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The averaging principle and periodic solutions for nonlinear evolution equations at resonance. (English) Zbl 1292.34059
The author considers semilinear differential equations of the form $u'(t)=-Au(t)+\lambda u(t)+F(t,u(t))\quad\text{for }\;t>0, \tag{1}$ where $$\lambda$$ is a real number, $$A: X\supset D(A) \to X$$ is a sectorial operator acting on a Banach space $$X$$ and $$F: [0,+ \infty) \times X^{\alpha} \to X$$ is a continuous mapping (here, $$X^{\alpha}$$ for $$\alpha \in (0,1)$$ is a fractional power space defined by $$X^{\alpha}:=D((A+\delta I)^{\alpha})$$, where $$\delta >0$$ is such that the operator $$A+\delta I$$ is positive definite).
The author begins his examination by considering a special direct sum decomposition of a Banach space $$X:=X_{-}\oplus X_{0}\oplus X_{+}$$. Next, assuming that the mapping $$F$$ depends also on a parameter $$s\in [0,1]$$, he proves, under suitable assumptions on $$F$$ and $$A$$, that the equation ({1}) possesses a unique mild solution, and he discusses the continuity of mild solutions, with respect to initial data and parameters.
Further, assuming that the nonlinearity in the equation ({1}) is of the form $$\epsilon F$$, where $$\epsilon \in[0,1]$$ is a parameter, he proves the resonant version of the averaging principle for such equation, expressing the fixed point index of the translation along trajectories in terms of the averaging of the right-hand side of the equation under consideration.
In Section 5, which seems to be the main part of this paper, the author, imposing some geometrical conditions on the nonlinearity $$F$$, proves the index formula for periodic solutions to the equation ({1}) being at resonanse at infinity, which determines the Leray-Schauder degree of the vector field $$I-\Phi_{T}$$, where $$\Phi_{T}: X^{\alpha} \to X^{\alpha}$$ denotes the translation along trajectories, with respect to a ball with sufficiently large radius. Obviously, this theorem is a tool for looking for fixed points of the operator $$\Phi_{T}$$. As an application, the author studies the existence of $$T$$-periodic solutions for partial differential equation of the form $u_{t}(t,x)=-{\mathcal A}u(t,x)+\lambda u(t,x)+f(t,x,u(t,x),\nabla u(t,x)),\quad t>0,\;x\in\Omega,$ being at resonanse at infinity, where $$\lambda \in \mathbb R$$, $$f: [0,+\infty) \times \mathbb R \times \mathbb R \to \mathbb R$$ is a continuous mapping satisfying suitable assumptions ($$\Omega\in\mathbb R^n$$ denotes here an open bounded set with the boundary $$\partial \Omega$$ of the class $$C^{\infty}$$) and $${\mathcal A}$$ is a second order differential operator.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C29 Averaging method for ordinary differential equations 47N20 Applications of operator theory to differential and integral equations 34C25 Periodic solutions to ordinary differential equations
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##### References:
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