وجود نقاط ثابت براي نگاشت هاي α -پذيرفتني گرختي تعميم يافته و كاربرد آن در حل معادلات ديفرانسيل غير خطي

Existence of fixed points for generalized α-admissible Geraghty and application to solution of nonlinear differential equations

اﺧﯿﺮا، ﺳﺎﻣﺖ و ﻫﻤﮑﺎران ﺗﻌﻤﯿﻢ ﺟﺎﻟﺒﯽ از اﺻﻞ اﻧﻘﺒﺎض ﺑﺎﻧﺎخ را اراﺋﻪ ﮐﺮده اﻧﺪ. در اﯾﻦ ﻣﻘﺎﻟﻪ ﺑﺎ اﻟﻬﺎم ﮔﺮﻓﺘﻦ از اﯾﺪه اﺻﻠﯽ ﺳﺎﻣﺖ و ﻫﻤﮑﺎران، ﻧﮕﺎﺷﺖﻫﺎي ﮔﺮﺧﺘﯽ α-ﭘﺬﯾﺮﻓﺘﻨﯽ θ − α -ﺗﻌﻤﯿﻢ ﯾﺎﻓﺘﻪ در ﻓﻀﺎﻫﺎي ﻣﺘﺮي را ﻣﻌﺮﻓﯽ و ﭼﻨﺪﯾﻦ ﻗﻀﯿﻪ وﺟﻮد و ﯾﮑﺘﺎﯾﯽ ﻧﻘﻄﻪ ﺛﺎﺑﺖ در ﻓﻀﺎﻫﺎي ﻣﺘﺮي ﮐﺎﻣﻞ را ﺑﺮاي ﭼﻨﯿﻦ ﻧﮕﺎﺷﺘﻬﺎﯾﯽ ﻣﻄﺮح و ﺛﺎﺑﺖ ﻣﯽﮐﻨﯿﻢ. ﻧﺘﺎﯾﺞ ﺑﺪﺳﺖ آﻣﺪه در اﯾﻦ ﭘﮋوﻫﺶ، ﺑﺴﯿﺎري از ﻧﺘﺎﯾﺞ ﻣﻮﺟﻮد در اﯾﻦ زﻣﯿﻨﻪ ﺑﺨﺼﻮص ﻧﺘﺎﯾﺞ ﻣﻮﺟﻮد در ﻣﻘﺎﻟﻪ ﺟﻠﯿﻠﯽ و ﻫﻤﮑﺎران و ﮐﺎر اﻧﺠﺎم ﺷﺪه ﺗﻮﺳﻂ ﮔﺮﺧﺘﯽ را ﺗﻌﻤﯿﻢ ﻣﯽدﻫﺪ. در اداﻣﻪ، ﺑﺎ اراﺋﻪ ﻣﺜﺎﻟﯽ ﻧﺸﺎن ﻣﯽدﻫﯿﻢ ﮐﻪ ﻧﺘﺎﯾﺞ ﻣﺎ ﺗﻌﻤﯿﻢ واﻗﻌﯽ از ﻧﺘﺎﯾﺞ ﻣﻮﺟﻮد ﻗﺒﻠﯽ در اﯾﻦ زﻣﯿﻨﻪ اﺳﺖ. ﺳﭙﺲ، ﻧﺘﺎﯾﺞ ﺟﺪﯾﺪي در ﻓﻀﺎﻫﺎي ﻣﺘﺮي ﻣﺮﺗﺐ ﺟﺰﺋﯽ و ﻓﻀﺎﻫﺎي ﻣﺘﺮي ﮔﺮاﻓﯿﮏ ﺑﺪﺳﺖ ﻣﯽآورﯾﻢ. در ﭘﺎﯾﺎن، ﮐﺎرﺑﺮدي از ﻧﺘﺎﯾﺞ ﺑﺪﺳﺖ آﻣﺪه را در زﻣﯿﻨﻪ وﺟﻮد و ﯾﮑﺘﺎﯾﯽ ﺟﻮاب ﻣﻌﺎدﻻت دﯾﻔﺮاﻧﺴﯿﻞ ﻣﻌﻤﻮﻟﯽ ﻣﺮﺗﺒﻪ اول ﻏﯿﺮ ﺧﻄﯽ و ﻣﺴﺎﺋﻞ ﻣﻘﺪار ﻣﺮزي ﻣﺘﻨﺎوب اراﺋﻪ ﻣﯽدﻫﯿﻢ.

Recently, samet et al. introduced an interesting extension of the Banach contraction principle. In this paper, motivated by the main idea of Samet et al., we introduce the concept of α-admissible α-θ-generalized mappings in metric spaces and give and prove several theorems of the existence and uniqueness of a fixed point in complete metric spaces for such mappings. The results obtained in this study, generalize many of the results in this field, especially, the results presented by Jleli et al. and the work done by Geraghty. By presenting an example, we show that our results are real generalization of the previous results. Next, we get new results in ordered metric spaces and graphical metric spaces using the concept of α-admissible α-θ-generalized mappings. Finally, we present an application of our obtained results for the existence and uniqueness of the solution of nonlinear first-order ordinal differential equations and periodic boundary value problems.