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Special Session 105: Nonlinear Differential Problems on Flat and Curved Structures: Variational and Topological Methods

A Cauchy problem and a semigroup of positive operators

AUGUSTA RATIU

LUCIAN BLAGA UNIVERSITY OF SIBIU
Romania

Co-Author(s): Augusta Ratiu, Mihai Ilina

Abstract:

Let

$j\in {\mathbb Z}$ . Motivated by Swiderski's result [4], the following Cauchy problem

$\left\{ \begin{array}{l} u_t^{\prime}=xu_{xx}^{\prime\prime}-(j-1)u_x^{\prime},\, x\geq 0,\, t>0, \nonumber \ \displaystyle\lim_{t\to 0^{+}}u(t,x)=f(x),\,\, x\geq 0, \nonumber\end{array}\right.$ was considered in the paper [2]. A semigroup of positive operators was investigated in [3] and it was proved that it provides a solution of the Cauchy problem. Direct approaches to find solutions were also considered in [3]. The Cauchy problem $\left\{ \begin{array}{l} u_t^{\prime}=\dfrac{x}{2}u_{xx}^{\prime\prime},\, x\geq 0,\, t>0, \nonumber \ \displaystyle\lim_{t\to 0^{+}}u(t,x)=f(x),\,\, x\geq 0, \nonumber\end{array}\right.$ was investigated in [1] using the theory of $C_0$ -semigroups. We will present the connections between the solutions of the two problems. References [1] F. Altomare, I. Carbone, On Some Degenerate Differential Operators on Weighted Function Spaces, J. Math. Anal. Appl. 213 (1997), 308-333. [2] U. Abel, A.M. Acu, M. Heilmann, I. Ra\c sa, On some Cauchy problems and positive linear operators (manuscript) [3] U. Abel, A.M. Acu, M. Heilmann, I. Ra\c sa, Commutativity and spectral properties for a general class of Sz\`asz-Mirakjan-Durrmeyer operators, arXiv:2407.21722 [4] T. Swiderski, Global approximation theorems for the generalized modified Sz\`asz-Mirakyan operators in polynomial weight spaces, Demo. Math., 36(2), 2003, 383-392.

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