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.
[1] A. Aghajani, R. Allahyari, M. Mursaleen. A generalization of Darbos theorem with application to the solvability of systems of integral equations. Journal of Computational and Applied Mathematics
[2] A. Aghajani, J. Bans, N. Sabzali. Some generalizations of Darbo fixed point theorem and applications. Bulletin of the Belgian Mathematical Society- Simon Stevin
[3] A. Aghajani, N. Sabzali. Existence of coupled fixed points via measure of noncompactness and applications. Journal of Nonlinear and Convex Analysis 14(5):
[4] R. Allahyari, R. Arab, A. Shole Haghighi. Measure of noncompactness in a Sobolev space and integro-differential equations. Bulletin of Australian Mathematical Society
[5] R. Allahyari, R. Arab, A. Shole Haghighi. Construction of measures of noncompactness of and with application to the solvability of th-order integro-differential equations in Banach spaces. Advances in Difference Equations
[6] R. Arab. The existence of fixed points via the measure of noncompactness and its application to functional-integral equations. Mediterranean Journal of Mathematics13:759-773 (2016),
[7] J. Bana . One Measures of noncompactness in Banach spaces. Commentationes Mathematicae Universitatis Carolinae
[8] J. Bana , K. Goebel. Measures of noncompactness in Banach Space. Lecture Notes in pure and Applied Mathematics vol. Marcel Dekker, New York
[9] J. Bana , R. Rzepka. An application of a measure of noncompactness in the study of asymptotic stability. Applied Mathematics Letters :
[10] S. S. Chang, Y. J. Cho, N. J. Huang. Coupled fixed point theorems with applications. Journal of the Korean Mathematical Society (3):
[11] G. Darbo. Punti uniti in transformazioni a codomio non compatto. Rendiconti del Seminario Matematico della Universita di Padova :
[12] J. Garcia-Falset, K. Latrach. On Darbo-Sadovskii’s fixed point theorems type for abstract measures of (weak) noncompactness. Bulletin of the Belgian Mathematical Society- Simon Stevin
[13] L. S. Goldenstein, I. T. Gohberg, A. S. Markus. Investigations of somme properties of bounded linear operators with their q-norms. Ucen. Zap. Kishinovsk
[14] M. Mursaleen, S.A. Mohiuddine. Applications of measures of noncompactness to the infinite system of differential equations in spaces. Nonlinear Analysis
[15] M. Mursaleen, A. Alotaibi. Infinite system of differential equations in some spaces. Abstract and Applied Analysis Volume 2014: Article ID
[16] J. R. Roshan. Existence of solutions for a class of system of functional integral equation via measure of noncompactness. Journal of Computational and Applied Mathematics 313: 129-141(2017)
[17] A. Samadi, M. B. Ghaemi. An Extension of Darbo’s Theorem and Its Application. Abstract and Applied Analysis Volume 2014: Article ID 852324(2014)
[18] A. Samadi, M. B. Ghaemi. An Extension of Darbo Fixed Point Theorem and its Applications to Coupled Fixed Point and Integral Equations. Filomat
[19] N. Khodabakhshi, S.M. Vaezpour. Common fixed point theorems via measure of noncompactness. Fixed Point Theory 17(2): 381-386(2016)
[20] K. Kuratowski. Sur les espaces complets. Fundamenta Mathematicae
[21] B.N. Sadovskii. Limit compact and condensing operators. Russian Mathematical Surveys 27: 86-144 (1972)
