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### Open Access Journals

MBE

Symmetric evolutionary games, i.e., evolutionary games with symmetric fitness matrices, have important applications in population genetics, where they can be used to model for example the selection and evolution of the genotypes of a given population. In this paper, we review the theory for obtaining optimal and stable strategies for symmetric evolutionary games, and provide some new proofs and computational methods. In particular, we review the relationship between the symmetric evolutionary game and the generalized knapsack problem, and discuss the first and second order necessary and sufficient conditions that can be derived from this relationship for testing the optimality and stability of the strategies. Some of the conditions are given in different forms from those in previous work and can be verified more efficiently. We also derive more efficient computational methods for the evaluation of the conditions than conventional approaches. We demonstrate how these conditions can be applied to justifying the strategies and their stabilities for a special class of genetic selection games including some in the study of genetic disorders.

PROC

The authors study a type of nonlinear fractional boundary value problem with nonlocal boundary conditions. An associated Green's function is constructed. Then a criterion for the existence of at least one positive solution is obtained by using fixed point theory on cones.

PROC

In this paper, by using the dual least action principle, the authors investigate the existence of solutions to a multi-point boundary value problem for a second-order differential system with $p$-Laplacian.

PROC

Sufficient conditions are obtained for the existence of multiple solutions to the
discrete fourth order periodic boundary value problem
\begin{equation*}
\begin{array}{l}
\Delta^4 u(t-2)-\Delta(p(t-1)\Delta u(t-1))+q(t) u(t)=f(t,u(t)),\quad t\in [1,N]_{\mathbb{Z}},\\
\Delta^iu(-1)=\Delta^iu(N-1),\quad i=0, 1,2, 3.
\end{array}
\end{equation*}
Our analysis is mainly based on the variational method and critical point theory.
One example is included to illustrate the result.

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