Existence theory for a third-order ordinary differential equation with non-separated multi-point and nonlocal Stieltjes boundary conditions

1.   Introduction

In recent years, solution examinations of boundary value models have increasingly become an interesting area of study. In fact, many real-world applications are modeled through boundary value problems of ordinary differential equations, for instance, problems arising from thermoelasticity, elasticity, fluid dynamics, quantum, and optical physics, chemical engineering, and population dynamics to mention a few, see ^[1,2,3,4,5] and the references therewith. Moreover, the expensive relevance of these problems entices vast a number of researchers to continuously develop ways to solve the models from the analytical, theoretical, and numerical aspects. In fact, the classical boundary conditions, including the Neumann, Dirichlet, and Robin have relatively become old-fashioned nowadays due to the development of more sophisticated conditions, which perfectly model complicated scenarios. In his regard, we mention some of the most significant boundary conditions comprising nonlocal boundary conditions ^[6,7,8,9,10], nonlocal multipoint integral boundary conditions ^[11,12], discontinuous type integral boundary conditions ^[13], four-point nonlocal integral boundary conditions ^[14], multi-strip integral boundary conditions ^[15], integral boundary conditions ^[16,17,18] and the non-classical boundary condition ^[19] among other. Indeed, these types of conditions take into account all the physical or chemical changes that can occur inside the domain. Also, they can construct boundary data that contain random shapes such as fluid flow problems in the blood vessels. For more developments in the boundary value problems and their related boundary conditions, we refer the reader(s) to ^{[20,21,22,23,24]} and the given references therein.

In particular, the boundary conditions featuring integral terms (the nonlocals) are by far more suitable for describing irregularities in parts of curved boundary structures. Indeed, the Riemann-Stieltjes integral boundary condition ^[6,7,8,9,10] serves as a chief in this class of boundary conditions. Furthermore, they have many applications in physics and statistics (in the area of stochastic processes). We, therefore, refer the reader(s) to see ^{[25,26,27,28,29,30,31,32,33,34]} that dealt with Riemann-Stieltjes integral boundary conditions. These recently published articles discussed the single-valued case of differential equations equipped with Riemann-Stieltjes integral boundary conditions. The novelty in this study is to study the single and multi-valued case of differential equations with Riemann-Stieltjes integral boundary conditions using modern theories in the arena.

Consequently, this research discusses some existence and uniqueness results for the third-order boundary value problem with nonlocal Stieltjes typed boundary conditions. More explicitly, we make consideration to the following model

where $f$ is a given continuous function from $[a, T]\times\mathbb{R}$ to $\mathbb{R}, \; a < \sigma_1 < \sigma_2 < \dots < \sigma_r < T, \; \phi$ is a bounded variation function. The values $\lambda_{j} \in \mathbb{R} \; (j = 1, 2, 3),$ while $\alpha_j, \beta_j, \delta_j \in \mathbb{R} \; (j = 1, 2),$ and $\gamma_i, \rho_i, \nu_i \in \mathbb{R}^+ \; (i = 1, 2, \dots, r)$.

More precisely, we aim in this study to extend the traditional third-order boundary value problems to feature non-separated and multi-point Stieltjes boundary data over an interval of choice. Thus, the existence theory for boundary value problems of third-order ordinary differential equations (and inclusions) in the presence of these new boundary conditions will be established in the present study. Besides, Stieltjes' conditions act in the same way with several types of boundary conditions, such as the multi-point and integral boundary conditions. For details on Riemann-Stieltjes integral conditions, we refer the reader(s) to Whyburn ^[35] and Conti ^[36]. In addition, the standard tools of functional analysis would be used to prove the existence theory for nonlinear boundary value problems as derived in ^{[37,38,39,40,41]}.

Lastly, we arrange the paper as follows: In Section 2, we prove an auxiliary lemma related to the linear variant of the problem (1.1). The existence and uniqueness of results for the boundary value problem (1.1), together with illustrative examples are proved in Section 3. Section 4 presents the existence of solutions for the parallel multi-valued problem of the problem (1.1), while Section 5 gives some concluding notes.

2.   Preliminary result

The solution for the linear variants of the problem (1.1) is defined in the lemma that follows.

Lemma 2.1. Assume that $\xi \in C[a, T],$ and

is satisfied. Then, the following linear problem

is equivalent to the integral equation

where

with

Proof. Integrating $y'''(\tau) = \xi(\tau)$ from $a$ to $\tau$ trice, one gets

where $c_0, \; c_1$ and $c_2$ are chosen real unknown constants. Then, upon combining the third boundary condition of (2.1) with (2.5), we get

Using the second boundary condition of (2.1) together with (2.6) in

we obtain

Finally, upon using the first boundary condition of (2.1) in (2.5) together with (2.6) and (2.7), one gets

where $\Gamma$ and $E_j \; (j = 1, \dots, 6)$ are given by (2.4). Therefore, substituting the values of $c_0, c_1,$ and $c_2$ into (2.5) and using the notations expressed in (2.3), the solution of (2.2) is thus obtained. Moreover, the converse of the lemma follows by direct computation. This completes the proof.

3.   Main results

Let ${\mathcal{H}} = C([a, T], {\mathbb R})$ denote the Banach space, which further contains continuous functions from $[a, T] \to {\mathbb R}$ with the norm introduced by $\|y\| = \sup \{|y(\tau)|, \tau \in [a, T]\}$. Then, on using Lemma 2.1, we transform problem (1.1) into an analogue fixed point problem as

where $\mathcal{P}: {\mathcal{H}} \to {\mathcal{H}}$ is defined by

If the operator expressed in $(2.1)$ has fixed points, then we can say that problem (1.1) has solutions.

Additionally, let us consider the following values, which would be used in the result of this study,

and

where $\ \max_{\tau \in[a, T]}|K_1(\tau)| = k_1$ and $\ \max_{\tau \in[a, T]}|K_2(\tau)| = k_2.$

3.1.   Existence of solutions

In this regard, Krasnoselskii's fixed point theorem ^[42] would be used to prove the existence of solutions for the problem (1.1) in what follows.

Lemma 3.1. (Krasnoselskii's fixed point theorem). Let $L$ be a closed bounded, convex and nonempty subset of a Banach space $X.$ Let $\phi_1$ and $\phi_2$ be operators such that

(i) $\phi_1 l_1+\phi_2 l_2 \in L$ whenever $l_1, l_2 \in L,$

(ii) $\phi_1$ is compact and continuous,

(iii) $\phi_2$ is a contraction mapping. Then, there exists $z \in L$ such that

Theorem 3.1. Let $f : [a, T]\times \mathbb{R} \to \mathbb{R}$ be a continuous function such that

${(C_{1})} \; |f(\tau, y)-f(\tau, x)| \le \ell |y-x|, \; \forall \tau \in [a, T], \; \ell > 0, \; y, x\in \mathbb{R}.$

${(C_{2})}$ There exist a map $\kappa \in C([a, T], {\mathbb R}^+)$ with $\|\kappa\| = \sup_{\tau\in [a, T]}|\kappa(\tau)|$ such that $|f(\tau, y)|\le \kappa(\tau), \; \; \forall (\tau, y) \in [a, T] \times \mathbb{R}.$

In addition, it is assumed that

where $\Theta$ is defined by $(3.3).$ Then, the problem $(1.1)$ has at least one solution on $[a, T].$

Proof. Consider a closed ball $B_{w} = \{y \in {\mathcal{H}}: \|y\|\le w \}$ for fixed $w \ge \Theta_{1} \|\kappa\|,$ where $\Theta_{1}$ is given by (3.4). Let us define the operators $\mathcal{P}_1$ and $\mathcal{P}_2$ on $B_{w}$ as follows:

and

Notice that $(\mathcal{P} y)(\tau) = (\mathcal{P}_1 y)(\tau)+(\mathcal{P}_2 y)(\tau).$ For $\tau \in [a, T]$ and $y, x \in B_{w},$ we find that

where $\Theta_{1}$ is given by (3.4). This shows that $\mathcal{P}_1 y+ \mathcal{P}_2 x \in B_{w}$. Thus, condition $(i)$ in Lemma 3.1 is satisfied. Next, on using the assumption of $(C_1)$ and (3.5), we obtain

which implies that $\mathcal{P}_2$ is a contraction operator. Then, we prove that $\mathcal{P}_1$ is compact and continuous. Notice also that, the continuity of $f$ means that the operator $\mathcal{P}_1$ is continuous. In addition, $\mathcal{P}_1$ is uniformly bounded on $B_{w}$ as

Let us fix $\sup_{(\tau, y) \in [a, T] \times B_{w}}|f(\tau, y)| = \bar{f},$ and take $a < \tau_1 < \tau_2 < T.$ Then,

independently of $y \in B_{w}.$ This implies that ${\mathcal{P}_1}$ is relatively compact on $B_{w}.$ Hence, it follows by the Arzelá-Ascoli theorem that the operator ${\mathcal{P}_1}$ is compact on $B_{w}.$ Thus, all the hypotheses of Lemma 3.1 have been fulfilled. Consequently, by Krasnoselskii's fixed point theorem, the problem $(1.1)$ has at least one solution on $[a, T].$

Remark 3.1. If we switch the role of the operators $\mathcal{P}_1$ and $\mathcal{P}_2$ in the theorem above, then the condition $(3.5)$ is replaced with $\ell \frac{(T-a)^{3}}{3!} < 1.$

In the following theorem, we prove the uniqueness result of solutions for the problem $(1.1)$ by implementation of the Banach's contraction mapping principle ^[43].

3.2.   Uniqueness of solutions

Theorem 3.2. Assume that $f: [a, T]\times \mathbb{R} \to \mathbb{R}$ satisfies $(C_{1})$ condition. Then, the boundary value problem (1.1) has a unique solution on $[a, T]$ if $\ell < 1/\Theta,$ where $\Theta$ is given by $(3.3)$.

Proof. Consider a set $B_\iota = \{y \in {\mathcal{H}}: \|y\|\le \iota \},$ where $\iota \ge \frac{\Theta_{1} q}{1-\ell \Theta_{1}}, \; \sup_{\tau \in [a, T]}|f(\tau, 0)| = q$. In the first step, we show that $\mathcal{P} B_\iota \subset B_\iota,$ where the operator $\mathcal{P}$ is defined by (2.2). For any $y\in B_\iota, \tau \in[a, T]$, we find that

Then, for $y \in B_\iota,$ we obtain

where $\Theta_{1}$ is given by (3.4). This shows that $\mathcal{P} B_\iota \subset B_\iota.$

Now we show that the operator $\mathcal{P}$ is a contraction. Then, for $y, x \in {\mathcal{H}},$ we have

where we have used (3.3). By the given assumption: $\ell < 1/\Theta,$ it follows that the operator $\mathcal{P}$ is a contraction. Thus, by Banach's contraction mapping principle, we deduce that the operator $\mathcal{P}$ has a fixed point, which corresponds to a unique solution of the problem (1.1) on $[a, T].$

Example 3.1. Consider the following non-separated multi-point and Stieltjes boundary value problem:

where $a = 0, \, T = 2, \, r = 4, \, \alpha_{1} = 2/9, \, \alpha_{2} = 4/9, \, \beta_{1} = 1/7, \, \beta_{2} = 3/7, \, \delta_{1} = 1/8, \, \delta_{2} = 1/4, \, \gamma_{1} = 1/9$, $\gamma_{2} = 1/3, \, \gamma_{3} = 5/9, \, \gamma_{4} = 2/3, \, \rho_{1} = 2/7, \, \rho_{2} = 3/7, \, \rho_{3} = 4/7, \, \rho_{4} = 5/7, \, \nu_{1} = 3/8, \, \nu_{2} = 1/2, \, \nu_{3} = 5/8$, $\nu_{4} = 3/4, \, \sigma_{1} = 1/3, \sigma_{2} = 2/3, \, \sigma_{3} = 1, \, \sigma_{4} = 4/3, \, \lambda_{1} = 3/2, \, \lambda_{2} = 3, \, \lambda_{3} = \frac{1}{2}, \, \phi(s) = \frac{(s-a)^3}{3}.$ Clearly, $|f(\tau, y)|\leq \frac{3}{\sqrt{\tau^3+441}}+\Big|\frac{e^{-\tau}}{21(1+e^{\tau})}\Big|+|\cos \tau|,$ $|f(\tau, y)-f(\tau, x)|\leq \ell |y-x|$, with $\ell = 1/6$. Using the given data, we find that $|E_1| = 4.541667\neq0, \, \, |E_2| = 4.095238\neq0, \, \, |E_4| = 3.666667\neq0, \, \, |E_3| = 1.047619, \, \, |E_5| = 4.814815, \, \, |E_6| = 3.261729$ and $|\Gamma| = 68.197099$ ($\Gamma$ and $E_i \; (i = 1, \dots, 6)$ are given by $(2.4)$), $\Theta = 4.386978, \, \, \Theta-\frac{(T-a)^{3}}{3!} = 3.053645$ ($\Theta$ is defined by $3.3$).

Furthermore, we note that all the conditions of Theorem $3.1$ are satisfied with $\ell \Big(\Theta-\frac{(T-a)^{3}}{3!}\Big)\approx 0.508941 < 1.$ Hence the conclusion of Theorem $3.1$ applies to the problem (3.6).

We also observe that all the conditions of Theorem $3.2$ hold true with $\ell \Theta \approx 0.731163 < 1.$ Hence we deduce by the conclusion of Theorem 3.2 that there exists a unique solution for the problem (3.6) on $[0, 2].$

4.   The multi-valued case

Here we discuss the existence of solutions for the multi-valued analogue (inclusions) case of the problem (1.1) given by

where $F: [a, T]\times \mathbb{R} \to \mathcal{P}(\mathbb{R})$ is a multi-valued map, $\mathcal{P}(\mathbb{R})$ is the family of all nonempty subsets of $\mathbb{R}$ and the other quantities are the same as defined in the problem (1.1). We then apply Bohnenblust-Karlin fixed point theorem to prove the existence of solutions for the problem (4.1).

Furthermore, for the convenience of the reader, we outline some basic concepts about multi-valued analysis ^{[44,45,46,47]} as follows:

I) A multi-valued map $\mathcal{S} : U \to \mathcal{P}(U)$ is

(i) convex (closed) valued if $\mathcal{S}(u)$ is convex (closed) for all $u \in U,$ where $(U, \|.\|)$ is a Banach space,

(ii) bounded on a bounded set if $\mathcal{S}(\mathbb{Z}) = \cup_{u \in \mathbb{Z}}\mathcal{S}(u)$ is bounded in $U$ for all $Z \in \mathcal{P}_b(U)$ (that is, $\sup_{u \in \mathbb{Z}}\{\sup \{|v| : v \in \mathcal{S}(u)\}\} < \infty),$

(iii) upper semi-continuous (u.s.c.) on $U$ if for each $u_0 \in U,$ the set $\mathcal{S}(u_0)$ is a nonempty closed subset of $U$, and if for each open set $A$ of $U$ containing $\mathcal{S}(u_0),$ there exists an open neighborhood $\mathcal{A}_0$ of $u_0$ such that $\mathcal{S}(\mathcal{A}_0) \subseteq A,$

(iv) completely continuous if $\mathcal{S}(\mathbb{B})$ is relatively compact for every $\mathbb{B} \in \mathcal{P}_b(U)$.

II) If the multi-valued map $\mathcal{S}$ is completely continuous with nonempty compact values, then $\mathcal{S}$ is u.s.c. if and only if $\mathcal{S}$ has a closed graph; that is, $u_n \to u_*, \; v_n \to v_*, \; v_n \in \mathcal{S}(u_n)$ implies $v_* \in \mathcal{S}(u_*).$

III) A multi-valued map $\mathcal{S} : U \to \mathcal{P}(U)$ has a fixed point if there is $u \in U$ such that $u \in \mathcal{S}(u).$

IV) In the sequel, we denote the set of all nonempty bounded, closed and convex subset of $U$ by $BCC (U),$ and $L^{1}([a, T], \mathbb{R})$ denotes the Banach space of functions $u:[a, T]\rightarrow \mathbb{R},$ which are Lebesgue integrable, and normed by $\Vert u\Vert_{L^{1}} = \int_{0}^{1}|u(\tau)|d\tau.$

V) Consider the following assumptions, which are needed in the forthcoming analysis:

$\mathbf{(A_{1})}$ Let $F:[a, T]\times \mathbb{R}\rightarrow BCC(\mathbb{R}); (\tau, y)\rightarrow F(\tau, y)$ be measurable with respect to $\tau$ for each $y\in \mathbb{R}$, u.s.c. with respect to $y$ for a.e. $\tau \in [a, T]$, and for each fixed $y\in \mathbb{R},$ the set $S_{F, y}: = \{g \in L^{1}([a, T], \mathbb{R}):g(\tau)\in F(\tau, y)\; \mbox{for all}\; \tau \in [a, T]\}$ is nonempty.

$\mathbf{(A_{2})}$ For each $r > 0,$ there exists a function $\psi_{r}\in L^{1}([a, T], \mathbb{R}^{+})$ such that $\Vert F(\tau, y)\Vert = \sup \{|g|:g(\tau)\in F(\tau, y)\}\leq \psi_{r}(\tau)$ for each $(\tau, y)\in [a, T]\times \mathbb{R }$ with $|y|\leq r$ and

VI) Lastly, in relation to (4.1), we define

Now, we state the following lemmas, which are needed to prove the main result:

Lemma 4.1. (Bohnenblust-Karlin ^[48]) Let $D \subset U$ be nonempty bounded, closed, and convex. Let $\mathcal{S} : D \to \mathcal{B}(U)$ be u.s.c. with closed, convex values such that $\mathcal{S}(D) \subset D$ and $\overline{\mathcal{S}(D)}$ is compact. Then, $\mathcal{S}$ has a fixed point.

Lemma 4.2. ^[49] Let $F$ be a multi-valued map satisfying the condition $(A_{1})$ and $\phi$ is linear continuous from $L^{1}([a, T], \mathbb{R})\rightarrow C([a, T], \mathbb{R}).$ Then, the operator $\phi \circ S_{F}:C([a, T], \mathbb{R})\rightarrow BCC(C([a, T], \mathbb{R})), y \mapsto (\phi \circ S_{F})(y) = \phi (S_{F, y})$ is a closed graph operator in $C([a, T], \mathbb{R})\times C([a, T], \mathbb{R})$.

Theorem 4.1. Assume that $(A_{1})$ and $(A_{2})$ hold and that

where $\Theta$ and $\varepsilon$ are given by $(3.3)$ and $(4.2)$ respectively. Then, the problem $(4.1)$ has at least one solution on $[a, T].$

Proof. Transforming the problem (4.1) into a fixed point problem, define an operator ${\mathcal G}: {\mathcal{H}}\rightarrow {\mathcal B}({\mathcal{H}})$ by

for $g \in S_{F, y}.$ It is obvious that the fixed points of ${\mathcal G}$ are solutions of the boundary value problem (4.1).

We will show that ${\mathcal G}$ satisfies the assumptions of Lemma $3$ and hence it will have a fixed point which guarantees the existence of a solution for the problem $(4.1).$

In the first step, we show that ${\mathcal G}(y)$ is convex for each $y\in {\mathcal{H}}.$ For $h_{1}, h_{2}\in {\mathcal G},$ there exist $g_{1}, g_{2}\in S_{F, y}$ such that for all $t\in [a, T],$ we have

For $0\leq \varsigma \leq 1$ and all $\tau \in [a, T],$ we obtain

As $S_{F, y}$ is convex ($F$ has convex values), so one can deduce that $\Big(\varsigma h_{1}+(1-\varsigma)h_{2}\Big)\in {\mathcal G}(y).$

Now we show that there exists a positive number $r$ such that ${\mathcal G}(B_{r})\subseteq B_{r},$ where $B_r = \{y\in {\mathcal{H}}: \|y\|\le r \},$ and $B_{r}$ is a bounded closed convex set in ${\mathcal{H}}$. If it is not true, then we can find a function $y_{r}\in B_{r}, h_{r}\in {\mathcal G}(y_{r})$ with $\| {\mathcal G}(y_{r})\| > r$ for each positive number $r$ such that

for some $g_{r}\in S_{F, y_{r}}.$ On the other hand, using $(A_{2})$, we get

where $\Theta$ and $\Lambda$ are given by $(3.3)$ and (4.3) respectively. Dividing both sides of the above inequality by $r$ yields

Now, taking the limit inf as $r \rightarrow \infty$ together with the notation (4.2), we obtain

which contradicts (4.4). Consequently, there exists a positive number $r_{1}$ such that ${\mathcal G}(B_{r _{1}})\subseteq B_{{r _{1}}}.$

Next, we show that ${\mathcal G}(B_{{r _{1}}})$ is equicontinuous set of ${\mathcal{H}}.$ Let $\tau_1, \tau_2 \in [a, T]$ with $\tau_1 < \tau_2$, $y \in B_{{r _{1}}}$ and $h\in {\mathcal G}(y),$ there exists $g\in S_{F, y}$ such that for each $\tau \in [a, T],$ we have

independently of $y \in B_{{r _{1}}}.$ Subsequently, the Ascoli-Arzelá theorem applies since the above three conditions are satisfied. Thus the operator ${\mathcal G}: {\mathcal{H}} \to {\mathcal{B}}({\mathcal{H}})$ is compact multi-valued map. In order to prove that ${\mathcal G}$ is u.s.c. we have to show that ${\mathcal G}$ has a closed graph as follows, where ${\mathcal G}$ is completely continuous (see Proposition 1.2 ^[44]). Let $y_n \to y_*, \; h_n \in {\mathcal G} (y_n)$ and $h_n \to h_*.$ Then, we prove that $h_* \in {\mathcal G} (y_*).$ Associated with $h_n \in {\mathcal G} (y_n),$ there exists $g_n \in S_{F, y_n}$ for all $\tau \in [a, T],$ such that

Therefore, it is enough to prove that there exists $g_* \in S_{F, y_*}$ such that for all $\tau \in [a, T],$ we have

Set $Q : L^1([a, T], \mathbb{R}) \to {\mathcal{H}}$ as a continuous linear operator given by

Observe that

Thus, by Lemma $4.2$, $Q \circ S_F$ is a closed graph operator and $h_n(\tau) \in Q(S_{F, y_n})$ as $y_n \to y_*.$ Consequently, we have

for some $g_* \in S_{F, y_*}$. The hypothesis of Lemma 4.1 holds true and we can conclude that ${\mathcal G}$ is a compact multi-valued map, u.s.c. with convex closed values. Thus, the operator ${\mathcal G}$ has a fixed point $y$ which is indeed a solution of problem (4.1). This completes the proof.

Example 4.1. Consider the following boundary value problem

where all the constants take the same values as in Example 3.1.

So, $\|F(\tau, y)\| \leq \frac{3}{(2+\tau)^4} |y|+\cos \tau$ in this case, with condition (4.4) satisfied, since $\varepsilon\Theta \approx 0.482568 < 1.$ Hence, it is deduced from the conclusion of Theorem $4.1$ that there exists at least one solution for the problem (4.5) on $[0, 2].$

5.   Conclusions

In conclusion, the present study made use of the fixed point theorems to develop and further prove the existence and uniqueness results for the generalized nonlinear third-order ordinary differential equation with non-separated multi-point and nonlocal Riemann-Stieltjes (integral) boundary conditions. Additionally, the existence of solutions for the multi-valued case has equally been established through the application of Bohnenblust-Karlin's version of the fixed point theorem. Lastly, some supportive examples are supplied to clarify the applicability of the proven results. Besides, the present study is relevant to a variety of physical models in science and engineering applications.

Conflict of interest

The author declares that she has no conflict of interest.