بعضی از تعمیم های قضیه داربو برای حل یک دستگاه از معادلات انتگرال تابعی با استفاده از اندازه نافشردگی
محورهای موضوعی : آمار
1 - استاد یار دانشکده ریاضی و آمار، دانشگاه آزاد اسلامی واحد قائم شهر
کلید واژه: System of functional integral equations, Banach space, Measure of noncompactness, fixed point,
چکیده مقاله :
در این مقاله با استفاده از مفهوم اندازه نافشردگی، که یک ابزار بسیار مفید و قدرتمند در آنالیز تابعی غیرخطی و نظریه نقطه ثابت متریک و معادلات انتگرال است، یک انقباض جدید در فضای باناخ معرفی می‌‌‌کنیم. برای این منظور با استفاده از یک اندازه نافشردگی روی یک فضای حاصل ضرب متناهی، تعمیم هایی از قضیه نقطه ثابت داربو بدست می‌آوریم. آنگاه با استفاده از نتایج حاصله، چند قضیه در وجود زوج نقطه ثابت برای رده‌ای از عملگرها در فضای باناخ ارائه می دهیم. نتایج حاصله بسیاری از نتایج قابل مقایسه را در پیشینه تحقیق بسط و توسعه می دهد. همچنین به عنوان یک کاربرد به مطالعه وجود جواب برای یک رده از دستگاه معادلات انتگرال تابعی غیر خطی می‌پردازیم که توابع و عملگرها در عملگرهای انتگرال وابسته، در یک شرط انقباض خاص صدق می‌کنند. سرانجام یک مثال ملموس نیز گنجانده شده است که کاربرد نتایج بدست آمده را نشان می‌دهد.
In this paper, using the concept of measure of noncompactness, which is a very useful and powerful tools in nonlinear functional analysis, metric fixed point theory and integral equations, we introduce a new contraction on a Banach space. For this purpose by using of a measure of noncompactness on a finite product space, we obtain some generalizations of Darbo’s fixed-point theorem. Then, with the obtained results, we present some theorems on the existence of coupled fixed point for a class of operators in a Banach space. Our results generalize and extend a lot of comparable results in the literature. Also as an application, we study the existence of solution for a class of the system of nonlinear functional integral equations, which the functions and operators in the related integral operators, satisfies in a particular contraction. Finllay a concrete example is also included, which demonstrates the applicability of the obtaind results.
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