Mortici, Cristinel The distance between fixed points of some pairs of maps in Banach spaces and applications to differential systems. (English) Zbl 1164.47358 Czech. Math. J. 56, No. 2, 689-695 (2006). Summary: Let \(T\) be a \(\gamma \)-contraction on a Banach space \(Y\) and let \(S\) be an almost \(\gamma \)-contraction, i.e. the sum of an \(\left ( \varepsilon ,\gamma \right ) \)-contraction with a continuous bounded function which is less than \(\varepsilon \) in norm. According to the contraction principle, there is a unique element \(u\) in \(Y\) for which \(u=Tu\). If, moreover, there exists \(v\) in \(Y\) with \(v=Sv\), then we will give estimates for \(\| u-v\| \). Finally, we establish some inequalities related to the Cauchy problem. MSC: 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations 34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations Keywords:contraction principle; Cauchy problem × Cite Format Result Cite Review PDF Full Text: DOI EuDML Link References: [1] C. Mortici: Approximate methods for solving the Cauchy problem. Czechoslovak Math. J. 55 (2005), 709–718. · Zbl 1081.34009 · doi:10.1007/s10587-005-0058-1 [2] C. Mortici and S. Sburlan: A coincidence degree for bifurcation problems. Nonlinear Analysis, TMA 53 (2003), 715–721. · Zbl 1028.47046 · doi:10.1016/S0362-546X(02)00308-5 [3] C. Mortici: Operators of monotone type and periodic solutions for some semilinear problems. Mathematical Reports 54 (1/2002), 109–121. · Zbl 1062.47062 [4] C. Mortici: Semilinear equations in Hilbert spaces with quasi-positive nonlinearity. Studia Cluj. 4 (2001), 89–94. · Zbl 1027.47044 [5] D. Pascali and S. Sburlan: Nonlinear Mappings of Monotone Type. Alphen aan den Rijn, Sijthoff & Noordhoff International Publishers, The Netherlands, 1978. · Zbl 0423.47021 [6] S. Sburlan, L. Barbu and C. Mortici: Ecuaţii Diferenţiale. Integrale şi Sisteme Dinamice. Editura Ex Ponto, Constanţa, Romania, 1999. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.