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Approximating Monotone Positive Solutions of a Nonlinear Fourth-Order Boundary Value Problem via Sum Operator Method

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Abstract

In this article, the authors investigate the existence and uniqueness as well as approximations of monotone positive solutions for a nonlinear fourth-order boundary value problem of the form \( u^{(4)}(t)=f(t,u(t)),\ 0< t<1; u(0)=u'(0)= u'(1)=0, u^{(3)}(1)+g(u(1))=0,\) where \(f\in C([0,1]\times [0,+\infty ),[0,+\infty )),\ g\in C([0,+\infty ),[0,+\infty )).\) It is shown that the above boundary value problem has a unique monotone positive solution and the sequence of successive approximations converges to the monotone positive solution under some proper conditions. These results are based upon two fixed point theorems of a sum operator in partial ordering Banach space. Finally, two examples are also given to illustrate the main abstract results.

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanxi University, Taiyuan, 030006, Shanxi, People’s Republic of China

    Chengbo Zhai, Chunrong Jiang & Shunyong Li

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  1. Chengbo Zhai
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  2. Chunrong Jiang
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  3. Shunyong Li
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Correspondence to Chengbo Zhai.

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Zhai, C., Jiang, C. & Li, S. Approximating Monotone Positive Solutions of a Nonlinear Fourth-Order Boundary Value Problem via Sum Operator Method. Mediterr. J. Math. 14, 77 (2017). https://doi.org/10.1007/s00009-017-0844-7

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  • DOI: https://doi.org/10.1007/s00009-017-0844-7

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