Global existence and stability results for Hadamard--Volterra--Stieltjes integral equations

Yıl 2019, Cilt: 68 Sayı: 2, 1387 - 1400, 01.08.2019

Said Baghdad Mouffak Benchohra

https://doi.org/10.31801/cfsuasmas.530865

Cited By: 2

Öz

In this paper we prove the existence and stability of solutions of a class of Hadamard--Volterra--Stieltjes integral equations in the Banach space of continuous and bounded functions on unbounded interval. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of measure of noncompactness in conjunction with the Darbo and Mönch fixed point theorems.

Anahtar Kelimeler

Hadamard-Volterra-Stieltjes integral equations, fixed-point theorems, measure of noncompactness, asymptotic behavior

Kaynakça

• Abbas, S., Alaidarous, E., Benchohra, M., Nieto, J.J., Existence and Ulam stabilities for Hadamard-Stieltjes fractional integral equations. Discrete Dyn. Nat. Soc. 2015, Art. ID 317094, 6 pp.
• Abbas, S. and Benchohra, M., Nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations on unbounded domains, Commun. Math. Anal. 14 (1) (2013), 104--117.
• Abbas, S., Benchohra, M., Graef, J. and Henderson, J., Implicit Fractional Differential and Integral Equations; Existence and Stability, De Gruyter, Berlin, 2018.
• Abbas, S., Benchohra, M. and Henderson, J., Asymptotic behavior of solutions of nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations, Int. Elect. J. Pure Appl. Math. 4 (3) (2012), 195--209.
• Abbas, S., Benchohra, M. and Nieto, J.J., Global attractivity of solutions for nonlinear fractional order Riemann-LiouvilleVolterra-Stieltjes partial integral equations, Electron. J. Qual. Theory Differ. Equ. 81 (2012), 1--15.
• Abbas, S., Benchohra, M. and N'Guérékata, G. M., Topics in Fractional Differential Equations, Springer, New York, 2012.
• Abbas, S., Benchohra, M. and N'Guérékata, G. M., Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
• Agarwal, R. P., O'Regan, D., Infinite Interval Problems for Differential, Difference and Integral Equations , Kluwer Academic Publishers, Dordrecht, 2001.
• Ahmad, B., Ntouyas, S.K., Initial value problems of fractional order Hadamard-Type functional differential equations Electron. J. Differential Equations 2015, No. 77, 9 pp.
• Appell, J., Banas, J., Merentes, N., Bounded Variation and Around, De Gruyter Series in Nonlinear Analysis and Applications 17. Walter de Gruyter, Berlin, 2014.
• Banas, J., Dhage, B. C., Global asymptotic stability of solutions of a functional integral equation, Nonlinear Anal. 69 (2008), no. 7, 1945--1952.
• Banas, J., Cabrera, I. J., Sadarangani, K., On existence and asymptotic behaviour of solutions of a functional integral equation, Nonlinear Anal. 66 (2007), no. 10, 2246--2254.
• Banas, J., Dhage, B. C., Monotonic solutions of a class of quadratic integral equations of Volterra type, Comput. Math. Appl. 49 (2005), no. 5-6, 943--952.
• Banas, J., Geobel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in pure and appl. Math. 60, Marcel Dekker, New York and Basel, 1980.
• Banas, J., O'Regan, D., On existence and local attractivity of solutions of a quadratic Volterra integral equations of fractional order J. Math. Anal. Appl 345 (2008), no. 1, 573--582.
• Banas, J., Rzepka, B., Nondecreasing solutions of a quadratic singular Volterra integral equation, Math. Comput. Modelling 49 (2009), no. 3-4, 488--496.
• Benchohra, M., Henderson, J., Seba, D., Measure of noncompactness and fractional differential equations in Banach spaces PanAmer. Math. J. 20 (2010), no. 3, 27--37.
• Butzer, P. L., Kilbas, A. A. and Trujillo, J. J., Fractional calculus in the mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269 (2002),1-27.
• Butzer, P. L., Kilbas, A. A. and Trujillo, J. J., Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270 (2002), 1-15.
• Castillo, M., Rlvas, S., Sanoja, M., Zea, I., Functions of bounded variation in the sense of Riesz-Korenblum, J. Funct. Spaces Appl. 2013, Art. ID 718507, 12 pp.
• Corduneanu, C., Integral Equations and Stability of Feedback Systems, Acedemic Press, New York, 1973.
• González, C., Melado, A. J., Fuster, E. L., A Mönch type fixed point theorem under the interior condition J. Math. Anal. appl 352 (2009), 816-821.
• Kilbas, A., Hadamard-Type fractional calculus, J. Korean Math. Soc. 38 (2001), no. 6, 1191--1204.
• Mcfadden, C., Riemann--Stieltjes Integration, Theses and Dissertations - Virginia University (2011).
• Natanson, I. P., Theory of Functions of a Real Variable, Ungar, New York, 1960.
• Pooseh, S., Almeida, R. and Torres, D., Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative. Numer. Funct. Anal. Optim. 33 (3) (2012), 301-319.
• Rudin, W., Real and Complex Analysis, McGraw-Hill, New York, 1970. SKM : Samko, S., Kilbas, A., Marichev, O. I., Fractional integrals and derivatives (Theory and Applications) , Gordon and Breach Science Publishers. Yverdon, 1993.
• Schwabik, S., Tvrdy, M., Vejvoda, O. Differential and integral equations ( Boundary Value Problems and Adjoints), C. A. S. Praha (1979).