Results of Semigroup of Linear Operator Generating a Quasilinear Equations of Evolution

Abstract

In this paper, results of $\omega$-order preserving partial contraction mapping generating a quasilinear equation of evolution were presented. In general, the study of quasilinear initial value problems is quite complicated. For the sake of simplicity we restricted this study to the mild solution of the initial value problem of a quasilinear equation of evolution. We show that if the problem has a unique mild solution $v\in C([0,T]: X)$ for every given $u\in C([0,T]:X)$, then it defines a mapping $u\to v=F(u)$ of $C([0,T]:X)$ into itself. We also show that under the suitable condition, there exists always a $T',\ 0<T'\leq T$ such that the restriction of the mapping $F$ to $C([0,T']:X)$ is a contraction which maps some ball of $C([0,t']:X)$ into itself by proving the existence of a local mild solution of the initial value problem.

Results of Semigroup of Linear Operators Generating a General Class of Semilinear Initial Value Problems

“…Omosowon et al [13], generated some analytic results of semigroup of linear operator with dynamic boundary conditions, and also in [11], Omosowon et al, introduced dual Properties of ω-order Reversing Partial Contraction Mapping in Semigroup of Linear Operator. Omosowon et al [10], established a regular weak*-continuous semigroup of linear operators, and also in [9], Omosowon et al, obtained a quasilinear equations of evolution on semigroup of linear operator.reversing partial contraction mapping generating a differential operator. Omosowon et al [12], deduced results of semigroup of linear equation generating a wave equation.…”

Results of semigroup of linear operators in extrapolation spaces

“…Neerven [8], presented some results on adjoint of semigroup of linear operators. Both in [9] and [10], Omosowon et al built a regular weak*-continuous semigroup of linear operators and presented quasilinear equations of evolution on the semigroup of linear operator, a differential operator is produced by partially contraction mapping in reverse.…”

Results of Hankel Semigroup of Linear Operator as Eventually Uniform Continuous Semigroup

“…Neerven [7], established the asymptotic behavior of semigroup of linear operator. Omosowon et al [8], proved some analytic results of semigroup of linear operator with dynamic boundary conditions, and also in [9], Omosowon et al, deduced dual Properties of ω-order Reversing Partial Contraction Mapping in Semigroup of Linear Operator. Partington [10], introduced Hankel operators.…”