### Positive solution of asymptotically linear elliptic eigenvalue problems for certain differential equations of the fourth order

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This paper is devoted to the investigation of the abstract semilinear initial value problem du/dt + A(t)u = f(t,u), u(0) = u₀, in the "parabolic" case.

This paper is devoted to the investigation of the abstract semilinear initial value problem $du/dt=A\left(t\right)u+f(t,u),u\left(0\right)={u}_{0}$, in the “hyperbolic” case.

We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence ${A}_{n}\left(t\right)$ of bounded operators.

By using the theory of strongly continuous cosine families of linear operators in Banach space the existence of solutions of some semilinear second order Volterra integrodifferential equations in Banach spaces is proved. The results are applied to some integro-partial differential equations.

By using the theory of strongly continuous cosine families of linear operators in Banach space the existence of solutions of a semilinear second order differential initial value problem (1) as well as the existence of solutions of the linear inhomogeneous problem corresponding to (1) are proved. The main result of the paper is contained in Theorem 5.

The purpose of this paper is to give theorems on continuity and differentiability with respect to (h,t) of the solution of the initial value problem du/dt = A(h,t)u + f(h,t), u(0) = u₀(h) with parameter $h\in \Omega \subset {\mathbb{R}}^{m}$ in the “hyperbolic” case.

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