Open Access

A fixed point theorem for weakly inward A-proper maps and application to a Picard boundary value problem

Fixed Point Theory and Applications20162016:71

https://doi.org/10.1186/s13663-016-0564-x

Received: 22 January 2016

Accepted: 17 June 2016

Published: 29 June 2016

Abstract

A fixed point theorem for weakly inward A-proper maps defined on cones in Banach spaces is established using a fixed point index for such maps. The result generalizes a theorem in Deimling (Nonlinear Functional Analysis, 1985) for weakly inward maps defined on a cone in \(\mathbb{R}^{n}\). We then apply the theorem to a Picard boundary value problem and obtain the existence of a positive solution.

Keywords

fixed point indexconepositive solutionsboundary value problem

MSC

34B1834B15

1 Introduction

The purpose of this paper is to establish a fixed point theorem for weakly inward A-proper maps defined on cones in Banach spaces that generalizes a result in Deimling [1], p.254, for weakly inward maps defined on a cone in \(\mathbb{R}^{n}\). We use the fixed point index for weakly inward A-proper maps introduced by Lan and Webb [2] to obtain our new result. As an application, we obtain a positive solution to the Picard boundary value problem
$$- x''(t) = f\bigl(t,x(t),x'(t),x''(t) \bigr),\quad \mbox{where } x(0) = x(1) = 0, $$
under suitable conditions on f. This problem has been extensively studied; in particular, we refer to [3], where the concept of \(P_{\gamma} \)-compact maps and quasinormal cones is used, [4], where the problem is formulated as a semilinear equation, [5], where f is allowed to take negative values, and [6], where positive solutions for three-point boundary value problems are obtained. As mentioned in [5], in [3] and [4], examples were provided with conflicting hypotheses; our theorem will allow a different approach, which corrects the hypotheses of the analogous examples.

2 Preliminaries

Let X be a Banach space, \(X_{n} \subset X\) a sequence of oriented finite-dimensional subspaces, and \(P_{n}:X \to X_{n}\) a sequence of continuous linear projections such that \(P_{n} x \to x\) for each \(x \in X\).

Then X is called a Banach space with projection scheme \(\Gamma = \{ X_{n},P_{n} \}\).

A map \(f:\operatorname{dom} f \subset X \to X\) is said to be A-proper with respect to Γ if \(P_{n}f:X_{n} \to X_{n}\) is continuous for each n and for any bounded sequence \(\{ x_{n_{j}}|x_{n_{j}} \in X_{n_{j}} \}\) such that \(f_{n_{j}}(x_{n_{j}}) \to y\), there exists a subsequence \(\{ x_{n_{j_{k}}} \}\) such that \(x_{n_{j_{k}}} \to x\) and \(f(x) = y\).

A closed convex set K in a Banach space X is called a cone if \(\lambda K \subset K\) for all \(\lambda \ge 0\) and \(K \cap \{ - K \} = 0\).

Let \(K \subset X\) be a closed convex set. For each \(x \in K\), the set \(I_{K}(x) = \{ x + c(z - x):z \in K, c \ge 0 \}\) is called the inward set of x with respect to K. A map \(f:K \to X\) is called inward (respectively, weakly inward) if for all \(x \in K\), \(f(x) \in I_{K}(x)\) (\(f(x) \in \bar{I}_{K}(x)\)).

A map \(f:\overline{\Omega}_{K} \to X\) is said to be inward (respectively, weakly inward) on \(\overline{\Omega}_{K}\) relative to K if \(f(x) \in I_{K}(x)\) (respectively, \(f(x) \in \bar{I}_{K}(x)\)) for \(x \in \overline{\Omega}_{K}\), where \(\Omega \subset X\) is open and bounded with \(\Omega_{K} = \Omega \cap K \ne \emptyset\).

For the definition and properties of the Lan-Webb fixed point index, see [2].

3 An existence theorem for weakly inward A-proper maps

Theorem 3.1

Let K be a closed convex set, and \(f:K \to X\) be weakly inward on K, where \(I - f\) is A-proper. Suppose that
(a): 

\(f(x)\not\le x\) on \(\Vert x \Vert = r\), and

(b): 

there exists \(\rho \in (0, r)\) such that \(\lambda x \not\le f (x)\) for \(\Vert x \Vert = \rho\) and \(\lambda > 1\).

Then f has a fixed point in \(\{ x \in K:\rho < \Vert x \Vert < r\}\).

Proof

Let \(B_{r} = \{ x \in X:\Vert x \Vert < r\}\), \(B_{r_{K}} = B_{r} \cap K\), \(B_{\rho} = \{ x \in X:\Vert x \Vert < \rho \}\), and \(B_{\rho_{K}} = B_{\rho} \cap K\). We show that \(i_{K}(f,B_{r_{K}}) = \{ 0\}\) and \(i_{K}(f,B_{\rho_{K}}) = \{ 1\}\), so that by the additivity property of the index \(i_{K}(f,B_{r_{K}}\backslash B_{\rho_{K}}) = i_{K}(f,B_{r_{K}}) - i_{K}(f,B_{\rho_{K}}) = \{ 0\} - \{ 1\} = \{ - 1\} \ne \{ 0\}\), which implies the existence of a fixed point \(x \in K\) such that \(\rho < \Vert x \Vert < r\).

To show that \(i_{K}(f,B_{r_{K}}) = \{ 0\}\), suppose instead that \(i_{K}(f,B_{r_{K}}) \ne \{ 0\}\). Then we choose an a with \(\Vert f(x) \Vert \le a\) on \(\overline{B}_{r_{K}}\) and an \(e \in K\) with \(\Vert e \Vert > r + a\). Define the weakly inward A-proper homotopy \(H(x,t) = f(x) + te\). Now if \(H(x,t) = x\) for some \((x,t) \in \partial B_{r_{K}} \times [0, 1]\), then \(f(x) + te = x\), so that \(x \in K\) and \(x - f(x) = te \in K\) so \(f (x) \leq x\), which contradicts (a). Thus, H is an admissible homotopy, and \(i_{K}(H(x, 1),B_{r_{K}}) = i_{K}(f,B_{r_{K}}) \ne \{ 0\}\). Then there exists \(x \in B_{r_{K}}\) with \(f(x) + e = x\), so that \(\Vert e \Vert = \Vert x - f(x) \Vert \le \Vert x \Vert + \Vert f(x) \Vert \le r + a\), which contradicts \(\Vert e \Vert > r + a\). Hence, \(i_{K}(f,B_{r_{K}}) = \{ 0\}\).

Now we show that \(i_{K}(f,B_{\rho_{K}}) = \{ 1\}\). Define the weakly inward A-proper homotopy \(H(x,t) = tf(x)\).

If \(H(x,t) = x\) for some \((x,t) \in \partial B_{\rho_{K}} \times [0, 1]\), then \(t \ne 0\) (this would give \(0 = x\) on \(\partial B_{r_{K}}\)) and \(tf(x) = x\) and \(x \in K\), so that \(f(x) = \frac{1}{t} x \ge x\), which contradicts (b).

Thus \(H(x,t) \ne x\) on \(\partial B_{\rho_{K}} \times [0, 1]\).

By the homotopy property of the index, \(i_{K}(H(x,0),B_{\rho_{K}}) = i_{K}(H(x,1),B_{\rho_{K}}) = i_{K}(f,B_{\rho_{K}}) = \{ 1\}\).

Consequently, \(i_{K}(f,B_{r_{K}}\backslash B_{\rho_{K}}) = i_{K}(f,B_{r_{K}}) - i_{K}(f,B_{\rho_{K}}) = \{ 0\} - \{ 1\} = \{ - 1\}\).

Since the index is not 0, the existence property implies that there exists a fixed point \(x \in K\) such that
$$\rho < \Vert x \Vert < r. $$
 □

Remark 3.1

The conclusion of Theorem 3.1 is valid if condition (a) holds for \(\Vert x \Vert = \rho\) and condition (b) holds for \(\Vert x \Vert = r\), that is,
(a): 

\(f(x) \not\le x\) on \(\Vert x \Vert = \rho\), and

(b): 

\(\lambda x \not\le f (x)\) for \(\Vert x \Vert = r\) and \(\lambda > 1\).

We shall use these conditions in the following application.

4 Application

We formulate the Picard boundary value problem
$$ - x''(t) = f\bigl(t,x(t),x'(t),x''(t) \bigr),\quad \mbox{where } x(0) = x(1) = 0 $$
(1)
as a fixed point equation of the operator \(T:\overline{K}_{r} \to K\), \(K_{r} = \{ x \in K:\Vert x \Vert < r \}\),
$$Ty(t) = f\biggl(t,L^{ - 1}y,\frac{d}{dt}\bigl(L^{ - 1}y \bigr), - y\biggr), $$
where \(L:X \to Y\) is defined by \(Lx = - x''(t)\). Observe that (1) is equivalent to \(y = Ty\).

Let \(X = \{ x \in C^{2}[0, 1]:x(0) = x(1) = 0\}\), \(Y = C[0, 1]\), and \(K = \{ y \in C[1, 0]:y(t) \ge 0\}\) with norms \(\Vert x \Vert _{X} = \max \{ \Vert x \Vert _{Y},\Vert x' \Vert _{Y},\Vert x'' \Vert _{Y}\}\) and \(\Vert x \Vert _{Y} = \max_{t \in [0,1]}\{ \vert x(t) \vert \}\). Then L is a linear bounded isometric homeomorphism.

Theorem 4.1

Under the above assumptions, suppose also that
(a′): 

there exist \(r > 0\) and \(k \in (0, 1)\) such that \(f:[0, 1] \times [0, r] \times [ - r, r] \times R^{ -} \to R^{ +}\) is continuous with \(\vert f(t,p,q,s_{1}) - f(t,p,q,s_{2}) \vert \le k\vert s_{1} - s_{2} \vert \) for \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [ - r, r]\), \(s_{1},s_{2} \in R^{ -}\);

(b′): 

\(f(t,p,q,s) < r\) for every \(t \in [0, 1]\), \(p \in [0, r]\), \(q \in [ - r, r]\), \(s = - r\);

(c′): 

there are \(\rho \in (0, r)\), \(t_{0} \in [0, 1]\) such that \(f(t_{0},p,q,s) > \rho\) for \(p \in [0, \rho ]\), \(q \in [ - \rho, \rho ]\), \(s = - \rho\).

Then there exists a positive solution \(x \in K\) to equation (1) with \(\rho < \Vert x \Vert _{X} < r\).

Proof

Since T maps K to K, T is weakly inward. Condition (a′) implies that T is \((\beta_{K})k\)-ball contractive, where \(\beta_{K}\) is the ball measure of noncompactness associated with K, and thus \(\lambda I - T\) is A-proper with respect to the projection scheme \(\Gamma = \{ X_{n},P_{n}\}\) for every \(\lambda \ge \gamma\), \(\gamma \in (k, 1)\) (cf. [3]). To verify the remaining hypotheses of Remark 3.1, we first show that (b′) implies (b). Let r be as in (b′) and \(y \in K\) such that \(\Vert y \Vert _{Y} = r\). Then there exists \(x \in L^{ - 1}(K)\) such that \(Lx = y\) and \(\Vert x \Vert _{X} = \Vert y \Vert _{Y} = \Vert x'' \Vert _{Y}\), so that \(r = \Vert x'' \Vert _{Y} = \Vert x \Vert _{X}\) and there exists \(t_{0} \in [0, 1]\) such that \(y(t_{0}) = r\). Now since \(y = Lx\) for some \(x \in L^{ - 1}(K)\), we have that \(x(t) \in [0, r]\), \(x'(t) \in [ - r, r]\) for all \(t \in [0, 1]\) and \(r = - x''(t_{0})\). Then if \(Ty = \lambda y\) for some \(\lambda > 1\) and \(y \in K\) with \(\Vert y \Vert _{Y} = r\), we would have \(f(t,x(t),x'(t),x''(t)) = \lambda y(t)\) for all \(t \in [0, 1]\), including \(t_{0}\), but then this implies \(\lambda r < r\), a contradiction. So (b) holds.

To show that (c′) implies (a) of Remark 3.1, let \(x \in K\) with \(\Vert x \Vert _{X} = \rho\). Then \(\Vert Lx \Vert _{Y} = \Vert - x'' \Vert _{Y} = \rho\), and there exists \(t_{1} \in [0, 1]\) such that \(- x''(t_{1}) = \rho\) or \(x''(t_{1}) = - \rho\). So we have for \(t \in [0, 1]\) that \(x(t) \in [0, \rho ]\), \(x'(t_{1}) \in [ - \rho, \rho ]\), and \(x''(t_{1}) = - \rho\). By (c′) we have \(Ty(t_{1}) = f(t_{1},x(t_{1}),x'(t_{1}),x''(t_{1})) > \rho\), and so (a) is satisfied.

Thus, there exists a solution to equation (1) with \(x \in K\) and \(\rho < \Vert x \Vert < r\). □

Example 4.1

The function \(f(t,x,x',x'') = 1 + \frac{3}{4}\sin x''\) with \(r = \frac{3\pi}{2}\) and \(\rho = \frac{\pi}{2}\) shows that the class of maps that satisfy the conditions of Theorem 4.1 is nonempty.

Declarations

Acknowledgements

The author is grateful to the referees for their useful comments that have improved this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, University of Maryland

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© Cremins 2016