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

## 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.

## 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 Î“ 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

$$\rho < \Vert x \Vert < r.$$

â€ƒâ–¡

### Remark 3.1

The conclusion of TheoremÂ 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

(aâ€²):

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 $$t \in [0, 1]$$, $$p \in [0, r]$$, $$q \in [ - r, r]$$, $$s_{1},s_{2} \in R^{ -}$$;

(bâ€²):

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

(câ€²):

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 (aâ€²) 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 RemarkÂ 3.1, we first show that (bâ€²) implies (b). Let r be as in (bâ€²) 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 (câ€²) implies (a) of RemarkÂ 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 (câ€²) 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 $$\rho < \Vert x \Vert < r$$.â€ƒâ–¡

### 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 TheoremÂ 4.1 is nonempty.

## References

1. Deimling, K: Nonlinear Functional Analysis. Springer, Berlin (1985)

2. Lan, K, Webb, JRL: A fixed point index for weakly inward A-proper maps. Nonlinear Anal. 28, 315-325 (1997)

3. Lafferriere, B, Petryshyn, WV: New positive fixed point and eigenvalue results for $$P_{\gamma}$$-compact maps and applications. Nonlinear Anal. 13, 1427-1440 (1989)

4. Cremins, CT: Existence theorems for semilinear equations in cones. J. Math. Anal. Appl. 265, 447-457 (2002)

5. Lan, K, Webb, JRL: A-Properness of contracting and condensing maps. Nonlinear Anal. 49, 885-895 (2002)

6. Infante, G: Positive solutions of some three point boundary value problems via fixed point index for weakly inward A-proper maps. Fixed Point Theory Appl. 2005, 177-184 (2005)

## Acknowledgements

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

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Correspondence to Casey T Cremins.

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The author declares that he has no competing interests.

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Cremins, C.T. A fixed point theorem for weakly inward A-proper maps and application to a Picard boundary value problem. Fixed Point Theory Appl 2016, 71 (2016). https://doi.org/10.1186/s13663-016-0564-x