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Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

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Abstract

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of “bang–bang” principle for control systems is given as well by utilizing the relaxation theorem.

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Acknowledgements

The author is indebted to Professor Yong Li for his encouragement and helpful discussion. The author sincerely thanks the anonymous referees for their valuable suggestion and helpful comments that improved the paper. This work is partially supported by NSFC Grant (11301541).

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Authors and Affiliations

  1. Department of Mathematics, Bohai University, Jinzhou, 121013, People’s Republic of China

    Yi Cheng

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  1. Yi Cheng
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Correspondence to Yi Cheng.

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Communicated by Bernard Dacorogna.

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Cheng, Y. Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions. J Optim Theory Appl 162, 13–33 (2014). https://doi.org/10.1007/s10957-013-0446-x

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