حلپذیری معادلات انتگرال-دیفرانسیل تابعی در فضای سوبولوف W^(k,∞) (R^n)
محورهای موضوعی : آمارمعصومه حسینی فرهی 1 , محمود حسنی 2 , رضا الهیاری 3
1 - گروه ریاضی، واحد مشهد، دانشگاه آزاد اسلامی، مشهد، ایران
2 - گروه ریاضی، واحد مشهد، دانشگاه آزاد اسلامی، مشهد، ایران
3 - گروه ریاضی، واحد مشهد، دانشگاه آزاد اسلامی، مشهد، ایران
کلید واژه: Measures of noncompactness, Integral-differential equations, Sobolev spaces, Caratheodory condition, Darbo's fixed point theorem,
چکیده مقاله :
در سال 1930، کراتوسکی مفهوم اندازه نا فشردگی را معرفی کرد. سپس، بنس وگوبل این مفهوم را تعمیم دادند که کارایی بیشتری دارد. کاربرد اصلی اندازه‌های نافشردگی در نظریه نقطه ثابت، در قضیه نقطه ثابت داربو است. این یک ابزار برای بررسی وجود و رفتار جواب تعدادی معادلات انتگرال مانند انواع ولترا ، فردهولم و اورایسون است. روش اندازه‌های‌نافشردگی‌اغلب‌درچندین شاخه‌آنالیزغیرخطی‌قابل‌اجرااست. به ویژه، این روش به عنوان ابزاری بسیار مفید برای چندین نوع از انواع معادلات انتگرالی و انتگرال-دیفرانسیلی است. علاوه بر این، اندازه نافشردگی در معادلات تابعی، معادلات دیفرانسیل جزئی کسری، معادلات دیفرانسیل معمولی و جزئی، نظریه عملگر و نظریه کنترل بهینه نیز استفاده می شود. هدف این مقاله معرفی یک اندازه نافشردگی جدید در فضای سوبولف W^(k,∞) (R^n) است. نتایج بدست آمده در حل معادلات انتگرال-دیفرانسیلی بکار می رود. در پایان نیز با ارائه یک مثال کارایی نتایج حاصل می شود.
In 1930, Kuratowski introduced the concept of measure of noncompactness. Later, Banas and Goebel generalized this concept axiomatically, which is more convenient in applications. The principal application of measures of noncompactness in fixed point theory is contained in the Darbo'sfixed point theorem. This is a tool to investigate the existence and behaviour of solutions of manyclasses of integral equations such as Volterra, Fredholm and Uryson types.The technique of measure of noncompactness is applicable in several branches of nonlinear analysis. In particular, it is a very useful tool for several types of integral and integral-differential equations. In addition, the measure of noncompactness is also used in functional equations, fractional partial differential equations, ordinary and partial differential equations, operator theory and optimal control theory. The purpose of this article is to introduce a new measure of noncompactness in the Sobolev space W^(k,∞) (R^n). The results are obtained to solve integral-differential equations. Finally, by providing an example to show the efficiency of our results.
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