Electronic Journal of Qualitative Theory of Differential Equations

In this paper, we prove the existence of extremal positive, concave and pseudo-symmetric solutions for a general three-point second order p−Laplacian integro-differential boundary value problem by using an abstract monotone iterative technique.

Ahmad and Nieto [1] studied a three-point second order p−Laplacian integrodifferential boundary value problem with the non-integral term of the form f (t, x(t)).
In this paper, we allow the nonlinear function f to depend on x ′ along with x and consider a more general three-point second order p−Laplacian integro-differential boundary value problem of the form where p > 1 , ψ p (s) = s|s| p−2 .Let ψ q be the inverse of ψ p .
We apply an abstract monotone iterative technique due to Amann [2] to prove the existence of extremal positive, concave and pseudo-symmetric solutions for (1.1)-(1.2).For the details of the abstract monotone iterative method, we refer the reader to the papers [1, 4-5, 14-15, 18].The importance of the work lies in the fact that integro-differential equations are encountered in many areas of science where it is necessary to take into account aftereffect or delay.Especially, models possessing hereditary properties are described by integro-differential equations in practice.Also, the governing equations in the problems of biological sciences such as spreading of disease by the dispersal of infectious individuals, the reaction-diffusion models in ecology to estimate the speed of invasion, etc. are integro-differential equations.
EJQTDE, 2010 No. 3, p. 2 Definition 2.1.Let us define an operator G : P → E as follows By the definition of G, it follows that Gx ∈ C 1 [0, 1] and is nonnegative for each x ∈ P , and is a solution of (1.1) and (1.2) if and only if Gx = x.
In order to develop the iteration schemes for (1.1) and (1.2), we establish some properties of the operator Gx.
Next, we show that G : P θ → P θ .For u ∈ P θ , it follows that |u| ≤ θ and By the assumptions (A 1 ), (A 2 ) and (A 4 ), we have (2.1) By the definition of Gx and (2.1), we obtain Consequently, we have Gx ≤ θ.Hence we conclude that G : P θ → P θ .

Conclusions
The extremal positive, concave and pseudo-symmetric solutions for a nonlocal three-point p−Laplacian integro-differential boundary value problem are obtained by applying an abstract monotone iterative technique.The consideration of p−Laplacian boundary value problems is quite interesting and important as it covers a wide range of problems for various values of p occurring in applied sciences (as indicated in the introduction).The nonlocal three-point boundary conditions further enhance the scope of p−Laplacian boundary value problems as such boundary conditions appear in certain problems of thermodynamics and wave propagation where the controller at the end t = 1 dissipates or adds energy according to a censor located at a position t = η (0 < η < 1) where as the other end t = 0 is maintained at a constant level of energy.The results presented in this paper are new and extend some earlier results.For p = 2, our results correspond to a three-point second order quasilinear integro-differential boundary value problem.The results of [1] are improved as the nonlinear function f is allowed to depend on x ′ together with x in this paper whereas it only depends on x in [1].