Geodesic
Existence of solutions for nonlinear singular $q$-Sturm–Liouville problems
Ufa mathematical journal, Tome 12 (2020) no. 1, pp. 91-102
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In this paper, we study a nonlinear $q$-Sturm–Liouville problem on the semi-infinite interval, in which the limit-circle case holds at infinity for the $q$-Sturm–Liouville expression. This problem is considered in the Hilbert space $L_{q}^{2}\left( 0,\infty\right)$. We study this problem by using a special way of imposing boundary conditions at infinity. In the work, we recall some necessary fundamental concepts of quantum calculus such as $q$-derivative, the Jackson $q$-integration, the $q$-Wronskian, the maximal operator, etc. We construct the Green function associated with the problem and reduce it to a fixed point problem. Applying the classical Banach fixed point theorem, we prove the existence and uniqueness of the solutions for this problem. We obtain an existence theorem without the uniqueness of the solution. In order to get this result, we use the well-known Schauder fixed point theorem.
Keywords: Nonlinear $q$-Sturm–Liouville problem, singular point, Weyl limit-circle case, completely continuous operator, fixed point theorems. 
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B. P. Allahverdiev; H. Tuna. Existence of solutions for nonlinear singular $q$-Sturm–Liouville problems. Ufa mathematical journal, Tome 12 (2020) no. 1, pp. 91-102. http://geodesic.mathdoc.fr/item/UFA_2020_12_1_a6/

[1] V. Kac, P. Cheung, Quantum calculus, Springer-Verlag, New York, NY, USA, 2002 | MR | Zbl

[2] T. Ernst, The History of $q$-calculus and a new method, U. U. D. M. Report, Department of Math., Uppsala Univ., Uppsala, Sweden, 2000

[3] M.H. Annaby, Z.S. Mansour, $q$-Fractional Ccalculus and equations, Lecture Notes Math., 2056, Springer-Verlag, Heidelberg, Germany, 2012 | DOI | MR

[4] B. Ahmad, J.J. Nieto, “Basic theory of nonlinear third-order $q$-difference equations and inclusions”, Math. Model. Anal., 18:1 (2013), 122–135 | DOI | MR | Zbl

[5] B. Ahmad, S. K. Ntouyas, “Boundary value problems for $q$-difference inclusions”, Abstr. Appl. Anal., 2011 (2011), 292860 | MR | Zbl

[6] B. Ahmad, S. K. Ntouyas, “Boundary value problems for $q$-difference equations and inclusions with nonlocal and integral boundary conditions”, Math. Model. Anal., 19:5 (2014), 647–663 | DOI | MR

[7] B. Ahmad, S. K. Ntouyas, I. K. Purnaras, “Existence results for nonlinear $q$-difference equations with nonlocal boundary conditions”, Comm. Appl. Nonl. Anal., 19:3 (2012), 59–72 | MR | Zbl

[8] T. Saengngammongkhol, B. Kaewwisetkul, T. Sitthiwirattham, “Existence results for nonlinear second-order $q$-difference equations with $q$-integral boundary conditions”, Diff. Equat. Appl., 7:3 (2015), 303–311 | MR | Zbl

[9] T. Sitthiwirattham, J. Tariboon, S.K. Ntouyas, “Three-point boundary value problems of nonlinear second-order $q$-difference equations involving different numbers of $q$”, J. Appl. Math., 2013 (2013), 763786 | DOI | MR | Zbl

[10] W. Sudsutad, S. K. Ntouyas, J. Tariboon, “Quantum integral inequalities for convex functions”, J. Math. Inequal., 9:3 (2015), 781–793 | DOI | MR | Zbl

[11] P. Thiramanus, J. Tariboon, “Nonlinear second-order $q$-difference equations with three-point boundary conditions”, Comput. Appl. Math., 33:2 (2014), 385–397 | DOI | MR | Zbl

[12] C. Yu, J. Wang, “Existence of solutions for nonlinear second-order $q$-difference equations with first-order $q$-derivatives”, Adv. Difference Equat., 2013 (2013), 124 | DOI | MR

[13] M. El-Shahed, H.A. Hassan, “Positive solutions of $q$-difference equation”, Proc. Amer. Math. Soc., 138:5 (2010), 1733–1738 | DOI | MR | Zbl

[14] F.H. Jackson, “On $q$-difference equations”, Amer. J. Math., 32:4 (1910), 305–314 | DOI | MR | Zbl

[15] Z. Hefeng, L. Wenjun, “Existence results for a second-order $q$-difference equation with only integral conditions”, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 79:4 (2017), 221–234 | MR

[16] Z. Mansour, M. Al-Towailb, “$q$-Lidstone polynomials and existence results for $q$-boundary value problems”, Bound. Value Probl., 2017 (2017), 178 | DOI | MR | Zbl

[17] M.J. Mardanov, Y.A. Sharifov, “Existence and uniqueness results for $q$-difference equations with two-point boundary conditions”, AIP Conf. Proc., 1676 (2015), 020065 | DOI

[18] R. P. Agarwal, G. Wang, B. Ahmad, L. Zhang, A. Hobiny, S. Monaquel, “On existence of solutions for nonlinear $q$-difference equations with nonlocal $q$-integral boundary conditions”, Math. Model. Anal., 20:5 (2015), 604–618 | DOI | MR

[19] G.Sh. Guseinov, I. Yaslan, “Boundary value problems for second order nonlinear differential equations on infinite intervals”, J. Math. Anal. Appl., 290:2 (2004), 620–638 | DOI | MR | Zbl

[20] B.P. Allahverdiev, H. Tuna, “Nonlinear singular Sturm-Liouville problems with impulsive conditions”, Facta Univ., Ser. Math. Inf., 34:3 (2019), 439–457 | MR

[21] P.N. Swamy, “Deformed Heisenberg algebra:origin of $q$-calculus”, Physica A: Statist. Mechan. Appl., 328:1-2 (2003), 145–153 | DOI | MR | Zbl

[22] W. Hahn, “Beitraäge zur Theorie der Heineschen Reihen”, Math. Nachr., 2 (1949), 340–379 (in German) | DOI | MR | Zbl

[23] M.H. Annaby, Z.S. Mansour, I.A. Soliman, “$q$-Titchmarsh-Weyl theory: series expansion”, Nagoya Math. J., 205 (2012), 67–118 | DOI | MR | Zbl

[24] A.N. Kolmogorov, S.V. Fomin, Introductory Real Analysis, Dover Publ., New York, 1970 | MR

[25] Pergamon Press, New York, 1964 | MR | Zbl