@article{IBRAHIM_1998, title={ON $L^2_W$ QUASI-DERIVATIVES FOR SOLUTIONS OF PERTURBED GENERAL QUASI-DIFFERENTIAL EQUATIONS}, volume={29}, url={https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4264}, DOI={10.5556/j.tkjm.29.1998.4264}, abstractNote={<div class="page" title="Page 1"> <div class="layoutArea"> <div class="column"> <p><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">This paper is concerned with square integrable quasi-derivatives for any solution of a general quasi-differential equations of nth order with complex coefficients $M[y] </span><span style="font-size: 10.000000pt; font-family: ’DFKaiShu-SB-Estd-BF’;">-\lambda </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">wy </span><span style="font-size: 14.000000pt; font-family: ’TimesNewRomanPSMT’;">= </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">w f(t, y^{[0]}, \cdots ,y^{[n- 1]})$, $t \in</span><span style="font-size: 8.000000pt; font-family: ’TimesNewRomanPSMT’;"> [</span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">a,b)$ provided that all $r$th quasi-derivatives o</span><span style="font-size: 9.000000pt; font-family: ’TimesNewRomanPSMT’;">f </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">solutions of </span><span style="font-size: 8.000000pt; font-family: ’ArialMT’;"><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">$M[y] </span><span style="font-size: 10.000000pt; font-family: ’DFKaiShu-SB-Estd-BF’;">-\lambda </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">wy </span><span style="font-size: 14.000000pt; font-family: ’TimesNewRomanPSMT’;">= </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">0$</span> </span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">and all solutions of its formal adjoint $M^+[z] <span style="font-size: 10.000000pt; font-family: ’DFKaiShu-SB-Estd-BF’;">-\lambda </span>wz <span style="font-size: 14.000000pt; font-family: ’TimesNewRomanPSMT’;">= </span>0$</span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;"> are in </span><span style="font-size: 11.000000pt; font-family: ’DFKaiShu-SB-Estd-BF’;">$L _W^2</span><span style="font-size: 11.000000pt; font-family: ’TimesNewRomanPSMT’;">(a, b)$ and under suitable conditions on the function $f$. </span></p> </div> </div> </div>}, number={3}, journal={Tamkang Journal of Mathematics}, author={IBRAHIM, SOBHY EL-SAYED}, year={1998}, month={Sep.}, pages={175–185} }