# zbMATH — the first resource for mathematics

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:
 [1] Diethelm K. The analysis of fractional differential equations, Lecture Notes in Mathematics, 2010. · Zbl 1215.34001 [2] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, () · Zbl 1092.45003 [3] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Scientific Publishers · Zbl 1188.37002 [4] Miller, K.S.; Ross, B., An introduction to the fractional calculus and differential equations, (1993), John Wiley New York · Zbl 0789.26002 [5] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [6] Tarasov, V.E., Fractional dynamics: application of fractional calculus to dynamics of particles, fields and media, (2010), Springer HEP · Zbl 1214.81004 [7] 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 [8] 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 [9] Bai, Z., On positive solutions of a nonlocal fractional boundary value problem, Nonlinear anal: TMA, 72, 916-924, (2010) · Zbl 1187.34026 [10] 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 [11] 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 [12] Wang, J.; Zhou, Y., A class of fractional evolution equations and optimal controls, Nonlinear anal: RWA, 12, 262-272, (2011) · Zbl 1214.34010 [13] 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 [14] 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 [15] 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 [16] Zhou, Y.; Jiao, F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal: RWA, 11, 4465-4475, (2010) · Zbl 1260.34017 [17] 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 [18] 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 [19] 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 [20] 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 [21] Sneddon, I.N., The use in mathematical analysis of erdélyi – kober operators and some of their applications, (), 37-79 [22] Prudnikov, A.P.; Brychkov, Y.A.; Marichev, O.I., Integral and series, () · Zbl 0511.00044 [23] Kiryakova, V., Generalized fractional calculus and applications, Pitman res notes math ser, vol. 301, (1994), Longman New York · Zbl 0882.26003 [24] Deng, W., Smoothness and stability of the solutions for nonlinear fractional differential equations, Nonlinear anal: TMA, 72, 1768-1777, (2010) · Zbl 1182.26009 [25] Li, Y.; Chen, Y.; Podlubny, I., Mittag – leffler stability of fractional order nonlinear dynamic systems, Automatica, 45, 1965-1969, (2009) · Zbl 1185.93062 [26] 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 [27] 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 [28] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag · Zbl 0559.47040
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.