Geodesic
Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations
Zhang, Xinqiu1 ; Liu, Lishan2 ; Wu, Yonghong3
1 School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China
2 School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China;Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia
3 Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia
Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3364-3380.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this article, we study the existence and the uniqueness of iterative positive solutions for a class of nonlinear singular integral equations in which the nonlinear terms may be singular in both time and space variables. By using the fixed point theorem of mixed monotone operators in cones, we establish the conditions for the existence and uniqueness of positive solutions to the problem. Moreover, we derive various properties of the positive solutions to the equation and establish their dependence on the model parameter. The theorem obtained in this paper is more general and complements many previous known results including singular and nonlinear cases. Application of the results to the study of differential equations are also given in the article.
DOI : 10.22436/jnsa.010.07.01
Classification : 34B16, 34B18
Keywords: Mixed monotone operator, fixed point theorem, iterative positive solution, singular integral equations, boundary value problem, cone.

Zhang, Xinqiu 1 ; Liu, Lishan 2 ; Wu, Yonghong 3

1 School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China
2 School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, China;Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia
3 Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia 
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Zhang, Xinqiu; Liu, Lishan; Wu, Yonghong. Existence and uniqueness of iterative positive solutions for singular Hammerstein integral equations. Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 7, p. 3364-3380. doi : 10.22436/jnsa.010.07.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.010.07.01/

[1] Agarwal, R. P. On fourth order boundary value problems arising in beam analysis, Differential Integral Equations, Volume 2 (1989), pp. 91-110

[2] Agarwal, R. P.; Chow, Y. M. Iterative methods for a fourth order boundary value problem, J. Comput. Appl. Math., Volume 10 (1984), pp. 203-217 | DOI

[3] Cabada, A.; Wang, G.-T. Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl., Volume 389 (2012), pp. 403-411 | DOI

[4] Cao, Z.-W.; Jiang, D.-Q.; Yuan, C.-J.; O’Regan, D. Existence and uniqueness of solutions for singular integral equation, Positivity, Volume 12 (2008), pp. 725-732

[5] Cui, Y.-J.; Liu, L.-S.; Zhang, X.-Q. Uniqueness and existence of positive solutions for singular differential systems with coupled integral boundary value problems, Abstr. Appl. Anal., Volume 2013 (2013), pp. 1-9 | Zbl

[6] Guo, D.-J.; Cho, Y. J.; Zhu, J. Partial ordering methods in nonlinear problems, Nova Science Publishers, Inc., Hauppauge, NY, 2004

[7] Guo, D.-J.; Lakshmikantham, V. Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, Academic Press, Inc., , Boston, MA, 1988

[8] Hao, X.-A.; Liu, L.-S.; Wu, Y.-H.; Sun, Q. Positive solutions for nonlinear nth-order singular eigenvalue problem with nonlocal conditions, Nonlinear Anal., Volume 73 (2010), pp. 1653-1662 | DOI

[9] Hashem, H. H. G. On successive approximation method for coupled systems of Chandrasekhar quadratic integral equations, J. Egyptian Math. Soc., Volume 23 (2015), pp. 108-112 | Zbl | DOI

[10] Jiang, W.-H.; Zhang, J.-L. Positive solutions for (k, n-k) conjugate boundary value problems in Banach spaces, Nonlinear Anal., Volume 71 (2009), pp. 723-729 | DOI | Zbl

[11] Jleli, M.; Samet, B. Existence of positive solutions to an arbitrary order fractional differential equation via a mixed monotone operator method, Nonlinear Anal. Model. Control, Volume 20 (2015), pp. 367-376

[12] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J. Theory and applications of fractional differential equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006

[13] Lan, K. Q. Multiple positive solutions of conjugate boundary value problems with singularities, Appl. Math. Comput., Volume 147 (2004), pp. 461-474 | Zbl | DOI

[14] Latrach, K.; Taoudi, M. A. Existence results for a generalized nonlinear Hammerstein equation on \(L_1\) spaces, Nonlinear Anal., Volume 66 (2007), pp. 2325-2333 | Zbl | DOI

[15] Lei, P.-D.; Lin, X.-N.; Jiang, D.-Q. Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems, Nonlinear Anal., Volume 69 (2008), pp. 2773-2779 | DOI

[16] Li, F.-Y.; Li, Y.-H.; Liang, Z.-P. Existence of solutions to nonlinear Hammerstein integral equations and applications, J. Math. Anal. Appl., Volume 323 (2006), pp. 209-227 | DOI

[17] Li, H.-D.; Liu, L.-S.; Wu, Y.-H. Positive solutions for singular nonlinear fractional differential equation with integral boundary conditions, Bound. Value Probl., Volume 2015 (2015), pp. 1-15 | DOI

[18] Lin, X.-N.; Jiang, D.-Q.; Li, X.-Y. Existence and uniqueness of solutions for singular (k, n - k) conjugate boundary value problems, Comput. Math. Appl., Volume 52 (2006), pp. 375-382 | Zbl | DOI

[19] Lin, X.-N.; Jiang, D.-Q.; Li, X.-Y. Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. Comput. Appl. Math., Volume 196 (2006), pp. 155-161 | DOI

[20] Liu, L.-S.; Guo, F.; C.-X.Wu; Y.-H.Wu Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., Volume 309 (2005), pp. 638-647 | DOI

[21] Liu, L.-S.; Wu, C.-X.; Guo, F. Existence theorems of global solutions of initial value problems for nonlinear integrodifferential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., Volume 47 (2004), pp. 13-22 | Zbl | DOI

[22] Liu, L.-S.; Zhang, X.-Q.; Jiang, J.; Wu, Y.-H. The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems, J. Nonlinear Sci. Appl., Volume 9 (2016), pp. 2943-2958 | Zbl

[23] Lomtatidze, A.; Malaguti, L. On a nonlocal boundary value problem for second order nonlinear singular differential equations, Georgian Math. J., Volume 7 (2000), pp. 133-154 | DOI

[24] Pei, M.-H.; Chang, S. K. Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem, Math. Comput. Modelling, Volume 51 (2010), pp. 1260-1267 | DOI | Zbl

[25] Podlubny, I. Fractional differential equations, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering, Academic Press, Inc., San Diego, CA, 1999

[26] Samko, S. G.; Kilbas, A. A.; Marichev, O. I. Fractional integrals and derivatives, Theory and applications, Edited and with a foreword by S. M. Nikol'skii, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon, 1993

[27] Sun, Y.-P.; Zhao, M. Positive solutions for a class of fractional differential equations with integral boundary conditions, Appl. Math. Lett., Volume 34 (2014), pp. 17-21 | DOI

[28] Wang, Y.-Q.; Liu, L.-S.; Wu, Y.-H. Positive solutions for a class of fractional boundary value problem with changing sign nonlinearity, Nonlinear Anal., Volume 74 (2011), pp. 6434-6441 | DOI | Zbl

[29] Wang, Y.-Q.; Liu, L.-S.; Wu, Y.-H. Positive solutions for a nonlocal fractional differential equation, Nonlinear Anal., Volume 74 (2011), pp. 3599-3605 | DOI

[30] Webb, J. R. L. Uniqueness of the principal eigenvalue in nonlocal boundary value problems, Discrete Contin. Dyn. Syst. Ser. S, Volume 1 (2008), pp. 177-186

[31] Webb, J. R. L. Nonlocal conjugate type boundary value problems of higher order, Nonlinear Anal., Volume 71 (2009), pp. 1933-1940 | DOI

[32] Webb, J. R. L. Existence of positive solutions for a thermostat model, Nonlinear Anal. Real World Appl., Volume 13 (2012), pp. 923-938 | Zbl | DOI

[33] Webb, J. R. L. Positive solutions of nonlinear differential equations with Riemann-Stieltjes boundary conditions, Electron. J. Qual. Theory Differ. Equ., Volume 2016 (2016), pp. 1-13 | Zbl | DOI

[34] Wong, P. J. Y. Triple positive solutions of conjugate boundary value problems, II, Comput. Math. Appl., Volume 40 (2000), pp. 537-557 | DOI | Zbl

[35] Yang, Z.-L. Positive solutions for a system of nonlinear Hammerstein integral equations and applications, Appl. Math. Comput., Volume 218 (2012), pp. 11138-11150 | DOI

[36] Yang, B. Upper estimate for positive solutions of the (p, n - p) conjugate boundary value problem, J. Math. Anal. Appl., Volume 390 (2012), pp. 535-548 | DOI | Zbl

[37] Yuan, C.-J.; Wen, X.-D.; Jiang, D.-Q. Existence and uniqueness of positive solution for nonlinear singular 2mth-order continuous and discrete Lidstone boundary value problems, Acta Math. Sci. Ser. B Engl. Ed., Volume 31 (2011), pp. 281-291 | DOI

[38] Zhai, C.-B.; Song, R.-P.; Han, Q.-Q. The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl., Volume 62 (2011), pp. 2639-2647 | DOI | Zbl

[39] Zhang, H.-E. Iterative solutions for fractional nonlocal boundary value problems involving integral conditions, Bound. Value Probl., Volume 2016 (2016), pp. 1-13 | Zbl | DOI

[40] Zhang, X.-G.; Liu, L.-S.; Wiwatanapataphee, B.; Wu, Y.-H. The eigenvalue for a class of singular p-Laplacian fractional differential equations involving the Riemann-Stieltjes integral boundary condition, Appl. Math. Comput., Volume 235 (2014), pp. 412-422 | Zbl | DOI

[41] Zhang, X.-G.; Liu, L.-S.; Wu, Y.-H. The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., Volume 37 (2014), pp. 26-33 | Zbl | DOI

[42] Zhang, X.-Q.; Liu, L.-S.; Wu, Y.-H. Fixed point theorems for the sum of three classes of mixed monotone operators and applications, Fixed Point Theory Appl., Volume 2016 (2016), pp. 1-22 | Zbl | DOI

[43] Zhang, X.-Q.; Wang, L.; Sun, Q. Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter, Appl. Math. Comput., Volume 226 (2014), pp. 708-718 | DOI

[44] Zhang, M.-C.; Yin, Y.-M.; Wei, Z.-L. Existence of positive solution for singular semi-positone (k, n-k) conjugate m-point boundary value problem, Comput. Math. Appl., Volume 56 (2008), pp. 1146-1154 | DOI | Zbl

[45] Zhao, Y.-L.; Chen, H.-B.; Huang, L. Existence of positive solutions for nonlinear fractional functional differential equation, Comput. Math. Appl., Volume 64 (2012), pp. 3456-3467 | DOI

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