Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 3, 1-9; http://www.math.u-szeged.hu/ejqtde/

Existence of extremal solutions of a three-point boundary

value problem for a general second order p−Laplacian integro-differential equation

Bashir Ahmad^{1}, Ahmed Alsaedi

Department of Mathematics, Faculty of Science, King Abdulaziz University,

P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail: bashir_{−}qau@yahoo.com, aalsaedi@hotmail.com

Abstract 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. Keywords and Phrases: extremal solutions, integro-differential equations, p−Laplacian, nonlocal conditions. AMS Subject Classifications (2000): 34B15, 45J05.

1 Introduction

The subject of multi-point second order boundary value problems, initiated by Il’in and Moiseev [9, 10], has been extensively addressed by many authors, for instance, see [6, 7, 13, 16, 17]. There has also been a considerable attention on p−Laplacian boundary value problems [3, 8, 12, 19] as p−Laplacian appears in the study of flow through porous media (p = 3/2), nonlinear elasticity (p ≥ 2), glaciology (1 ≤ p ≤ 4/3), etc. Recently, Sun and Ge [18] discussed the existence of positive pseudosymmetric solutions for a second order three-point boundary value problem involving p−Laplacian operator given by

(ψ_{p}(u′(t)))′ + q(t)f(t, u(t), u′(t)) = 0, t ∈ (0, 1),
u(0) = 0, u(η) = u(1), 0 < η < 1.

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

(ψ_{p}(x′(t)))′ + a(t)(f(t, x(t), x′(t)) + ∫_{t}^{(1+η)/2} K(t, ζ, x(ζ))dζ) = 0, t ∈ (0, 1), (1.1)

1Corresponding author This research was supported by Deanship of Scientific Research, King Abdulaziz University (Project No. 427/176).

x(0) = 0,  x(η) = x(1), 0 < η < 1,  (1.2)

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.

2 Preliminaries

Let E = C^{1}[0, 1] be the Banach space equipped with norm ∥x∥ = max_{0≤t≤1}[x^{2}(t) + (x′(t))^{2}]^{1/2} and let P be a cone in E defined by P = {x ∈ E : x is nonnegative, concave on [0, 1] and pseudo-symmetric about (1 + η)/2 on [0, 1]}. Further, for θ > 0, let \overline{P}_{θ} = \{x ∈ P : ∥x∥ ≤ θ\}. A functional γ is said to be concave on [0, 1] if

γ(tx + (1 − t)y) ≥ tγ(x) + (1 − t)γ(y), ∀x, y ∈ [0, 1] and t ∈ [0, 1].

A function x is said to be pseudo-symmetric about (1 + η)/2 on [0, 1] if x is symmetric on the interval [η, 1], that is, x(t) = x(1 − (t − η)) for t ∈ [η, 1].

Throughout the paper, we assume that

(A_{1}) f(t, x, y) : [0, 1] × [0, ∞) × R → [0, ∞) is continuous with f(t, x_{1}, y_{1}) ≤ f(t, x_{2}, y_{2}), for any 0 ≤ t ≤ 1, 0 ≤ x_{1} ≤ x_{2} ≤ θ, 0 ≤ |y_{1}| ≤ |y_{2}| ≤ θ (f is nondecreasing in x and |y|) and f(t, x, y) is pseudo-symmetric in t about (1 + η)/2 on (0, 1) for any fixed x ∈ [0, ∞), y ∈ R. Moreover, f(t, 0, 0) is not identically equal to zero on any subinterval of (0, 1).

(A_{2}) K(t, ζ, x) : [0, 1] × [0, 1] × [0, ∞) → [0, ∞) is continuous, nondecreasing in x and for any fixed (ζ, x) ∈ [0, 1] × [0, ∞), K(t, ζ, x) is pseudo-symmetric in t and ζ about (1 + η)/2 on (0, 1). Further, K(t, ζ, 0) is not identically equal to zero for 0 ≤ t, ζ ≤ 1.

(A_{3}) a(t) ∈ L(0, 1) is nonnegative on (0, 1) and pseudo-symmetric in t about (1 + η)/2 on (0, 1). Further, a(t) is not identically zero on any nontrivial compact subinterval of (0, 1).

(A_{4}) max_{0≤t≤1} {f(t, θ, θ) + ∫_{t}^{(1+η)/2} K(t, ζ, θ)dζ} ≤ ψ_{p}(θ/Θ),

where Θ = max\{\sqrt{2}Θ_{1}, \sqrt{2}Θ_{2}\},
Θ_{1} = ∫_{0}^{(1+η)/2} ψ_{q}( ∫_{w}^{(1+η)/2} a(ν)dν)dw, Θ_{2} = ψ_{q}( ∫_{0}^{(1+η)/2} a(ν)dν).

Definition 2.1. Let us define an operator G : P → E as follows

(Gx)(t) = \{ \begin{matrix}∫_{0}^{t} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw, \\ t ∈ [0, (1 + η)/2]; \\ ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw \\ + ∫_{t}^{1} ψ_{q}[ ∫_{(1+η)/2}^{w} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw, \\ t ∈ [(1 + η)/2, 1]. \end{matrix}

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. Since

(Gx)′(t) = \{ \begin{matrix}ψ_{q}[ ∫_{t}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν], \\ t ∈ [0, (1 + η)/2]; \\ −ψ_{q}[ ∫_{(1+η)/2}^{t} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν], \\ t ∈ [(1 + η)/2, 1], \end{matrix}

is continuous and nonincreasing on [0, 1] with (Gx)′((1 + η)/2) = 0, therefore, it follows that Gx is concave. The nondecreasing nature of Gx in x and |x′| follows from the assumptions (A_{1}) and (A_{2}). Now, we show that Gx is pseudo-symmetric about (1 + η)/2 on [0, 1]. For that, we note that (1 − (t − η)) ∈ [(1 + η)/2, 1] for all t ∈ [η, (1 + η)/2]. Thus,

(Gx)(1 − (t − η))
= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
+ ∫_{1−(t−η)}^{1} ψ_{q}[ ∫_{(1+η)/2}^{w} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw

= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
− ∫_{t}^{η} ψ_{q}[ ∫_{(1+η)/2}^{1−(w−η)} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw

= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
+ ∫_{η}^{t} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{1−(ν−η)} K(ν, ζ, x(ζ))dζ)dν]dw

= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw

+ ∫_{η}^{t} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
= ∫_{0}^{t} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
= (Gx)(t).

Now, ∀t ∈ [(1 + η)/2, 1], we have (1 − (t − η)) ∈ [η, (1 + η)/2]. Thus,

(Gx)(1 − (t − η))

= ∫_{0}^{1−(t−η)} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
+ ∫_{η}^{1−(t−η)} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
+ ∫_{t}^{1} ψ_{q}[ ∫_{1−(w−η)}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw

= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw

+ ∫_{t}^{1} ψ_{q}[ ∫_{(1+η)/2}^{w} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{1−(ν−η)}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw

= ∫_{0}^{η} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw
+ ∫_{t}^{1} ψ_{q}[ ∫_{(1+η)/2}^{w} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ)dν]dw
= (Gx)(t).

So (Gx) is pseudo-symmetric about (1 + η)/2 on [0, 1]. Hence we conclude that G : P → P. Also, it follows by the standard arguments [1, 18] that G : P → P is completely continuous.

Next, we show that G : \overline{P}_{θ} → \overline{P}_{θ}. For u ∈ \overline{P}_{θ}, it follows that |u| ≤ θ and

0 ≤ u(t) ≤ max_{0≤t≤1} |u(t)| ≤ ∥u∥ ≤ θ, 0 ≤ |u′(t)| ≤ max_{0≤t≤1} |u′(t)| ≤ ∥u∥ ≤ θ.

By the assumptions (A_{1}), (A_{2}) and (A_{4}), we have

0 ≤ f(t, x(t), x′(t)) + ∫_{t}^{(1+η)/2} K(t, ζ, x(ζ))dζ

≤ f(t, θ, θ) + ∫_{t}^{(1+η)/2} K(t, ζ, θ)dζ

≤ max_{0≤t≤1} {f(t, θ, θ) + ∫_{t}^{(1+η)/2} K(t, ζ, θ)dζ} ≤ ψ_{p}(θ/Θ), t ∈ [0, 1].

(2.1)

Clearly

∥Gx∥ = max_{0≤t≤1}[(Gx)^{2}(t) + ((Gx)′)^{2}(t)]^{1/2}

≤ \sqrt{2} max_{0≤t≤1}\{|Gx(t)|, |(Gx)′(t)|\} = \sqrt{2} max\{(Gx)((1 + η)/2), max_{0≤t≤1} |(Gx)′|(t)\}.

By the definition of Gx and (2.1), we obtain

Gx((1 + η)/2)

= ∫_{0}^{(1+η)/2} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν]dw
≤ ∫_{0}^{(1+η)/2} ψ_{q}[ ∫_{w}^{(1+η)/2} a(ν)dνψ_{p}(θ/Θ)]dw = θΘ_{1}/Θ ≤ θ/\sqrt{2},

max_{0≤t≤1} |(Gx)′(t)| = max{(Gx)′(0), −(Gx)′(1)}

= max {ψ_{q}[ ∫_{0}^{(1+η)/2} a(ν)(f(ν, x(ν), x′(ν)) + ∫_{ν}^{(1+η)/2} K(ν, ζ, x(ζ))dζ)dν],

ψ_{q}[ ∫_{(1+η)/2}^{1} a(ν){f(ν, x(ν), x′(ν)) + ∫_{(1+η)/2}^{ν} K(ν, ζ, x(ζ))dζ}dν]}

≤ max {ψ_{q}[ ∫_{0}^{(1+η)/2} a(ν)dνψ_{p}(θ/Θ)], ψ_{q}[ ∫_{(1+η)/2}^{1} a(ν)dνψ_{p}(θ/Θ)]}
= θΘ_{2}/Θ ≤ θ/\sqrt{2}.

Consequently, we have ∥Gx∥ ≤ θ. Hence we conclude that G : \overline{P}_{θ} → \overline{P}_{θ}.

3 Existence of extremal solutions

Theorem 3.1 Assume that (A_{1}) − (A_{4}) hold. Then there exist extremal positive, concave and pseudo-symmetric solutions α^{∗}, β^{∗} of (1.1) and (1.2) with 0 < α^{∗} ≤ θ_{1}/\sqrt{2}, 0 < |(α^{∗})′| ≤ θ_{1}/\sqrt{2} and lim_{n→∞} α_{n} = lim_{n→∞} G^{n}α_{0} = α^{∗}, lim_{n→∞}(α_{n})′ = lim_{n→∞}(G^{n}α_{0})′ = (α^{∗})′, where

α_{0}(t) = \{ \begin{matrix}θ_{1}t/\sqrt{2}, & if 0 ≤ t ≤ (1 + η)/2, \\ θ_{1}|1 − (t − η)|/\sqrt{2}, & if (1 + η)/2 ≤ t ≤ 1, \end{matrix}

and 0 < β^{∗} ≤ θ_{1}, 0 < |(β^{∗})′| ≤ θ_{1} with lim_{n→∞} β_{n} = lim_{n→∞} G^{n}β_{0} = β^{∗}, lim_{n→∞}(β_{n})′ = lim_{n→∞}(G^{n}β_{0})′ = (β^{∗})′, where β_{0}(t) = 0, t ∈ [0, 1].

Proof. We define the iterative schemes

α_{1} = Gα_{0}, α_{n+1} = Gα_{n} = G^{n+1}α_{0}, n = 1, 2, ...,
β_{1} = Gβ_{0} = G0, β_{n+1} = Gβ_{n} = G^{n+1}β_{0}, n = 1, 2, ....

In view of the fact that G : \overline{P}_{θ} → \overline{P}_{θ}, it follows that α_{n} ∈ G\overline{P}_{θ} ⊆ \overline{P}_{θ} for n = 1, 2, .... As G is completely continuous, therefore {α_{n}}_{n=1}^{∞} is sequentially compact. By the assumption (A_{3}), it easily follows that α_{1}(t) = Gα_{0}(t) ≤ α_{0}(t), t ∈ [0, 1]. Also

|α′_{1}(t)| = |(Gα_{0})′(t)| ≤ max\{(G(θ_{1}t/\sqrt{2}))′(0), −(G(θ_{1}(1 − (t − η))/\sqrt{2}))′(1)\}
≤ θ_{1}/\sqrt{2} = |α′_{0}(t)|, t ∈ [0, 1].

Now, by the nondecreasing nature of (Gx), we obtain

α_{2}(t) = Gα_{1}(t) ≤ Gα_{0}(t) = α_{1}(t), |α′_{2}(t)| = |(Gα_{1})′(t)| ≤ |(Gα_{0})′(t)| = |α′_{1}(t)|,

for t ∈ [0, 1]. Thus, by induction, we have

α_{n+1}(t) ≤ α_{n}(t), |α′_{n+1}(t)| ≤ |α′_{n}(t)|, n = 1, 2, ...., t ∈ [0, 1].

Hence there exists α^{∗} ∈ \overline{P}_{θ} such that α_{n} → α^{∗}. Applying the continuity of G and α_{n+1} = Gα_{n}, we get Gα^{∗} = α^{∗} [18]. In view of the fact that f(t, 0, 0) and K(t, ζ, 0) are not identically equal to zero for 0 ≤ t, ζ ≤ 1, we find that the zero function is not the solution of (1.1) and (1.2). Thus, ∥α^{∗}∥ > 0 and hence α^{∗} > 0, t ∈ (0, 1). Now, let β_{1} = Gβ_{0} = G0, β_{2} = G^{2}β_{0} = G^{2}0 = Gβ_{1}. Since G : \overline{P}_{θ} → \overline{P}_{θ}, it follows that β_{n} ∈ G\overline{P}_{θ} ⊆ \overline{P}_{θ} for n=1,2,.... As G is completely continuous, therefore \{β_{n}\}_{n=1}^{∞} is sequentially compact. Since β_{1} = Gβ_{0} = G0 ∈ G\overline{P}_{θ}, we have β_{1}(t) = Gβ_{0}(t) = (G0)(t) ≥ 0 and |β′_{1}(t)| = |(Gβ_{0})′(t) = |(G0)′(t)| ≥ 0, t ∈ [0, 1]. Thus, β_{2}(t) = Gβ_{1}(t) ≥ (G0)(t) = β_{1}(t) and |β′_{2}(t)| = |(Gβ_{1})′(t)| = |(G0)′(t)| = |β′_{1}(t)|, t ∈ [0, 1]. As before, by induction, it follows that

β_{n+1}(t) ≥ β_{n}(t), |β′_{n+1}(t)| ≥ |β′_{n}(t)|, n = 1, 2, ...., t ∈ [0, 1].

Hence there exists β^{∗} ∈ \overline{P}_{θ} such that β_{n} → β^{∗} and Gβ^{∗} = β^{∗} with β^{∗}(t) > 0, t ∈ (0, 1). Now, using the well known fact that a fixed point of the operator G in P must be a solution of (1.1) and (1.2) in P, it follows from the monotone iterative technique [11] that α^{∗} and β^{∗} are the extremal positive, concave and pseudo-symmetric solutions of (1.1) and (1.2). This completes the proof.

Example. Consider the boundary value problem

x′′(t) + f(t, x(t), x′(t)) + ∫t^{\frac{3}{4}} K(t, ζ, x(ζ))dζ = 0, t ∈ (0, 1),  (3.1)

x(0) = 0,  x(\frac{1}{2}) = x(1),  (3.2)

where a(t) = 1, f(t, x, x′) = −\frac{16}{3}t^{2} + 8t + \frac{1}{2\sqrt{2}}x + \frac{1}{36}(x′)^{2}, K(t, ζ, x) = −\frac{4}{3}t^{2} + 2t + x. Clearly the functions f(t, x, x′) and K(t, ζ, x) satisfy the assumptions (A_{1}) and (A_{2}). In relation to (A_{3}), we choose θ = 6\sqrt{2} so that Θ = 3/(2\sqrt{2}) and max_{0≤t≤1}{f(t, θ, θ) + ∫_{t}^{(1+η)/2} K(t, ζ, θ)dζ} = 8 = ψ_{2}(θ/Θ). Hence the conclusion of Theorem 3.1 applies to the problem (3.1)-(3.2).

4 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].

Acknowledgement. The authors are grateful to the anonymous referee for his/her valuable comments.

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(Received July 20, 2009)