24020

MAT2 699

كارشناسي ارشد

رياضيات و كاربردها- آناليز

رياضي و آمار

1403/06/27

96 ص.

عليرضا اميني هرندي

ﻓﻀﺎﯼ متريك ﮐﺎﻣﻞ , ﻓﻀﺎﯼ ﺑﺎﻧﺎﺥ , ﻧﻘﻄﻪ ﺛﺎﺑﺖ , ﺩﻧﺒﺎﻟﻪ ﺗﮑﺮﺍﺭﺷﻮﻧﺪﻩ , ﻧﮕﺎﺷﺖ ﻧﺎﺍﻓﺰﺍ , ﻧﮕﺎﺷﺖ ﺍﻧﻘﺒﺎضي , ﻓﻀﺎﯼ ﻫﯿﻠﺒﺮﺕ

ﺍﺑﺘﺪﺍ ﺑﺮﺍﯼ يك ﺭﺩﻩ ﺟﺪﯾﺪ ﺍﺯ ﻧﮕﺎﺷﺖﻫﺎﯼ ﻏﯿﺮﺧطي ﻧﺎﺍﻓﺰﺍﯼ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ، يك ﻗﻀﯿﻪ ﻧﻘﻄﻪ ﺛﺎﺑﺖ ﺩﺭ ﻓﻀﺎﻫﺎﯼﺑﺎﻧﺎﺥ ﺭﺍ ﺍﺛﺒﺎﺕ مي‌كنيم. ﺳﭙﺲ ﺑﺮﺍﯼ يك ﺭﺩﻩ ﺍﺯ ﻧﮕﺎﺷﺖﻫﺎﯼ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ ﺗﻌﺮﯾﻒ ﺷﺪﻩ ﺑﺮ ﺯﯾﺮ ﻣﺠﻤﻮﻋەﻫﺎﯼ ﻣﺤﺪﺏ، ﺑﺴﺘﻪ ﻭ ﮐﺮﺍﻧﺪﺍﺭ ﺍﺯ يك ﻓﻀﺎﯼ ﺑﺎﻧﺎﺥ ﻣﻔﻬﻮﻡ ﻧﮕﺎﺷﺖ ﺍﻧﻘﺒاضي ﺭﺍ ﺗﻌﺮﯾﻒ ﮐﺮﺩﻩ ﻭ ﻧﺸﺎﻥ مي‌دهيم ﮐﻪ ﻫﺮ ﻧﮕﺎﺷﺖ ﺍﻧﻘﺒﺎضي يك ﻧﻘﻄﻪ ﺛﺎﺑﺖ يكتا ﺩﺍﺭﺩ، ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ ﺩﻧﺒﺎﻟﻪ ﺗﮑﺮﺍﺭﻫﺎﯼ ﻧﮕﺎﺷﺖ همگراي يكنواخت )همگرايي ﺩﻗﯿﻖ( ﺑﻪ ﻧﻘﻄﻪ ﺛﺎﺑﺖ مي‌باشد. ﺳﭙﺲ ﻧﺘﺎﯾﺞ ﺧﻮﺩ ﺭﺍ ﺑﻪ ﻣﺠﻤﻮﻋەﻫﺎﯼ ﻏﯿﺮﮐﺮﺍﻧﺪﺍﺭ ﮔﺴﺘﺮﺵ مي‌دهيم ﻭ ﺑﺎ ﺩﺭ ﻧﻈﺮ ﮔﺮﻓﺘﻦ ﺧﻄﺎﻫﺎﯼ محاسباتي همگرايي ﻧﺎﺩﻗﯿﻖ ﺗﮑﺮﺍﺭﻫﺎﯼ ﻧﮕﺎﺷﺖﺭﺍ ﻧﺘﯿﺠﻪ ﻣͬﮔﯿﺮﯾﻢ. ﺩﺭ ﺍﻧﺘﻬﺎ يك ﻣﺜﺎﻝ ﺍﺯ يك ﺭﺩﻩ ﺟﺪﯾﺪ ﻧﮕﺎﺷﺖﻫﺎﯼ ﻧﺎﺍﻓﺰﺍﯼ ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ ﺍﺯ ﻣﺠﻤﻮﻋەﻫﺎﯼ ﻣﺤﺪﺏ، ﺑﺴﺘﻪ ﻭ ﮐﺮﺍﻧﺪﺍﺭ ﺩﺭ يك ﻓﻀﺎﯼ ﻫﯿﻠﺒﺮﺕ ﺳﺎﺧﺘﻪ ﺑﻪ ﻃﻮﺭﯼ ﮐﻪ ﺍﺯ ﻟﺤﺎﻅ ﮐﻼسيك ناافزا نمي‌باشد.

Complete metric space , Banach space , Fixed point , Iterate sequence , Nonexpansive mapping , Contractive mapping , Hilbert space

Fixed points of a class of generalized nonexpansive and contractive mappings

رياضي محض

First we stablish a fixed point theorem for a new class of generalized nonexpansive nonlinear mappings in a Banach space. Then for a class of generalized mappings of a bounded, closed and convex subset of a Banach space, we introduce a new notion of contractivity mapping and show that every such contractive mapping has a uniqe fixed point which uniformly attracts all the iterats of the mapping (exact convergence). Finally we construct an example of a new class of generalized nonexpansive mappings of a bounded, closed and convex set in a Hilbert space, which is not nonexpansive in the classical sense.

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