Finding the Fixed Points Inside Large Mapping Sets: Integral Equations
Keywords:
compact maps, repeated mappings, integral equations, fixed points, limit setsAbstract
Let xf(t,x) > 0 for x 6= 0 and let A(t−s) satisfy some classical properties yielding a nice resolvent. Using repeated application of a fixed point mapping and induction we develop an asymptotic formula showing that solutions of the Caputo equation cDqx(t) = −f(t,x(t)), 0 < q < 1, x(0) ∈<, x(0) 6= 0, and more generally of the integral equation x(t) = x(0)−Zt 0 A(t−s)f(s,x(s))ds,x(0) 6= 0, all satisfy x(t) → 0 as t →∞.
References
L. C. Becker, T. A. Burton, and I. K. Purnaras, Integral and fractional equations, positive solutions, and Schaefer’s fixed point theorem, Opuscula Math. 36 (2016), 431-458. 2
T. A. Burton, Fractional differential equations and Lyapunov functionals, Nonlinear Anal.:TMA 74, (2011), 5648-5662.
T. A. Burton, Fractional equations and a theorem of Brouwer-Schauder type, Fixed Point Theory, 14 No. 1 (2013), 91-96.
T. A. Burton, Correction of "Fractional equations and a theorem of Brouwer-Schauder type", Fixed Point Theory 16 No. 2 (2015), 233-236.
T. A. Burton and Bo Zhang, Fixed points and fractional differential equations:Examples, Fixed Point Theory 14 (2013), 313-326.
K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg, 2010.
D. P. Dwiggins, Fixed point theory and integral equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23 (2016), 47-57.
G. Gripenberg, On positive, nonincreasing resolvents of Volterra equations, J. Differential Equations 30 (1978), 380-390.
R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, Menlo Park, CA, 1971.
D. R. Smart, Fixed Point Theorems, Cambridge, 1980.
