Positive solutions to second-order differential equations with dependence on the first-order derivative and nonlocal boundary conditions

involving Stieltjes integrals with suitable functions A and B of bounded variation on J. It is not assumed that ${\lambda }_1$, ${\lambda }_2$ are positive to all positive x. As we see later, the measures dA, dB can be signed measures.

We introduce the following assumptions:

H₁: $f\in C(J\times {\mathbb{R}}_{+}\times \mathbb{R},{\mathbb{R}}_{+})$, $\alpha ,\beta \in C(J,J)$, A and B are functions of bounded variation;

H₂: $h\in C(J,{\mathbb{R}}_{+})$ and h does not vanish identically on any subinterval;

H₃: $1-\gamma -{\lambda }_1[p]>0$ or $1-\xi -{\lambda }_2[p]>0$ for $p(t)=1$, $t\in J$, $\gamma ,\xi \ge 0$.

Recently, the existence of multiple positive solutions for differential equations has been studied extensively; for details, see, for example, [1–31]. However, many works about positive solutions have been done under the assumption that the first-order derivative is not involved explicitly in nonlinear terms; see, for example, [3, 6, 8–14, 17, 20, 25–27, 30]. From this list, only papers [9–12, 14, 20, 30] concern positive solutions to problems with deviating arguments. On the other hand, there are some papers considering the multiplicity of positive solutions with dependence on the first-order derivative; see, for example, [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 28, 29, 31]. Note that boundary conditions (BCs) in differential problems have important influence on the existence of the results obtained. In this paper, we consider problem (1) which is a problem with dependence on the first-order derivative with BCs involving Stieltjes integrals with signed measures of dA, dB appearing in functionals ${\lambda }_1$, ${\lambda }_2$; moreover, problem (1) depends on deviating arguments.

For example, in papers [2, 4, 15, 18, 22, 24], the existence of positive solutions to second-order differential equations with dependence on the first-order derivative (but without deviating arguments) has been studied with various BCs including the following:

or $1-\alpha \eta >0$, respectively.

For example, in papers [8–11, 20, 22, 30], the existence of positive solutions to second-order differential equations including impulsive problems, but without dependence on the first-order derivative, has been studied with various BCs including the following:

here ${\lambda }_1$ has the same form as in problem (1) with signed measure dA appearing in functional ${\lambda }_1$.

Positive solutions to second-order differential equations with boundary conditions that involve Stieltjes integrals have been studied in the case of signed measures in papers [25, 26] with BCs including, for example, the following:

The main results of papers [25, 26] have been obtained by the fixed point index theory for problems without deviating arguments. The study of positive solutions to boundary value problems with Stieltjes integrals in the case of signed measures has also been done in papers [3, 7, 13, 14, 27] for second-order differential equations (also impulsive) or third-order differential equations by using the fixed point index theory, the Avery-Peterson fixed point theorem or fixed point index theory involving eigenvalues.

Note that BCs in problem (1) with functionals ${\lambda }_1$, ${\lambda }_2$ cover some nonlocal BCs, for example,

for some constants $a_i$, $b_i$ and some functions $g_1$, $g_2$. In our paper, the assumption that the measures dA, dB in the definitions of ${\lambda }_1$, ${\lambda }_2$ are positive is not needed. More precisely, one needs to choose the above functions $g_1$, $g_2$ in such a way that the assumption H₄ holds. It means that $g_1$, $g_2$ can change sign on J.

In our paper, we eliminate ${\lambda }_1$ and ${\lambda }_2$ from problem (2) to obtain the equation $x=\overline{\mathcal{W}}x$ with a corresponding operator $\overline{\mathcal{W}}$, and then we seek solutions as fixed points of this operator $\overline{\mathcal{W}}$.

Note that if we put $\gamma =\xi =0$ in the BCs of problem (1), then this new problem is more general than the previous one because in this case someone, for example, can take ${\lambda }_1[x]=\gamma x(\eta )$, ${\lambda }_2[x]=\xi x(\eta )$. In this paper, we try to explain why for some cases we have to discuss problem (1) with constants $\gamma >0$ or $\xi >0$.

for ζ, ϱ such that $\zeta +\varrho <1$, $0<\zeta <\varrho <1$ with $\kappa =min(\zeta ,1-\varrho )$; see Section 5.

see Section 3. For the case $\alpha (t)\ge t$ on J, we can put $\gamma =0$ to work similarly as in Section 3; see Section 4. Note that in the above three cases for the argument β, we need only the assumption $\beta \in C(J,J)$, which means that β can change the character in J.

Note that in cited papers, positive solutions to differential equations with dependence on the first-order derivative have been investigated only for problems without deviating arguments, see [2, 4, 5, 7, 15, 16, 18, 19, 21–24, 28, 29, 31]. Moreover, BCs in problem (1) cover some nonlocal BCs discussed earlier.

Motivated by [25–27], in this paper, we apply the fixed point theorem due to Avery-Peterson to obtain sufficient conditions for the existence of multiple positive solutions to problems of type (1). In problem (1), an unknown x depends on deviating arguments which can be both of advanced or delayed type. To the author’s knowledge, it is the first paper when positive solutions have been investigated for such general boundary value problems with functionals ${\lambda }_1$, ${\lambda }_2$ and with deviating arguments α, β in differential equations in which f depends also on the first-order derivative. It is important to indicate that problems of type (1) have been discussed with signed measures of dA, dB appearing in Stieltjes integrals of functionals ${\lambda }_1$, ${\lambda }_2$.

The organization of this paper is as follows. In Section 2, we present some necessary lemmas connected with our main results. In Section 3, we first present some definitions and a theorem of Avery and Peterson which is useful in our research. Also in Section 3, we discuss the existence of multiple positive solutions to problems with delayed argument α, by using the above mentioned Avery-Peterson theorem. At the end of this section, an example is added to verify theoretical results. In Section 4, we formulate sufficient conditions under which problems with advanced argument α have positive solutions. In the last section, we discuss problems of type (1) when $\alpha (t)=t$ on J.

Here, VarA denotes the variation of a function A on J.

This proves case (i).

we get the result in case (ii). This ends the proof. □

Let us introduce the assumption.

for

We require the following result.

with

Solving this system with respect to ${\lambda }_1[u]$, ${\lambda }_2[u]$ and then substituting to (6), we have the assertion of this lemma. This ends the proof. □

Let us introduce the following assumption.

Lemma 3 Let the assumptions H₁-H₄ hold. Then $T:K\to K$.

Note that ${\lambda }_1[Fu]\ge 0$, ${\lambda }_2[Fu]\ge 0$ in view of the assumptions H₁, H₂, H₄ and the positivity of Green’s function G.

Note that ${(Tu)}^{″}\le 0$. Moreover,

Hence, Tu is concave and $Tu(t)\ge 0$ on J.

We next show that ${\lambda }_1[Tu]\ge 0$, ${\lambda }_2[Tu]\ge 0$. Indeed,

To do it, we consider two steps. Let ${\parallel Tu\parallel }_1=Tu(t^{\ast })$.

Step 1. Let $Tu(0)<Tu(\eta )$. Then $t^{\ast }\in (0,\eta )$ or $t^{\ast }\in (\eta ,1)$ and ${min}_{[0,\eta ]}Tu(t)=Tu(0)$.

It shows $T:K\to K$. This ends the proof. □

If we assume that $b\ge 3$, then $B_1>0$, $B_2>0$, ${\mathcal{G}}_2(s)\ge 0$, $s\in J$.

If we assume that $a\ge 6$, then $A_1>0$, $A_2>0$, ${\mathcal{G}}_1(s)\ge 0$, $s\in J$.

It means that the assumption H₃ holds. Moreover,

It proves that the assumption H₄ holds.

By a similar way, we prove the assertion in case (ii) or (iii).

Now, we present the necessary definitions from the theory of cones in Banach spaces.

Note that every cone $P\subset E$ induces an ordering in E given by $x\le y$ if $y-x\in P$.

for all $x,y\in P$ and $t\in [0,1]$.

for all $x,y\in P$ and $t\in [0,1]$.

Definition 3 An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Let φ and Θ be nonnegative continuous convex functionals on P, let Φ be a nonnegative continuous concave functional on P, and let Ψ be a nonnegative continuous functional on P. Then, for positive numbers a, b, c, d, we define the following sets:

We will use the following fixed point theorem of Avery and Peterson to establish multiple positive solutions to problem (1).

Theorem 1 (see [1])

is completely continuous and there exist positive numbers a, b, c with $a<b$ such that

(S₁): $\{x\in P(\phi ,\mathrm{\Theta },\mathrm{\Phi },b,c,d):\mathrm{\Phi }(x)>b\}\ne 0$ and $\mathrm{\Phi }(Tx)>b$ for $x\in P(\phi ,\mathrm{\Theta },\mathrm{\Phi },b,c,d)$,

(S₂): $\mathrm{\Phi }(Tx)>b$ for $x\in P(\phi ,\mathrm{\Phi },b,d)$ with $\mathrm{\Theta }(Tx)>c$,

(S₃): $0\notin R(\phi ,\mathrm{\Psi },a,d)$ and $\mathrm{\Psi }(Tx)<a$ for $x\in R(\phi ,\mathrm{\Psi },a,d)$ with $\mathrm{\Psi }(x)=a$.

Then T has at least three fixed points $x_1,x_2,x_3\in \overline{P(\phi ,d)}$ such that

Note that $\mathrm{\Phi }(x)\le {\parallel x\parallel }_1$. Put $\mathrm{\Psi }(x)=\mathrm{\Theta }(x)={\parallel x\parallel }_1,\phi (x)={\parallel x^{\prime }\parallel }_1$.

Now, we can formulate the main result of this section.

Theorem 2 Let the assumptions H₁-H₄ hold with $\xi =0$, $\gamma >0$. Let $\alpha (t)\le t$, $t\in J$. In addition, we assume that there exist positive constants a, b, c, d, M, $a<b$ and such that

with

and

(A₁): $f(t,u,v)\le \frac{d}{\mu }$ for $(t,u,v)\in J\times [0,Md]\times [-d,d]$,

(A₂): $f(t,u,v)\ge \frac{b}{L}$ for $(t,u,v)\in [0,\eta ]\times [b,\frac{b}{\rho }]\times [-d,d]$,

(A₃): $f(t,u,v)\le \frac{a}{\mu }$ for $(t,u,v)\in J\times [0,a]\times [-d,d]$.

and ${\parallel x_3\parallel }_1<a$.

Proof Basing on the definitions of T, we see that $T\overline{P}$ is equicontinuous on J, so T is completely continuous.

Let $x\in \overline{P(\phi ,d)}$, so $\phi (x)={\parallel x^{\prime }\parallel }_1\le d$. By Lemma 1, ${\parallel x\parallel }_1\le Md$, so $0\le x(t)\le Md$, $t\in J$. Assumption (A₁) implies $f(t,x(\alpha (t)),x^{\prime }(\beta (t)))\le \frac{d}{\mu }$.

Moreover, in view of (7),

This proves that $T:\overline{P(\phi ,d)}\to \overline{P(\phi ,d)}$.

Then $x_0(t)>0$, $t\in J$, and

for $p(t)=1$, $t\in J$. Moreover,

Let $b\le x(t)\le \frac{b}{\rho }$ for $t\in [0,\eta ]$. Then $0\le \alpha (t)\le t\le \eta$ for $t\in [0,\eta ]$, so $b\le x(\alpha (t))\le \frac{b}{\rho }$, $t\in [0,\eta ]$. Assumption (A₂) implies $f(t,x(\alpha (t)),x^{\prime }(\beta (t)))\ge \frac{b}{L}$. Hence,

Moreover,

This proves that condition (S₁) holds.

so condition (S₂) holds.