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Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator. (English) Zbl 1298.45011
This paper initiates the investigation of the following nonlinear integral equation with Erdélyi-Kober fractional operator
$y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}f(s,y(s))ds,\qquad t\in J=[0,T],\quad T>0,$
where $$a,b:J\to \mathbb R$$, $$f:J\times \mathbb R\to \mathbb R$$ and $$\alpha,\beta,\gamma$$ are positive parameters.
Using the Schauder fixed point theorem, the authors obtain the existence results of the solution for this equation with some chosen parameters $$\alpha,\beta$$ and $$\gamma$$. Then the uniqueness is derived by virtue of a weakly singular integral inequality due to Q.-H. Ma and J. Pečarić [J. Math. Anal. Appl. 341, No. 2, 894–905 (2008; Zbl 1142.26015)].
The local stability of the solutions for the generalized equation
$y(t)=a(t)+\frac{b(t)}{\Gamma(\alpha)}\int\limits_{0}^{t}(t^{\beta}-s^{\beta})^{\alpha-1}s^{\gamma}g(t,s,y(s))ds, \qquad t\in \mathbb R_{+}=[0,\infty),$
is also studied. “Meanwhile, three certain solutions sets $$Y_{K,\sigma }$$, $$Y_{1,\lambda }$$ and $$Y_{1,1}$$, which tend to zero at an appropriate rate $$t^{ - \nu }$$, $$0 < \nu = \sigma$$ (or $$\lambda$$ or 1) as $$t\to +\infty$$, are constructed and local stability results of the solutions are obtained based on these sets respectively under some suitable conditions. Two examples are given to illustrate the results.”

MSC:
 45G10 Other nonlinear integral equations 34A08 Fractional ordinary differential equations
Zbl 1142.26015
Full Text:
References:
  Diethelm K. The analysis of fractional differential equations, Lecture Notes in Mathematics, 2010. · Zbl 1215.34001  Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, () · Zbl 1092.45003  Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002  Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002  Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010  Tarasov, V.E., Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2010), Springer HEP · Zbl 1214.81004  Agarwal, R.P.; Benchohra, M.; Hamani, S., A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl math, 109, 973-1033, (2010) · Zbl 1198.26004  Ahmad, B.; Nieto, J.J., Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via leray – schauder degree theory, Topol methods nonlinear anal, 35, 295-304, (2010) · Zbl 1245.34008  Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal: TMA, 72, 916-924, (2010) · Zbl 1187.34026  Benchohra, M.; Henderson, J.; Ntouyas, S.K.; Ouahab, A., Existence results for fractional order functional differential equations with infinite delay, J math anal appl, 338, 1340-1350, (2008) · Zbl 1209.34096  Mophou, G.M.; N’Guérékata, G.M., Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay, Appl math comput, 216, 61-69, (2010) · Zbl 1191.34098  Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear anal: RWA, 12, 262-272, (2011) · Zbl 1214.34010  Wang, J.; Zhou, Y.; Wei, W., A class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces, Commun nonlinear sci numer simulat, 16, 4049-4059, (2011) · Zbl 1223.45007  Wang, J.; Dong, X.; Zhou, Y.; Existence, Attractiveness and stability of solutions for quadratic Urysohn fractional integral equations, Commun nonlinear sci numer simulat, 17, 545-554, (2012) · Zbl 1257.45004  Zhang, S., Existence of positive solution for some class of nonlinear fractional differential equations, J math anal appl, 278, 136-148, (2003) · Zbl 1026.34008  Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal: RWA, 11, 4465-4475, (2010) · Zbl 1260.34017  Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal: TMA, 71, 2724-2733, (2009) · Zbl 1175.34082  Zhou, Y.; Jiao, F.; Li, J., Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal: TMA, 71, 3249-3256, (2009) · Zbl 1177.34084  Ibrahim, R.W.; Momani, S., On the existence and uniqueness of solutions of a class of fractional differential equations, J math anal appl, 334, 1-10, (2007) · Zbl 1123.34302  Ma, Q.H.; Pec˘arić, J., Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations, J math anal appl, 341, 894-905, (2008) · Zbl 1142.26015  Sneddon, I.N., The use in mathematical analysis of erdélyi – kober operators and some of their applications, (), 37-79  Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I., Integral and series, () · Zbl 0511.00044  Kiryakova, V., Generalized fractional calculus and applications, Pitman res notes math ser, vol. 301, (1994), Longman New York · Zbl 0882.26003  Deng, W., Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear anal: TMA, 72, 1768-1777, (2010) · Zbl 1182.26009  Li, Y.; Chen, Y.; Podlubny, I., Mittag – leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969, (2009) · Zbl 1185.93062  Li, Y.; Chen, Y.; Podlubny, I., Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized mittag – leffler stability, Comput math appl, 59, 1810-1821, (2010) · Zbl 1189.34015  Ma, Q.H.; Yang, E.H., Estimates on solutions of some weakly singular Volterra integral inequalities, Acta math appl ser A, 25, 505-515, (2002) · Zbl 1032.26009  Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag · Zbl 0559.47040
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