Existence theorems for nonlinear problems by continuation methods

Continuation methods of Leray-Schauder type represent an important tool in the theory of differential and integral equations. Roughly speaking, by means of a continuation theorem we can obtain a solution of a given equation starting from one of the solutions of a more simpler equation. There are two main approaches to the theory of continua,tion methods. One uses the subtle notion of Leray-Schauder degree for compact perturbations of t.he identity and its extensions to various classes of mappings. We only mention a few names of contributors as follows: J. Leray, J. Schauder, E.H. Rothe, H. Amann, J. Mawhin, F.E. Browder, R.D. Nussbaum and W.V. Petryshyn. The other approach is based upon the fixed point theorem of Schauder and its generalizations, and on the notion of an essential mapping. In this direction, we mention the names of H. Schafer, A. Granas, M. Furi, M. Martelli, A. Vignoli, I. Massabb and W. Krawcewicz.
In this report, we adopt the second approach and we describe recent developments both in theory and applications. We first present an abstract continuation principle which makes possible to understand unitary particular continuation theorems for a great variety of single and set-valued mappings in metric, locally convex or Banach spaces. We then present several existence principles of coincidence type which complement the results obtained by K. Geba, A. Granas, T. Kaczynski and W. Krawcewicz [8] and by A. Granas, R.B. Guenther and J.W. Lee [lo]. Using these principles we can give no degree versions to some continuation theorems in the absence of a priori bounds, recently obtained by A. Capietto, J. Mawhin and F. Zanolin [2]. Finally, following the ideas from [2] and [3], we describe two applications to the existence of periodic solutions of impulsive differential equations and of singular superlinear differential equations.

Radu Precup

R. Precup, Existence theorems for nonlinear problems by continuation methods, Nonlinear Anal. 30 (1997), 3313-3322. https://doi.org/10.1016/S0362-546X(96)00333-1

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MR: 99a:47097

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