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An abstract existence theorem at resonance and its applications. (English) Zbl 0940.34056
The authors consider the operator equation in the form: $(1) \quad Lx =Nx$ , where $L$ is a Fredholm mapping of index zero and $N$ is $L$-completely continuous. By using Brouwer degree theory and a continuation theorem based on Mawhin’s coincidence degree theory there is developed an abstract existence theorem at resonance for the equation (1). As application of this result sufficient conditions are proved for the existence of $2 {\pi}$-periodic solutions to semilinear equations at resonance, where the kernel of the linear part has dimension $\geq 2$. Finally, some ilustration examples on the theory are given.

##### MSC:
 34K13 Periodic solutions of functional differential equations 34K30 Functional-differential equations in abstract spaces 34C25 Periodic solutions of ODE
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##### References:
 [1] Hale, J. K.: Ordinary differential equations. (1969) · Zbl 0186.40901 [2] Nagle, R. K.: Nonlinear boundary value problems for ordinary differential equations with a small parameter. SIAM J. Math. anal. 9, 719-729 (1978) · Zbl 0389.34017 [3] Cesari, L.; Kannan, R.: Solutions in large of Liénard systems with forcing term. Ann. mat. Pura appl. 111, 101-124 (1976) · Zbl 0429.34038 [4] Rouche, N.; Mawhin, J.: Ordinary differential equations. (1980) · Zbl 0433.34001 [5] Fucik, S.: Solvability of nonlinear equations and boundary value problems. (1980) · Zbl 0453.47035 [6] Lazer, A. C.; Leach, D. E.: Bounded perturbations of forced harmonic oscillations at resonance. Ann. mat. Pura. appl. 82, 49-68 (1969) · Zbl 0194.12003 [7] Cesar, L.: Nonlinear problems across a point of resonance for non-self-adjoint systems. (1978) [8] Schuur, J. D.: Perturbation at resonance for a fourth order ordinary differential equation. J. math. Anal. appl. 65, 20-25 (1978) · Zbl 0406.34026 [9] Nagle, R. K.; Sinkala, Z.: Semilinear equations at resonance where the kernel has dimension two. (1991) · Zbl 0711.34053 [10] Nagle, R. K.; Sinkala, Z.: Existence of $2{\pi}$. Nonlinear anal. 25, 1-16 (1995) · Zbl 0826.34038 [11] S. W. Ma, Z. C. Wang, J. S. Yu, Coincidence degree and periodic solutions of Duffing equations, Nonlinear Anal. · Zbl 0931.34048 [12] Drabek, P.; Invernizzi, S.: Periodic solutions for systems of forced coupled pendu lum-like equations. J. differential equations 70, 390-402 (1987) · Zbl 0652.34049 [13] Ding, T. R.: Nonlinear oscillations at a point of resonance. Sci. China ser. A 1, 1-13 (1982) [14] D. Y. Hao, S. W. Ma, Semilinear Duffing equations crossing resonance points, J. Differential Equations · Zbl 0877.34036 [15] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040 [16] Mawhin, J.: Topological degree methods in nonlinear boundary value problems. Cbms 40 (1979) · Zbl 0414.34025 [17] Mawhin, J.: Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces. J. differential equations 12, 610-636 (1972) · Zbl 0244.47049