Second Order Iterative Dynamic Boundary Value Problems with Mixed Derivative Operators with Applications

• Jehad Alzabut ^[2] ; Mahammad Khuddush ^[3] ; A. George Maria Selvam ^[1] ; D. Vignesh ^[1]
  1. [1] Sacred Heart College

     Sacred Heart College

     India

     
  2. [2] Prince Sultan University & OST˙IM Technical University
  3. [3] Dr. Lankapalli Bullayya College of Engineering

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 22, Nº 1, 2023
• Idioma: inglés
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• Resumen

  • In this paper, we derive sufficient conditions for the existence and uniqueness of solutions of the iterative dynamic boundary value problem of second order with mixed derivative operators. For the existence, we utilize Schauder’s fixed point theorem while for uniqueness we apply contraction mapping principle. Further, a continuous dependence of bounded solutions to the addressed problem is studied. Finally, we demonstrate the validity of our findings by constructing examples as applications to beam deflection due to thermal stress and temperature distribution along the wire

• Referencias bibliográficas
  • 1. Anderson, D., Hoffacker, J.: Green’s function for an even order mixed derivative problem on time scales. Dyn. Syst. Appl. 12(1/2), 9–22...
  • 2. Atici, F.M., Guseinov, GSh.: On Green’s functions and positive solutions for boundary value problems on time scales. J. Comput. Appl. Math....
  • 3. Aulbach, B., Hilger, S.: Linear dynamic processes with inhomogeneous time scale. Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig,...
  • 4. Baitiche, Z., Derbazi, C., Alzabut, J., Samei, M.E., Kaabar, M.K.A., Siri, Z.: Monotone Iterative Method for Langevin Equation in Terms...
  • 5. Berinde, V.: Existence and approximation of solutions of some first order iterative differential equations. Miskolc Math. Notes 11, 13–26...
  • 6. Bohner, M., Luo, H.: Singular second-order multipoint dynamic boundary value problems with mixed derivatives. Adv. Differ. Equ. 1, 1–15...
  • 7. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston Inc, Boston (2001)
  • 8. Bohner, M., Peterson, A.C.: Advances in Dynamic Equations on Time Scales. Birkhäuser Boston Inc, Boston (2003)
  • 9. Bouakkaz, A., Ardjouni, A., Khemis, R., Djoudi, A.: Periodic solutions of a class of third-order functional differential equations with...
  • 10. Bouakkaz, A., Khemis, R.: Positive periodic solutions for a class of second-order differential equations with state dependent delays....
  • 11. Bouakkaz, A., Khemis, R.: Positive periodic solutions for revisited Nicholson’s blowflies equation with iterative harvesting term. J....
  • 12. Cannon, J.: The solution of the heat equation subject to the specification of energy. Quart. Appl. Math. 21, 155–160 (1963)
  • 13. Chegis, R.: Numerical solution of a heat conduction problem with an integral boundary condition. Litovsk. Mat. Sb. 24, 209–215 (1984)
  • 14. Cheraiet, S., Bouakkaz, A., Khemis, R.: Bounded positive solutions of an iterative three-point boundary-value problem with integral boundary...
  • 15. Chouaf, S., Khemis, R., Bouakkaz, A.: Some existence results on positive solutions for an iterative second order boundary value problem...
  • 16. Duke, E.R.: Solving higher order dynamic equations on time scales as first order systems. Theses, Dissertations and Capstones, Marshall...
  • 17. Feckan, M.: On a certain type of functional differential equations. Math. Slovaca 43, 39–43 (1993)
  • 18. Feckan, M., Wang, J., Zhao, H.Y.: Maximal and minimal nondecreasing bounded solutions of iterative functional differential equations....
  • 19. Ionkin, N.I.: Solution of a boundary-value problem in heat conduction with a non-classical boundary condition. Differ. Equ. 13, 204–211...
  • 20. Khuddush, M., Prasad, K.R.: Nonlinear two-point iterative functional boundary value problems on time scales. J. Appl. Math. Comput. (2022)....
  • 21. Lakshmikantham, V., Sivasundaram, S., Kaymakçalan, B.: Dynamic Systems on Measure Chains, Mathematics and Its Applications, vol. 370....
  • 22. Messer, K.R.: A second-order self-adjoint equation on a time scale. Dyn. Syst. Appl. 12(1–2), 201–215 (2003). Special issue: dynamic equations...
  • 23. Si, J.G., Wang, X.P.: Analytic solutions of a second order iterative functional differential equation. J. Comput. Appl. Math. 126, 277–285...
  • 24. Turab, A.: A unique solution of the iterative boundary value problem for a second-order differential equation approached by fixed point...
  • 25. Williams, P.A.: Unifying fractional calculus with time scales, [Ph.D. thesis], University of Melbourne (2012)
  • 26. Yang, D., Zhang, W.: Solution of equivalence for iterative differential equations. Appl. Math. Lett. 17, 759–765 (2004)
  • 27. Zhou, J., Shen, J.: Positive solutions of iterative functional differential equations and application to mixed-type functional differential...
  • 28. Usmani, A.S., Rotter, J.M., Lamont, S., Sanad, A.M., Gillie, M.: Fundamental principles of structural behaviour under thermal effects....
  • 29. Volz, S.: Microscale and nanoscale heat transfer. Top. Appl. Phys. 107, 181–238 (2007). https://doi. org/10.1007/11767862_9