Existence of positive solutions for a nonlinear semipositone boundary value problems on a time scale

• Autores: Saroj Panigrahi, Sandip Rout
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 24, Nº. 3, 2022, págs. 413-437
• Idioma: inglés
• DOI: 10.56754/0719-0646.2403.0413
• Enlaces
  • Texto completo
• Resumen
  • español

    Resumen En este artículo estudiamos la existencia de soluciones positivas del siguiente problema de valor de frontera semipositón en escalas de tiempo: (ψ(t)y∆(t))∇+λ1g(t, y(t))+λ2h(t, y(t)) = 0, t ∈ [ρ(c), σ(d)]T, con condiciones de frontera mixtas αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c)) = 0, γy(σ(d)) + δψ(d)y∆(d) = 0, donde ψ : C[ρ(c), σ(d)]T, ψ(t) > 0 para todo t ∈ [ρ(c), σ(d)]T; ambas g y h: [ρ(c), σ(d)]T × [0, ∞) → R son continuas y semipositón. Hemos establecido la existencia de al menos una solución positiva o múltiples soluciones positivas del problema de valor en la frontera anterior usando un teorema de punto fijo en un cono en un espacio de Banach, cuando g y h son ambas superlineales o sublineales o una es superlineal y la otra es sublineal para λi > 0; i = 1, 2 suficientemente pequeños.

  • English

    Abstract In this paper, we are concerned with the existence of positive solution of the following semipositone boundary value problem on time scales: (ψ(t)y∆(t))∇+λ1g(t, y(t))+λ2h(t, y(t)) = 0, t ∈ [ρ(c), σ(d)]T, with mixed boundary conditions αy(ρ(c)) − βψ(ρ(c))y∆(ρ(c)) = 0, γy(σ(d)) + δψ(d)y∆(d) = 0, where ψ : C[ρ(c), σ(d)]T, ψ(t) > 0 for all t ∈ [ρ(c), σ(d)]T; both g and h: [ρ(c), σ(d)]T × [0, ∞) → R are continuous and semipositone. We have established the existence of at least one positive solution or multiple positive solutions of the above boundary value problem by using fixed point theorem on a cone in a Banach space, when g and h are both superlinear or sublinear or one is superlinear and the other is sublinear for λi > 0; i = 1, 2 are sufficiently small.

• Referencias bibliográficas
  • Anderson, D. R.,Wong, P. J. Y.. (2009). Positive solutions for second-order semipositone problems on time scales. Comput. Math. Appl.. 58....
  • Anderson, D. R.,Zhai, C.. (2010). Positive solutions to semi-positone second-order three-point problems on time scale. Appl. Math. Comput.....
  • Aris, R.. (1965). Introduction to the analysis of chemical reactors. Prentice Hall, Englewood Cliffs,. New Jersey.
  • Bai, D.,Xu, Y.. (2008). Positive solutions for semipositone BVPs of second-order difference equations. Indian J. Pure Appl. Math.. 39. 59-68
  • Bohner, M.,Peterson, A.. (2001). Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser. Boston.
  • Bohner, M.,Peterson, A.. (2003). Advances in Dynamic Equations on Time Scales. Birkhäuser. Boston.
  • Dahal, R.. (2011). Positive solutions for a second-order, singular semipositone dynamic boundary value problem. Int. J. Dyn. Syst. Differ....
  • Erbe, L.,Peterson, A.. (2000). Positive solutions for a nonlinear differential equations on a measure chain. Math. Comput. Modelling. 32....
  • Giorgi, C.,Vuk, E.. (2013). Steady-state solutions for a suspension bridge with intermediate supports. Bound. Value Probl..
  • Goodrich, C. S.. (2013). Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale. Comment. Math....
  • Hilger, S.. (1990). Analysis on measure chains-a unified approach to continuous and discrete calculus. Results Math.. 18. 18-56
  • Infante, G.,Pietramala, P.,Tenuta, M.. (2014). Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor...
  • Krasnosel’skii, M. A.. (1964). Positive Solutions of Operator Equations. P. Noordhoff,. Groningen.
  • Kreyszig, E.. (1978). Introductory Functional Analysis With Applications. John Wiley & Sons, Inc.. New York.
  • Selgrade, J.. (1998). Using stocking and harvesting to reverse period-doubling bifurcations in models in population biology. J. Differ. Equations...
  • Sun, J. P.,Li, W. T.. (2007). Existence of positive solutions to semipositone Dirichlet BVPs on time scales. Dynam. Systems Appl.. 16. 571
  • Yang, Y.,Meng, F.. Positive solutions of the singular semipositone boundary value problem on time scales. Math. Comput. Modelling. 52. 481