP n using perturbation theory.
Theorem 1. Let A ∈ ω − OCP n be the generator of a one-parameter semigroup T(t) z⩾0 on the Banach space X and suppose that ∥T(t)∥ ⩽ Me at for all t ⩾ 0. If B is a bounded operator on X, then (A + B) is the generator of a one-parameter semigroup S(t) t⩾0 on X such that for all t ⩾ 0 and B ∈ ω − OCP n .
Proof. We define the operators S(t) by The nth term is an n-fold integral whose integrand is a norm continuous function of the variables. It is easy to verify that the series is norm convergent and that for all f ∈ X, t ⩾ 0 and B ∈ ω − OCP n . Since S(s)S(t) = S(s + t) and if f ∈ X, then lim t→0 Me at ∥ f ∥(tM∥B∥) n /n! ⩾ 0 , so that s(t) is a one-parameter semigroup. If f ∈ X and B ∈ ω − OCP n , then It follows that f lies in the domain of the generator Y of S(t) if and only if it lies in the domain of A, and that for such f . As well as being illuminating in its own right, (2) easily leads to the identities Hence the proof is complete.

Theorem 2. Suppose B is a class P perturbation of the generator A, then
Dom(B) ⊇ Dom(A).

Proof. Combining (1) with the bound
valid for all t ⩾ 1, we then see that for all λ > a. Suppose ε > 0 and A, B ∈ ω − OCP n , then for all large enough λ we have for all f ∈ X, so by the closedness of B, we see that R(λ, A) f ∈ Dom(B) and as required to prove (7). If g ∈ Dom(A) and we put f := (λI − A)g, then we deduce from (7) that for all large enough λ > 0. This implies the last statement of the theorem and hence the proof is complete.
Theorem 3. Assume B is a class P perturbation of the generator A of the one-parameter semigroup T(t) on X, then B + A is the generator of a one-parameter semigroup S(t) on X and A, B ∈ ω − OCP n .

Proof.
Let a be small enough that We may define S(t) by the convergent series (2) for 0 ⩽ t ⩽ 2a, and verify as in the proof of Theorem 1 that S(s)S(t) = S(s + t) for all s, t ⩾ 0 such that s + t ⩽ 2a. We now extend the definition of S(t) inductively for t ⩾ 2a by putting S(t) := (S(a)) n S(t − na) , if n ∈ N and na < t ⩽ (n + 1)a. It is straight forward to verify that S(t) is a semigroup. Now suppose that Assume f ∈ X and B ∈ ω − OCP n , then and S(t) is a one-parameter semigroup on X. It is an immediate consequence of the definition that for all f ∈ X, B ∈ ω − OCP n and all 0 ⩽ t ⩽ a. Suppose that this holds for all t such that 0 ⩽ t ⩽ na. If na ⩽ u ⩽ (n + 1)a, then By induction, (12) holds for all t ⩾ 0. We finally have to identify the generator Y of S(t). The subspace is contained in Dom(A) and is invariant under T(t) and so is a core for A. If f ∈ D, then there exists g ∈ Dom(A) where A ∈ ω − OCP n and ε > 0 such that f = T(ε)g. Hence, Therefore, Dom(Y) contains D and Y f (B + A) for all f ∈ D and A, B ∈ ω − OCP n . If f ∈ Dom(A), then there exists a sequence f n ∈ D such that ∥ f n − f ∥ → 0 and ∥A f n − A f ∥ → 0 as n → ∞. It follows by Theorem 2 that ∥B f n − B f ∥ → 0 and hence that Y f n converges. Since Y is a generator that is closed, then we deduce that for all f ∈ Dom(A) and A, B ∈ ω − OCP n . Multiplying (12) by e −λt and integrating over (0, ∞), we see as in the proof of Theorem 2 that if λ > 0 is large enough, then If λ is also large enough that ∥BR(λ, A)∥ < 1 , Hence, Dom(Y) = Ran(R(λ, Y)) = Ran(R(λ, A)) = Dom(A) , and Y = A + B, and this achieve the proof.
If a α ∈ L P α (R N ) + L ∞ (R N ) for each α, where P α ⩾ 2 and P α > N/(2n − |α|), the A + B is the generator of a one-parameter semigroup and B has relative bound 0 with respect to A where A, B ∈ ω − OCP n .
Proof. Suppose A ∈ ω − OCP n is the generator of holomorphic semigroup T(t) such that for all t ∈ (0, 1). And also the operator B ∈ ω − OCP n has domain containing Dom(A) and there exists α ∈ (0, 1), such that for all f ∈ Dom(a) and 0 < ε ⩽ 1. Then for all t ∈ (0, 1) so that B is a class P perturbation of A and by Theorem 3 under the stated conditions on t and ε, we have By putting ε = t 1−α , then we obtain (16). Assume α ∈ (0, 1), H is a non-negative self-adjoint operator on P and B is a linear operator with for all ε > 0 if and only if there is a constant c 4 such that for all f ∈ Dom(A) and A, B ∈ ω − OCP n . By Theorem 3, it is sufficient to prove that for each α there exists β < 1 for which If a α ∈ L ∞ (R N ), then ∥X∥ ⩽ ∥a α ∥ ∞ ∥b α ∥ ∞ < ∞ provided |α|/2n < β < 1. On the other hand, if a α ∈ L P (R N ) where P ⩾ 2 and P > N/(2n − |α|), then there exists β such that N + |α|P 2np < β < 1.

Conclusion
In this paper, it has been established that ω-order preserving partial contraction mapping generates a one-parameter semigroup using a perturbation theory on Banach space by showing that the semigroup of a linear operator is bounded, that B has a relative bound 0 with respect to A, and also that B + A is a generator of the one-parameter semigroup.