Finding the Fixed Points Inside Large Mapping Sets: Integral Equations

Öz

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 →∞.

Anahtar Kelimeler

• compact maps,repeated mappings,integral equations,fixed points,limit sets

Kaynakça

1. 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
2. T. A. Burton, Fractional differential equations and Lyapunov functionals, Nonlinear Anal.:TMA 74, (2011), 5648-5662.
3. T. A. Burton, Fractional equations and a theorem of Brouwer-Schauder type, Fixed Point Theory, 14 No. 1 (2013), 91-96.
4. T. A. Burton, Correction of "Fractional equations and a theorem of Brouwer-Schauder type", Fixed Point Theory 16 No. 2 (2015), 233-236.
5. T. A. Burton and Bo Zhang, Fixed points and fractional differential equations:Examples, Fixed Point Theory 14 (2013), 313-326.
6. K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Heidelberg, 2010.
7. D. P. Dwiggins, Fixed point theory and integral equations, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis 23 (2016), 47-57.
8. G. Gripenberg, On positive, nonincreasing resolvents of Volterra equations, J. Differential Equations 30 (1978), 380-390.

9. R. K. Miller, Nonlinear Volterra Integral Equations, Benjamin, Menlo Park, CA, 1971.
10. D. R. Smart, Fixed Point Theorems, Cambridge, 1980.